论文标题
固体力学的自动差异化
Automatic differentiation for solid mechanics
论文作者
论文摘要
自动分化（AD）是一组技术，允许评估以计算机编程语言表达的数学函数的准确数值导数。在本文中，我们使用广告来说明和解决固体力学问题。鉴于域的有限元离散化，我们将固体的自由能视为其应变能密度的组成部分，并且我们利用AD直接获得问题的残留力矢量和问题的切线刚度矩阵，分别为梯度和自由能的Hessian。结果是在陈述和解决涉及非琐碎约束系统以及几何和材料非线性的复杂问题中的明显简化和解决方案。该论文与连续力学理论基础以及对所采用的特定AD技术的描述一起说明了许多固体力学问题的解决方案，目的是向近期编程语言易于获得的方便数值实施方法，向固体机械师社区易于使用。
Automatic differentiation (AD) is an ensemble of techniques that allow to evaluate accurate numerical derivatives of a mathematical function expressed in a computer programming language. In this paper we use AD for stating and solving solid mechanics problems. Given a finite element discretization of the domain, we evaluate the free energy of the solid as the integral of its strain energy density, and we make use of AD for directly obtaining the residual force vector and the tangent stiffness matrix of the problem, as the gradient and the Hessian of the free energy respectively. The result is a remarkable simplification in the statement and the solution of complex problems involving non trivial constraints systems and both geometrical and material non linearities. Together with the continuum mechanics theoretical basis, and with a description of the specific AD technique adopted, the paper illustrates the solution of a number of solid mechanics problems, with the aim of presenting a convenient numerical implementation approach, made easily available by recent programming languages, to the solid mechanics community.
