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.

Figures (13)

• Figure 1: Illustrating PSI method's key ideas. (a) Scheme of a one-element system whose phase-space is the simplest possible (will be used in \ref{['sec:1D_closed_form']}). (b) $E$ represents the physically-admissible set (equilibrium and compatibility), $D$ the materially-admissible one (constitutive law). Phase-space possible projections: from a point satisfying equilibrium but not the constitutive law, $[0 \quad F/A ]^\top$, to different material-admissible points dependening on choice of distance, i.e., parameter $C$ (see \ref{['eq:distance_general', 'eq:distance']} for details). (c) Phase-space iterations in normalized space: every search/projection is equivalent to finding the point where the smallest possible circle centered in the previous iteration is tangent to the other set (only the first and the last circles are fully shown for clarity).
• Figure 2: Simple illustration of the PSIs method: scheme of a one-element phase-space featuring the two subsets, see also the two projections that intervene in \ref{['eq:convrate']} to define the convergence rate. In this case, the convex sets are understood to be $D = \{ (\varepsilon', \sigma') \in Z : \sigma' - m(\varepsilon') \le 0 \}$ and $E = \{ (\varepsilon', \sigma') \in Z : \sigma' - F/A \ge 0 \}$. This setting was also considered in Ref. Ladeveze:2012.
• Figure 3: Sketch of the Friedrichs angle in the 1D one-element case. The left vertical axis depicts stresses, while the right vertical one depicts the slope angle of the constitutive; both share horizontal axis (strains). The black cross marks the solution, while the blue one marks the "local" Friedrichs angle value controlling the performance of the PSI method in this example.
• Figure 4: Visual representations of the key observations. (a) First: projections onto linear envelope preserve ordering: if $\varepsilon_1^{(n)}<\varepsilon_2^{(n)}$ then $\varepsilon_1^{(n+1)}<\varepsilon_2^{(n+1)}$. (b) Second: projection onto the actual constitutive law move faster to the solution than the projection onto the linear envelope: $\varepsilon_{1,L}^{(n+1)}<\varepsilon_{1,NL}^{(n+1)}$.
• Figure 5: Truss example. (a) Structure: red arrows indicate applied forces, blue ones correspond to imposed displacements. (b) Constitutive law used for the all bars in the truss.

Theorems & Definitions (19)

• Definition 3.1: Non-expansiveness
• Definition 3.2: Asymptotic regularity
• Definition 3.3: Set of fixed points $\mathcal{T}_T$
• Theorem 3.1: Opial
• Theorem 3.2: Hilbert projection theorem
• Definition 3.4: Projection Operator
• Definition 3.5: Intersection of convex subsets $\bar{\mathcal{C}}$
• Definition 3.6: Global projection $P(\boldsymbol{x})$
• Proposition 3.1
• Proposition 3.2