2024/10/10 by Chao Wang, Shaofan Li, Wang, Chao +1
Decision Sciences · Engineering · #Computer Vision and Pattern Recognition (cs.CV) #FOS: Computer and information sciences #FOS: Mathematics #Machine Learning (cs.LG) #Numerical Analysis (math.NA) #Probabilistic and Robust Engineering Design #Structural Health Monitoring Techniques
paper · pdf · doi:10.48550/arxiv.2410.07605
openalex publication_date 2024/10/10 · openalex created_date 2024/10/13 · openalex updated_date 2026/07/28
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.