Combining Set Propagation with Finite Element Methods for Time Integration in Transient Solid Mechanics Problems

TL;DR

This paper introduces a novel set propagation approach combined with finite element methods for efficient and accurate time integration in transient solid mechanics, especially under uncertain initial conditions.

Contribution

The paper proposes a new method that uses set propagation and reachability analysis to enclose all possible solutions of PDEs in solid mechanics problems, improving efficiency for uncertain initial states.

Findings

Accurate for problems with known initial conditions.

More efficient than traditional methods with uncertain initial conditions.

Validated on five numerical examples.

Abstract

The Finite Element Method (FEM) is the gold standard for spatial discretization in numerical simulations for a wide spectrum of real-world engineering problems. Prototypical areas of interest include linear heat transfer and linear structural dynamics problems modeled with partial differential equations (PDEs). While different algorithms for direct integration of the equations of motion exist, exploring all feasible behaviors for varying loads, initial states and fluxes in models with large numbers of degrees of freedom remains a challenging task. In this article we propose a novel approach, based in set propagation methods and motivated by recent advances in the field of Reachability Analysis. Assuming a set of initial states and inputs, the proposed method consists in the construction of a union of sets (flowpipe) that enclose the infinite number of solutions of the spatially discretized PDE. We present the numerical results obtained in five examples to illustrate the capabilities of our approach, and compare its performance against reference numerical integration methods. We conclude that, for problems with single known initial conditions, the proposed method is accurate. For problems with uncertain initial conditions included in sets, the proposed method can compute all the solutions of the system more efficiently than numerical integration methods.