Phase-space iterative solvers

Abstract

We introduce an iterative method to solve problems in small-strain non-linear elasticity, termed ``Phase-Space Iterations'' (PSIs). The method is inspired by recent work in data-driven computational mechanics, which reformulated the classic boundary value problem of continuum mechanics using the concept of ''phase space'' associated with a mesh. The latter is an abstract metric space, whose coordinates are indexed by strains and stress components, where each possible state of the discretized body corresponds to a point. Two subsets are then defined: an affine space termed ``physically-admissible set'' made up by those points that satisfy equilibrium and a ``materially-admissible set'' containing points that satisfy the constitutive law. Solving the boundary-value problem amounts to finding the intersection between these two subdomains. In the linear-elastic setting, this can be achieved through the solution of a set of linear equations; when material non-linearity enters the picture, such is not the case anymore and iterative solution approaches are necessary. Our iterative method consists on projecting points alternatively from one set to the other, until convergence. To evaluate the performance of the method, we draw inspiration from the ''method of alternative projections'' and the ''method of projections onto convex sets'', both of which have a robust mathematical foundation in terms of conditions for the existence of solutions and guarantees convergence. This foundation is leveraged to analyze the simplest case and to establish a geometric convergence rate. We also present a realistic case to illustrate PSIs' strengths when compared to the classic Newton-Raphson method, the usual tool of choice in non-linear continuum mechanics. Finally, its aptitude to deal with constitutive laws based on neural network is also showcased.