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

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.

An Artificial-intelligence/Statistics Solution to Quantify Material Distortion for Thermal Compensation in Additive Manufacturing

In this paper, we introduce a probabilistic statistics solution or artificial intelligence (AI) approach to identify and quantify permanent (non-zero strain) continuum/material deformation only based on the scanned material data in the spatial configuration and the shape of the initial design configuration or the material configuration. The challenge of this problem is that we only know the scanned material data in the spatial configuration and the shape of the design configuration of three-dimensional (3D) printed products, whereas for a specific scanned material point we do not know its corresponding material coordinates in the initial or designed referential configuration, provided that we do not know the detailed information on actual physical deformation process. Different from physics-based modeling, the method developed here is a data-driven artificial intelligence method, which solves the problem with incomplete deformation data or with missing information of actual physical deformation process. We coined the method is an AI-based material deformation finding algorithm. This method has practical significance and important applications in finding and designing thermal compensation configuration of a 3D printed product in additive manufacturing, which is at the heart of the cutting edge 3D printing technology. In this paper, we demonstrate that the proposed AI continuum/material deformation finding approach can accurately find permanent thermal deformation configuration for a complex 3D printed structure component, and hence to identify the thermal compensation design configuration in order to minimizing the impact of temperature fluctuations on 3D printed structure components that are sensitive to changes of temperature.