When using the finite element method (FEM) in inverse problems, its discretization error can produce parameter estimates that are inaccurate and overconfident. The Bayesian finite element method (BFEM) provides a probabilistic model for the epistemic uncertainty due to discretization error. In this work, we apply BFEM to various inverse problems and compare its performance to the random mesh finite element method (RM-FEM) and the statistical finite element method (statFEM), which serve as a frequentist and inference-based counterpart to BFEM. We find that by propagating this uncertainty to the posterior, BFEM can produce more accurate parameter estimates and prevent overconfidence compared to FEM. Because the BFEM covariance operator is designed to leave uncertainty only in the appropriate space, orthogonal to the FEM basis, BFEM is able to outperform RM-FEM, which does not have such a structure to its covariance. Although it is also possible to use a model misspecification formulation such as statFEM to infer the discretization error downstream rather than model it at the source, the feasibility of such an approach is contingent on the availability of sufficient data. We find that the BFEM is the most robust way to consistently propagate uncertainty due to discretization error to the posterior of a Bayesian inverse problem.

In this work, the uncertainty associated with the finite element discretization error is modeled following the Bayesian paradigm. First, a continuous formulation is derived, where a Gaussian process prior over the solution space is updated based on observations from a finite element discretization. To avoid the computation of intractable integrals, a second, finer, discretization is introduced that is assumed sufficiently dense to represent the true solution field. A prior distribution is assumed over the fine discretization, which is then updated based on observations from the coarse discretization. This yields a posterior distribution with a mean that serves as an estimate of the solution, and a covariance that models the uncertainty associated with this estimate. Two particular choices of prior are investigated: a prior defined implicitly by assigning a white noise distribution to the right-hand side term, and a prior whose covariance function is equal to the Green’s function of the partial differential equation. The former yields a posterior distribution with a mean close to the reference solution, but a covariance that contains little information regarding the finite element discretization error. The latter, on the other hand, yields posterior distribution with a mean equal to the coarse finite element solution, and a covariance with a close connection to the discretization error. For both choices of prior a contradiction arises, since the discretization error depends on the right-hand side term, but the posterior covariance does not. We demonstrate how, by rescaling the eigenvalues of the posterior covariance, this independence can be avoided.

The Thick Level Set method (TLS) is an approach for non-local damage modeling in which the damage evolution is linked to the movement of a damage front described with the level set method. More recently, a new version of the TLS, designated as the TLSV2, has been proposed as a new concept for coupling continuum damage modeling and discrete cohesive crack modeling for failure analysis in solids. The main objective of this new framework is to profit from both modeling approaches. The continuum part allows for handling crack initiation, branching and merging, whereas the cohesive part brings the capability to handle discrete cracks with large crack opening or sliding without heavily distorted elements, and with the possibility to model stiffness recovery upon contact. In this paper, a generalized framework for the TLSV2 is introduced. Two major issues with the TLSV2 method that have not been dealt with since its inception are addressed in this study, and solutions are proposed. Firstly, the method depends on the location of skeleton curve of the level set field, on which the discontinuity in the displacement field is evaluated. The problem of locating the skeleton curve can be a complicated task, mainly because topological events may emerge as the analysis progresses, such as crack branching. The skeleton curve is determined through a combination of ball-shrinking and graph-based algorithms and then mapped onto the finite element mesh. Secondly, the cohesive forces and displacement discontinuity of the TLSV2 are modeled using the phantom node method. Furthermore, a new approach to compute the averaged values of local quantities is introduced, and model calibration is discussed. The degree of stiffness recovery under compression that is still needed for the continuum part is investigated. Numerical experiments demonstrate the accuracy and ability of the proposed model to handle simulation of failure analysis presenting complex topological crack patterns.