The a posteriori finite element method (APFEM), a framework for efficient parametric study and Bayesian inferences

Stochastic methods have recently been the subject of increased attention in Computational Mechanics for their ability to account for the stochasticity of both material parameters and geometrical features in their predictions. Among them, the Galerkin Stochastic Finite Element Method (GSFEM) was shown to be particularly efficient and able to provide accurate output statistics, although at the cost of intrusive coding and additional theoretical algebraic efforts. In this method, distributions of the stochastic parameters are used as inputs for the solver, which in turn outputs nodal displacement distributions in one simulation. Here, we propose an extension of the GSFEM-termed the A posteriori Finite Element Method or APFEM-where uniform distributions are taken by default to allow for parametric studies of the inputs of interest as a postprocessing step after the simulation. Doing so, APFEM only requires the knowledge of the vertices of the parameter space. In particular, one key advantage of APFEM is its use in the context of Bayesian inferences, where the random evaluations required by the Bayesian setting (usually done through Monte Carlo) can be done exactly without the need for further simulations. Finally, we demonstrate the potential of APFEM by solving forward models with parametric boundary conditions in the context of (i) metamaterial design and (ii) pitchfork bifurcation of the buckling of a slender structure; and demonstrate the flexibility of its use for Bayesian inference by (iii) inferring friction coefficient of a half plane in a contact mechanics problem and (iv) inferring the stiffness of a brain region in the context of cancer surgical planning. (c) 2023 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY license

Automated summary

Stochastic methods, especially the Galerkin Stochastic Finite Element Method (GSFEM), have gained attention in Computational Mechanics for considering the stochasticity of material parameters and geometrical features. We propose an extension of GSFEM called A posteriori Finite Element Method (APFEM), which uses uniform distributions for parametric studies as a postprocessing step after the simulation. APFEM is advantageous in Bayesian inference as it allows for exact random evaluations without further simulations. Examples demonstrate APFEM's potential in metamaterial design, buckling analysis, friction coefficient inference, and cancer surgical planning.