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In this work, we develop a Bayesian variational’ inference theory for nonlinear hyperelastic materials, which is accomplished by using a mixed Galerkin variational Bayesian inference nonlinear finite element method (VBI-NFEM). In the proposed Bayesian statistical continuum mechanics theory, the nonlinear elastic potential energy is used as a prior in a Bayesian inference network, which can intelligently and inversely recover the detailed continuum deformation mappings with only the information on the shapes of the deformed and undeformed continuum body, without knowing the actual boundary conditions, including both traction and displacement boundary conditions, and the fine scale material constitutive relation. To solve the mixed variational problem, we developed an operator splitting or staggered algorithm that consists of a finite element (FE) step and a Bayesian learning (BL) step, analogous to the well-known Expectation-Maximization (EM) algorithm. By solving the probabilistic Galerkin variational finite element problem, we demonstrated in several examples that the proposed method can inversely predict continuum nonlinear deformation mappings without requiring knowledge of the external load conditions. This long-sought-after inverse problem solution has been a significant challenge in the forensic pattern analysis of structural failures over the past several decades, and the proposed method provides a robust machine-intelligent solution.
Wang et al. (Sat,) studied this question.