Automatic Differentiation for Solid Mechanics

TLDR: Automatic differentiation (AD) is an ensemble of techniques that allows to evaluate accurate numerical derivatives of a mathematical function expressed in a computer programming language as discussed by the authors, and it has been used for stating and solving solid mechanics problems.

Abstract: Automatic differentiation (AD) is an ensemble of techniques that allows 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.