Robot de opción binaria Robot de opción binaria Cómo funciona el software Robot de opción binaria analiza la tendencia del mercado en tiempo real y calcula el valor de cada indicador de negociación. Los indicadores dan una señales de comercio de automóviles para llamar o poner. La opción binaria Robot ejecuta instantáneamente las operaciones en el corredor de opciones binarias siguiendo las señales y el sistema de comercio. Opción binaria Historia del robot El Robot de opción binaria original (que sólo está disponible en este sitio web) fue publicado por primera vez en enero de 2013 por una empresa francesa y con la ayuda de comerciantes profesionales. El objetivo de este software es automatizar el comercio de comerciantes profesionales. Mediante el uso de los mejores métodos e indicadores para generar señales binarias, la opción binaria Robot permite obtener ganancias en los mercados automáticamente. La opción binaria Robot ha sido copiada varias veces e incluso por productos con el mismo nombre pero el verdadero es el francés. La empresa francesa que creó el robot de opción binaria posee derechos de autor en EE. UU. y en la UE. Tan apenas toma cuidado y no sea scam por otros productos comerciales auto que usan el mismo nombre. Últimos Trades Inicie el Auto Trading 1- Abra una Cuenta 2- Login 3- Auto Trade Haga clic en AUTO TRADE y el robot inicia las opciones binarias de negociación automática. Este sitio y los productos y servicios ofrecidos en este sitio no están asociados, afiliados, respaldados o patrocinados por Google, ClickBetter, eBay, Amazon, Yahoo o Bing ni han sido examinados probados o certificados por Google, ClickBetter, Yahoo, eBay, Amazon, o Bing. La opción binaria Robot no garantiza ingresos o éxitos, y los ejemplos que se muestran en esta presentación no representan una indicación de éxito futuro o ganancias. La compañía declara que la información compartida es verdadera y exacta. Robot de opción binaria Robot de opción binaria Cómo funciona el software Robot de opción binaria analiza la tendencia del mercado en tiempo real y calcula el valor de cada indicador de negociación. Los indicadores dan una señales de comercio de automóviles para llamar o poner. La opción binaria Robot ejecuta instantáneamente las operaciones en el corredor de opciones binarias siguiendo las señales y el sistema de comercio. Opción binaria Historia del robot El Robot de opción binaria original (que sólo está disponible en este sitio web) fue publicado por primera vez en enero de 2013 por una empresa francesa y con la ayuda de comerciantes profesionales. El objetivo de este software es automatizar el comercio de comerciantes profesionales. Mediante el uso de los mejores métodos e indicadores para generar señales binarias, la opción binaria Robot permite obtener ganancias en los mercados automáticamente. La opción binaria Robot ha sido copiada varias veces e incluso por productos con el mismo nombre pero el verdadero es el francés. La empresa francesa que creó el robot de opción binaria posee derechos de autor en EE. UU. y en la UE. Tan apenas toma cuidado y no sea scam por otros productos comerciales auto que usan el mismo nombre. Últimos Trades Inicie el Auto Trading 1- Abra una Cuenta 2- Login 3- Auto Trade Haga clic en AUTO TRADE y el robot inicia las opciones binarias de negociación automática. Este sitio y los productos y servicios ofrecidos en este sitio no están asociados, afiliados, respaldados o patrocinados por Google, ClickBetter, eBay, Amazon, Yahoo o Bing ni han sido examinados probados o certificados por Google, ClickBetter, Yahoo, eBay, Amazon, o Bing. La opción binaria Robot no garantiza ingresos o éxitos, y los ejemplos que se muestran en esta presentación no representan una indicación de éxito futuro o ganancias. La compañía declara que la información compartida es verdadera y exacta. Robot de opción binaria Robot de opción binaria Cómo funciona el software Robot de opción binaria analiza la tendencia del mercado en tiempo real y calcula el valor de cada indicador de negociación. Los indicadores dan una señales de comercio de automóviles para llamar o poner. La opción binaria Robot ejecuta instantáneamente las operaciones en el corredor de opciones binarias siguiendo las señales y el sistema de comercio. Opción binaria Historia del robot El Robot de opción binaria original (que sólo está disponible en este sitio web) fue publicado por primera vez en enero de 2013 por una empresa francesa y con la ayuda de comerciantes profesionales. El objetivo de este software es automatizar el comercio de comerciantes profesionales. Mediante el uso de los mejores métodos e indicadores para generar señales binarias, la opción binaria Robot permite obtener ganancias en los mercados automáticamente. La opción binaria Robot ha sido copiada varias veces e incluso por productos con el mismo nombre pero el verdadero es el francés. La empresa francesa que creó el robot de opción binaria posee derechos de autor en EE. UU. y en la UE. Tan apenas toma cuidado y no sea scam por otros productos comerciales auto que usan el mismo nombre. Últimos Trades Inicie el Auto Trading 1- Abra una Cuenta 2- Login 3- Auto Trade Haga clic en AUTO TRADE y el robot inicia las opciones binarias de negociación automática. Este sitio y los productos y servicios ofrecidos en este sitio no están asociados, afiliados, respaldados o patrocinados por Google, ClickBetter, eBay, Amazon, Yahoo o Bing ni han sido examinados probados o certificados por Google, ClickBetter, Yahoo, eBay, Amazon, o Bing. La opción binaria Robot no garantiza ingresos o éxitos, y los ejemplos que se muestran en esta presentación no representan una indicación de éxito futuro o ganancias. La compañía declara que la información compartida es verdadera y exacta. LS-DYNA R8.0.0 (R8.95309) lanzado Nueva versión de LS-DYNA se publica para todas las plataformas comunes. Notas de la versión para R8.0.0 Aquí se resumen nuevas características, mejoras y algunas correcciones de errores seleccionadas no incluidas en la última versión anterior de LS-DYNA, versión R7.1.2. Estas notas se organizan por tema. Entender que en muchos casos, una nota en particular puede tener aplicabilidad a varios temas, pero en aras de la brevedad, cada nota aparece sólo bajo un encabezado de tema. Airbag Corregir el error de truncamiento de mensajes al escribir datos AIRBAG PARTICLE en el archivo binout. Agregue la opción RDT para AIRBAG SHELL REFERENCE GEOMETRY. Cuando la opción RDT está activa, el tamaño del paso de tiempo se basará en la geometría de referencia una vez que el tiempo de solución exceda el tiempo de nacimiento que puede ser definido por RGBRTH de MAT FABRIC. LCIDM y LCIDT de AIRBAG HYDRID ahora se pueden definir a través de DEFINE CURVE FUNCTION. Nueva variable RGBRTH en MAT FABRIC para introducir el tiempo de activación dependiente de la pieza para la geometría de referencia del airbag. Corrige un error que podría resultar en energía interna distinta de cero cuando se utilice la opción BIRTH de AIRBAG REFERENCE GEOMETRY junto con AIRBAG SHELL REFERENCE GEOMETRY. El PID negativo de AIRBAG INTERACTION considera el bloqueo del área de partición debido al contacto. Mejoras a AIRBAG PARTICULA: Nueva opción de bloqueo (IBLOCK) para respiraderos. eq.0: no. eq.1: sí. eq.2: sí, excluir salidas externas. eq.3: sí, excluir salidas internas. eq.4: Sí, excluir todos los respiraderos Después de dos palabras clave MPP se aplicará automáticamente a la bolsa CPM si no se utilizan para obtener un mejor rendimiento MPP. DECOMPOSICIÓN BAGREF DECOMPOSICIÓN DE LAS PARTES DE ARRANQUE El trabajo externo realizado por el gas inflador a la estructura se informa a glstat. Mejore la comprobación de la orientación del segmento de la bolsa y las cámaras CPM. Permita que el usuario excluya algunas partes de la superficie de las partículas de aire iniciales. Esto ayudará a prevenir partículas atrapadas entre dos capas atadas de telas. Soporte que comprime el respiradero del sello que actúa como la ventilación de la aleta. Apoyo a la ecuación de porosidad de Anagonye y Wang mediante MAT FABRIC. Añadir palabra clave opción MOLEFRACTION. Si se utiliza la opción, el usuario debe proporcionar una curva de flujo total de masa total e introducir la curva de fracción molar de la especie para cada gas. El código generará la curva de caudal másico individual para cada gas dinámicamente. Agregar palabra clave de ID a AIRBAG REFERENCE GEOMETRY y AIRBAG SHELL REFERENCE: El ID de variable distingue una bolsa de otra cuando se postprocesa cuando hay múltiples bolsas. También son variables opcionales para escalar la geometría de referencia. Habilite DEFINE CURVE FUNCTION para AIRBAG SIMPLE AIRBAG MODEL. PARTICULA DE AIRBAG: Calcule la convección de calor (HCONV) entre el ambiente y el airbag de manera consistente cuando TSW se utiliza para cambiar de un airbag de partículas a un volumen de control. Para AIRBAG PARTICLE. Añada la opción ENH V 2 para el orificio de ventilación de tal manera que se pueda producir un flujo bidireccional, es decir, fluir con o contra el gradiente de presión. Por el contrario, ENH V 1 sólo permite el flujo unidireccional a través de los orificios de ventilación, de alta a baja presión. ALE MAPA DEL AIRE LÍMITE. Agregue las siguientes asignaciones: 1d-3d. SET POROUS ALE: nueva palabra clave para definir las propiedades de un medio poroso ALE mediante un conjunto de elementos. Las fuerzas porosas se calculan mediante CARGA DEL CUERPO PORO. ALE FSI SWITCH MMG: se aplica también ahora a 2D. ALE SWITCH MMG: nueva palabra clave para cambiar grupos de materiales múltiples basados en criterios definidos por el usuario con DEFINE FUNCTION. CONTROL ALE: Permite definir PREF (presión de referencia) por los materiales. Implemente el ALE COUPLING NODAL DRAG para modelar el acoplamiento de fuerza de arrastre entre las esferas de elementos discretos o las partículas SPH y los fluidos ALE. Este comando incluye una opción para calcular el coeficiente de arrastre usando DEFINE FUNCTION. Implementar ALE COUPLING RIGID BODY como una alternativa eficiente para el acoplamiento de tipo de restricción entre fluidos ALE y un cuerpo rígido Lagrangiano. Boundary Fijar error de redondeo de entrada cuando se lee la quinta curva de carga de DIRECCIONES RIGIDAS DE ORIENTACIÓN PRESCRIBIDA DE LÍMITE. Que fue leído y almacenado como real, causando errores en una sola precisión para curvas de carga con más de 7 dígitos. BOUNDARY PAP podría calcular la presión incorrecta en MPP cuando CVMASS ahora fija. Corregir el trabajo externo incorrecto cuando se utiliza BOUNDARY PRESCRIBED MOTION con o sin opción RIGID. El DOF especificado no fue considerado al computar el trabajo externo. Corregir las velocidades incorrectas cuando se utiliza BOUNDARY PRESCRIBED MOTION RIGID LOCAL e INICIAL VELOCITY RIGID BODY para cuerpos rígidos. El error finaliza si se aplica a un nodo que pertenece a un cuerpo rígido. Fix BOUNDARY SPC SET NACIMIENTO DE MUERTE que no estaba funcionando para MPP cuando el tiempo de nacimiento y muerte se establecieron a cero. Corregir el número de ID incorrecto en d3hsp para BOUNDARY CYCLIC. Corrija la salida a bndout (BASE DE DATOS BNDOUT) para RIGIDO DE ORIENTACIÓN PRESCRIBIDA LÍMITE. Modifique el VECTOR DE ORIENTACIÓN PRESCRIBIDA DEL LÍMITE para acomodar los cuerpos que no experimentan ningún cambio en la orientación. Corregir el error en el que typeID en BOUNDARY PRESCRIBED MOTION no se podría introducir como una etiqueta alfanumérica. Agregar una nueva palabra clave BOUNDARY SPC SYMMETRY PLANE: Propósito: Definir restricciones para imponer la simetría planar para nodos en o cerca de un plano especificado. Estas restricciones se aplicarán incluso durante la adaptación. Este comando es similar a CONSTRAINED LOCAL pero permite la selectividad a través de un ID de pieza. Blast La parte sólida o el conjunto de piezas sólidas están permitidos ahora para PARTICLE BLAST. Añada el indicador de condición de límite de presión ambiente BC P para la EXPLOTACIÓN DE PARTÍCULAS. Si se invoca, la presión ambiente, cableada a 1 bar, resiste el escape de las partículas del dominio global. Active el tiempo de muerte antes inactivo BTEND para PARTICULA BLAST. La partícula de explosión se omite en la pantalla después del tiempo de muerte. El nuevo comando DEFINE PBLAST GEOMETRY permite que el dominio de alto explosivo para PARTICLE BLAST sea definido por varias formas geométricas. Permitir múltiples definiciones de PARTICLE BLAST Añada DATABASE PBSTAT para generar estadísticas de explosiones de partículas. Se obtiene el volumen inicial y la masa inicial de las partıculas de HE y las partıculas de aire para la PARTICULA BLAST a d3hsp. Compressible Flow Solver CESE Añada el comando CESE BOUNDARY BLAST LOAD para permitir que una explosión descrita por el comando LOAD BLAST ENHANCED sea utilizada como una condición de límite en CESE. Está diseñado para usarse con el solver de límite FSI inmerso en CESE. Corregir muchos problemas con la transferencia de calor conjugado involucrando CESE y CESE FSI solvers. Modifique el tratamiento de la presión de la condición de contorno reflectante de la interfaz FSI en algunos cálculos para la malla móvil y los solucionadores de límites sumergidos. Cambiar el método de cálculo de derivados de CESE para utilizar los valores actuales de las variables de flujo. Agregue dos nuevos comandos MAT para el solver de CESE, CESE MAT 000 y CESE MAT 002. Agregue un solver de marco de referencia no inercial para problemas de fluido y FSI usando el método de malla móvil. Para el solucionador CESE de malla móvil, reemplace la comunicación all-to-all para calor conjugado y cantidades FSI con un mecanismo de comunicación escaso. En combinación con el modelo de chemkin referenciado por los comandos de CHEMISTRY, corrija los problemas relacionados con el uso del comando CONTROL UNITS en problemas de flujo químicamente reactivos con el solver de CESE. Evite la situación de interbloqueo de MPP causada por la inclusión de nodos sin masa con una malla de entrada de CFD-only (CESE). Añada capacidad de erosión de elementos estructurales al método de límite sumergido. Solver de CESE FSI (sólo capacidad en serie). Agregue capacidad de condiciones de contorno cíclico 2D. Corrija los tiempos finales d3plot y dump para un solver independiente de CESE. Agregue una capacidad de detección de NaN para el solver de CESE. Cambie todas las condiciones de límite CESE que utilizan una parte de superficie de malla para definir el límite para utilizar la cadena de caracteres MSURF en lugar de PART en la parte de opción del nombre de palabra clave. Corrija la búsqueda de elementos en LSO para la salida de puntos. Corregir el CESE dt para acoplamiento de cuerpo rígido. Tenga en cuenta que la escala de masa no funcionará en un acoplamiento FSI con el solver de CESE. Añada la interpolación de temperatura que falta a tiempo para imponer temperaturas sólidas como condición de contorno en el solver de CESE. Optimice el movimiento de malla basado en IDW para el solver de malla móvil de CESE. Trate la malla de entrada como 3D de forma predeterminada. Química Todas las características químicas que se mencionan a continuación se acoplan únicamente al solver compresible de flujo de CESE cuando se trata de cálculos en 2D o 3D. Se han añadido fuentes Jacobianas de origen químico. Modelo implícito completo modelo simplificado implícito modelo explícito: sólo iso-combustión. Introduzca dos nuevos comandos para las aplicaciones de airbag: CONTROL DE QUÍMICA PIROTÉCNICA y PROPIEDADES PROPIEDADES QUÍMICAS Conjuntamente con estas órdenes, se implementan modelos básicos de infladores de airbag. El modelo de inflador pirotécnico que utiliza propelente NaN3 / Fe2O3 se implementa recientemente. Para conectarse con el solucionador Aribag ALE existente, se guardan dos curvas de carga, caudal másico y temperatura, en el archivo de inflador en función del tiempo. Este modelo calcula tres sub-regiones: cámara de combustión, plenum de gas y tanque de descarga. Cada región se puede inicializar con diferentes modelos de CHEMISTRY COMPOSION, lo que significa que el usuario puede calcular el modo híbrido de gas propulsor. Se han mejorado los siguientes problemas de combustión tridimensional: volumen constante, presión constante y CSP. Para la iso-combustión. La temperatura y las fracciones de masa de especies como un fucntion de tiempo se muestran en la pantalla y se guardan en isocom. csv para trazar con LS-PrePost. Se ha implementado otro método de integración de ODE químico. El archivo de salida del inflador pirotécnico se actualiza para que este archivo pueda ser leído del ALE solver para una simulación de airbag. Mejorar los explosivos de explosión gaseosos TNT 2-D y 3-D, categorizados como TBX (explosivos termobaricales), se implementan para los sistemas de ecuación de Euler (sólo CESE). Además, la combustión de aluminio de explosión TNT 3-D para problemas en serie se implementa ahora. Los datos de termodinámica especiales (póngase en contacto con el distribuidor local de LS-DYNA) Sólo se incluye un modelo de combustión sólida en esta versión. Para el modelo de cinética química, se implementa el modelo de equilibrio estequiométrico ya que no hay cinética química disponible hasta el momento. Implementar un método de modelado de mezcla para su uso con solucionadores de CESE. Estos solucionadores de mezcla de componentes múltiples se implementan en: Solucionadores 2-D Euler y N-S. Euler de 2-D y Euler y N-S solvers. Solucionadores 3D de Euler y N-S Todos los solucionadores requieren especificación de composiciones de especies iniciales, lo que significa que los modelos de COMPOSICIÓN DE QUÍMICA deben ser introducidos, y se deben proporcionar ficheros de datos de termodinámica y transporte. Corregir la salida variable de química D3PLOT cuando se utilizan mallas basadas en MESH. Comandos. Modifique los comandos de palabras clave relacionados con CHEMISTRY para permitir múltiples modelos de química en el mismo problema. Básicamente, los comandos químicos cambiados se deben a la necesidad de identificar un modelo químico (modelo chemkin) en varias de las órdenes de palabras clave, y eventualmente, permitir que múltiples modelos químicos se utilicen en el mismo problema (pero en subdominios separados del problema que No mezclar). Esto incluye un mecanismo para la selección de entrada de un método de cálculo Jacobiano. Agregue el comando CHEMISTRY MODEL que identifica los archivos que definen un modelo químico de Chemkin. Modifique los siguientes comandos para que los archivos relacionados con el modelo de química se hayan eliminado. Estos comandos sólo se utilizan para seleccionar el tipo de solución química: CONTROL DE QUÍMICA CONTROL DE QUÍMICA CSP CONTROL DE QUÍMICA COMPLETA 1D Modificar INICIACIÓN DE DET EN QUÍMICA donde se han eliminado los archivos relacionados con el modelo de química y el ID de modelo se infiere mediante una referencia a un ID de composición química. Modifique la COMPOSICIÓN DE QUÍMICA y la DIMENSIÓN DE CESE D3PLOT para agregar el ID del Modelo. Contacto Añada CONTACTO TORNILLO CONJUNTO A SÓLIDO para transferir momentos de las cáscaras a sólidos. Añadir cálculo de energía de fricción para vigas en CONTACTO AUTOMÁTICO GENERAL. Fuerza EROSOP 1 para todos los contactos del tipo ERODING, con una advertencia al usuario si ellos lo han introducido como 0. Hacer correcciones MPP relacionadas con el contacto s intfor base de datos (DATABASE BINARY INTFOR): Corregir posible colgar durante la salida de intfor. Corrección de la salida de la brecha de contacto en intfor para nodos compartidos. Fijar la salida de intfor para el contacto MPP agrupa - ble de SURFACE TO SURFACE, que no escribía el lado esclavo de la interfaz aunque pudiera tener. No agrupable no puede porque trata CONTACTO LÍNEA DE SUPERFICIE A SUPERFICIE como si fuera CONTACTO NODOS ATADOS A SUPERFICIE. Contacto MPP: corrige una rutina que actualiza el espesor de los segmentos de los elementos sólidos (cada ciclo) para que honre el parámetro OPTT en PART CONTACT. Fijar la adaptabilidad MPP para modelos muy grandes: la lectura del archivo de volcado adaptable no funcionaba correctamente para archivos de más de 8 GB. Corregir el problema al utilizar INTERFACE LINKING junto con los contactos vinculados basados en restricciones. Los nodos involucrados en ambos son eliminados de la interfaz de enlace, pero esto dejaba algunas estructuras internas fuera de sincronía, causando a veces que los nodos se movieran incorrectamente. Añadir comprobación de error en caso de una definición de contacto con un conjunto de nodos vacío dado para el lado esclavo. Modifique la salida de ncforc (DATABASE NCFORC) para soportar la salida en un sistema de coordenadas local. Si se define CID RCF para una interfaz de contacto, envíe el archivo ncforc en este sistema de coordenadas (para que coincida con el archivo rcforc). Para los contactos ERODING, reduzca la memoria asignada para los segmentos de modo que cada segmento interior sólo se asigna una vez. Solucionar el problema generando nodos de visualización RIGIDWALL si también se generan nodos de orientación de haz. Agregar palabra clave DEFINE CONTACT EXCLUSION (MPP solamente) para permitir que los nodos atados en algunos contactos sean ignorados en ciertos otros contactos. Solucionar el problema con el TRANSDUCTOR DE FUERZA DE RÍGIDA: las fuerzas de penalización para los nodos rígidos que impactan la pared se sobrescribían durante el proceso de restricción de la pared rígida. Solucionar problema ENTIDAD de CONTACTO cuando se utiliza con INCLUDE TRANSFORM: las entidades no estaban siendo correctamente transformadas. Reescriba el acoplamiento de CONTACT ENTITY para utilizar la memoria dinámica, que elimina el límite anterior de 100 entidades de contacto en malla. Ahora no hay límite. Elimine la condición de liberación indocumentada para las opciones 5 y mayores de MPP CONTACT AUTOMATIC TIEBREAK a petición del cliente. Mejorar la inicialización del contacto de desempate. Si hay más de 10 nodos maestros, utilice un método bucketsort. BT y DT en CONTACTO se pueden configurar para definir más de un par de horas de nacimiento / muerte para el contacto apuntando a una curva o tabla. Estos pares pueden ser únicos para la fase de relajación dinámica y la fase normal de la simulación. Añada la opción EDGEONLY a CONTACTO AUTOMÁTICO GENERAL para excluir el contacto de nodo a segmento y considere sólo el contacto borde a borde y haz a haz. VDC define el coeficiente de restitución cuando se define CORTYP en la Tarjeta Opcional E. Disponible para CONTACTAR NODOS AUTOMÁTICOS A LA SUPERFICIE. CONTACTE LA SUPERFICIE AUTOMATICA A LA SUPERFICIE. Y CONTACTO AUTOMÁTICO SOLA SUPERFICIE SOFT 0 o 1 solamente. Mejoras para CONTACT AUTOMATIC GENERAL: Añada la opción de contacto viga a viga CPARM8 en PART CONTACT (MPP solamente). Añada la opción por la cual el haz generado en el borde de la envoltura exterior será desplazado a la envoltura por la mitad del espesor de la cubierta. De esta manera, el contacto del borde de la cáscara con el borde de la cáscara comienza directamente en el borde de la cáscara y no en una extensión del borde de la cáscara. (Véase OPT2 10, CPARM8, MPP Card 1). Fijar el error de segmentación cuando se utiliza el CABLE GUÍA DE CONTACTO. Fijar error de segmentación cuando se utiliza CONTACT AUTOMATIC SINGLE SURFACE TIED (sólo SMP, ncpu 0 (modo de coherencia)). Fijar los contactos incorrectos al utilizar CONTACT AUTOMATIC GENERAL INTERIOR para vigas con grandes diferencias de grosor y cuando las vigas más delgadas están más próximas entre sí que con las vigas más gruesas. Afecta únicamente al SMP. Agregar advertencia si número de segmentos / nodos de contacto 999999999 (i9). Fijar el espesor de contacto incorrecto utilizado para los tipos CONTACTO ENTIDAD 8 y 9 cuando el factor de escala del espesor de contacto, g2, es distinto de cero. Fijar RIGIDWALL GEOMETRIC FLAT DISPLAY que muestra incorrectamente la pared rígida que está siendo penetrada por los nodos esclavos. Implementar CONTROL CONTACTO PENOPT 3 opción de CONTACTO NOMBRES AUTOMÁTICOS PARA SUPERFICIE y CONTACTO NODOS ERODIFICACIÓN A SUPERFICIE para SMP. Fijar la SUPERFICIE CONECTADA DE CONTACTO A LA SUPERFICIE que no ató los elementos de la cáscara usando MAT ORTHO ELASTIC PLASTIC / MAT 108. Corregir mensaje de error no válido cuando se utiliza DEFINE FRICTION donde PTYPEi y PTYPEj no son los mismos. Fijar el mensaje de error espurio, CLAVE 141, cuando se usa RIGIDWALL GEOMETRIC. DISPLAY y la tarjeta opcional 6 para la opción DISPLAY no está definida. Fijar el CONTACTO AUTOMÁTICO AUTOMÁTICO..TIEBREAK si la OPCIÓN 4 y el PARAM 1 y el coeficiente de amortiguamiento viscoso, VDC 0.0. Actualiza el contacto basado en el segmento (SOFT 2) para mejorar la precisión en puntos alejados del origen. Los cálculos finales se realizan ahora con ubicaciones nodales y de segmentos que se han desplazado hacia el origen para que los valores de coordenadas sean pequeños. Habilitar fricción definida por el usuario (USER INTERFACE FRICTION subrutine usrfrc) para contacto MPP SOFT 4. Corregir la rigidez de la penetración de contacto para elementos cohesivos. Unifique los mensajes automáticos del tiebreak para el comienzo del daño y el fallo final. SMP y MPP ahora deben dar la misma salida a d3hsp y messag. Esto afecta a CONTACT AUTOMATIC. TIEBREAK. OPCIONES 6, 7, 8, 9, 10 y 11. CONTACTO ADD WEAR: Asocian los cálculos de desgaste a una interfaz de contacto formadora cuyas cantidades se pueden contabilizar en el archivo de base de datos intfor. Se admite la adaptación. CONTACTO. MORTERO: Las salidas de aviso detalladas activadas para contacto con mortero, también aclaran los datos de eco en d3hsp. Espesor de contacto hecho en consonancia con otros contactos en términos de prioridad entre ISTUPD en CONTROL SHELL, SST en CONTACT y OPTT en PART CONTACT. Mejora de la eficacia del tipo de cuchara en el contacto con el mortero permitiendo una aceleración significativa en simulaciones de contacto a gran escala. CONTACTO. MORTERO. DEFINE LA FRICCIÓN. CONTACTO DE LA PARTE: El contacto con el mortero soporta FS -1.0, lo que significa que los coeficientes de fricción se toman de los parámetros PART CONTACT. El contacto de mortero soporta FS. EQ.-2 lo que significa que la fricción se toma de DEFINE FRICTION. MORTERO AUTOMÁTICO DE SUPERFICIE AUTOMÁTICA: Utilizando IGNORE. LT.0 para el contacto de mortero superficial simple, se ignorarán las penetraciones de segmentos que pertenezcan a la misma pieza. Esto facilitará con gusto la eliminación de penetraciones iniciales en modelos grandes. Los factores de fricción son ahora una función de la temperatura para el CONTACTO. FRICCIÓN TÉRMICA. Constrained SET POROUS LAGRANGIAN: nueva palabra clave para definir la porosidad de los elementos Lagrangianos en un conjunto de elementos. Las fuerzas porosas se calculan por medio de la concentración limitada en el tipo sólido 11 o 12. LAGRANGE CONSTRAINED EN SÓLIDO: CTYPE 12 está ahora también disponible en 2D. Agregue la opción del ángulo de la hélice para los ENGRANAJES CONJUNTOS CONSTRAINED. Cambie la palabra clave de RODAMIENTO CONSTRAINED a ELEMENT BEARING. Mejorar explícitamente el uso de las restricciones implícitas de alivio de inercia. Esto permite la conmutación implícita-explícita para tales problemas. Agregue nuevas opciones de entrada para CONTROL IMPLICIT INERTIA RELIEF. Usuario especificado número de nodos usuario especificó la lista de modos de restringir a cabo. Aplicar el VIGO CONSTRAINED EN SÓLIDO. Esta característica es básicamente una restricción revisada de corte entre vigas y sólidos de Lagrange que incluye características que lo hacen más atractivo en algunos casos que LAGRANGE CONSTRAINED EN SÓLIDO. Por ejemplo, en el modelado del acoplamiento de barras de refuerzo en hormigón. Corregir un mensaje de error no válido cuando se utiliza un ID de curva de carga grande para FMPH, FMT, FMPS en la tarjeta 3 de RENUNCIA CONJUNTA CONSTRAINED con opciones GENERALIZADAS o TRANSLACIONALES. Fijar la salida de coordenadas / desplazamientos a d3plot para el nodo PADRON NODAL RIGIDO CONSTRAINED. Fijar el error de segmentación si se utilizan tablas para FMPH de RASGÓN CONJUNTA CONSTRAINED y si el ángulo de rotación es menor que la abscisa de la tabla o curvas de carga. Fijar la velocidad inicial incorrecta de los nodos EXTRA NODES CONSTRAINED cuando se utiliza IVATN 1 e IRIGID 1 en GENERACION DE VELOCIDAD INICIAL. Fijar la masa / inercia incorrectas si se utilizan CUERPOS RIGIDOS CONSTRAINED con IFLAG 1 y la parte maestra, PIDM, está utilizando PART INERTIA. Fijar la inicialización incorrecta con drdisp. sif cuando se utilizan NOMBRES EXTRAS CONSTRAINED. Fijar la inercia de masa incorrecta para la parte del cuerpo rígida cuando IFLAG 1 en NODOS EXTRACTIVOS CONSTRAINED. Permita que INTERPOLACIÓN CONSTRAINED utilice el conjunto de nodos para definir los nodos independientes. Añada la nueva función MODEL. GE.10 a INTERRUPTOR CONSTRAINED SPOTWELD (SPR3). Esto permite que los parámetros STIFF, ALPHA1, RN, RS y BETA se definan como FUNCIONES DE DEFINICIÓN de espesores y tensiones máximas de rendimiento de ingeniería de hojas conectadas. Se agregaron informes de fallos para CONSTRAINED SPR2. Agregue más salida d3hsp para INTERPOLATION CONSTRAINED SPOTWELD y SPR2 CONSTRAINED. Se puede desactivar ajustando NPOPT 1 en CONTROL OUTPUT. Añadir opción a CONJUNTO CONSTRAINED: La rigidez de penalización relativa ahora se puede definir como función del tiempo cuando RPS 0 se refiere a una curva de carga. Trabajos para SPHERICAL, REVOLUTE, CYLINDRICAL en análisis explixit. Añadir nueva opción a INTERPOLACIÓN CONSTRAINED SPOTWELD: SPR4. Hay un nuevo parámetro MODEL para seleccionar el nuevo método. ENGRANAJES CONJUNTAS CONJUNTA: La rueda dentada soporta ahora engranajes cónicos y tipos similares, es decir, el punto de contacto no tiene necesariamente que estar en el eje entre los centros de engranajes. Fijar la imposibilidad de manejar identificaciones duplicadas para SPOTWELD CONSTRAINED. NODE SET, PUNTOS y SPR2. CONSTRAINED MULTIPLE GLOBAL: Soporta múltiples restricciones definidas en los DOFs adicionales de los elementos definidos por el usuario. Control Haga que la opción CONTROL SHELL PSNFAIL funcione con el criterio de eliminación de W-MODE para los shells. Fijar para INCLUDE PATH RELATIVE cuando se utiliza con adaptabilidad. Corregir el comportamiento transitorio incorrecto si IDRFLG 6 para CONTROL DYNAMIC RELAXATION. Arreglar el mensaje de error no válido cuando se utiliza CONTROL ADAPTIVE con elementos de shell y NÚMEROS SPC BOUNDARY y contacto de superficie única. También corrija el tipo de contacto reportado para el mensaje de error MPP 185. Elimine la activación inapropiada de la relajación dinámica después de reajuste adaptativo. SUBCYCLE DE CONTROL: Nuevo esquema de subciclo activado para CONTROL SUBCYCLE y CONTROL SUBCYCLE MASS SCALED PART. Por defecto, la relación entre el paso de tiempo más grande y más pequeño es ahora 16 y las fuerzas externas se evalúan cada paso de tiempo. El esquema anterior tenía una relación cableada dura de 8. Las relaciones se pueden cambiar opcionalmente por el SUBCÍCULO DE CONTROL K L donde K es la relación máxima entre etapas de tiempo para fuerzas internas y L es también la relación para fuerzas externas. Corrección de fallo para CONTROL IMPLICIT MODAL DYNAMIC si se utiliza jobid Corregir error de colapso de soldadura por puntos sólidos cuando se utiliza CONTROL SPOTWELD BEAM para convertir una soldadura por puntos de haz en una soldadura por puntos sólida. ADAPTADOR DE CONTROL: Corrige un error por el cual los identificadores de elementos de piezas que no se adaptaban cambiaban cuando se invocaba la adaptabilidad 3D. El remesador adaptativo en 3D, que crea una malla de elementos tetraédricos, soporta el remezclado de partes que inicialmente comprendían tetraedros de 4 nodos, pentaedros de 6 nodos, hexaedros de 8 nodos o una mezcla de peniques y hexágonos. CONTROL REMESHING: Puede especificar un sistema de coordenadas local (CID) para la adaptabilidad orbital (ADPOPT 3 en PART) con la limitación de que el eje z local es paralelo al eje z global. El eje z local es el eje orbital para la parte que se adapta. CID se convierte por defecto en el sistema de coordenadas global. Base de datos PERFIL DE BASE DE DATOS: perfiles de energía cinética e interna de salida, perfiles de fracción de volumen de salida, agregue un parámetro MMG para especificar el grupo ALE para el cual se pueden generar datos de elementos. BASE DE DATOS ALE MAT: ahora puede usar DEFINE BOX para calcular las energías materiales, volúmenes y masas de los elementos dentro de las cajas (en lugar de toda la malla). GENERAR TRACER DE BASE DE DATOS: nueva palabra clave para crear partículas trazadoras ALE a lo largo de iso-superficies. BASE DE DATOS FSI: agrega una opción a los momentos de salida creados por fuerzas FSI sobre cada nodo en un conjunto de nodos. Estos momentos acerca de los nodos se informan en dbfsi. Al escribir INITIAL STRESS SOLID en el archivo dinámico, IALEGP se escribe ahora como cero de forma predeterminada. Anteriormente se escribió como 1, lo que sugiere un grupo de material no existente. Esto era inofensivo en LS-DYNA pero podría ser confuso para los usuarios. Añada ROLLOS DE BASE DE DATOS para escribir los datos correspondientes a los RODAMIENTOS DE ELEMENTOS. BASE DE DATOS SECFORC. BASE DE DATOS SECCIÓN CRUZADA: Los datos secforc para secciones transversales a través de elementos de cinturón de seguridad 2D se recodifican para proporcionar resultados más robustos y precisos. Corrija la salida incorrecta de los haces discretos al formato binario de disbout (DATABASE DISBOUT). Afecta únicamente al SMP. Fijar el número de nodo incorrecto en eloutdet para la tensión de los sólidos cuando se envía a binout. Ruptura en la revisión 89209. Corrija la tensión / deformación de salida a elout que estaban en el sistema local en lugar de global cuando EOCS 2 en CONTROL OUTPUT y CMPLG 1 en BASE DATABASE EXTENT. Resolver la entrada en conflicto dando prioridad a EOCS. Corrija la salida incorrecta 0.0 para la variable de historia 3, la presión de la viscosidad a granel, de los sólidos a la trama d3 cuando HYDRO. gt.0 en DATABASE EXTENT BINARY y cuando se usa la ecuación de estado. DATABASE SBTOUT y DATABASE DEFORC funcionan ahora. Estos comandos fueron ignorados en algunas revisiones del código. Fije la salida a curvout para DEFINIR FUNCIONES DE CURVA TM, TX, TY, TZ. Fijar el número incorrecto del retractor del cinturón de seguridad en la salida del cinturón de seguridad (DATABASE SBTOUT). Afecta sólo al MPP. Fijar el sistema de coordenadas incorrecto utilizado para la salida de deformación a d3plot cuando CMPFLG 1 en DATABASE EXTENT BINARY y MAXINT 0 para materiales ortotrópicos y anisotrópicos. Corrija la salida de deformación inadvertida a las variables de historial de d3plot cuando NODOUT ALL / ALL GL / STRAIN / STRAIN GL o INTOUT ALL / STRAIN en BATERIA DE BASE DE BASE DE DATABASE. Agregar mensaje de advertencia, SOL 1264, si el usuario especifica itype 0 (cuerpo rígido) para DATABASE CROSS SECTION y el ID de DEFINE COORDINATE NODES especifica nodos que no pertenecen a un cuerpo rígido. Corregir archivo intfor dañado cuando RESPLT 1 en DATABASE EXTENT BINARY. Corrige el número de elemento incorrecto en el archivo eloutdet para elementos sólidos. Fijar las fuerzas de jntforc (DATABASE JNTFORC) que se emitió en el sistema de coordenadas incorrecto para cuerpos rígidos cuando LMF 1 en CONTROL RIGID. Evite que el archivo d3drlf se sobrescriba si la relajación dinámica está activada y la adaptabilidad activada. Fijar los desplazamientos incorrectos de la rotación de los acelerómetros en el nodout cuando se ingiere 1 en ELEMENT SEATBELT ACCELEROMETER. Corregir los momentos incorrectos en secforc (DATABASE SECFORC) cuando se utiliza la resolución de haz 5. Fijar NaN en el archivo secforc cuando se utiliza la sección transversal de DATABASE. ITYPE 2 y el sistema de coordenadas se especifica usando DEFINE COORDINATE VECTOR con un NID diferente de cero. (MPP solamente). Corregir mensaje de error no válido al configurar BEAM 1 en DATABASE BINARY D3PLOT para elementos discretos. Incluye energía de reloj de arena erosionada en energía de reloj de arena en el archivo glstat para ser coherente con los cálculos de IE KE de modo que la energía total de la energía cinética de energía interna energía de reloj de arena de energía rigidwall. Agregar soporte para la nueva base de datos pbstat (DATABASE PBSTAT) para PARTICLE BLAST. Energía interna y energía de traslación de aire y productos de detonación fuerza / presión de aire y productos de detonación para cada parte BASE DE DATOS INTFOR: El nuevo parámetro NWEAR en tarjeta opcional rige la salida de profundidad de desgaste a la base de datos intfor. DATABASE EXTENT BINARY. MAT TELA. MAT FABRIC MAP: El uso de CMPFLG -1 funcionará igual que CMPFLG 1, excepto que para MAT FABRIC (forma 14 y forma -14) y MAT FABRIC MAP las tensiones y tensiones locales serán cantidades de ingeniería en lugar de Green-Lagrange strain y 2nd Piola - Kirchhoff estrés. DATABASE EXTENT BINARY. ESTERA. SECCIÓN: Para algunos materiales y elementos, los tensores de deformación térmica y plástica pueden ser enviados a la base de datos d3plot, véase STRFLG. Corregir el error en el archivo d3plot si se utiliza LOAD THERMAL D3PLOT. Permitir PSETID en DATABASE BINARY D3PLOT y D3PART para elementos SPH Elimine el error emitido cuando DATABASE HISTORY SHELL SET hace referencia a un conjunto de shell definido utilizando una caja y se invoca la adaptabilidad del shell. Añadir INTEGRACIÓN. Datos a la base de datos d3prop (DATABASE BINARY D3PROP). Corrija los ID escritos en swforc y rcforc. Los tiempos de salida para intfor (DATABASE BINARY INTFOR) estaban desactivados por un paso de tiempo. Esto era particularmente notable para las ejecuciones implícitas. Este problema ahora está solucionado. Agregar opción para la salida de mensajes de advertencia / error detallados (o largos) a d3msg. Véase MSGFLG en CONTROL OUTPUT. Solamente algunas versiones largas de advertencias / errores en este tiempo pero esa lista se espera que crezca. Agregar opciones para la compresión de datos en d3plot, consulte DCOMP en DATABASE EXTENT BINARY. Agregar opción para escribir la leyenda revisada a los archivos jntforc, secforc, rcforc, deforc y nodout a través de la bandera de entrada NEWLEG en CONTROL OUTPUT. Esto ayuda a evitar la confusión sobre los ID no asignados y los identificadores duplicados. Si se codifica cualquier dato de entrada y se solicita dynain, el código emite un mensaje de error y detiene el trabajo. Método de Elemento Discreto - DEM Ahora se permite la parte sólida o el conjunto de piezas sólidas para DEFINE DE TO SURFACE COUPLING. El factor de desgaste basado en la ecuación de desgaste de Archard se envía a la base de datos de salida de DEM. Implementar DELETE PART para la Esfera de Elemento Discreto. La unidad de ángulo de contacto cambió de radiano a grado para el elemento DISCRETO DE CONTROL. Implementar la ley de desgaste de Archard para DEFINE DE TO SURFACE COUPLING para esferas de elementos discretos. Agregar energía de amortiguación y energía de fricción de elementos discretos a energía de amortiguación y términos de energía de interfaz deslizante en glstat. Introduzca una pequeña perturbación en la posición inicial de los elementos discretos recién generados para DEFINE DE INJECTION. Esto permite una distribución espacial más aleatoria de las partículas generadas. INTERFACE DE HBOND sustituye a INTERFACE DE BOND. Se utiliza para definir los modelos de fallo para enlaces que enlazan varias partes de elementos discretos (DE) dentro de una definición de enlace heterogéneo (DEFINE DE HBOND). DEFINE SÓLIDOS ADAPTATIVOS A DES: Insertar y / o transformar elementos sólidos fallidos en partículas DES (ELEMENT DISCRETE SPHERE). Las partículas del DES heredan las propiedades materiales de los elementos sólidos. Todas las características basadas en DES están disponibles a través de esta transformación, incluyendo los modelos de enlace y algoritmos de contacto. Este comando es esencialmente a DES lo que DEFINE ADAPTIVE SOLID TO SPH es a SPH partículas. EFG Corrige un error cuando se utilizan los depósitos EFG en un modelo de cinturón de seguridad. Agregue la salida del parámetro a d3hsp para sólidos EFG. Electromagnetic Solver - EM Agregue los materiales ortotrópicos EM donde la conductividad eléctrica es un tensor 3x3, vea la nueva tarjeta EM MAT 003. Agregue nueva familia de palabras clave, EM DATABASE. Que activa la salida de las variables y variables EM. Todas las salidas ASCII relacionadas con EM ahora empiezan con em. Añada capacidad para trazar líneas de campo magnético en y alrededor de los conductores en momentos determinados, vea EM DATABASE FIELDLINE. Los archivos de salida ASCII se generan (lspp fieldLine xx) y son legibles por LSPP para trazar las líneas de campo. En el futuro, LSPP será capaz de generar directamente las líneas de campo. Añadir las cantidades EM en DEFINE CURVE FUNCTION: Añadir EM EOS TABULATED2 donde una curva de carga define la conductividad eléctrica vs tiempo. Introduce capability to use the EM solver on (thin) shells: An underlying solid mesh (hexes and prisms) is built where the EM is solved and the EM fields are then collapsed onto the corresponding shell. The EM mat for shells is defined in EM MAT 004 This works for EM solvers 1, 2 and 3 and the EM contact is available for shells. Add different contact options in the EM CONTACT card. Add new methods to calculate electric contact resistance between two conductors for Resistive Spot Welding applications (RSW). See EM CONTACT RESISTANCE . Add Joule Heating in the contact resistance (see card EM CONTACT RESISTANCE ). The Joule heating is evenly spread between the elements adjacent to the faces in contact. Add new circuit types 21 and 22 (see EM CIRCUIT ) allowing users to put in their own periodic curve shape when using the inductive heating solver. This is useful in cases where the current is not a perfect sinusoidal. Provide default values for NCYCLEBEM and NCYCLEFEM ( 5000) and set default value of NUMLS to 100 in EM CIRCUIT . Element Fix bug affecting DAMPING FREQUENCY RANGE DEFORM used with beams with ELFORM 2. Damping forces were included in the forces and moments output to the d3plot and elout files, but are now excluded. For all other element types and formulations, the damping forces and stresses were already excluded from the output. This change has no effect on the solution, only on the output forces and moments. Add two additional formulations, FORM 3 and 4, to PART MODES . Add 20-node solid element, ELFORM 23 in SECTION SOLID . Add H8TOH20 option to ELEMENT SOLID to convert 8-node to 20-node solids. Add option SOLSIG to CONTROL OUTPUT which will permit stresses and other history variables for multi-integration point solids to be extrapolated to nodes. These extrapolated nodal values replace the integration point values normally stored in d3plot. NINTSLD must be set to 8 in DATABASE EXTENT BINARY when a nonzero SOLSIG is specified. Supported solid formulations are solid elements are: -1, -2, 2, 3, 4, 18, 16, 17, 23. Warning: Do not use Setting - Extrapolate in LS-PrePost when SOLSIG is nonzero. Activate contact thickness input from PART CONTACT for solids. Fix possible problems in ELEMENT MASS DISTRIBUTE when some beam types are used. Enhance FORM 3 of PART MODES to properly set rigid body constraints. Also set FORM 3 when using input from d3mode database. Made many enhancements for PART MODES for robustness and MPP implementation. Correct the application of mass damping when used with superelements ( ELEMENT DIRECT MATRIX INPUT ). Mass damping on the superelement attachment nodes had to be moved from non-superelement side to the superelement side. Works for both DAMPING GLOBAL and DAMPING PART MASS . Issue error message and terminate the simulation when ELEMENT SEATBELT RETRACTOR or ELEMENT SEATBELT PRETENSIONER refers to an undefined seatbelt sensor. Fix a bug regarding contact thickness of seatbelt element: THICK of SECTION SEATBELT was ignored but is now active. Add new cohesive shell element (elform 29) for edge-to-edge connectivity between shells. This element type takes bending into account and supports MPP and implicit solvers. Fix incorrect initial velocity and also mass output to d3hsp for shell types 23 24. Fix termination due to mass increase error when using mass scaling with ELEMENT MASS PART . Fixes pertaining to thick shell elements ( ELEMENT TSHELL ): Fix incorrect INITIAL STRESS TSHELL output to dynain. Fix INITIAL STRAIN TSHELL output to dynain. Fix incorrect strains written to elout det and also INTOUT NODOUT options in DATABASE EXTENT BINARY . Fix incorrect results when using MAT JOHNSON COOK / MAT 15 with ELFORMs 3 and 5. Fix initialization error for MAT 108 / MAT ORTHO ELASTIC PLASTIC affecting ELFORM 2. Fix incorrect INITIAL STRAIN TSHELL output to d3plot if STRFLG 1, EPSFLG 2, ENGFLG 2 in DATABASE EXTENT BINARY . Fix incorrect stresses output to eloutdet for ELFORM 5. Fix incorrect reading of TIME in card 3 of ELEMENT SEATBELT SENSOR SBSTYP 3 when long s in command line. Error terminate with message, STR 1296, if same node is defined multiple times in ELEMENT MASS MATRIX . Fix input read error when using ELEMENT SHELL SHL4 TO SHL8 . Add warning message, STR 1286, when no mid-side nodes are defined for shell element formulations 23 and 24. Fix invalid part not found error during keyword input phase when using ELEMENT SHELL COMPOSITE . Add support for negative MAXINT option in DATABASE EXTENT BINARY for thick shell elements. ELEMENT TSHELL : Add BETA as option for ELEMENT TSHELL to provide an orthotropic material angle for the element. Add Rayleigh damping ( DAMPING PART STIFFNESS ) for triangular shell element types 3 and 17. Add new keyword ELEMENT BEAM SOURCE. Purpose: Define a nodal source for beam elements. This feature is implemented for truss beam elements ( SECTION BEAM. ELFORM 3) with material MAT 001 or for discrete beam elements (ELFORM 6) with material MAT 071 . Add new option to DEFINE ELEMENT DEATH. New variable IDGRP defines a group id for simultaneous deletion of elements. SECTION SOLID. MAT COHESIVE : Convert solid type 20 and 22 to incremental formulation to properly handle large rotations. Also use consistent mass for presumed stability. Add Smoothed Particle Galerkin (SPG) method for solid analysis (ELFORM 47) and corresponding keyword option SECTION SOLID SPG. SPG is a true particle method in Galerkin formulation that is suitable for severe deformation problems and damage analysis. Forming Enhance ELEMENT LANCING by supporting PARAMETER. PARAMETER EXPRESSION. This is especially useful to define birth and death time for lancing by using the distance from punch bottom and by using PARAMETER EXPRESSION . Add a new feature, CONTROL FORMING TRIMMING. for 2D and 3D trimming of a 3-layer, sandwich laminate blank via DEFINE CURVE TRIM . Add 3D normal trimming of solid elements via DEFINE CURVE TRIM 3D : Allow normal (to sheet plane) trimming. Trimming is done by using the local element normal directions. Add new features for solid elements 2D trimming DEFINE CURVE TRIM NEW : Allow support of arbitrary trimming vector (previously only global z direction was allowed). Improve trimming algorithm for speed up. Allow trimming curves to project to either the top or bottom surface. Add a new AUTO CONSTRAINT option to CONTROL FORMING ONESTEP. Add new features to CONTROL FORMING UNFLANGING : The incoming flange mesh will be automatically checked for mesh quality and bad elements fixed. Allow thickness offset of deformable flange to use the blank thickness from user s input. Allow definition any node ID in the outer boundary of the flange, to speed up the search when holes are present in the part. Add a new parameter CHARLEN: default 150 to limit the search region. It should be set bigger than the biggest width of the flange. Allow holes to exist in the flange regions. Output a suggested flange part after unflanging simulation, with the failed elements deleted from the unflanged part. Define automatically a node set and constraints for the flange boundary nodes through the user definition of three nodes: NB1, NB2, and NB3. Add output of forming thickness, effective strain and trim curves after unflanging simulation. Add a new keyword CONTROL FORMING TRIM to replace ELEMENT TRIM . Add a new keyword: CONTROL FORMING UNFLANGING OUTPUT : In unflanging simulation, it s important to obtain xa correct trimming curves. In this keyword, failed elements are removed to come up with the trim curves. Enhance CONTROL FORMING OUTPUT : If NOUT is larger than the number of states specified by either Yi s or LCID, the remaining states are evenly distributed between TBEG and the time corresponding to the biggest Yi from the home position. If NOUT is zero or blank. then the number of states is automatically determined based on either Yi s or LCID. Add a new keyword CONTROL FORMING TRIM MERGE. - Purpose: Close a user specified (gap) value in the trim curves, so each trim curve will form a closed loop, which is required for a successful trimming. Add a new keyword: CONTROL FORMING MAXID : - Purpose: Set a maximum node ID and element ID for the incoming dynain file (typically the blank) in the current simulation. These give a starting point for newly added nodes and elements due to adaptivity of the blank. This is typically used in line die simulation when many sets of tools are used in several simulation processes. Frequency Domain FREQUENCY DOMAIN ACOUSTIC BEM : Update the boundary condition definition for BEM acoustics so that impedance and other user defined boundary conditions can be combined with time domain velocity boundary condition. Implement Burton-Miller BEM to MPP. Implement impedance boundary condition to Burton-Miller BEM. Implement half space option ( FREQUENCY DOMAIN ACOUSTIC BEM HALF SPACE ) to variational indirect BEM. Implement half space option to acoustic scattering problems. Extend acoustic ATV computation to elements, in addition to nodes. Support element based ATV output in d3atv. Add an option ( MATV) to run modal acoustic transfer vector. Implement MATV to MPP. Implement running BEM Acoustics based on modal ATV (SSD excitation only). FREQUENCY DOMAIN ACOUSTIC FEM : Enable running FEM acoustics based on restarting SSD ( FREQUENCY DOMAIN SSD ). FREQUENCY DOMAIN ACOUSTIC INCIDENT WAVE : Implement this keyword to define the incident waves for acoustic scattering problems. To be used with FREQUENCY DOMAIN ACOUSTIC BEM . FREQUENCY DOMAIN ACOUSTIC SOUND SPEED : Implement this keyword to define frequency dependent complex sound speed, which can be used in BEM acoustics. By using complex sound speed, the damping in the acoustic system can be considered. To be used with FREQUENCY DOMAIN ACOUSTIC BEM . FREQUENCY DOMAIN FRF : Implement mode dependent rayleigh damping to frf and ssd (DMPMAS and DMPSTF). A curve can be used to define the mass proportional damping constant or stiffness proportional damping constant vs. freq (or mode number), depending on LCTYP. FREQUENCY DOMAIN RESPONSE SPECTRUM : Add output of nodout spcm and elout spcm, to get nodal results and element results at user specified nodes and elements. Add von Mises stress computation. FREQUENCY DOMAIN RANDOM VIBRATION : Add semi-log, and linear-linear interpolation on PSD curves (parameter LDFLAG). Add a new keyword MAT ADD FATIGUE. to define the SN fatigue curve for material models. Enabled this keyword to define SN curve in random vibration fatigue. FREQUENCY DOMAIN SSD : Add strain computation. Add parameter LC3 to define the duration of excitation for each frequency. This parameter is optional and is only needed for simulating sine sweep vibration. Implement fatigue analysis based on ssd (sine sweep). The fatigue analysis feature is activated by option FATIGUE. The material s SN fatigue curve is defined by MAT ADD FATIGUE. Results of fatigue analysis are reported in binary database D3FTG. Add an option to use DAMPING PART MASS and DAMPING PART STIFFNESS in SSD (DMPFLG 1). MAT ADD FATIGUE : Implement this keyword to define material s SN fatigue curve, which can be used in random vibration fatigue and SSD fatigue analysis. To use MAT ADD FATIGUE in random vibration fatigue, set NFTG -999 in FREQUENCY DOMAIN RANDOM VIBRATION FATIGUE . Add linear-linear interpolation to the SN curve (parameter LTYPE). FREQUENCY DOMAIN ACCELERATION UNIT : Implement this keyword to facilitate the acceleration unit conversion. In industry, sometimes people use g (gravity acceleration, 9.81 m/s 2) as unit of acceleration (this is very common for vibration analysis). But in LS-DYNA, consistent units are required and the acceleration unit length unit / time unit 2. This keyword automatically converts g to the consistent acceleration unit. Incompressible Flow Solver - ICFD The icfd mstats. dat file now outputs the ten worst quality element locations. Add option in ICFD CONTROL OUTPUT allowing terminal output to be written to messag file. Add keyword ICFD CONTROL OUTPUT SUBDOM to output only part of the domain. Available for vtk, dx and gmv formats. Add new keyword family, ICFD DATABASE . which triggers the output of ICFD variables. All ICFD related output files now start with icfd . Add new keyword family ICFD SOLVER TOL . which allows the user to control tolerances and iteration number for the fractional step solve, the mesh movement solve, and the heat equation solve Curves identified on Card 2 of ICFD BOUNDARY PRESCRIBED VEL each provide a scaling factor vs. x, y, or z coordinate, respectively. These scaling factors are applied to the velocity boundary condition. Enable free-slip condition for FSI walls. Add new variable IDC to ICFD CONTROL FSI that allows the modification of the scaling parameter that multiplies the mesh size to detect contact. Add automatic squeezing to the elements of the boundary layer when there are two very close surfaces with poor (coarse) mesh resolution. Add the initialization for all nodes using ICFD INITIAL with PID 0. Add a curve (LCIDSF in ICFD CONTROL TIME ) that scales the CFL number as a function of time. Add a Heaviside function that allows the solution of simple multiphase problems. Add the computation of the heat convection coefficient. Add MPP support for y and shear for output. Add uniformity index. Add ICFD CONTROL TAVERAGE to control the restarting time for computing the time average values. By default there is no restarting and the variables are average from t 0. Implement the XMl format for vtk. The vtk Legacy format has issues with the reader for large output files. See ICFD CONTROL OUTPUT . Improve temperature stabilization for thermal problems. Add the Generalized Flow Through Porous Media model monolithically coupled to the incompressible Navier-Stokes model. See keyword ICFD MAT for the new options. Add the capability to define the porous properties using the Pressure-Velocity (P-V) experimental curves. See ICFD MAT for details: Users must define the experimental data using a load curve and the thickness of the probe (deltax). Permeability and Forchheimer factor are computed using the Least Squares method on the data cloud. Add the Anisotropic version of the Generalized Flow in Porous Media. See ICFD MAT for details. Compute drag forces around anisotropic/isotropic porous domains. ICFD bug fixes: Fixed the bug for when embedded shells and mesh interf were not working correctly when both were part of the same input deck. Fixed support for triangular elements on user defined meshes. Fixed and corrected the RANS K-e model to support the wall functions. Fixed a memory problem when a 3-D geometry had more than 3 internal holes. Implicit Extend implicit debug checking when LPRINT 3 on CONTROL IMPLICIT SOLVER . Add option for implicit dynamic relaxation so that only a subset of parts is active during the dynamic relaxation phase. See IDRFLG 6 and DRPSET in CONTROL DYNAMIC RELAXATION. This allows element formulations which are not supported by implicit, e. g. ALE and SPH, to be included in the model and active for the normal phase of the simulation but ignored during the dynamic relaxation phase. Fix bug involving MPP contact and CONTROL IMPLICIT FORMING . Extend implicit time step control via IAUTO 0 in CONTROL IMPLICIT AUTO (step size driven by load curve) to linear analysis (NSOLVR 1 in CONTROL IMPLICIT SOLUTION ). Add self piercing rivet capability to implicit mechanics, both SMP and MPP. This is CONSTRAINED SPR2 and CONSTRAINED INTERPOLATION SPOTWELD . Can now dump the damping matrix from implicit mechanics. See MTXDMP in CONTROL IMPLICIT SOLVER. In addition to mass damping and stiffness damping terms, beam damping terms are also dumped. Extend matrix dumping capability (MTXDMP) to MPP. MTXDMP 0 will terminate the run after dumping of matrices. Improve stress and strain computation induced by mode shapes. This computation is invoked by MSTRES in CONTROL IMPLICIT EIGENVALUE. This vastly improves the robustness of the stress/strain computation in the presence of nonlinear elements and materials. Add variable MSTRSCL to CONTROL IMPLICIT EIGENVALUE for user control of geometry scaling for the stress computation. Make SMP and MPP treatment of autospc constraints consistent. AUTOSPC on CONTROL IMPLICIT SOLVER controls the application of autospc constraints. Fix reading of d3kil file to allow for implicit switches with more than 4 characters. Fix MPP implicit s output of forces to bndout due to prescribed motion on nodes. Forces on shared nodes were being counted multiple times. Enhance output for ELEMENT DIRECT MATRIX INPUT (superelements) to describe how they are attached to the LS-DYNA model and to refer to their element ID from the input deck instead of an internal elmeent ID. Enhance superelement computation ( CONTROL IMPLICIT MODES or CONTROL IMPLICIT STATIC CONDENSATION ): The computation of the inertia matrix in the presense of rigid bodies is correct. Adjust superelement computation to accept initial velocities. Add null beams for the visualization of superelements. This allows users to observe each superelement where it is connected to the nonlinear model in LS-PrePost by the use of null beam elements connecting those nodes. Enhance implicit to allow the use of CONSTRAINED RIVET in conjunction with axisymmetric shell element problems. Remove application of stiffness damping ( DAMPING PART STIFFNESS ) during the implicit assembly of elemental stiffness matrices. This damping was being applied twice. Add output of performance statistics for the MPP implicit eigensolver to mes0000. Enhancements to modal dynamics ( CONTROL IMPLICIT MODAL DYNAMIC ): Correct reading of d3eigv database in the presence of higher order elements and tshell elements. Add Stress computation to modal dynamics. Fix computation of time step for modal dynamics in the presence of CONTROL IMPLICIT MODAL DYNAMIC DAMPING . Fix damping factor ZETA in CONTROL IMPLICIT MODAL DYNAMIC . Allow unsymmetric terms to the assembled stiffness matrix from some implicit features. This is available in both SMP and MPP. See LCPACK 3 in CONTROL IMPLICIT SOLVER. There are an increasing number of unsymmetric terms enhancing the physical modeling. These terms are listed under the remarks for LCPACK. Use of this unsymmetric feature adds those terms but at a cost of twice the storage and twice the computation for the linear algebra, a substantial cost. Output of midside nodes (e. g. 10 noded tets) data to d3eigv and d3mode databases was incorrect in MPP for implicit. Code was add to correct this issue. Enhance implicit-explicit switching (IMFLAG 0 in CONTROL IMPLICIT GENERAL ) so that curve IMFLAG can be defined using DEFINE CURVE FUNCTION . Fix implicit treatment of constraint-based tied contacts that feature offsets. Upgrade the implicit implementation of rack and pinion and screw joints ( CONSTAINED JOINT RACK AND PINION. CONSTRAINED JOINT SCREW ) so the joint is driven by relative motion of the assembly instead of absolute motion. Fix MPP implementation of LaGrange Multiplier treatment of joints (LMF 1 in CONTROL RIGID. CONSTRAINED JOINT option) to properly handle prescribed motion constraints. Fix handling of multiple cyclic boundary conditions ( BOUNDARY CYCLIC ) by implicit mechanics in SMP (not applicable to MPP). Correct the computation of external work for implicit mechanics problems where the external work is coming from prescribed motion on rigid bodies ( BOUNDARY PRESCRIBED MOTION RIGID ). Fix the computation of forces associated with constraints in implicit. This affects forces output to bndout, jntforc, rcforc and intfor (for tied contact), and spcforc. Fix final output state in spcforc data ( DATABASE SPCFORC ) for implicit mechanics. Enhance implicit to properly output forces for CONSTRAINED RIVET to swforc ( DATABASE SWFORC ). Add CONTACT 1D to implicit mechanics. Fix segmentation fault when using shell formulation 18,20,21 for implicit eigenvalue analysis and when triangular shell elements are present. Fix segmentatin fault when using CONTROL IMPLICIT GENERAL with IMFLAG 4 for implicit/explicit switching. CONTROL IMPLICIT ROTATIONAL DYNAMICS : This new keyword is added to study Rotordynamics using the implicit time integrator. Applications for this feature include the transient and vibration analysis of rotating parts such as turbine blades, propellers in aircraft, and rotating disks in hard disk drives. It is available for beam, shell, solid and thick shell elements. The current implementation requires a double-precession SMP version of LS-DYNA. An MPP implementation is under development. Fix a problem with implicit solutions using CONTACT 2D AUTOMATIC . THERMAL. The thermal gap option (LMIN, LMAX) was not working because the contact pairs data needed to calculate the heat transfer across the gap was being purged of pairs that were not currently generating stiffness. This purging is now done after the thermal step. MAT SEATBELT is supported for implicit by introducing bending stiffness, preferrably used with NSOLVR 12 Nonsymmetric linear solver can be used by specifying LCPACK 3 in CONTROL IMPLICIT SOLVER . CONTROL IMPLICIT SOLVER. SECTION SHELL. SECTION SOLID : User resultant elements (ELFORM 101,102,103,104,104 and NIP 0) can be used with nonsymmetric implicit solver. CONTROL IMPLICIT SOLVER. LOAD SEGMENT NONUNIFORM : Non-symmetric contribution to stiffness matrix for LOAD SEGMENT NONUNIFORM on 4- 6- and 8-noded segments considered. Initial INITIAL LAG MAPPING : new keyword to initialize a 3D Lagrangian mesh from the last cycle of a 2D Lagrangian simulation. Fix incorrect intialization of velocities if using INITIAL VELOCITY GENERATION with STYP 1, i. e. part set for shells with formulation 23 24. Fix incorrect initial velocities when using INITIAL VELOCITY GENERATION with irigid 1 and PART INERTIA with xc yc zc 0 and nodeid 0 with DEFINE TRANSFORMATION . Fix incorrect initial velocity when using INITIAL VELOCITY GENERATION with NX, NY, NZ defined and also PART INERTIA is used. Fix incorrect start time for initialization of velocities when PHASE 0 in INITIAL VELOCITY GENERATION and INITIAL VELOCITY GENERATION START TIME is present. Fix stress initialization with INITIAL STRESS SECTION for tetahedron formulation 17. Assign initial velocities ( INITIAL VELOCITY ) to beam nodes that are generated when release conditions are defined (RT1, RT2, RR1, RR2 on ELEMENT BEAM .) Fix a problem where angular speeds for body-fixed axes were not properly applied for INITIAL VEHICLE KINEMATICS . Isogeometric Elements ELEMENT SHELL NURBS PATCH : Correct stress computation in interpolation elements for isogeometric shells. Correct the thickness change output (istupd. ne.0). Add support for dumping of strain tensor (STRFLG. eq.1) for isogeometric shells via interpolation shells. Add support for dumping of shell internal energy density for isogeometric shells via interpolation shells. Add conventional mass-scaling for isogeometric shells. Improved writing of IGAPLOT-file. Load LOAD BODY POROUS : applies also now to 1D and 2D problems. Add capability LOAD SEGMENT CONTACT MASK. which currently works in MPP only. This feature masks the pressure from a LOAD SEGMENT SET when the pressure segments are in contact with another material. Curve LCID of LOAD NODE can be defined by DEFINE CURVE FUNCTION . USER LOADING : pass more data to user-defined loading subroutine loadud including. nodal moment nodal rotational displacement and velocity nodal translational mass and rotational inertia of each node Add load curves for dynamic relaxation for LOAD THERMAL VARIABLE . Fix invalid error message when using LOAD MOVING PRESSURE with 100 card 2s. Fix incorrect stress initialization with dynamic relaxation when using LOAD SEGMENT SET with AT. ne.0. Fix incorrect loading when using LOAD MASK for triangular 3-D shells. LOAD SEGMENT NONUNIFORM. LOAD SEGMENT SET NONUNIFORM : By specifying a negative load curve ID the applied load becomes a follower force, i. e. the direction of the load is constant with respect to a local coordinate system that rotates with the segment. LOAD SEGMENT CONTROL ADAPTIVE : Fix bug whereby loads on non-adapting parts were affected when 3D adaptivity was invoked. Material Fix bug in MAT 174 whereby the code could crash when input parameter EUR 0 and FRACR 0. Fix bug in MAT ARUP ADHESIVE. When the power-law terms PWRS, PWRT were not both equal to 2, the plastic strain was calculated incorrectly, leading to a post-yield stress-strain response that did not match the description in the manual. This was especially noticeable with rate effects (EDOT2, SDFAC), when the peak stress generated did not match the theoretical rate-enhanced yield stress. Make several enhancements to MAT 172 : Add option for concrete behaviour from Eurocode 2 Part 1-1 (general structural engineering). The default options are taken from Part 1-2 (fire engineering). There are small differences in the compressive stress-strain curve and some other details. This is selected by TYPEC 9. A common design code assumption is that concrete has zero tensile strength. However, the actual (non-zero) tensile strength is used when calculating the maximum shear stress that can be carried across a crack. To allow for this in LS-DYNA models, it is now possible to input the two tensile strengths separately: FT for tension, and FTSHR for use in the shear calculation. These are input on the new optional Card 9. If the user inputs an artificially-low tensile strength (it is a common design assumption that concrete cannot carry any tension), models can crack in an uncontrolled way due to small stress oscillations when loads are first applied. An option has been added (LCFTT on new optional Card 9) to scale down the tensile strength as a function of time. This allows cracks to develop in a more realistic manner. Small correction to the action of TAUMXF: this property is no longer scaled down in response to previous crushing or high temperature. MAT HYPERELASTIC RUBBER ( MAT 077 H ) has new thermal option for material properties. Add MAT ORTHOTROPIC PHASE CHANGE. MAT ELASTIC PHASE CHANGE. and MAT MOONEY-RIVLIN PHASE CHANGE whereby elements change phase as they cross a plane in space. Add P1DOFF to 2D seatbelt material, MAT SEATBELT 2D. to specify a part ID offset for the internally created 1D seatbelt elements. All load curves for MAT 067 can be defined via DEFINE FUNCTION . Enhance MAT CWM : Add support for shell elements. Add support for hardening curves. Yield stress can be supplied as table depending on plastic strain and temperature. Fix bug in MAT HILL 3R 3D where wrong shear stress was calculated. Fix erosion due to damage, maximum shear critical temperature in elastic state for MAT MODIFIED JOHNSON COOK / MAT 107 for solids. Check diagonal elements of C-matrix of MAT 002 / MAT TROPIC ELASTIC and error terminate with message, STR 1306, if any of them are negative. Fix plastic strain tensor update for MAT 082 / MAT PLASTICITY WITH DAMAGE ORTHO . Fix incorrect stress initialization of MAT 057 / MAT LOW DENSITY FOAM using dynain file with INITIAL STRESS SOLID when NHISV is equal to the number of history variables for this MAT 057 . Fix error when using MAT 144 / MAT PITZER CRUSHABLE FOAM with solid tetahedral ELFORM 10. Fix incorrect writing of material data to dyna. str for MAT SEATBELT when using long s. Fix out-of-range forces after dynamic relaxation when using VP 1 for MAT PIECEWISE LINEAR PLASTICITY and non-zero strain rate parameters, C P, and the part goes into plastic deformation during dynamic relaxation. Fix incorrect MAT PLASTICITY COMPRESSION TENSION / MAT 124 yield stresses for shells when SRFLAG 2. The yield stress was not scaled correctly according to strain rate curves, LCSRC and LCSRT. Fix MAT ADD THERMAL EXPANSION which did not work for hyperelastic materials in 2D analysis. Fix incorrect results when using DEFINE TABLE for LCSS of MAT 089 / MAT PLASTICITY POLYMER . Fix incorrect results when using MAT MUSCLE / MAT 156 with PART AVERAGED for the truss elements. Fix segmentation fault when using MAT NONLOCAL with nhv than the number of stored history variables (including plastic strain). Fix effect of shear correction factor, SHRF, in SECTION SHELL and PART COMPOSITE on MAT ORTHO ELASTIC PLASTIC / MAT 108 . Add warning message, INI 382, for MAT HILL FOAM / MAT 177 when fitting is not performed because no load curve, LCSR, for the stretch data is defined. Fix incorrect transformation of load curves and tables by INCLUDE TRANSFORM for MAT 089 / MAT PLASTICITY POLYMER with LCSS set to a table ID. Fix spurious internal energy when using MAT MOONEY-RIVLIN RUBBER with type -2 for CONTROL BULK VISCOSITY and dynamic relaxation enabled. Fix spurious deletion of beam elements when using MAT ADD EROSION with an exclusion number EXCL and either MXEPS not set to the exclusion number, or all failure criteria set to the exclusion number. Fix spurious deletion of beam elements when using MAT ADD EROSION and all failure criteria are set to the exclusion number, EXCL. Fix incorrect stresses/strains when using MAT PIECEWISE LINEAR PLASTIC THERMAL / MAT 255 with LOAD THERMAL OPTION . Fix divide by zero for MAT HILL FOAM / MAT 177 during the stress tensor eigenvector iterations which resulted in elements disappearing in d3plot. Fix MAT SIMPLIFIED JOHNSON COOK ORTHOTROPIC DAMAGE / MAT 099 shell elements which were not deleted despite reaching the rupture strain and number of integration points failed. Fix MAT PLASTIC NONLINEAR KINEMATIC / MAT 165 for implicit analysis. Fix incorrectly large time step when using MAT MODIFIED PIECEWISE LINEAR PLASTICITY without any failure criteria, FAIL/TDEL/EPSMAJ, for solid elements. Fix incorrect viscous force to elout in binout for MAT MUSCLE / MAT 156 . Fix non-zero z-strain when using MAT USER DEFINED MATERIAL MODELS and shell element type 13 with nip 4. Fix zero stress/strains when using IORTHO 1 for MAT USER DEFINED MATERIAL MODELS and shell element type 13. Also fix incorrect strains output in general for shell type 13 and using MAT USER DEFINED MATERIAL MODELS . Add a keyword option called MIDFAIL for MAT 024. ( MAT PIECEWISE LINEAR PLASTICITY ). When MIDFAIL appears in the keyword, failure by plastic strain will only be checked at the mid-plane. If the mid-plane fails, then the element fails. If there are an even number of integration points through the thickness, then the two points closest to the middle will check for failure and the element fails when both layers fail. Enable solid and solid assembly spot welds ( MAT SPOTWELD ) to use the NF parameter for force filtering. Fix the resultant force and moment calculation that is used by solid spot weld assemblies. We were calculating moments that were too small because they omitted the contribution of hourglass control. This was particularly noticable in torsion loading. Add the shear angle in degrees as the first history variable for shell material MAT 214 ( MAT DRY FABRIC ). It starts as 90 and updates if the fibers scissor. Modify fabric material ( MAT FABRIC. MAT 034 ) so that it does not suffer from excessive high frequency noise in the strain and stress at points far from the origin. If fabric damping is used, this noise is controlled, but then strain energy from from the viscous damping may grow throughout the solution. This was observed in single precision but not double precision. A fix is now made which makes the strain calculation more accurate and eliminates this effect entirely. This fix should have little effect on airbag behavior since the high frequency noise does not have much affect on the solution. Correct the calculation of viscous strain energy in fabric material ( MAT FABRIC. MAT 034 ). This effects output only and could result in slightly higher reported strain energy. Expand from 2 to 5 the number of additional cards that can be used for the user defined weld failure, OPT 12 or OPT 22 on MAT SPOTWELD. Now a total of 46 user variables are possible. Add a solid spot weld material option in MAT SPOTWELD to treat the stress state as uniaxial. This option is available for solid assemblies also. The uniaxial option is used by setting the elastic modulus to a negative number on MAT SPOTWELD where the absolute value of E is the elastic modulus. The nodes of solid and solid assemblies are tied to shell elements with constraints that allow almost no movement relative to each other. As a consequence, the strain state is effectively uniaxial meaning no transverse strains are allowed. A weld loaded in tension will therefore develop significant tensile stress in the transverse directions, particularly after a weld has yielded when the strain state is nearly incompressible. This can lead to huge stresses and very slow growth of plastic strain based damage. The uniaxial option assumes that the transverse stress is always zero. In other words, the stress state is uniaxial. The shear stress and axail stress are unmodified, but the two transverse terms are set to zero. These stress terms really contribute nothing since the welds are tied at each end. By zeroing them, plastic strain grows much more quickly and plastic strain based damage is a reasonable option. Add MAT FABRIC form 24 which is a modified version of form 14. The main improvement is that the Poisson s effects work correctly with the nonlinear curves for fiber stress. Also, the output of stress and strain to d3plot are engineering stress and strain instead of 2nd PK stress and Green s strain. Added an option to input curves in engineering stress and strain rather than 2nd PK stress vs. Green s strain. To use this, set DATYP -1 on DEFINE CURVE Add support for up to 24 plys in a sublaminate with MAT CODAM2. Before the change, only 8 plys were possible. MAT THERMAL CHEMICAL REACTION : This is a beta release. The intent of this material is to model a material undergoing a chemical reaction such as an epoxy used in manufacturing composite materials. Epoxies are modeled by an endothermic induction reaction followed by an exothermic autocatlytic reaction. The heat of reaction is an input parameter. MAT 058 : Add possibility to use nonlinear (elastic) stress-strain curves instead of constant stiffnesses (EA, EB, GAB) in MAT 58 ( MAT LAMINATED COMPOSITE FABRIC ). If a negative value is input for EA/EB/GAB, it is assumed that a corresponding elastic stress-strain curve is defined. Add possibility to use strain-rate dependent nonlinear (elastic) stress-strain curves instead of constant stiffnesses (EA, EB, GAB) in MAT 58 ( MAT LAMINATED COMPOSITE FABRIC ) using a table definition (a negative value needs to be defined for EA, EB, GAB to point to the corresponding table ID). Add possibility to define proper poisson ratios PRCA and PRCB. If PRCA and PRCB are not defined, they are set to PRBA. This feature is also added to MAT 158. MAT 100 (solids): Add possibility to use yield curve or table in MAT 100 ( MAT SPOTWELD ) for solid elements if SIGY. lt.0 is used. Add MAT 157 for solid elements. This includes an optional variable IHIS that invokes INITIAL STRESS SOLID to initialize material properties on an element-by-element basis This allows the user to map/initialize anisotropic material properties from an injection molding simulation via INITIAL STRESS SOLID MAT 157 (shells): Add anisotropic scale factor for plastic strain rate to MAT 157. This affects only the viscoplastic formulation VP 1. Improve local stress projection for VP 1. Add optional variable IHIS, similar to that described for solids above. Add new option to MAT 103 for solids: When setting FLAG. eq.4, a table ID may be defined in LCSS which defines for each strain rate value a load curve ID giving the stress versus effective plastic strain for that rate. Only isotropic hardening is implemented for this option and it is only available for solid elements. MAT 136 ( MAT CORUS VEGTER ): Fixed input for N. gt.5. Implemented an alternative, implicit plasticity algorithm (define N. lt.0). In some examples, this enhances stability of the computation significantly. MAT 244 ( MAT UHS STEEL ): In plasticity with non-linear hardening, temperature effects and strain rate effects are now dealt with the same way they are implemented in MAT 106. In particular, strain rate now refers to the plastic strain rate. Modifications are included to avoid NaNs for TRIP algorithm and phase computation. Allow for the definition of start temperatures for each phase change, not only for cooling but also for heating in MAT 244. Parameters FS, PS, BS and MS now accept negative values that point to a load curve defining start temperatures for cooling (first value) and heating (last value). Account for elastic transformation strains, given as a curve wrt temperature. Add feature to MAT 244 for welding simulations. Similar to MAT 270. material can be initialized in a quiet (ghost) state. Material parameters for the ghost material are defined in an additional input card. Material is activated when temperature reaches birth temperature. Furthermore, annealing is accounted for. Feature is active when new parameter CWM 1. Modified formula for Pearlite phase kinetics based on Kirkaldy and Venugoplan (1983). MAT 249 ( MAT REINFORCED THERMOPLASTIC ): Implement new material formulation for shells, which is based on additive split of stress tensor. For the thermoplastic matrix, a thermo-elasto-plastic material is implemented, where the temperature dependence is defined by load curves/tables in the input file. Includes hyperelastic fiber contribution. For any integration point, up to three different fiber directions can be defined. Their (non-linear) response to elongation and shear deformations can also be defined with load curves. Includes input parameters for anisotropic transverse shear stiffness. MAT T07 ( MAT THERMAL CWM ): Add HBIRTH and TBIRTH as user input parameters Fix for combination of MAT USER with GISSMO. Arrangement of history variables was not always correct in that case. One additional parameter (exponent GAMMA) for B-K law of MAT 138 . MAT 187 : Speed-up of load curve lookup for curves with many points. Add new option MAGNESIUM to MAT 233 : MAT CAZACU BARLAT MAGNESIUM. This material model is available for shell and solid elements. Differences between tension and compression are included. Add enhanced damage model with crack closure effects to MAT 104. It is activated by setting FLAG 10 and includes some new parameters. Some improvements for MAT 075 ( MAT BILKHU/DUBOIS FOAM ): Volumetric strain rate can now be averaged over NCYCLE cycles, original input curve LCRATE is used instead of a rediscretized curve, and averaged strain rate is stored as history variable 3. Add new history variables to MAT 123 : A mixed failure indicator as history variable 10 and triaxiality as 11. Decrease memory requirements for MAT ADD EROSION by 50 . Add MAT 098 for tetrahedral solid type 13. Add new history variable 8 to MAT 157 for shell elements: Anisotropic equivalent plastic strain . Add tangent stiffness to MAT 224 for implicit analyses with solid and shell elements. Fix minor single precision issue in strain calculation of MAT FABRIC with FORM 12, 13, or 14. Put internal enery on plastic strain location for MAT 027 solids. Add new option for parameter BETA of MAT 224 : BETA. LT. 0: strain rate dependent amount given by load curve ID - BETA Add d3hsp output of additional user material parameters (LMCA) of MAT USER DEFINED MATERIAL MODELS . Fix for combination of MAT USER. with RYLEN 2 on CONTROL ENERGY. and DAMPING PART STIFFNESS . Add new flag to switch off all MAT ADD EROSION definitions globally. This will be the 1st parameter MAEOFF on new keyword CONTROL MAT . Add option to define a load curve for isotropic hardening in MAT 135. If SIGMA0 0, then SIGMA0 refers to a stress-strain curve. Parameters QR1, CR1, QR2, and CR2 are ignored in that case. MAT CDPM : MAT 273. MAT CONCRETE DAMAGE PLASTIC MODEL (CDPM), has been reimplemented by its original developers (Peter Grassl and Dimitros Xenos at University of Glasgow) for enhanced robustness. A new parameter EFC is introduced governing damage in compression and the bilinear law is exchanged for an exponential one. This model should be used in double precision only. MAT 3-PARAMETER BARLAT : In MAT 036. HR 7 is complemented with biaxial/shear hardening curves. MAT FABRIC MAP : A stress map material for detailed stress response in fabrics, stress can be prescribed through tables PXX and PYY corresponding to functions of biaxial strain states. A compaction effect due to packing of yarns in compression is obtained by specifying BULKC (bulk modulus) and JACC (critical jacobian for the onset of compaction effect). This results in increasing pressure that resists membrane elements from collapsing and/or inverting. Strain rate effects can be obtained by specifying FXX and FYY which in effect scales the stress based on engineering strain rate. A smoothing effect is applied by using a time window DT. A hysteresis option TH is implemented for stability, given in fraction dissipated energy during a cycle. Can also depend on the strain state through a table. MAT GENERAL HYPERELASTIC RUBBER. MAT OGDEN RUBBER : By specifying TBHYS. LT.0 a more intuitive interpolation of the damage vs. deviatoric strain energy is obtained. It requires however that the damage and strain energy axes are swapped. MAT SIMPLIFIED RUBBER : For AVGOPT. LT.0 the absolute value stands for a time window over which the strain rates are averaged. This is for suppressing extensive noise used for evaluating stress from tables. MAT FABRIC : The bending stiffness contribution in MAT 034. ECOAT/SCOAT/TCOAT, is now supported in implicit calculations. Enhance d3hsp to include possible values of ONEMPA for MAT 272 . Add MAT 122 3D which an extension of MAT 122 to solid elements. This material model combines orthotropic elastic behavior with Hill s 1948 anisotropic plasticity theory and its applicability is primarily to composite materials. MPP Fix MPP for table-based friction ( DEFINE FRICTION ) in non-groupable contact. MPP groupable tied contact: Output messages about initial node movement due to projection, like non-groupable routines do. MPP tied contact initialization: Change a tolerance in groupable tied contact bucketsort to match the non-groupable code, and fix the slave node thickness used for beam nodes during initial search in non-groupable contact to match groupable contact. Also, update the slave node from beam thickness calculation for type 9,11, and 12 beams. Fix decomposition compatibility problem, where decomposing in double precision and running in single (or bigendian then littleendian) was not working correctly due to the way ASCII legend data was being stored during decomposition. Fix CONSTRAINED BUTT WELD to behave better if there are no master nodes on my processor. Make fixes for MPP input processing of DEFINE HAZ TAILOR WELDED BLANK . Redo MPP support for CONSTRAINED MULTIPLE GLOBAL which before only worked for some specific problems. Fix MPP s input processing bug in input files that contain both DEFORMABLE TO RIGID and MAT ADD AIRBAG POROSITY LEAKAGE For MPP, set a last known location flag to give some indication of where the processors were if an error termination happens. Each writes a message to their own message file. Look for a line that says When error termination was triggered, this processor was . Fix MPP contact forces to shell element formulations that allow for through thickness compression. There was a summation error leading to incorrect behavior along decomposition boundaries. MPP BEAMS TO SURFACE contact: Remove beam node mass from the penalty stiffness calculation when soft 1 is used, which matches SMP behavior Make sure the pfile. log file gets created in case of termination due to CONTROL STRUCTURED TERM . Fix MPP decomposition problem if there are 2D sliprings. Fix MPP initialization problem for adaptivity when used with tied contacts, which could have resulted in a segmentation fault. Make new MPP eroding contact algorithms the default, but still able to be turned off (for now) Add two new decomposition region-related pfile options nproc and proc so that any given decomposition region can be assigned to some subset of all the processors. nproc takes a single argument, which is a specific number of processors. proc takes a single argument, which is a percentage of processors to use. The old modifies lump and distribute are still available, and are mapped to the new options thusly: MPP ncforc output: sum forces for slave nodes (and ALL of them) to fix possible problems with shared nodes and symmetric contacts. Fix MPP problem of creating a pre-decomposition file while using a jobid. Tweak MPP beam-to-beam contact routine for better handling of parallel beams. MPP: support for new solid and shell cost routines. Not yet suggested for regular use, as they need some filling out to include beams, thick shells, and additional materials. But the framework is all there. MPP tied contact: don t skip force calculation on the very first cycle. MPP contact: improve treatment of initial penetrations for solid elements, which were sometimes being offset in the wrong direction. Fix MPP groupable contact issue with adaptive constraints. MPP force transducers: fix possible memory clobber when some processors in a contact have force transducers and others don t. MPP tied contact with adaptivity: if any master segments get adapted, we were dropping any slave nodes. Now flag these, and do a bucketsort for just these nodes, to find their new tied location. This allows tied contacts to work across adaptive steps. MPP guided cable contact: modify initial bucketsort to better handle beams of very different sizes. Fix MPP CONTACT AUTOMATIC GENERAL in case a processor has beams but no shells: beam contact thicknesses and stiffnesses were not being properly set. Fix MPP for dynamic relaxation in polar coordinates. Part numbers for shared nodes were not consistent, resulting in some nodes not being properly treated. MPP contact: add support for IGAP 2 added to the SINGLE SURFACE. AUTOMATIC GENERAL. Y. TO SURFACE contacts. Fix bug in MPP non-groupable single surface contact when used with force transducers, which was causing memory overwrites in some cases. This problem has existed since r83944. Add MPP support for LOAD ERODING PART SET. which was not properly working before. Fix MPP groupable contact which wasn t working with implicit if there were both constraint and penalty groupable contacts. Change to the way MPP computes slave node areas for AUTOMATIC TIEBREAK contacts (and other that use areas). The calculation was being done every cycle, which is probably not a good idea. Now it is only being done at time 0. Also, a different calculation is being done which I think treats triangles much better, and should result in less mesh dependency in the failure condition of AUTOMATIC TIEBREAK contacts. MPP: - synchronize rigid body flags for shared nodes during rigid-to-deformable switching so that these nodes are handled consistently across processors. MPP spotweld thinning: - skip tied contacts with non-zero birthtime. Fix MPP adaptivity error termination in the case where processors do not all share the same local directory. Add new pfile decomposition region option: partsets Takes a list of partsets (SET PART) from the keyword input and uses them to define a region, like this: region would take those two partsets, scale y by 1000, and decompose them and distribute them to all processors. MPP: change to the decomposition behavior of CONTROL MPP DECOMPOSITION PARTS DISTRIBUTE. CONTROL MPP DECOMPOSITION PARTSET DISTRIBUTE. CONTROL MPP DECOMPOSITION ARRANGE PARTS in the case where a decomposition transformation is also used. Previously, any such regions were distributed without the transformation being applied. This has been fixed so that any given transformation applies to these regions also. Fix sleout output for two-sided force transducers in MPP: No energy calculations were being done for force transducers, so although they looked OK in the rcforc file, the sleout file had all 0s for them. Energies (including frictional energy) should now be OK for two-sided force transducers in MPP. Their energies are NOT included in the total energies, since that would be redundant. A new output column is added to the sleout file indicating which interfaces are from force transducers. These changes are all MPP only, and only for the two-sided force transducers. MPP spotweld deletion: fix bug whereby flags were not properly allocated/set if there were no solid spotwelds in the model, causing beam failure to be missed. This bug was introduced at r86335. Honor TIEDPRJ flag on CONTROL CONTACT for MPP groupable tied interfaces. Increase initial search distance in MPP tied contact to include slave and master thicknesses. Fix summation of master node forces for MPP contact output to ncforc file -- constraint-based contacts only. Tweak MPP INTERFERENCE contact to better handle deep initial penetrations. MPP: reorganize how RIGIDWALL PLANAR FORCES is handled, which greatly improves scaling. Add new experimental square edge option to MPP groupable single surface contact Ignore 32ieee flag when writing interface linking file, since this file may be used as an INPUT file to future double precision runs, and we can t read a single precision file as if it were double precision. This was an MPP-only problem. Add new MPP pfile option: directory will assign different local working directories to different processors, to balance the I/O load. Fix for MPP selective mass scaling (IMSCL 1 in CONTROL TIMESTEP ) when used together with tie-breaking contact. MPP contact with BEAM OFFSET : solids were in some cases still unstable, particularly if the user specified a large contact thickness (positive or negative) for searching purposes. So switch to using 10 of the characteristic edge length instead of 100 of the segment thickness. Speed up input processing of the d3part option for MPP. Change MPP treatment of two-sided force transducers so that proper mass and moment values can be output to the rcforc file. MPP support for non-zero birthtime for CONTACT SINGLE EDGE . Miscellaneous MPP enhancements: Restructure and reduce memory usage of 3D ALE searching of neighboring algorithm. Now, the code can handle hundreds of millions ALE elements during decomposition. Support PARTICLE BLAST . Support SPH 2D contact. Greatly speed up reconstruction of eroding contact surface, (soft 0,1) when using large number of cores. Tune Xeon 64 AVX2 executable for better performance. AVX2 is about 10 better than SSE2 exe. Fix hang up when using DEFINE CURVE FUNCTION with element function BEAM(id, jflag, comp, rm) and running MPP with np 1. Fix end-of-file error for MPP when using more than one LOAD ERODIING PART SET . Fix memory error messages after Normal Termination for MPP when using DAMPING RELATIVE . Restart Add the following options for small restarts: CHANGE VELOCITY GENERATION , CHANGE RIGIDWALL option, PSNFAIL option to CONTROL SHELL Make fixes for adaptivity with small deck restart. Fix MPP for full deck restart in case some processors have no contact interfaces. Fix possible MPP hang during full deck restart if unimplemented contact types are used (e. g. AUTOMATIC GENERAL ). Also, change the resulting warning message to indicate that the contact may not be supported yet. Fix a couple of minor issues for MPP full deck restarts: If the new run specifies an output file (e. g. rcforc) that was not output in the original run, it was not being output in the new run. If a jobid was used, an extra empty copy of the d3full input file was being created with the jobid prepended. MPP full deck restart: Restore behavior consistent with SMP which is that only the nodes of materials being initialized (not all nodes) are initialized from d3full. Fix a bug related to PARAMETER and restart that was resulting in parameters not being properly redefined on the first restart in MPP on processors other than 0, or on the second and subsequent restarts in SMP and MPP (all processors). This is only a problem if there are DEFINE CURVE FUNCTION definitions that use parameters. Free some memory that might have leaked during restarts. MPP: add full deck restart support for AUTOMATIC TIEBREAK contact types. Make change related to missing binout file on restarts: Previously, if the binout file that the code wanted to append to did not exist, it created a brand new series named binout , which caused confusion for some users. Now, it opens whatever would have been the next binout in the series. For example, if the user was running a long problem and already had binout, binout 001 and binout 002, if binout 002 is not available when the restart occurs, the file binout 003 will be created (instead of binout ). So the user can just copy the newly generated binout files in with their old ones, and all the post processing routines will automatically see them. Allow rcforc output in MPP for full deck restart. Implement DELETE PART for seatbelt parts. The associated slipring, retractors and pretensioners will be deactivated as well. Fix ineffective boundary condition for MAT RIGID when using CHANGE RIGID BODY CONSTRAINT with RIGID DEFORMABLE R2D for small restart. Fix internal energy oscillation after full deck restart when using CONTACT TIED SURFACE TO SURFACE OFFSET with TIEDID 1 in optional card D. This affects SMP only. DELETE ELEMENT BEAM / SOLID / SHELL / TSHELL - fix maximum number of elements that can be deleted in small deck restart due to field width limitation. Increase field width from i5 to i10. Fix zero time step after full deck restart. Affects models with discrete elements ( ELEMENT DISCRETE ). Fix input phase error when using Q REMAP on the execution line. Fix small deck restart for blstfor database ( DATABASE BINARY BLSTFOR ). Additonal data was not written out to blstfor after small deck restart. Fix bug when using dynain data from a previous run. Shell formulations 13, 14, 15 with MAT NULL and EOS IGNITION AND GROWTH OF REACTION IN HE were giving incorrect pressure in this situation. Fix invalid error while reading in structured file during restart and using long s. Fix full deck restart for CONTACT AUTOMATIC ONE WAY SURFACE TO SURFACE TIEBREAK for SMP only. Add support for MPP restarts with USA coupling. Fix bug in which curve LCDT from DATABASE BINARY D3PLOT was violated after full deck restart. Fix small restart bug in which INTERFACE COMPONENT FILE ignores the filename FNAME and writes to infmak. Fix bug in which SET NODE LIST GENERATE did not work in a small restart. Sensor Add NREP option to SENSOR CONTROL to repeat NREP cycles of switches given on Card 2. Implement SENSOR CONTROL TYPE s BELTPRET. BELTRETRA and BELTSLIP control the pretensioners, retractors and sliprings of a 2D seatbelt. Add function SENSORD to DEFINE CURVE FUNCTION to return the value of a sensor. Replace SENSOR DEFINE ANGLE with more general SENSOR DEFINE MISC. MTYPEs include ANGLE, RETRACTOR, RIGIDBODY, and TIME. Fix failure of sensor switch to turn off when using SENSOR DEFINE FORCE with ftype PRESC-MOT when running implicit. SPH MAT MOHR COULOMB now available for 2D SPH elements (IDIM 2 or -2 on CONTROL SPH ). This material was already available for 3D SPH. Add rcforc output for CONTACT 2D NODE TO SOLID (supported for ASCII output only not binout). Add temperature output (when applicable) to sphout file ( DATABASE SPHOUT ). Add support of MAT ALE VISCOUS for SPH particles. This allows modeling of non-viscous fluids with constant or variable viscosity, i. e, non-newtonian type fluid using SPH. Add support of EOS for MAT 272 with SPH particles. Add support of MAT 255. MAT 126. and MAT 26 (with AOPT 2 only) for SPH particles. Add new keyword command SECTION SPH INTERACTION : Combined with CONT 1 in CONTROL SPH card, this keyword is used to define the partial interaction between SPH parts through normal interpolation method and partial interaction through the contact option. All the SPH parts defined through this keyword will interact with each other through normal interpolation method automatically. Add support for DATABASE TRACER for axisymmetric SPH (IDIM -2 in CONTROL SPH ). ICONT in CONTROL SPH now affects DEFINE SPH TO SPH COUPLING in the sense of enabling or disabling the coupling for deactivated particles. Storastic The commands STOCHASTIC TBX PARTICLES and CHEMISTRY CONTROL TBX are no available for use (along with the CESE solver) in TBX-based explosives simulations. Multi-nozzle injection mode is implemented for spray injection. Thermal Fix MPP for possible deadlock in adaptivity when used with thermal coupling. Fix bug that IAUTO affected the thermal time step when IAUTO 0 in CONTROL IMPLICIT AUTO and CASE was used. IAUTO should have no bearing on the thermal time step. LOAD THERMAL D3PLOT : This keyword is used to read a d3plot file to define node point temperatures. The format for reading d3plot was changed to be compatibile with the new d3plot data structure. The 1st d3plot family member contains control words, geometry, and other control entities. Time state data begins in the 2nd family member. This change is not backward compatible. The old d3plot data structure, which may be written by 3rd party sofware, will not be read correctly by ls-dyna. CONTROL DYNAMIC RELAXATION : Logic was added to skip thermal computations during dynamic relaxation for a coupled thermal-stress problem (i. e. when SOLN 2 on the CONTROL SOLUTION keyword). This does not affect the use of LOAD THERMAL keywords during dynamic relaxation. Implement DEFINE CURVE FUNCTION for convection, flux, radiation boundary conditions in thermal-only analyses, both 2D and 3D. Combination of LOAD THERMAL VARIABLE BEAM and MAT ADD THERMAL EXPANSION is now working properly. BOUNDARY CONVECTION. BOUNDARY FLUX. and thermal dynamics are implemented for 20 node brick element. Added input error detection for loc 1/-1 for radiation, flux, and temperature boundary conditions when THSHEL is set to a thin or thick thermal shell in PART COMPOSITE . Friction energy distribution to triangular shells is updated. Include the reading of thermal data to INCLUDE BINARY . Miscellaneous Fix bug in long format input for DEFINE STAGED CONSTRUCTION PART . Fix problem of LOCAL parameters and adaptivity. Because the adaptive process merges all the include files, the locality of these was being lost. Two new keywords are introduced ( PARAMETER PUSH and PARAMETER POP ) and are used to work around this issue. Fix problem with collection of contact forces for USER INTERFACE FORCES . Fix bug in LSDA option of INTERFACE COMPONENT FILE : the titles of the interfaces were clobbered before being written to the linking file. Also, add general job information to the LSDA linking file (date of run, title of job, executable information). Allow DEFINE FUNCTION TABULATED to be used in any place that requires a function of 1 variable. Specifically, as a displacement scale factor with INTERFACE LINKING NODE . Make fixes to NODE TRANSFORM. which was not working correctly with SET NODE GENERAL Add new MUTABLE option for PARAMETER and PARAMETER EXPRESSION to indicate that it is OK to redefine a specific parameter even if PARAMETER DUPLICATION says redefinition is not allowed. Also, only honor the first PARAMETER DUPLICATION card. Add functions DELAY and PIDCTL to DEFINE CURVE FUNCTION for simulating PID (proportional-integral-derivative) controllers. Fix incorrect dyna. inc file when using MAT FU CHANG FOAM / MAT 83. DEFINE COORDINATE NODES. and CONSTRAINED JOINT STIFFNESS GENERALIZED with INCLUDE TRANSFORM . Fix incorrect shell set generated when using SET SHELL GENERAL with OPTION PART. Fix segmentation fault when reading dynain. bin ( INCLUDE BINARY ( TRANSFORM )). Fix invalid error message node set for nodal rigid body not found when using PART INERTIA with CONTROL SUBCYCLE . Fix NODE MERGE SET which gives incorrect set of nodes. Fix invalid error message for null part set when using SET PART LIST GENERATE and PART MOVE together. The delay time, TIME3, in DEFORMABLE TO RIGID AUTOMATIC was not correctly implemented. The TIME3 in the last DEFORMABLE TO RIGID AUTOMATIC was used instead. This is now fixed. Fix incorrect sign of result returned from DEFINE CURVE FUNCTION s JOINT(id, jflag, comp, rm) if jflag 1. Fix invalid error reported when mesh adapted due to CONTROL ADAPTIVITY and PART ANNEAL was also invoked. Fix invalid error triggered by having a parameter (as previously defined by PARAMETER ) included anywhere in a section of input commented out via COMMENT . Activate DEFINE COORDINATE NODES axis flag, DIR, even when FLAG 0. Formerly, DIR was only activated when FLAG 1. Fix incorrect nodes generated by SET NODE GENERAL with OPTION PART when used together with PART MOVE and IFSET 1, i. e. a part set id is used. When using TERMINATION DELETED SOLIDS SET or TERMINATION DELETED SHELLS SET only the first part in the part set was counted and used to determine the termination. This is now fixed. Fix output of zeros/NaNs to curvout for DEFINE CURVE FUNTIONS AX/AY/AZ/WX/WY/WZ. Fixed invalid error during keyword input phase when using alphanumeric label for SECTION and SET NODE GENERAL . Fix invalid error during keyword input phase when PLOTEL 1 in CONTROL RIGID and number of CONSTRAINED NODAL RIGID BODIES exceeds 500. DEFINE TABLE : Add table curves check for mismatching origin or end points. Update ANSYS library to version 16.0. INTERFACE COMPONENT : Fix bug in which command line option z isf1 produces unusable binary data. Enhance report of Elapsed time in d3hsp. The format is as follows for SMP, MPP, and Hybrid. Add keyword INCLUDE UNITCELL to create a keyword file containing user-defined unit cell information with periodic boundary conditions. Add a new keyword INCLUDE AUTO OFFSET. With this keyword, the node and element IDs of the include file will be checked with the node and element IDs of the previous included/read files to see if there is any duplication. If duplicate element or node ID is found, it will be replaced with another unique id. Required input is a incoming file name. Advances in mathematical programming models for enterprise-wide optimization Ignacio E. Grossmann . Center for Advanced Process Decision-making, Department of Chemical Engineering, Carnegie Mellon University, Pittsburgh, PA 15213, United States Received 1 April 2012. Revised 27 June 2012. Accepted 28 June 2012. Available online 5 July 2012. Abstract Enterprise-wide Optimization (EWO) has become a major goal in the process industries due to the increasing pressures for remaining competitive in the global marketplace. EWO involves optimizing the supply, manufacturing and distribution activities of a company to reduce costs, inventories and environmental impact, and to maximize profits and responsiveness. Major operational items include planning, scheduling, real-time optimization and control. We provide an overview of EWO in terms of a mathematical programming framework. We first provide a brief overview of mathematical programming techniques (mixed-integer linear and nonlinear optimization methods), as well as decomposition methods, stochastic programming and modeling systems. We then address some of the major issues involved in the modeling and solution of these problems. Finally, based on the EWO program at the Center of Advanced Process Decision-making at Carnegie Mellon, we describe several applications to show the potential of this area. Keywords Planning Scheduling Supply chain optimization Mixed-integer programming Stochastic programming Fig. 1. La Fig. 2. La fig. 3. La fig. 4. Fig. 5. Table 3. Fig. 6. La Fig. 7. La fig. 8. La fig. 9. La fig. 10. Table 4. Fig. 11. La Fig. 12. La Fig. 13. La fig. 14. Table 5. Fig. 15. Fig. 16. Fig. 17. Fig. 18. Corresponding author. Tel. 1 412 268 3642 fax: 1 412 268 7139. Copyright 2012 Elsevier Ltd. All rights reserved. Citing articles ( ) Table of Contents for the GuideBooks Programming Volume 1.0 Remarks 1.1 Basics of Mathematica as a Programming Language 1.1.1 General Background In and Out Numbering General Naming, Spelling, and Capitalization Conventions for Symbols Options and Option Settings Messages Add-On Packages 1.1.2 Elementary Syntax Common Shortcuts Parentheses, Braces, and Brackets Comments Inside Code Font Usage Referring to Outputs Functional Programming Style Ideal Formatting 1.2 Introductory Examples 1.2.0 Remarks 1.2.1 Numerical Computations Periodic Continued Fractions Pisot Numbers Fast Integer Arithmetic Digit Sums Numerical Integration Numerical ODE Solving Burridge-Knopoff Earthquake Model Trajectories in a Random Two-Dimensional Potential Numerical PDE Solving Benney PDE Sierpinski Triangle-Generating PDE Monitoring Numerical Algorithms Hilbert Matrices Distances between Matrix Eigenvalues Special Functions of Mathematica l Physics Sums and Products Computing a High-Precision Value for Euler s Constant Numerical Root-Finding Roots of Polynomials Jensen Disks De Rham s Function Logistic Map Built-in Pseudo-Compiler Forest Fire Model Iterated Digit Sums Modeling a Sinai Billiard 1.2.2 Graphics Gibbs Phenomena Fourier Series of Products of Discontinuous Functions Dirichlet Function Counting Digits Apollonius Circles Generalized Weierstrass Function 3D Plots Plotting Parametrized Surfaces Plotting Implicitly Defined Surfaces Graphics-Objects as Mathematica Expressions Kepler Tiling Fractal Post Sign Polyhedral Flowers Gauss Map Animation Random Polyehdra 1.2.3 Symbolic Calculations Differentiation Integration Symbolic Solutions of ODEs Vandermonde Matrix LU Decomposition of a Vandermonde Matrix Redheffer Matrix Symbolic Representations of Polynomial Roots Solving Systems of Polynomials Eliminating Variables from Polynomial Systems Series Expansions L Ho spital s Rule Radical Expressions of Trigonometric Function Values Prime Factorizations Symbolic Summation Proving Legendre s Elliptic Integral Identity Geometric Theorem Proofs Using Gr bner Bases Medial Parallelograms Inequality Solving Symbolic Description of a Thickened Lissajous Curve Simplifications under Assumptions Numbers with Identical Digits in the Decimal and Continued Fraction Expansions Conformal Map of a Square to the Unit Disk Vortex Motion in a Rectangle Magnetic Field of a Magnet with Air Gap Localized Propagating Solution of the Maxwell Equation Customized Notations Schmidt Decomposition of a Two-Particle State 1.2.4 Programming Large Calculations Partitioning Integers Binary Splitting-Based Fast Factorial Bolyai Expansion in Nested Radicals Defining Pfaffians Bead Sort Algorithm Structure of Larger Programs Making Platonic Solids from Tori Equipotential Surfaces of a Charged Icosahedral Wireframe Tube along a 3D Hilbert Curve 1.3 What Computer Algebra and Mathematica 5.1 Can and Cannot Do What Mathematica Does Well What Mathematica Does Reasonably Well What Mathematica Cannot Do Package Proposals What Mathematica Is and What Mathematica Not Is Impacts of Computer Algebra Relevant Quotes Computer Algebra and Human Creativity New Opportunities Opened by Computer Algebra Computer Mathematics--The Joy Now and the Joy to Come Exercises Computing Wishes and Proposals Computer Algebra Systems Solutions 100 Proposals for Problems to Tackle Sources of Interesting and Challenging Problems ISSAC Challenge Problems 100 -100-Digit Challenge References CHAPTER 2 Structure of Mathematica Expressions 2.0 Remarks 2.1 Expressions Everything Is an Expression Hierarchical Structure of Symbolic Expressions Formatting Possibilities Traditional Mathematics Notation versus Computer Mathematics Notation Typeset Forms Heads and Arguments Symbols Nested Heads Input Form and the Formatting of Programs 2.2 Simple Expressions 2.2.1 Numbers and Strings Formatting Fractions Integers Autosimplifications Rational Numbers Approximate Numbers Real Numbers Complex Numbers Autonumericalization of Expressions Strings High-Precision Numbers Inputting Approximate Numbers Inputting High-Precision Numbers Approximate Zeros 2.2.2 Simplest Arithmetic Expressions and Functions Basic Arithmetic Operations Reordering Summands and Factors Precedences of Simple Operators Algebraic Numbers Domains of Numeric Functions Autoevaluations of Sums, Differences, Products, Quotients, and Powers 2.2.3 Elementary Transcendental Functions Exponential and Logarithmic Functions Trigonometric and Hyperbolic Functions Exponential Singularities Picard s Theorem Secants Iterations Exact and Approximate Arguments Postfix Notation Infix Notation 2.2.4 Mathematica l Constants Imaginary Unit pi Autoevaluations of Trigonometric Functions Base of the Natural Logarithm Golden Ratio Euler s Constant Directed and Undirected Infinities Indeterminate Expressions 2.2.5 Inverse Trigonometric and Hyperbolic Functions Multivalued Functions Inverse Trigonometric Functions Inverse Hyperbolic Functions Complex Number Characteristics Real and Imaginary Parts of Symbolic Expressions Branch Points and Branch Cuts Branch Cuts Not Found in Textbooks 2.2.6 Do Not Be Disappointed Real versus Complex Arguments Seemingly Missing Simplifications Principal Sheets of Multivalued Functions 2.2.7 Exact and Approximate Numbers Symbols and Constants Numericalization to Any Number of Digits Precision of Real Numbers Precision of Complex Numbers 2.3 Nested Expressions 2.3.1 An Example Constructing Nested Expressions Canonical Order Displaying Outlines of Expressions Displaying Nested Expressions 2.3.2 Analysis of a Nested Expression A Large Expression Parts of Expressions Recursive Part Extraction Depths of Expressions Extracting Multiple Parts Extracting Parts Hierarchically Locating Subexpressions in Expressions Level Specifications Length of Expressions Leaves of Expressions 2.4 Manipulating Numbers 2.4.1 Parts of Fractions and Complex Numbers Rational Numbers as Raw Objects Numerators and Denominators Complex Numbers as Raw Objects Real and Imaginary Parts 2.4.2 Digits of Numbers Digits of Integers Digits of Real Numbers Writing Numbers in Any Base Counting Digits of Numbers Fibonacci Chain Map Animation Overview Exercises Analyzing the Levels of an Expression Branch Cuts of Nested Algebraic Functions Analyzing the Branch Cut Structure of Inverse Hyperbolic Functions Strange Analytic Functions Solutions Principal Roots Analyzing a Large Expression Levels Counted from Top and Bottom Branch Cuts of (z 4) 1/4 Branch Cuts of sqrt(z 1/z)sqrt(z-1/z) Riemann Surface of arctan(tan(z/2)/2) Repeated Mappings of Singularities References CHAPTER 3 Definitions and Properties of Functions 3.0 Remarks 3.1 Defining and Clearing Simple Functions 3.1.1 Defining Functions Immediate and Delayed Function Definitions Expansion and Factorization of Polynomials Expansion and Factorization of Trigonometric Expressions Patterns Nested Patterns Patterns in Function Definitions Recursive Definitions Indefinite Integration Matching Patterns Definitions for Special Values Functions with Several Arguments Ordering of Definitions 3.1.2 Clearing Functions and Values Clearing Symbol Values Clearing Function Definitions Clearing Specific Definitions Removing Symbols Matching Names by Name Fragments Metacharacters in Strings 3.1.3 Applying Functions Univariate and Multivariate Functions Prefix Notation Postfix Notation Infix Notation 3.2 Options and Defaults Meaning and Usage of Options Lists as Universal Containers Options of Functions Plotting Simple Functions Extracting Option Values Setting Option Values 3.3 Attributes of Functions Meaning and Usage of Attributes Assigning Attributes to Functions Commutative Functions Associative Functions Functions Operating Naturally on Lists Numerical Functions Differentiation of Functions Protected Functions Preventing the Evaluation of Expressions Forcing the Evaluation of Expressions 3.4 Downvalues and Upvalues Function Definitions Associated with Heads Function Definitions Associated with Specific Arguments Downvalues and Upvalues Timing for Adding and Removing Definitions Caching Values of Symbols Numerical Values of Symbols 3.5 Functions that Remember Their Values Caching Function Values Multiple Assignments Simplification of Expressions Timings of Computations Takeuchi Function 3.6 Functions in the - Calculus - Calculus Functions as Mappings Functions without Named Arguments Self-Reproducing Functions Splicing of Arguments Sequences of Arguments Pure Functions with Attributes Nested Pure Functions 3.7 Repeated Application of Functions Applying Functions Repeatedly Iterative Maps Solving an ODE by Iterated Integration Iterated Logarithm in the Complex Plane Fixed Points of Maps Fixed Point Iterations Newton s Method for Square Root Extraction Basins of Attractions Cantor Series 3.8 Functions of Functions Compositions of Functions Applying Lists of Heads Inverse Functions Differentiation of Inverse Functions Overview Exercises Predicting Results of Inputs Nice Polynomial Expansions Laguerre Polynomials Puzzles Unexpected Outputs Power Tower Cayley Multiplication Solutions Matching Unevaluated Arguments Equality of Pure Functions Invalid Patterns Counting Function Applications References CHAPTER 4 Meta - Mathematica 4.0 Remarks 4.1 Information on Commands 4.1.1 Information on a Single Command Built-in Function Definitions as Outputs Information about Functions Listing of All Built-in Commands Messages Printing Text and Cells Warnings and Error Messages Wrong and Unexpected Inputs Suppressing Messages Carrying out Multiple Calculations in One Input 4.1.2 A Program that Reports on Functions Converting Strings to Expressions Converting Expressions to Strings String Form of Typeset Expressions 4.2 Control over Running Calculations and Resources 4.2.1 Intermezzo on Iterators Do Loops Multiple Iterators Possible Iterator Constructions Iterator Step Sizes 4.2.2 Control over Running Calculations and Resources Aborting Calculations Protecting Calculations from Aborts Interrupting and Continuing Calculations Collecting Data on the Fly Time-Constrained Calculations Memory-Constrained Calculations Time and Memory Usage in a Session Expressions Sharing Memory Memory Usage of Expressions 4.3 The - Commands 4.3.1 System-Related Commands Mathematica Versions The Date Function Smallest and Largest Machine Real Numbers 4.3.2 Session-Related Commands In and Out Numbering Input History Collecting Messages Display of Graphics Controlling Recursions and Iterations Deep Recursions Ackermann Function 4.4 Communication and Interaction with the Outside 4.4.1 Writing to Files Extracting Function Definitions Writing Data and Definitions to Files Reading Data and Definitions from Files File Manipulations 4.4.2 Simple String Manipulations Concatenating Strings Replacing Substrings General String Manipulations Case Sensitivity and Metacharacters A Program that Prints Itself 4.4.3 Importing and Exporting Data and Graphics Importing and Exporting Files Importing Web Pages Importing From and To Strings Making Low-Resolution JPEGs 4.5 Debugging Displaying Steps of Calculations Evaluation Histories as Expressions Recursion versus Iteration Interactive Inputs 4.6 Localization of Variable Names 4.6.1 Localization of Variables in Iterator Constructions Sums and Products Scoping of Iterator Variables 4.6.2 Localization of Variables in Subprograms Scoping Constructs Lexical Scoping Dynamic Scoping Local Constants Temporary Variables Variable Scoping in Pure Functions Creating Unique Variables Nonlocal Program Flow 4.6.3 Comparison of Scoping Constructs Delayed Assignments in Scoping Constructs Temporarily Changing Built-in Functions Variable Localization in Iterators Scoping in Nested Pure Functions Nesting Various Scoping Constructs Timing Comparisons of Scoping Constructs 4.6.4 Localization of Variables in Contexts Contexts Variables in Contexts Searching through Contexts Manipulating Contexts Beginning and Ending Contexts 4.6.5 Contexts and Packages Loading Packages General Structure of Packages Private Contexts Analyzing Context Changes 4.6.6 Special Contexts and Packages Developer Functions Special Simplifiers Bit Operations Experimental Functions Standard Packages 4.7 The Process of Evaluation Details of Evaluating an Expression Analyzing Evaluation Examples Standard Evaluation Order Nonstandard Evaluations Held Arguments Overview Exercises Frequently Seen Messages Unevaluated Arguments Predicting Results of Inputs Analyzing Context Changes Evaluated versus Unevaluated Expressions Solutions Shortcuts for Functions Functions with Zero Arguments Small Expressions that Are Large Localization of Iterator Variables Dynamical Context Changes Local Values References CHAPTER 5 Restricted Patterns and Replacement Rules 5.0 Remarks 5.1 Boolean and Related Functions 5.1.1 Boolean Functions for Numbers Truth Values Predicates Functions Ending with Q Numbers and Numeric Quantities Integer and Real Numbers Compound Numeric Quantities Exact and Inexact Numbers Primality Gaussian Primes Stating Symbolic and Verifying Numeric Inequalities Comparisons of Numbers Ordering Relations Positivity 5.1.2 Boolean Functions for General Expressions Testing Expressions for Being a Polynomial Vectors and Matrices Mathematica l Equality Equality and Equations Structural Equality Identity of Expressions Equality versus Identity Canonical Order Membership Tests 5.1.3 Logical Operations Boolean Operations And, Or, Not, and Xor Rewriting Logical Expressions Precedences of Logical Operators 5.1.4 Control Structures Branching Constructs The If Statement Undecidable Conditions While and For Loops Prime Numbers in Arithmetic Progression 5.1.5 Piecewise Functions Piecewise Defined Functions Canonicalization of Piecewise Functions Composition of Piecewise Functions Interpreting Functions as Piecewise Functions Specifying Geometric Regions Endpoint Distance Distribution of Random Flights 5.2 Patterns 5.2.1 Patterns for Arbitrary Variable Sequences Simple Patterns Patterns for Multiple Arguments Testing Patterns Named Patterns Trace of Products of Gamma Matrices Shortcuts for Patterns Avoiding Evaluation in Patterns Literal Patterns 5.2.2 Patterns with Special Properties Optional Arguments Default Values for Optional Arguments Repeated Arguments Excluding Certain Patterns Alternative Arguments Restricted Patterns Pattern Tests Conditional Patterns Recursive Definitions Pattern-Based Evaluation of Elliptic Integrals Generating Tables Selecting Elements from Lists All Syntactically Correct Shortcuts 5.2.3 Attributes of Functions and Pattern Matching Pattern Matching in Commutative and Associative Functions Arguments in Any Order Nested Functions Automatic Use of Defaults Analyzing Matchings and Recursions in Pattern and Attribute Combinations 5.3 Replacement Rules 5.3.1 Replacement Rules for Patterns Immediate and Delayed Rules One-Time and Repeated Replacements Unevaluated Replacements Common Pattern Matching Pitfalls Finding All Possible Replacements Scoping in Rules Replacements and Attributes Modeling Function Definitions Options and Rules Replacing Position-Specified Parts of Expressions 5.3.2 Large Numbers of Replacement Rules Optimized Rule Application Complexity of Optimized Rule Application 5.3.3 Programming with Rules Examples of Rule-Based Programs Splitting Lists Cycles of Permutations Sorting of Complex Numbers Cumulative Maxima Dividing Lists House of the Nikolaus Polypaths Rule-Based versus Other Programming Styles 5.4 String Patterns Strings with Pattern Elements Patterns for Character Sequences String-Membership Tests Shortest and Longest Possible Matches Overlapping Matches Counting Characters Replacing Characters All Possible Replacements Analyzing the Online Documentation Cumulative Letter Frequencies Overview Exercises Rule-Based Expansion of Polynomials All Possible Patterns from a Given Set of Shortcuts Extending Built-in Functions General Finite Difference Weights Zeta Function Derivatives Operator Products q-Binomial Theorem q-Derivative Ordered Derivatives Differentiating Parametrized Matrices Ferrer Conjugates Hermite Polynomial Recursions Peakons Puzzles Catching Arguments and Their Head in Calculations Nested Scoping Solutions Modeling Noncommutative Operations Campbell-Baker-Hausdorff Formula Counting Function Calls Using Side Effects q-Deformed Pascal Triangle Ordered Derivative Avoiding Infinite Recursions in Pattern Matchings Dynamically Generated Definitions References CHAPTER 6 Operations on Lists, and Linear Algebra 6.0 Remarks Prevalence of List Manipulations Building Polyhedra by Reflecting Polygons Iteratively Animating the Folding Process Based on Iterated Reflections 6.1 Creating Lists 6.1.1 Creating General Lists Lists and Nested Lists as Arrays, Tables, Vectors, and Matrices Timings of Creating Nested Lists Changing Heads of Expressions Summing Elements of Lists 6.1.2 Creating Special Lists Kronecker Symbol and Identity Matrix Levi-Civita Symbol and Antisymmetric Tensors Creating Multiple Iterators Stirling Numbers Subsets and Tuples 6.2 Representation of Lists 2D Formatting of Tables and Matrices Aligning Rows and Columns Formatting Higher-Dimensional Tensors Tensors and Arrays 6.3 Manipulations on Single Lists 6.3.1 Shortening Lists Extracting Elements from Lists Deleting Elements by Specifying Position, Pattern, or Property Prime Sieving 6.3.2 Extending Lists Prepending, Appending, and Inserting List Elements Working with Named Lists 6.3.3 Sorting and Manipulating Elements Rotating Lists Cyclically Sorting Lists Sorting Criteria Analyzing the Built-in Sorting Algorithm Splitting Lists Mapping Functions over Lists Listable Functions Mapping Functions to Expressions and Parts of Expressions Extracting Common Subexpressions Optimized Expressions 6.3.4 Arithmetical Properties of Lists Average Value of a List Sum of a List Variance of a List Quantiles of a List 6.4 Operations with Several Lists or with Nested Lists 6.4.1 Simple Operations Hadamard Arithmetic on Lists Transposing Tensors Permutations Using Side Effects for Monitoring List Algorithms Joining Lists Intersections and Complements of Lists Finding Approximately Identical Elements 6.4.2 List of All System Commands Working with Unevaluated Expressions Options and Attributes of All Built-in Functions Analyzing All Built-in Function Names Dependencies of Definitions 6.4.3 More General Operations Contractions and Kronecker Products--Inner and Outer Products Rotations in 3D Cross Products Threading Functions over Lists 6.4.4 Constructing a Crossword Puzzle A Large, List-Based Calculation Example Construction Manipulating Function Definitions through Downvalues Crossword Array of All Built-in Functions Crossword Array of All Package Functions Crossword Array of All Named Characters 6.5 Mathematica l Operations with Matrices 6.5.1 Linear Algebra Inverse Matrices Determinants Timing Comparisons for Various Element Types Traces of Matrices Modeling Trace Calculations Eigenvalues and Eigenvectors Pauli Matrices Properties of Eigenprojectors Power Method for Finding the Largest Eigenvalue Generalized Eigenvalue Problem Solving Systems of Linear Equations Siamese Sisters Lorentz Transformations in Matrix Form Moore-Penrose Inverse Best Solutions to Overdetermined Linear Systems Algorithms of Linear Algebra Quantum Cellular Automata Extending Linear Algebra Functions 6.5.2 Constructing and Solving Magic Squares Underdetermined Linear Systems Integer Solutions of Linear Systems Decoding and Encoding Magic Squares Finding All Solutions of a Magic Square 6.5.3 Powers and Exponents of Matrices Integer and Fractional Powers of Matrices Exponential Function of a Matrix Trigonometric Functions of Matrices Fractional Powers and Matrix Spectral Decompositions Matrix Evolution Equations Time-Development of a Linear Chain Cayley-Hamilton Theorem Characteristic Polynomials 6.6 The Top Ten Built-in Commands Finding Filenames Working with Unevaluated Expressions Counting Function Uses Reading Packages Zipf s Law Analyzing Notebooks, Cell Types, References, Typeset Structures, and Text Overview Exercises Benford s Rule Timing Comparisons for List Operations Sum-Free Sets Generating an Index for This Book Consistency of References Line Length Distribution Spacing Check Moessner s Process Ducci s Iterations Stieltjes Iterations Pseudorandom trees Levi-Civita Tensor Contractions Dirac Matrices Products Determinants of Multidimensional Arrays Mediants d Hondt Voting Identifying Approximate Vectors Efficiently Unsorted Complements All Arithmetic Expressions Ideal Formatting Functions with Method Options Functions with Level Specifications Changing Formatting by Programs Pattern Instances Matrix Identities Amitsur-Levitzky Identity Frobenius Formula for Block Matrices Iterative Matrix Square Root Differential Matrix Identities Matrix Derivatives Autoloaded Functions Precedences of All Operators One-Liners Changing 1 Meissel Formula Binary Bracketing Kolakoski Sequence Puzzles Cloning Functions Hash Values Permutation Digit Sets Solutions Chemical Element Data Population Data of US Cities and Villages Caching versus List-Lookup Electronic Publication Growth Statistics of Author Initials Analyzing Bracket Frequencies Word Neighbor Statistics Weakly Decreasing Sequences Finding All Built-in Symbols with Values Automated Custom Code Formatting Making Dynamically Formatted Inputs Working with Symbolic Matrices Downvalues and Autoloading Determining Precedence Automatically Permutation Polynomials Working with Virtual Matrices References Graphics Volume CHAPTER 1 Two-Dimensional Graphics 1.0 Remarks Role of Visualization in and of Mathematics 1.1 Fundamentals 1.1.1 Graphics Primitives Points, Lines, and Polygons Text in Graphics Creating and Displaying Graphics Complex Cantor Sets Dimension Transitions Animation Tree of Pythagoras Generalized Pythagoras Theorem 2D Graphics Sampler with 100 Examples Constructing a Caustic Pedal Curve Projection into 2D Pentagon Tree Meyer Quasicrystal Poincar Model of the Hyperbolic Plane B ttcher Function of the Quadratic Map Complex Continued Fractions From Graphics to Animations Phyllotaxis Spiral Julia Sets Farey Tree Deposition Modeling Rauzy Tessellations Islamic Wicker 1.1.2 Directives for Graphics Primitives Absolute and Relative Sizes of Points and Lines Color Schemes and Color Values Circles Rolling on Circles An Optical Illusion: The Bezold Effect 1.1.3 Options for 2D Graphics Max Bill s Picture of Nested n-gons Influence of Each Options Aspect Ratios Adding Axes to Graphics Labeling Axes Fonts and Typeset Expressions in Graphics Framing Graphics Adding Labels to Graphics Overlaying Graphics Specifying Tick Marks Repeatedly Displaying Graphics 1.1.4 A First Graphics Application: Voderberg Nonagon Polygons that Enclose Each Other Reinhardt s Conjecture Finding Matching Polygons 1.2 Plots of Functions 1.2.1 Plots of Functions Given Analytically The Process of Making a Plot Controlling Smoothness and Resolutions of Plots Iterated Trigonometric Functions Plotting Multiple Functions Absolute Value Approximation Distribution of Bend Angles Fooling the Plotting Function Visualizing High-Order Taylor Series Plotting Parametrized Curves Lissajous Figures Hedgehogs of Curve Families Astroid 1.2.2 Plots of Functions Defined Only at Discrete Points Digit Distributions in Various Bases Nowhere Differentiable Continuous Functions Riemann s Continuous Nondifferentiable Function Minkowski s Function Periodic Continued Fractions Made Continuous 1.3 Combining Several Images 1.3.1 Arrays of Graphics Spirals Arrays of Graphics Inverting Graphics Polyspirals Inscribing Graphics into Rectangles Graphing a Mouse Manipulating Given Graphics Puzzles Made from Subdivided Graphics Clipping Polygons Absolute Size of Text 1.3.2 Animations Vibrating Linear Chain Perron Tree Construction Circles on Circles Microscopic Moir Pattern Tangential Circles in Regular Polygons Julia Set Evolution from Pullbacks of the Quadratic Map Polygonal Radix Representation Lattice Interpolations Po lya s Orchard Problem Dragon Generation Animation 1.4 Packages Graphics Packages Visualizing Graphs Hypercube Wireframe Graphing Implicit Curves Graphing Vector Fields 1.5 Graphics of Iterative Mappings 1.5.0 Remarks 1.5.1 Sierpinski Triangle Iteratively Subdividing Triangles Overlaying Graphics Inverted Sierpinski Triangle Applying Nonlinear Transformations 1.5.2 Peano Curves Space-Filling Curves Filling a Triangle with a Curve Connecting Subdivided Triangles 1.5.3 Lebesgue s Mapping of the Cantor Set Curves Based on Digit Expansions Filling Fractal Curves General Digit Expansions 1.5.4 Subdivision of an L-Shaped Domain Aperiodic Tilings Applying Transformations to Graphics Triangle Subdivisions 1.5.5 Penrose and Substitution Tilings Tilings Using Rhombii Coloring and Painting Tilings Tilings Based on Kites and Darts Manipulating Existing Graphics Fractal Tilings Cut-and-Project Method 1.5.6 Barnsley s Fern, Mazes, and Other Random Images Random Numbers Random Number Generators Generating Random Expressions Law of the Iterated Logarithm Random Sums Random Replacements Bak-Sneppen Model Samples of 2D Graphics that Contain Randomness Eigenvalues of Random Matrices Randomly Nested Radicals Making Concave Polygons Convex Strange Nonchaotic Attractors Random Circle Segment Patterns Kaleidoscopes Mazes Square and Hexagonal Truchet Images Randomly Bent Ropes Iterated Function Systems Barnsley s Fern Searching for Iterated Function Systems Bahar Systems 1.5.7 Koch Curves Koch Curve Generator Random and Deterministic Koch Curves Filling Koch Curves Manipulating Koch Curves 1.5.8 Honeycombs and Escher Drawings Constructing and Coloring Hexagon Lattices Interlocking Lizards Hyperbolic Triangles and Hyperbolic Tilings Inversion on a Circle 1.5.9 Lindenmayer Systems, Monster Curves, Grasses, and Herbs L-System Syntax: Axioms and Replacement Rules Examples of L-Systems Space Filling Curves Filled Gosper Curve L-Systems with Branching L-Systems that Model Plants Random L-Systems 1.6 Coloring Closed Curves Coloring Plots Finding Curve Intersections Sorting 2D Line Segments Loop Construction Constructing the Clusters Checkerboard Coloring Some Examples Checking if Polygons are Disjoint Overview Exercises Game of Life Langton s Ant Brillouin Zones Maxwell-Helmholtz Color Triangle Conformal Maps Cornet Isogons Jarni k Polygons Light Ray Reflections in a Water Drop Warped Patterns Moir Patterns Triptych Fractal Multiple Reflected Pentagons Random Lissajous Figures Walsh Function Sorting Game Ball Moves Rectangle Packings Smoothed L-Systems Polygonal Billiards Random Walk on a Sierpinski Fractal Voronoi Tessellations L vy Flights Random Supersymmetric Potential Common Plotting Problems Nomogram for Quadratic Equation Clusters on Square Grids Aperiodic Triangle Tilings Solutions Random Cluster Generation Leath Clusters Midsector Lines Analyzing Mathematica Code Visualizing Piecewise Linear Approximations Cartesian Ray Kepler Cubes Modulated Sin-Curves Superimposed Lattices Triptych Fractals Two Superimposed Bumps Forming Three Bumps Repeatedly Mirrored Decagons Smoothly Connected Curves Randomly Deformed Graphics Random Expressions References CHAPTER 2 Three-Dimensional Graphics 2.0 Remarks 2.1 Fundamentals 2.1.1 Graphics Primitives Points, Lines, and Polygons Cuboids Projecting a Hypercube into 3D Nonplanar and Nonconvex Polygons Translating 3D Shapes Escher s Cube World 2.1.2 Directives for Three-Dimensional Graphics Primitives Absolute and Relative Sizes of Points and Lines Constructing an Icosahedron from Quadrilaterals Coloring Polygons in the Presence of Light Sources Diffuse and Specular Reflection Edges and Faces of Polygons Rotating 3D Shapes Random Rotations Stacked Tubes Text in 3D Graphics 2.1.3 Options for 3D Graphics The 34 Options of 3D Graphics Relative and Absolute Coordinate Systems Space Curves versus Space Tubes 2.1.4 The Structure of Three-Dimensional Graphics Resolving Automatic Option Settings Nested Primitives and Directives Converting 3D Graphics to 2D Graphics 2.1.5 Discussion of Selected Options Platonic Solids Choosing the Viewpoint Simple 3D Shapes Light Sources and Colored Polygons Cluster of Dodecahedra Views on an Octant Filled with Cubes Restricting the Plot Range The 3D Graphics Enclosing Box View Direction Sizing Identical Graphics Independently of the Viewpoint Rendering All versus Rendering Only Visible Polygons Intersecting Polygons Colliding Platonic Solids A Scale with Platonic Solids Diamond Faces Rolled Checkered Paper Woven Tubes Smooth Dodecahedron-Icosahedron Transition Platonic Solid Metamorphosis Slicing a Cube 2.2 Display of Functions 2.2.1 Functions Given Analytically Graphing Functions of Two Variables Special Plotting Options Wireframes Showing Multiple Plots Parametrized Vector Functions Cubed Torus Klein Bottle Parametrized Surfaces Samples Using Symmetries to Construct Graphics Constructing a Candelabra Surfaces of Revolution Emission of an Accelerated Point Charge Borromaen Rings Spiraling Spiral Constructing a Birthday Bow 2.2.2 Functions Given at Data Points Visualizing 2D Arrays of Data Visualizing Computation Timings Time Evolution on a Torus 3D Bar Charts Randomized Geode 2.3 Some More Complicated Three-Dimensional Graphics 2.3.0 Remarks 2.3.1 3D Graphics of Iterative Mappings Rauzy Fractal From a 3D Projection 3D Sierpinski Sponge Exercising a Sierpinski Sponge Kepler Tiling 3D Iterated Function System Random Clusters of Tetrahedra Quaquaversal Tiling 3D Truchet Graphics 3D Space Fillers 2.3.2 Tubular Curves Frenet Frame Tangents, Normals, and Binormals of Space Curves Tubes around Space Curves Knots Mapping Textures to Knots Tubes around Piecewise Straight Curves Biased Random Walk Osculating Circles of Curves 2.3.3 Recursively Colored Easter Eggs Recursively Subdividing Surfaces Deformed Spheres Mapping Patterns to Spheres Rough Surfaces 2.3.4 Klein Bottles Making Surfaces by Gluing the Edges of a Square Spine Curves Cross Sections of Klein Bottles Slicing and Coloring Klein Bottles Deformed Klein Bottles Cubistic Klein Bottles 2.3.5 A Hypocycloidal Torus Triangulating Quadrilaterals Rotating Curves to Sweep out Surfaces Triangulations Surfaces with Holes 2.3.6 The Penrose Tribar Constructing a Tribar Coordinate System Transformations Choosing the Right View Point Calculating the Optimal Viewpoint An Impossible Crate 2.3.7 Riemann Surfaces of Simple Functions Plotting Multivalued Functions Riemann Surfaces of Algebraic Functions Cutting Surfaces along Branch Cuts Surfaces Subdivided Using Tilings A Family of Polynomial Riemann Surfaces Implicit Parametrizations Riemann Surfaces of Nested Logarithms Riemann Surfaces over the Riemann Sphere 2.3.8 Interwoven Polygonal Frames Planes Intersecting Convex Bodies Calculating All Intersections Creating Frames Interweaving Frames Examples of Interwoven Frames 2.3.9 Selfintersecting Origami and 4D Hilbert Curves Paper Folding Models Goffinet Kite Folding Animation Hilbert Curves in Higher Dimensions 2.3.10 The Cover Image: Hyperbolic Platonic Bodies Triangulating Platonic Solids Symmetry Considerations Compact Code Evolution of the Cover Graphics from Version 2 to Version 5 Nonplanar Contraction and Expansion of Polyhedra 2.4 Brillouin Zones of Cubic Lattices Higher Degree Voronoi Regions Simple Cubic Lattice Bisector Planes Intersection of Planes Symmetry of a Cube Forming Brillouin Zones from Polygons Gluing Polygons Together Body-Centered Lattice Face-Centered Lattice Overview Exercises 3D Surface Sampler Warped, Twisted, and Interlocked Tori Dodecahedra Iteratively Reflected on its Faces Snail Trinomial Theorem Visualization Ball Blending Method Loop Subdivision sqrt(3)-Subdivision Algorithm Averaging Closed Curves Projective Plane Model Counting Surfaces for a Given Genus Lattice Pyramids Fractal Mountains Random Walk on a Sphere Projecting onto Polyhedra Alexander s Horned Sphere Polyhedral Caustic Sliced M bius Strip Perspective Modeling Displaying Hidden Edges Generating Platonic Solid Clusters A 4D Platonic Solid--The 120-Cell Folding a Dodecahedron Continuously Changing Polyhedra Inscribing Five Cubes in a Dodecahedron Interwoven Bands around a Dodecahedron Knot Made from Knots Knot with Escher Tiling Gear Chain Animation 3D Peano Polygon Tetraview Riemann Surface Animation Riemann Surface of Kepler Equation Sierpinski Plant Solutions Cayley Cusp Boy Surface M bius Strip Steiner s Cross Cap Henneberg Surface Flying Saucer Construction Random Parametrized Surfaces Dodecahedral Flowers Extruded Platonic Solids Smoothing through Graph Plotting Staggered Trefoil Knots Field Lines of Two Charged Spheres Random Symmetric Polyhedra Graphics of a Screw Arranging Worn Stones Tightly Random Cones Broken Tube Weaving a Torus Constructing Double and Triple Tori from Torus Pieces Massive Wireframes of Platonic Solids Smoothing a Cube Wireframe Smoothing a Stellated Icosahedron Pyramids on Lattices Closed Random Walks Slicing and Coloring a M bius Strip Coordinate System Transformations Kochen-Specker Theorem Smooth Random Functions Subdividing Concave Polygons References CHAPTER 3 Contour and Density Plots 3.0 Remarks 3.1 Contour Plots Contour Graphics Converting Contour Graphics Options of Contour Graphics Cassini Curve Various Sample Contour Plots Functions Varying Strongly Homogeneous Contour Line Density Coloring Contour Plots Contour Graphics in Nonrectangular Domains Speckles and Scarlets from Superimposing 2D Waves Smoothing Contour Lines Superimposed 2D Waves in Symmetric Directions Comparing Options and Option Settings of Plotting Functions Algebraic Description of Polygons Blaschke Products Charged Goffinet Dragon Square Well-Scattering Amplitude 3.2 Density Plots Density Graphics Converting Density Graphics Arrays of Gray or Color Values Lifting Color Value Arrays to 3D Earth Graphics Array Plots Gauss Sums Visualizing Difference Equation Solutions Visualizing Matrices Saunders Pictures Making Photomosaics from Density Plots 3.3 Plots of Equipotential Surfaces Visualizing Scalar Functions of Three Variables Marching Cubes Plots of Implicitly Defined Algebraic Surfaces Implicit Descriptions of Riemann Surfaces Gluing Implicitly Defined Surfaces Smoothly Together Using Reflection and Rotation Symmetries to Visualize Algebraic Surfaces Examples of Surfaces from Spheres, Tubes, and Tori Glued Together An Algebraic Candelabra Joining Three Cylinders Smoothly Zero-Velocity Surfaces Implicit Form of an Oloid Isosurfaces of Data Overview Exercises Clusters of Irreducible Fractions Chladny Tone Figures in Rectangles and Triangles Helmholtz Operator Eigenfunctions of a Tetrahedron Li nard-Wiechert Potential of a Rotating Point Charge Shallit-Stolfi-Barb Plots Random Fractals Functions with the Symmetry of Cubes and Icosahedra Icosahedron Equation Belye Functions Branch Cuts of Hyperelliptic Curves Equipotential Plots of Charged Letters Charged Random Polygon Gauss-Bonnet Theorem Interlocked Double and Triple Tori Inverse Elliptic Nome Contour Plots of Functions with Boundaries of Analyticity Isophotes on a Supersphere Structured Knots Textures on a Double Torus Solutions Visualizing Saddle Points Outer Products Repeatedly Mirrored Matrix Halley Map Generating Random Functions Weierstrass P Function Based Fractal Contour Plots in Non-Cartesian Coordinate Systems Spheres with Handles Cmutov Surfaces Random Surfaces with Dodecahedral Symmetry Polynomials over the Riemann Sphere Random Radial-Azimuthal Transition Contour Lines in 3D Plots Lines on Polygons Slicing Surfaces Euler-Poincar Formula Mapping Disks to Polygons Statistics of n-gons in 3D Contour Plots References Numerics Volume CHAPTER 1 Numerical Computations 1.0 Remarks Summing Machine Numbers Klein s Modular Function and Chazy Equation Discretizing the R ssler System Modeling the Ludwig-Soret Effect 1.1 Approximate Numbers 1.1.0 Remarks 1.1.1 Numbers with an Arbitrary Number of Digits Machine Arithmetic versus High-Precision Arithmetic Modified Logistic Map Numerical Calculation of Weierstrass Functions High-Precision Arithmetic System Parameters Fixed-Precision Arithmetic Random Fibonacci Recursion Smart Numericalization Precision and Accuracy of Real Numbers Precision and Accuracy of Complex Numbers Precision Loss and Gain in Calculations Error Propagation in Numerical Calculations Principles of Significance Arithmetic Error Propagation for Multivariate Functions Collapsing Numeric Expressions Setting Precision and Accuracy of Numbers Guard Digits in High-Precision Numbers The Bits of a Number Sum-Based Methods of Calculating pi Comparing High-Precision Numbers Automatic Switching to High-Precision Arithmetic 1.1.2 Interval Arithmetic Rigorous Arithmetic Notion of an Interval Joining and Intersecting Intervals Modeling Error Propagation Global Relative Attractor of Rationals Maps 1.1.3 Converting Approximate Numbers to Exact Numbers Rational Numbers from Approximate Numbers Continued Fractions Liouville Constant Periodic Continued Fractions Numbers with Interesting Continued Fraction Expansions Continued Fraction Convergents Pseudoconvergents Gauss-Kusmin Distribution Khinchin Constant Khinchin-L vy Theorem Lochs Theorem Canonical Continued Fractions Minkowski Function Generalized Expansions Rounding Numbers Frisch Function Egyptian Fractions 1.1.4 When N Does Not Succeed Using Extra Precision Undecidable Numerical Comparisons Caching High-Precision Results Recursive Prime Number Definition Sylvester Expansion 1.1.5 Packed Arrays Machine Numbers, Tensors, and Packed Arrays Developer Functions for Packed Arrays Invisibility of Packed Arrays Controlling Automatic Packed Array Generation Counting Sums and Products of Sets of Integers Long-Range Correlations in Natural Texts Analyzing Shakespeare s Hamlet Zipf s Law Mean Square Fluctuation of a Random Walk Analyzing a Chapter of This Book Analyzing a PostScript Graphic 1.2 Fitting and Interpolating Functions Fitting Data Least Squares and Pseudoinverses Approximate Solution of the Helmholtz Equation by Plane Wave Expansion Nonlinear Fits File Size Distribution Polynomial Interpolation of Data Neville Algorithm Convergence and Divergence of Polynomial Interpolations Runge Phenomena Newton-Cotes Weights Interpolating Functions Smoothness of Interpolating Functions Curvature Driven Evolution Dissecting an Interpolating Function Splines 1.3 Compiled Programs Compiling a Calculation Compiled Functions Julia Set of the Quadratic Map Timing Comparisons for Compiled Procedural and Functional Programs Randomized Fibonacci Iterations Products of partial Sums of Random Variables Hansen-Patrick Root-Finding Method Distances in Truchet Images Cycles in Iterated Exponentiation Ikeda Map 3D Period-Doubling Animation Sandpiles Identity Sandpile Nonlocal Cellular Automata Caustics from Refraction 1.4 Linear Algebra Finite Resistor Network Exact versus Approximate Solutions Avoiding Numericalization of Indicies Calculating Resistances Through Eigenvalues Tagaki Function Numerical Solution of a Functional Equation Fixed-Precision Arithmetic in Linear Algebra Modular Equation for Klein s Modular Function Null Spaces of Linear Systems Bound State in a Waveguide Crossing Sparse Matrices Square Network with Random Resistance Values Anderson Model 1.5 Fourier Transforms Discretized Periodic Functions Fourier Transform Amplitude and Frequency Modulation Approximating a Function Uncertainty Relations Strang s Strange Figures Timing Comparisons of Numerical Fourier Transforms Inverse Fourier Transforms Fourier Transforms of Arrays Approximating the Gosper Curve Fourier Transforms of Aperiodic Tilings Fractional Fourier Transform High-Precision Frequency Approximation of Data Approximating the Continuous Fourier transform List Convolutions and Correlations Manipulating Bitmap Graphics Visualizing Trigonometric Identities 1.6 Numerical Functions and Their Options Common Options of Numerical Functions Precision To Be Used in Calculations Machine Precision versus High-Precision Precision Goal for a Numerical Calculation Accuracy Goal for a Numerical Calculation Accuracy Goals for Independent and Dependent Variables Monitoring Numerical Calculations Evaluation Order in Numerical Function Avoiding the Evaluation of the First Argument Using Vector-Valued Variables Dummy Variable-Free Function Calls 1.7 Sums and Products Numerical Products Options of Numerical Product Calculations Compensated Summation Order Sensitivity in Floating Point Summations Numerical Sums Options of Numerical Summation Verifying Convergence Borel-Tanner Distribution Sequence Transformations Numerically Summing Divergent Series Continuous Integer Spiral 1.8 Integration Numerically Integrating a Function Introductory Examples Integrable Singularities Dealing with Singularities along the Integration Path Contour Integration Constructing Integration Path Iterators Monitoring Numerical Integration Matrix Functions Defined through Integrals Options of Numerical Integration Accuracy and Precision of Results Termination Conditions Methods of Numerical Integration Integrating Discontinuous Functions Comparison of Basic Integration Methods Visualization of the Sample Points Gauss Linking Number Area of a Supersphere Comparing Multidimensional Integration Methods Double Exponential Method Monte-Carlo and Quasi Monte-Carlo Integration Distribution of Monte Carlo Sample Points van Der Corput Sequences Integration of Piecewise Continuous Functions Using Symmetries of the Integrands Picard-Lindel f Iteration 1.9 Solution of Equations Numerical Solution of Polynomials, Polynomial Systems, and Arbitrary Functions Sensitivity of Polynomial Roots to Changes in a Coefficient Iterated Roots Distances between Polynomial Roots Hofstadter s Butterfly Schr dinger Equation for Periodic Potential and Applied Magnetic Field Farey Sequences Hofstadter Butterfly on a Finite Lattice Kohmoto Model B zout and Bernstein Bounds for the Number of Roots of Polynomial Equations Quadrature Weights Root Finding of General Functions Monitoring the Search Path Adapative Precision Raising Termination Conditions Root-Finding Methods Methods of Numerical Equation Solving Calculating Jacobians Multiple Roots and Roots of Noninteger Order Variable-Free Minimization Voderberg Spiral Nested Touching Circles 1.10 Minimization Finding the Minimum Methods of Numerical Minimization Visualizing Search Paths Method Option Choices for Numerical Optimization Minimizing Sums of Squares Sliding Down a Spiral Slide Finding Global Minima Minimum Energy Configuration of n Electrons in a Disk Iterative Minimizations 1.11 Solution of Differential Equations 1.11.1 Ordinary Differential Equations Boundary and Initial Value Problems Interpolating Functions as Solutions Differential-Algebraic Equations Pendulum ODE Anharmonic Oscillator with Random Forcing Squatting on a Swing Newton Vector Field Spiral Waves 4D Chaotic Attractor Energy Bands in a Random Complex Potential Stiff and Nonstiff Systems Precision Control Nonlinear Differential Equation with Isochronous Solutions Buchstab Function Higher Order ODEs Ablowitz-Ladik Chain Particle Motion in a Wave Field Chazy Equation Boundaries of Analyticity Generalized Airy Functions Monitoring Numerical Differential Equation Solving Stepsize Control Coupled Pendulums Restricting the Solutions Stopping the Solution Process Calculating and Visualizing Pursuits Finding the Initial Slope for the Thomas-Fermi Equation Forced Coupled Oscillators Chaotic Scattering on a Four-Hill Potential Events in Differential Equation Solving Vector and Matrix Differential Equations Method Option Choices Integrated Brownian Motion Modified Lorenz System Calculating Contour Curves Through Differential Equations Geodesics on a Triple-Periodic Surface Using Homotopies to Solve Polynomial Systems Modeling Newton s Cradle Trajectories in Central Force Fields Three-Body Scattering Interacting Vortices Periodic Orbits of the Restricted Three-Body Problem Combining Numerical Functions Periodic Orbits of the Lorenz System Bohm s Quantum Trajectories Continuous Time Random Walks on Graphs Sparse Arrays in Differential Equations 1.11.2 Partial Differential Equations Parabolic and Hyperbolic PDEs 1D Schr dinger Equation with Dirichlet Boundary Conditions Scattering on a Potential Wall 1D Wave Equation PDE-Specific Options Singular Initial Conditions Wave Function Shredding in an Infinite Well of Time-Dependent Width Fokker-Planck Equation for a Damped Anharmonic Oscillator Liouville Equation for an Anharmonic Oscillator Klein-Gordon Equation Differential Equations with Mixed Derivatives Nonlinear Schr dinger Equation Complex Ginzburg-Landau Equation Zakharov Equations Prague Reaction-Diffusion Model 1.12 Two Applications 1.12.0 Remarks 1.12.1 Visualizing Electric and Magnetic Field Lines Differential Equations for Field Lines Field Lines of 2D Charge Configurations Reusing Programs Stopping Criteria for Field Lines Field Lines for 3D Charge Configurations Field Lines as Tubes Field Lines of Magnetic Fields Biot-Savart Rule Magnetic Field Lines of a Peano Curve-Shaped Wire Nonclosed Magnetic Field Lines Field Lines of a Ring Coil 1.12.2 Riemann Surfaces of Algebraic Functions Algebraic Functions as Bivariate Polynomials Faithful Riemann Surfaces Implicit Parametrizations Branch Cuts and Branch Points Discriminant First Order ODEs for Algebraic Functions Sheets of Riemann Surfaces Samples of Riemann Surfaces Overview Exercises Logistic Map Randomly Perturbed Iterative Maps Functions with Boundaries of Analyticity q-Trigonometric Functions Franel Identity Bloch Oscillations Courtright Trick Hannay Angle Harmonic Nonlinear Oscillators Orbits Interpolating Between Harmonic Oscillator and Kepler Potential Shooting Method for Quartic Oscillator Eigenvalues of Symmetric Tridiagonal Matrices Optimized Harmonic Oscillator Expansion Diagonalization in the Schwinger Representation M bius Potential Bound States in the Continuum Wynn s Epsilon Algorithm Aitken Transformation Numerical Regularization Scherk s Fifth Surface Clebsch Surface Smoothed Dodecahedron Wireframe Standard Map Stochastic Webs Forced Logistic Map Web Map Strange Attractors H non Map Triangle Map Basins Trajectories in 2D Periodic Potentials Egg Crate Potential Pearcey Integral Charged Square and Hexagonal Grids Ruler on Two Fingers Branched Flows in Random Potentials Maxwell Line Iterated Secant Method Steps Unit Sphere Inside a Unit Cube Ising-Model Integral Random Binary Trees Random Matrices Iterated Polynomial Roots Weierstrass Root Finding Method Animation of Newton Basins Lagrange Remainder of Taylor Series Nodal Lines Bloch Equations Branch Cuts of Hyperelliptic Curves Strange 4D Attractors Billiard with Gravity Schwarz-Riemann Minimal Surface Jorge-Meeks Trinoid Random Minimal Surfaces Precision Modeling Infinite Resistor Networks Auto-Compiling Functions Card Game Modeling Charges With Cubical Symmetry on a Sphere Tricky Questions Very High-Precision Quartic Oscillator Ground State 1D Ideal Gas Odlyzko-Stanley Sequences Tangent Products Thompson s Lamp Parking Cars Seceder Model Avoided Patterns in Permutations Cut Sequences Exchange Shuffles Frog Model Second Arcsine Law Average Brownian Excursion Shape ABC-System Vortices on a Sphere Oscillations of a Triangular Spring Network Lorenz System Fourier Differentiation Fourier Coefficients of Klein s Function Singular Moduli Curve Thickening Random Textures Random Cluster Growth First Digit Frequencies in Mandelbrot Set Calculation Interesting Jerk Functions Initial Value Problems for the Schr dinger Equation Initial Value Problems for 1D, 2D, and 3D Wave Equation Continued Inverse Square Root Expansion L roth Expansion Lehner Expansion Brjuno Function Sum of Continued Fraction Convergents Errors Average Scaled Continued Fraction Errors Bolyai Expansion Symmetric Continued Fraction Expansion Solutions Solving Polynomials Using Differential Equations Stabilizing Chaotic Sequences Oscillator Clustering Transfer Matrices Avoided Eigenvalue Crossings Hellmann-Feynman Theorem Scherk Surface Along a Knot Time-Evolution of a Localized Density Under a Discrete Map Automatic Selection of Interesting Graphic Gradient Fields Static and Kinematic Friction Smoothing Functions Eigenvalues of Random Binary Trees Basins of Attraction Fractal Iterations Calculating Contour Lines Through Differential Equations Manipulating Downvalues at Runtime Path of Steepest Descent Fourier Series Arc Length Poincar Sections Random Stirring Heegner Numbers Quantum Random Walk Quantum Carpet Coherent State in a Quantum Well References CHAPTER 2 Computations with Exact Numbers 2.0 Remarks Using Approximate Numerics in Exact Calculations Integer Part Map Misleading Patterns Primes in Quadratic Polynomials 2.1 Divisors and Multiples Factoring Integers Number of Prime Factors Divisors Sum of Squares Derivative of an Integer mod Function Rotate and Mod nth Digit of a Proper Fraction Sch nberg s Peano Curve Greatest Common Divisors and Least Common Multiples Euclidean Algorithm Classical and Generalized Maurer Roses de Bruijn Medallions and Friezes 2.2 Number Theory Functions Prime Numbers Prime Number Spiral Prime Counting Function Euler s Totient Function Absolutely Abnormal Number M bius Function Redheffer Matrix M bius Inversion Calculating Fourier Transforms through M bius Inversion Jacobi Symbol Reciprocity Law 2.3 Combinatorial Functions Factorials Digits of Factorials Stirling s Formula Binomials and Multinomials Nested Triangle Patterns Stirling Numbers Counting Partitions Generating Partitions Partition Identities 2.4 Euler, Bernoulli, and Fibonacci Numbers Akiyama-Tanigawa Algorithm Euler-Maclaurin Formula Lidstone Approximations Boole Summation Formula Divide-and-Conquer Algorithm for Calculating Large Fibonacci Numbers Fibonacci-Binomial Theorem Discretized Cat Map Overview Exercises Sum of Divisor Powers Recurrence Relation for Primes Arcsin Law for Divisors Average Length of Continued Fractions of Rationals Isenkrahe Algorithm Prime Divisors Kimberling Sequence Cantor Function Integral Cattle Problem of Archimedes Mirror Charges in a Wedge Periodic Decimal Numbers Digit Sequences in Numbers Numbered Permutations Binomial Coefficient Values Smith s Sturmian Word Theorem Modeling a Galton Board Ehrenfest Urn Model Ring Shift Modeling Sandpile Model Longest Common Subsequence Riffle Shuffles Weekday from Date Easter Dates Lattice Points in Disks Binomial Digits Average of Partitions Partition Moments 15 and 6174 Selberg Identity Kluyver Identities Ford Circles Farey-Brocot Interval Coverings Sum of Primes Visualizing Eisenstein Series Magnus Expansion Rademacher Identity Goldbach Conjecture Zeckendorf Representation Sylvester-Fibonacci Expansion Ramanujan Tau Function Cross-Number Puzzle Cyclotomic Polynomials Generalized Bell Polynomials Online Bin Packings Composition Multiplicities Subset Sums Solutions Nested Iterators Being Prime Expressed Analytically Legendre Symbol Pell Equation Nested Radicals Identity Recognizing Algebraic Numbers Iterated Digit Sum of Divisors Guiasu Prime Counting Formula Divisor Sum Identities Choquet Approximation Optical Factoring Generalized Multinomial Theorem Sums with Constraints Fa di Bruno Formula Symbolic Tables References Symbolics Volume CHAPTER 1 Symbolic Computations 1.0 Remarks 1.1 Introduction General Assumptions about Variables Simplifying Expressions Type Declarations for Simplifications Evaluating Expressions Under Assumptions 1.2 Operations on Polynomials 1.2.0 Remarks 1.2.1 Structural Manipulations on Polynomials Expanding and Factoring Polynomials Factors of Random Polynomials Irreducible Polynomials Constructing Irreducible Polynomials from Primes Factorization over Extension Fields Reordering Multivariate Polynomials Indeterminates of Polynomials Extracting Coefficients from Polynomials Decomposing Polynomials 1.2.2 Polynomials in Equations Polynomial Division Resultants Sylvester Matrix Differential Equation for the Elliptic Nome Gr bner Bases Applications of Gr bner Bases Equation Solving Using Gr bner Basis Approximative Gr bner Bases Monomial Orders Showing Inconsistency of Equations Using Gr bner Bases Finite-Dimensional Representation of the Canonical Commutation Relations Eliminating Variables Using Gr bner Bases Geometric Theorem Proving All Square Roots of Square Matrices Bound States in Spherical Symmetric Potentials Gr bner Walks Reducing Polynomials 1.2.3 Polynomials in Inequalities Cylindrical Algebraic Decompositions Solving Inequalities Locally Parametrizing a Squeezed Torus Arnold Cat Map Generic Cylindrical Algebraic Decomposition Quantifier Elimination Generally Proving Inequalities Proving Triangle Inequalities Deriving New Geometry Theorems Restricting Polynomial Roots Proving the Sendov-Iliev Conjecture for Quadratic Polynomials Deriving Clauser-Horn Inequalities Algebraic Blending Minkowski Sums 1.3 Operations on Rational Functions Numerators and Denominators Expanding Parts of Nested Fractions Partial Fraction Decomposition Writing Rational Functions over Common Denominators Gale-Robinson Sequence The Power of Togethering Mapping of the Fundamental Domain 1.4 Operations on Trigonometric Expressions Expansion and Factorization of Trigonometric Expressions Addition Theorems for Trigonometric Functions Converting Trigonometric Functions to Exponential Form Real and Imaginary Parts of Symbolic Expressions 1.5 Solution of Equations The Notion of Generic Solutions Solving Univariate Polynomials in Radicals Cubic Polynomials with Three Real Roots Symbolic Roots as Solutions of Univariate Polynomials of Any Degree Exact Operations on Polynomial Roots Matrix Eigenvalues Canonicalization of Symbol-Free Algebraic Expression H lder s Theorem about Real Roots of Cubics Solving Systems of Polynomials Vieta Relations Solving Systems of Algebraic Equations Solving Nonpolynomial Equations Using Inverse Functions Solving Trigonometric Equations Solving Transcendental Equations Verifying Parametric Solutions Superposition of Damped Oscillations Finding Degenerate Solutions Elimination of Variables Universal Differential Equation Guidelines for Solving Equations and Systems of Equations 1.6 Classical Analysis 1.6.1 Differentiation Multivariate Differentiation Numericalization of Unevaluated Derivatives Numerical Differentiation Differentiating in the Complex Plane Schwarz Theorem Differential Algebraic Constants High-Order Derivatives Derivatives of Inverse Functions Differentiation With Respect to Vectors Derivatives of Pure Functions Adding New Differentiation Rules Differential Equations for n-Nomials Generalized Taylor Expansion Differentials Metric Tensors, Christoffel Symbols, and Geodesics Iterated Evolutes Phase Integral Approximation 1.6.2 Integration Algorithms for Symbolic Integration Assumptions on Variables Having Generic Values Integrating Abstract Functions Korteweg-deVries Equation Hierarchy Indefinite Integration Samples Integrals and Special Functions Integrating Rational Functions Integrating Algebraic Functions Assumptions of Parameter Variables Assumptions in Indefinite Integrals Generating Conditions for Convergence Divergent and Hadamard-Regularized Integrals Cauchy Principal Value Integrals Multidimensional Integrals Robbin s Integral Identity Definite Integrals from Indefinite Integrals Piecewise Continuous Antiderivatives Continuity of Indefinite Integrals Weierstrass Parametrization of Minimal Surfaces Infinite Resistor Network Timings of Indefinite versus Definite Integration d Alembert Solution of the One-Dimensional Wave Equation Schr dinger Equation with a Time-dependent Linear Potential Definite Integrals and Branch Cuts 1.6.3 Limits Indeterminate Expressions and Limits Limit Samples Direction Dependence of Limits Evaluating Limits Under Assumptions Limits of Analytic Functions Schwarz Derivative Extracting Leading Terms Limits of Iterative Function Applications Multiple Limits 1.6.4 Series Expansions Internal Structure of a Series-Object Taylor Series Continued Fraction with Three Limit Points Laurent Series Puiseux Series Series Expansions at Branch Points and Branch Cuts Series of Special Functions Essential Singularities Numerov-Mickens Scheme Multivariate Series Roots of Truncated Series q-Taylor Series Arithmetic of Series Change for 1 Iterated Constant Terms Inverse Series Higher-Order Newton and Chebyshev Methods Fractional Iterations Cumulant Expansions Laurent Series for Mandelbrot Set Approximating Linear Functionals 1.6.5 Residues Symbolic Residues at Poles Generalized Residues Residues of Special Functions 1.6.6 Sums Sum of Powers Numericalization of Symbolic Expressions Procedural versus Symbolic Finite Summations Riemann Surface of the Square Root Function Weierstrass s Method of Analytic Continuation 1.7 Differential and Difference Equations 1.7.0 Remarks 1.7.1 Ordinary Differential Equations Solutions as Rules Pure Functions as Solutions Degenerate Solutions Differential Equation for Free Fall Including the Coriolis Force Integration Constants Linear Inhomogeneous ODE with Constant Coefficients ODEs with Separated Variables Homogeneous ODEs Exact ODEs Bernoulli ODE Jacobi ODEs Special Riccati ODEs Abel ODEs of the First Kind Abel ODEs of the Second Kind Chini ODEs Lagrange ODEs Clairaut ODEs ODEs with Shifted Argument Cayley ODE Second Order ODEs Differential Equations of Special Functions Schr dinger Equations for Various Smooth Potentials Schr dinger Equations for Piecewise-Defined Potentials Higher-Order Differential Equations Implicit Solutions Monitoring Differential Equation Solving - Expansion 1.7.2 Partial Differential Equations Hamilton-Jacobi Equation Szebehely s Equation Solutions with Arbitrary Functions 1.7.3 Difference Equations Linear Difference Equations Calculating Casoratians Linear Difference Equations with Nonconstant Coefficients Some Nonlinear Difference Equations Difference Equations Corresponding to Differential Equations Systems of Difference Equations 1.8 Integral Transforms and Generalized Functions Generalized Functions and Linear Functionals Heaviside Theta Function and Dirac Delta Function Integrals Containing Generalized Functions Multivariate Heaviside Theta and Dirac Delta Function Time Dilation Derivatives of the Dirac Delta Function Simplifying Generalized Functions Sequence Representations of Generalized Functions Green s Function of Linear Differential Operators Generalized Solutions of Differential Equations Compactons Fourier Transforms Self-Fourier Transform Principle Value Distribution Sokhotsky-Plemelj Formula Poincar - Bertrand Identity Laplace Transforms Borel Summation of Divergent Sums Adomian Decomposition 1.9 Additional Symbolics Functions Variational Calculus Symbolic Series Terms Ramanujan s Master Theorem 1.10 Three Applications 1.10.0 Remarks 1.10.1 Area of a Random Triangle in a Square A Quote from M. W. Crofton Generalizations Generic Cylindrical Algebraic Decompositions Six-Dimensional Definite Integrals from Indefinite Integrals Monte Carlo Modeling Calculating the Probability Distribution of the Area 1.10.2 cos(2pi/257) la Gauss The Morning of March 29 in 1796 Gauss Periods Primitive Roots Splitting and Combining Periods Thousands of Square Roots cos(2pi/65537) Fermat Primes 1.10.3 Implicitization of a Trefoil Knot Parametric versus Implicit Description of Surfaces Envelope Surface of a Moving Ball Polynomialization of Trigonometric Expressions Calculating a Large Resultant Smoothing the Trefoil Knot Inflating a Trefoil Knot Implicit Klein Bottle Overview Exercises Heron s formula Tetrahedron Volume Apollonius Circles Proving Trigonometric Identities Icosahedron Inequalities Two-Point Taylor Expansion Horner Form Nested Exponentials and Logarithms Minimal Distance between Polynomial Roots Dynamical Determimants Appell-Nielsen Polynomials Scoping in Iterated Integrals Rational Solution of Painlev II Differential Equation for Products and Quotients of Linear Second Order ODEs Singular Points of First-Order ODEs Fredholm Integral Equation Inverse Sturm-Liouville Problem Graeffe Method Lagrange Interpolation in 2D Triangles Finite Element Matrices Hermite Interpolation-Based Finite Element Calculations Hylleraas-Undheim Helium Ground State Calculation Variational Calculations Hyperspherical Coordinates Constant Negative Curvature Surfaces Optimal Throw Angle Jumping from a Swing Normal Form of Sturm-Liouville Problems Noncentral Collisions Envelope of the Bernstein Polynomials Eigensystem of the Bernstein Operator A Sensitive Linear System Bisector Surfaces Smoothly Connecting Three Half-Infinite Cylinders Nested Double Tori Changing Variables in PDEs Proving Matrix Identities A Divergent Sum Casimir Effect Limit Generating Random Functions Numerical Techniques Used in Symbolic Calculations Series Solution of the Thomas-Fermi Equation Majorana Form of the Thomas-Fermi Equation Yoccoz Function Lagrange-B rmann Formula Divisor Sum Identities Eisenstein Series Product Representation of exp Multiple Differentiation of Vector Functions Expressing Trigonometric Values in Radicals First Order Modular Transformations Forced Damped Oscillations Series for Euler s Constant q-Logarithm Symmetrized Determinant High Order WKB Approximation Greenberger-Horne-Zeilinger State Entangled Four Particle State Integrating Polynomial Roots Riemann Surface of a Cubic Series Solution of the Kepler Equation Short Time-Series Solution of Newton s Equation Lagrange Points of the Three-Body Problem Implicitization of Lissajou Curves Evolutes Orthopodic Locus of Lissajous Curves Cissoid of Lisssajou Curves Multiple Light Ray Reflections Hedgehog Envelope Supercircle Normal Superpositions Discriminant Surface Periodic Surface 27 Lines on the Clebsch Surface 28 Bitangents of a Plane Quartic Pentaellipse Galilean Invariance of Maxwell Equations Relativistic Field Transformations X-Waves Thomas Precession Li nard-Wiechert Potential Expansion Spherical Standing Wave Ramanujan s Factorial Expansion q-Series to q-Products q-Binomial Multiplicative Series gcd-Free Partitions Single Differential Equations for Nonlinear Systems Lattice Green s Function Differential Equation Puzzles Newton-Leibniz Theorem in 2D Square Root of Differential Operator Polynomials with Identical Coefficients and Roots Amoebas Cartesian Leaf Area Average Distance between Random Points Series Solution for Duffing Equation Secular Terms Implicitization of Various Surfaces Kronig-Penney Model Riemann Surface Ellipse Secants Envelope Lines Intersecting Four Lines Shortest Triangle Billiard Path Weak Measurement Identity Logarithmic Residue Geometry Puzzle Differential Equations of Bivariate Polynomials Graph Eigenvalues Change of Variables in the Dirac Delta Function Probability Distributions for Sums Random Determinants Integral Representation of Divided Differences Fourier Transform and Fourier Series Functional Differentiation Operator Splitting Formula Tetrahedron of Maximal Volume Solutions ODE for Circles Modular Equations Converting Trigonometric Expressions into Algebraic Expressions Matrix Sign Function Integration with Scoping Collecting Powers and Logarithms Bound State in Continuum Element Vectors, Mass Matrices, and Stiffness Matrices Multivariate Minimization Envelopes of Throw Trajectories Helpful Warning Messages Using Ans tze Schanuel s Conjecture Matrix Derivatives Lewis-Carroll Identities Abel and H lder Summation Extended Poisson Summation Formula Integration Testing Detecting the Hidden Use of Approximate Numbers Functions with Nontrivial Derivatives Expressing ODEs as Integral Equations Finding Modular Null Spaces Canonicalizing Tensor Expressions Nonsorting Unioning Linear Diophantine Equations Ramanujan Trigonometric Identities Cot Identities Solving the Fokker-Planck Equation for the Forced Damped Oscillator Implementing Specialized Integrations Bras and Kets Density Matrices Recognizing Algebraic Numbers Differentiation of Symbolic Vectors Visualizing the Lagrange Points Gr bner Walk Piecewise Parametrizations of Implicit Surfaces Generalized Clebsch Surfaces Algorithmic Rewriting of Covariant Equations in 3D Vectors Darboux-Halphen System Cubed Sphere Equation Numerically Checking Integrals Containing Derivatives of Dirac Delta Functions Lagrange Multipliers Elementary Symmetric Polynomials References CHAPTER 2 Classical Orthogonal Polynomials 2.0 Remarks 2.1 General Properties of Orthogonal Polynomials Orthogonal Polynomials as Solutions of Sturm-Liouville Eigenvalue Problems General Properties of Orthogonal Polynomials Expansion of Arbitrary Functions in Orthogonal Polynomials 2.2 Hermite Polynomials Definition Graphs ODE Orthogonality and Normalization Harmonic Oscillator Eigenfunctions Density of States Shifted Harmonic Oscillator 2.3 Jacobi Polynomials Definition Graphs ODE Orthogonality and Normalization Electrostatic Interpretation of the Zeros P schl-Teller Potential 2.4 Gegenbauer Polynomials Laplace Equation in nD Definition Graphs ODE Orthogonality and Normalization Smoothing the Gibbs Phenomenon 2.5 Laguerre Polynomials Definition Graphs ODE Orthogonality and Normalization Expanding Riemann Spheres Summed Atomic Orbitals 2.6 Legendre Polynomials Definition Graphs ODE Orthogonality and Normalization Associated Legendre Polynomials Modified P schl-Teller Potential 2.7 Chebyshev Polynomials of the First Kind Definition Graphs ODE Orthogonality and Normalization Trigonometric Form Special Properties 2.8 Chebyshev Polynomials of the Second Kind Definition Graphs ODE Orthogonality and Normalization Trigonometric Form 2.9 Relationships Among the Orthogonal Polynomials Gegenbauer Polynomials as Special Cases of Jacobi Polynomials Hermite Polynomials as Special Cases of Associated Laguerre Polynomials Relations between the Chebyshev Polynomials Calogero-Sutherland Model Schmeisser Companion Matrix Iterated Roots of Orthogonal Polynomials 2.10 Ground-State of the Quartic Oscillator Harmonic and Anharmonic Oscillators Matrix Elements in the Harmonic Oscillator Basis High-Precision Eigenvalues from Diagonalizing the Hill Matrix Lagrange Interpolation-Based Diagonalization Complex Energy Surfaces Time-Dependent Schr dinger Equation PT-Invariant Oscillators Overview Exercises Mehler s Formula Addition Theorem for Hermite Polynomials Sums of Zeros of Hermite Polynomials Spherical Harmonics Sums of Zeros General Orthogonal Polynomials Gram-Schmidt Orthogonalization Power Sums Elementary Symmetric Polynomials Newton Relations Waring Formula Generalized Lissajous Figures Hyperspherical Harmonics Hydrogen Orbitals Zeros of Hermite Functions for Varying Order Ground State Energy of Relativistic Pseudodifferential Operator Moments of Hermite Polynomial Zeros Coherent States Smoothed Harmonic Oscillator States Darboux Isospectral Transformation Forming Wave Packets from Superpositions Multidimensional Harmonic Oscillator High-Order Perturbation Theory Differential Equation System for Eigenvalues Time-Dependent Sextic Oscillator Time Dependent Schr dinger Equation with Calogera Potential Solutions Bauer-Rayleigh Identity Parseval Identity Transmission through Periodic Structures Freud s Weight Function Wronski Polynomials Root-Finding Using Differential Equations Finding Ramification Indices Numerically Classical and Quantum Mechanical Probabilities for the Harmonic Oscillator Root Approximant Using Recursion Relations to Calculate Orthogonal Polynomials References CHAPTER 3 Classical Special Functions 3.0 Remarks Information Sources about Special Functions Experimental Mathematics Generalized Harmonic Numbers Position and Momentum Eigenfunctions and Wigner Function of the Liouville Potential Ramanujan Theta Functions Modular Identities 3.1 Introduction Simplifying Expressions Containing Special Functions Expressing Special Functions through Simpler Ones Indefinite Integrals of Compositions of Elementary Functions Volume of a Supersphere PT-Symmetric Oscillator Monitoring Simplifying Transformations 3.2 Gamma, Beta, and Polygamma Functions Definitions Exact Values Graphs Riemann Surface of the Incomplete Gamma Function Pochhammer Symbol 3.3 Error Functions and Fresnel Integrals Definitions Error Function in the Complex Plane Iterated Integrals of Error Functions Free Particle Schr dinger Equation with Piecewise Constant Initial Conditions Moshinsky Function Harmonic Oscillator Green s Function Fresnel Diffraction on a Half-Plane 3.4 Exponential Integral and Related Functions Definitions Graphs Logarithmic Integral and Prime Counting Function 3.5 Bessel and Airy Functions Definitions Random Walk on a 2D Square Lattice Fractal Based on Bessel Function Weber-Schafheitlin Integrals Bessel Zeros as a Function of the Index Oscillation of a Circular Drum Oscillation of a Drum of General Shape 2D Helmholtz Equation Eigenvalues and Eigenfunctions of the Stadium Billiard Free Nonspreading Wave Packet Airy Functions in the Uniform Approximation of Linear Turning Point Problem Harmonic Oscillator Approximations 3.6 Legendre Functions Definitions Graphs Electrostatic Potential in a Conducting Cone 3.7 Hypergeometric Functions Gauss Hypergeometric Function and Generalized Hypergeometric Functions Some Special Cases Closed Form of Partial Sums of Taylor Series for Trigonometric Functions Closed Form Pad Approximations of exp and sign Generalized Fresnel Integrals Generalized Exponential Functions Point Charge Outside a Dielectric Sphere Finding Contiguous Relations Regularized Hypergeometric Functions Solutions of the Hypergeometric Differential Equation Meijer G Function Eigenfunctions of the Inverse Harmonic Oscillator Bivariate Hypergeometric Functions 3.8 Elliptic Integrals Integrals Containing Square Roots of Cubics and Quartics Definitions Complete and Incomplete Elliptic Integrals Graphs Deriving Differential Equations for Incomplete Elliptic Integrals Green s Function of the Zeilon Operator Finding Modular Equations for Ratios of Elliptic Integrals 3.9 Elliptic Functions Inverting Elliptic Integrals Definitions Jacobi s Amplitude Function Minimal Surface in a Cube Wireframe Applications of Elliptic Functions Pendulum Oscillations Current Flow through a Rectangular Conducting Plate Arithmetic-Geometric Mean 3.10 Product Log Function Definition Solving Transcendental Equations Riemann Surface of the Product Log Function 3.11 Mathieu Functions Differential Equation with Periodic Coefficients Definition Characteristic Values Resonance Tongues Branch Cuts and Branch Points Oscillation of an Ellipsoidal Drum Degenerate Eigenfunctions Wannier Functions 3.12 Additional Special Functions Expressing Other Special Functions through Built-in Special Functions More Elliptic Functions Zeta Functions and Lerch Transcendents 3.13 Solution of Quintic Polynomials Solving Polynomials in Radicals Klein s Solution of the Quintic Tschirnhaus Transformation Principal Quintic Belyi Function and Stereographic Projection of an Icosahedron Projection Solving a Polynomial of Degree 60 through Hypergeometric Functions Numerical Root Calculation Based on Klein s Formula Overview Exercises Asymptotic Expansions of Bessel Functions Carlitz Expansion Meissel s Formula Rayleigh Sums Gumbel Distribution Generalized Bell Numbers Borel Summation Bound State in Continuum ODEs for Incomplete Elliptic Integrals Addition Formulas for Elliptic Integrals Magnetic Field of a Helmholtz Coil Identities, Expansions, ODEs, and Visualizations of the Weierstrass P Function Sutherland-Calogero Model Weierstrass Zeta and Sigma Functions Lam Equation Vortex Lattices ODEs, Addition Formulas, Series Expansions for the Twelve Jacobi Elliptic Functions Schr dinger Equations with Potentials that are Rational Functions of the Wave Functions Periodic Solutions of Nonlinear Evolution Equations Complex Pendulum Harmonic Oscillator Eigenvalues Contour Integral Representation of Bessel Functions Large Order and Argument Expansion for Bessel Functions Aperture Diffraction Circular Andreev Billiard Contour Integral Representation for Beta Functions Beta Distribution Euler s Constant Limit Time-Evolution in a Triple-Well Oscillator Eigenvalues of a Singular Potential Dependencies in the Numerical Calculation of Special Functions Hidden Derivative Definitions Perturbation Theory for a Square Well in an Electric Field Oscillations of a Pendulum with Finite Mass Cord Approximation and Asymptotics of Fermi-Dirac Integrals Sum of All 9-Free Reciprocal Numbers Green s Function for 1D Heat Equation Green s Function for the Laplace Equation in a Rectangle Addition Theorems for Theta Functions Series Expansion of Theta Functions Bose Gas in a 3D Box Scattering on a Conducting Cylinder Poincar Waves Scattering on a Dielectric Cylinder Coulomb Scattering Spiral Waves Scattering on a Corrugated Wall Random Helmholtz Equation Solutions Toroidal Coordinates Riemann-Siegel Expansion Zeros of the Hurwitz Zeta Function Zeta Zeta Function Harmonic Polylogarithms Riemann Surface of Gauss Hypergeometric Functions Riemann Surface of the Ratio of Complete Elliptic Integrals Riemann Surface of the Inverse Error Function Kummer s 24 Solutions of Gauss Hypergeometric Equation Differential Equation for Appell Function Gauss-Lucas Theorem Roots of Differentiated Polynomials Coinciding Bessel Zeros Ramanujan pi Formulas Force-Free Magnetic Fields Bessel Beams Gauge Transformation for a Square Riemann Surface of the Bootstrap Equation Differential Equations for Powers of Airy Functions Asymptotic Expansions for the Zeros of Airy Functions Map-Airy Distribution Dedekind ODE Darboux-Halphen System Ramanujan Identities for CurlyPhi and Functions Generating Identities in Gamma Functions Modular Equations for Dedekind Function Solutions Truncation of Asymptotic Series Contour Plots of the Gamma Function Series of a Gamma Function Ratio Partial Sums of Taylor Series for sin Area and Volume of a Hypersphere All Integrals of Three Compositions of Elementary Functions Binomial at Negative Integers Contour Lines of z z Weierstrass P Function over the Riemann Sphere Using Gr bner Bases to Derive ODEs Riemann Surface of Inverse Weierstrass P Function Rocket with Discrete Propulsion Monitoring All Internal Calculations Machine versus High-Precision Evaluations of Special Functions Checking Numerical Function Evaluation Zeta Regularized Divergent Products Fractional Derivatives Identifying Algebraic Numbers References A.0 Remarks A.1 References and Other Sources of Information A.1.1 General References on Algorithms for Computer Algebra General Computer Algebra Books, References, and Websites Sources of Algorithms Computer Algebra Journals and Conferences A.1.2 Comparison of Various Systems Benchmarks and Timing Comparisons A.1.3 References on Mathematica Books Journals and Websites Conferences Package Libraries Dedicated Newsgroups Timing Comparisions A.1.4 Applications of Computer Algebra Systems Article Samples Further Information Sources References B.0 Remarks B.1 Notebooks and Cells as Expressions B.8.3 Evaluating a Complete GuideBooks Chapter Programmatically References W.0 Remarks W.7 Additions to Chapter 1 of the Graphics Volume Repeated Breaking of a Stick Animation of Rotating Tiles of an Aperiodic Tiling Animation of Circles on Lissajou Figures W.8 Additions to Chapter 2 of the Graphics Volume Animation of Rotating Textured M bius Bands Animation Of Rotating Interlocked Tori Klein Bottle with Hexagonal Massive Wireframe Many Random Walkers in 3D Bivariate Minkowski Function Farey and Bary Addition Projections from 4D W.9 Additions to Chapter 3 of the Graphics Volume Animation of Equal-Eigenvalue Chladny Figures Animation of Moving Charged Regular Polygons Graphics of Charged Truchet Patterns W.10 Additions to Chapter 1 of the Numerics Volume Random Walks with Variable Stepsize Chaotic Scattering on Three Disks Vibrating 2D Hilbert Curve Optimal Projections of Hypercubes Currents Through a Penrose Tiling Numerical Solutions of Various Partial Differential Equations Brain Growth Modeling Step Bunching Modeling Swift-Hohenberg Equation Meinhardt Equations Complex Ginzburg-Landau Equation Hierarchy Splitting Localized Structures Wave Equation with Piecewise-Constant Phase Velocities Local Induction Approximation Born-Infeld Wave Equation Peakon Trains Vibrations of a Square Koch Drum Weyl-Berry Law Diverging Gradients at Inner Corners Classical and Quantum Mechanical Treatment of a Duffing Oscillator Calculating Wigner Functions Through Fractional Fourier Transforms Sub-h/(2pi) Structures in the Wigner Function Circular Aperture Diffraction Integral Checking the Cauchy-Born Hypothesis Schwarz-Christoffel Map for Some Symmetric Polygons Normalized Banzhaf Indices for the European Union Countries Wave Propagation on a Torus Surface W.11 Additions to Chapter 2 of the Numerics Volume A Special Infinite Product of Cosines A Special pi-Related Continued Fraction Plots of the Argument of Cyclotomic Polynomials W.12 Additions to Chapter 1 of the Symbolics Volume Convergence Radius of the Virial Series for the Bose Gas Midpoint Quadrature Formula MacMahon Master Theorem Adler-Moser Polynomials Differential Equation for Yablonskii-Vorob ev Polynomials Implicit Polynomial Description of A Hypocycloidal Torus Calculating the Second Feigenbaum Constant Green s Function for a Sequence of Delta Function Potentials Implicit Form of Poynting Vector Equisurfaces Symmetrically Arranged Points on Spheres The Isospectral Polygons Bilby and Hawk Probability Distribution of a Quotient Vibrations of Springs on a Gosper Curve Probability Distribution for the Distance Between Two Points from the Unit Square Animation of the Nodal Lines of a Dirichlet-Neumann Boundary Conditions Transition Checking Higher Order Generalized WKB Approximation for the Harmonic Oscillator Evaluating an Iterated Integral The Kobussen-Leubner-Lopez Lagrangian for the Harmonic Oscillator W.13 Additions to Chapter 2 of the Symbolics Volume Eigenfunctions of the H non-Heiles Potential W.14 Additions to Chapter 3 of the Symbolics Volume Rational Values of the Gauss Hypergeometric Function Eigenfunctions of the Discrete Harmonic Oscillator Average Length of Smallest Component of Multidimensional Unit Vectors Differential Equation of the Jacobi Elliptic Function sn with Respect to the Modul A Certain Sum of Zeta Functions High-Order Series Expansion of Harmonic Numbers of a Given Size Movement of a Sliding Spinning Disk Identities of Jacobi Theta Function for Special Argument Values References Website copyright by Michael Trott 2004-2007 printed book materials copyright Springer, 2004-2005.
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