Phase-space iterative solvers

We introduce an iterative method to solve problems in small-strain non-linear elasticity. 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". 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. Since the phase space is associated to the discretized body, it is finite dimensional. 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. The method is similar in spirit to the ``method of alternative projections'' and to the ``method of projections onto convex sets'', for which there is a solid mathematical foundation that furnishes conditions for existence of solutions and convergence, upon which we rely to assess the method's performance. We present a proof of geometric convergence rate and two examples: a fundamental one to illustrate the features of the method, and a realistic one to showcase its capacities and strengths when compared to the classic Newton-Raphson method, the usual tool of choice in non-linear continuum mechanics. Finally, its ability to deal with constitutive laws based on neural network is also showcased.