A Variational Bayesian Inference Theory of Elasticity and Its Mixed Probabilistic Finite Element Method for Inverse Deformation Solutions in Any Dimension

TL;DR

This paper introduces a novel variational Bayesian inference framework combined with a mixed finite element method to solve inverse deformation problems in continuum mechanics, capable of handling discontinuities without prior boundary or load information.

Contribution

It develops a new Bayesian inference theory for elasticity integrated with a finite element approach, enabling inverse deformation solutions without detailed boundary or material data.

Findings

Successfully predicts deformation mappings with discontinuities or fractures.

Operates without known boundary conditions or external loads.

Demonstrates robustness and potential for AI-based PDE solving.

Abstract

In this work, we have developed a variational Bayesian inference theory of elasticity, which is accomplished by using a mixed Variational Bayesian inference Finite Element Method (VBI-FEM) that can be used to solve the inverse deformation problems of continua. In the proposed variational Bayesian inference theory of continuum mechanics, the elastic strain energy is used as a prior in a Bayesian inference network, which can intelligently recover the detailed continuum deformation mappings with only given the information on the deformed and undeformed continuum body shapes without knowing the interior deformation and the precise actual boundary conditions, both traction as well as displacement boundary conditions, and the actual material constitutive relation. Moreover, we have implemented the related finite element formulation in a computational probabilistic mechanics framework. To numerically solve mixed variational problem, we developed an operator splitting or staggered algorithm that consists of the finite element (FE) step and the Bayesian learning (BL) step as an analogue of the well-known the Expectation-Maximization (EM) algorithm. By solving the mixed probabilistic Galerkin variational problem, we demonstrated that the proposed method is able to inversely predict continuum deformation mappings with strong discontinuity or fracture without knowing the external load conditions. The proposed method provides a robust machine intelligent solution for the long-sought-after inverse problem solution, which has been a major challenge in structure failure forensic pattern analysis in past several decades. The proposed method may become a promising artificial intelligence-based inverse method for solving general partial differential equations.