Abstract

Efficient solution via Newton’s method of nonlinear systems of equations requires an accurate representation of the Jacobian, corresponding to the derivatives of the component residual equations with respect to the degrees of freedom. In practice these systems of equations often arise from spatial discretization of partial differential equations used to model physical phenomena. These equations may involve domain motion or material equations that are complex functions of the systems’ degrees of freedom. Computing the Jacobian by hand in these situations is arduous and prone to error. Finite difference approximations of the Jacobian or its action are prone to truncation error, especially in multiphysics settings. Symbolic differentiation packages may be used, but often result in an excessive number of terms in realistic model scenarios. An alternative to symbolic and numerical differentiation is automatic differentiation (AD), which propagates derivatives with every elementary operation of a computer program, corresponding to continual application of the chain rule. Automatic differentiation offers the guarantee of an exact Jacobian at a relatively small overhead cost. In this work, we outline the adoption of AD in the Multiphysics Object Oriented Simulation Environment (MOOSE) via the MetaPhysicL package. We describe the application of MOOSE’s AD capability to several sets of physics that were previously infeasible to model via hand-coded or Jacobian-free simulation techniques, including arbitrary Lagrangian-Eulerian and level-set simulations of laser melt pools, phase-field simulations with free energies provided through neural networks, and metallic nuclear fuel simulations that require inner Newton loop calculation of nonlinear material properties.

摘要

通过牛顿非线性方程组方法的有效求解需要精确表示雅可比矩阵，对应于分量残差方程关于自由度的导数。在实践中，这些方程组通常来自用于模拟物理现象的偏微分方程的空间离散化。这些方程可能涉及域运动或材料方程，它们是系统自由度的复杂函数。在这些情况下手工计算雅可比矩阵是困难的并且容易出错。雅可比矩阵或其作用的有限差分近似容易出现截断误差，尤其是在多物理场设置中。可以使用符号微分包，但通常会导致现实模型场景中的术语数量过多。符号和数值微分的替代方法是自动微分 (AD)，它通过计算机程序的每个基本操作传播导数，对应于链规则的连续应用。自动微分以相对较小的开销成本保证了精确的雅可比行列式。在这项工作中，我们概述了通过 MetaPhysicL 包在多物理场面向对象仿真环境 (MOOSE) 中采用 AD。我们描述了 MOOSE 的 AD 功能在以前无法通过手动编码或无雅可比模拟技术进行建模的几组物理中的应用，包括激光熔池的任意拉格朗日-欧拉和水平集模拟、相场模拟与通过神经网络提供的自由能量，