In this dissertation, I present a computational framework to predict the inverse deformation mapping. Under this framework, two algorithms have been developed. The first one is called the material deformation finding (MDF) algorithm with a specific application in additive manufacturing. This method is designed to quantify the permanent (non-zero strain) continuum/material deformation. Different from physical-based modeling, the method developed here is based on a data-driven statistics approach, which solves the problem without needing information about the physical deformation process. The proposed method relies only on the scanned material data from the thermal distorted configuration as well as the shape of the initial design configuration. In this work, the MDF algorithm was first validated by a 2D synthetic example. We then demonstrate that the proposed MDF method can accurately find the permanent thermal distortion of a complex 3D printed structural component, and hence identify the thermal compensation design configuration. The results obtained in this work indicate that one can use this data-driven statistics approach to significantly mitigate the thermal distortion of 3D printed products in additive manufacturing.

The second algorithm developed is a mixed variational Bayesian learning finite element method (VBL-FEM), based on a Bayesian statistical continuum mechanics theory, in which elastic potential energy is used as a prior in a Bayesian regularization network, which can intelligently recover unknown continuum deformation mapping with only the information of the shapes of the deformed and undeformed continuum body without knowing actual boundary conditions, both traction as well as displacement boundary conditions and the precise material constitutive relation. Moreover, we also develop the related finite element formulation in a computational probabilistic mechanics framework. Using a data-driven likelihood function, we construct an entropic variational principle of continuum mechanics based on the maximum a posteriori (MAP) probability estimation and Bayesian regularization. By solving the probabilistic Galerkin variational problem, we also demonstrate in several examples that the proposed method is able to inversely predict continuum deformation mappings with strong discontinuity or fracture without knowing the external load conditions. This long-sought-after inverse problem solution has been a major challenge in structure failure forensic analysis in the past several decades, and the proposed method provides a machine intelligent solution for it.