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.

中文翻译：

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固体力学的自动微分

自动微分 (AD) 是一组技术，可以评估用计算机编程语言表达的数学函数的精确数值导数。在本文中，我们使用 AD 来陈述和解决固体力学问题。给定域的有限元离散化，我们将固体的自由能评估为其应变能密度的积分，我们利用 AD 直接获得问题的残余力矢量和切线刚度矩阵，作为分别是自由能的梯度和 Hessian。结果是对涉及非平凡约束系统以及几何和材料非线性的复杂问题的陈述和解决方案的显着简化。再加上连续介质力学的理论基础，