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Citations
... Let us consider the non-convex rotated L-shaped domain Ω = (−1, 1) 2 \ (−1, 0) 2 and use manufactured displacement and fluid pressure with sharp gradients near the domain re-entrant corner (see, e.g., [20] for the displacement and [16] for the fluid pressure) ...
We develop a family of mixed finite element methods for a model of nonlinear poroelasticity where, thanks to a rewriting of the constitutive equations, the permeability depends on the total poroelastic stress and on the fluid pressure and therefore we can use the Hellinger-Reissner principle with weakly imposed stress symmetry for Biot's equations. The problem is adequately structured into a coupled system consisting of one saddle-point formulation, one linearised perturbed saddle-point formulation, and two off-diagonal perturbations. This system's unique solvability requires assumptions on regularity and Lipschitz continuity of the inverse permeability, and the analysis follows fixed-point arguments and the Babuška-Brezzi theory. The discrete problem is shown uniquely solvable by applying similar fixed-point and saddle-point techniques as for the continuous case. The method is based on the classical PEERS elements, it is exactly momentum and mass conservative, and it is robust with respect to the nearly incompressible as well as vanishing storativity limits. We derive a priori error estimates, we also propose fully computable residual-based a posteriori error indicators, and show that they are reliable and efficient with respect to the natural norms, and robust in the limit of near incompressibility. These a posteriori error estimates are used to drive adaptive mesh refinement. The theoretical analysis is supported and illustrated by several numerical examples in 2D and 3D.
... It should be pointed out that different refinement criteria may have a nonneglectable impact on the final pattern of refined meshes. Therefore, there are alternatives to simple goal-oriented mesh adaptation strategies (González-Estrada et al., 2014, Bulle et al., 2023, which might be suitable for the UBFELA-RTME but are not studied. In addition, as mesh refinements in major collapse areas with high energy dissipation velocity discontinuities are performed in this study, which is slightly different from that in the traditional FEM with finite elements undergo plastic deformation. ...
This paper presents a stability study on the collapse mechanisms of a plane-strain tunnel face in c-ϕ soils using the upper bound finite element method with rigid translatory moving elements (UBFELA-RTME) and nonlinear programming technique. Practical considerations are given to the unlined length influence behind the tunnel face. An advanced mesh adaptive updating strategy is adopted, aiming to improve the computational efficiency, the accuracy of upper-bound solutions, as well as the produced collapse mechanisms. The unlined length influence on the face stability and collapse mechanism of the tunnel face are determined with various combinations of tunnel depth ratios, soil friction angles, and dilatancy angles. Using the UBFELA-RTME with the Davis’s approach and a mesh adapting strategy, the non-associated plasticity flow rule can be well approximated. The developed technique was validated against different numerical methods, and it is concluded that the tunnel face stability can be improved by increasing soil friction and dilatancy angles, and yet weakens as the unlined length increases where a mesh-liked collapse zone gradually appears on the tunnel vault top. It gradually evolves to a global collapse failure till the ground surface. The findings contribute to a better understanding of the ground surface failure under the unlined support length influence in tunnel construction.
... Most multi-purpose FEM packages have a huge code-base (often combining several languages) and, necessarily, a cleverly designed class hierarchy that may require significant effort to understand and get it running [20][21][22][23][24] . Since one of our aims is that our code is easy to modify and extend, we strive for a relatively small code-base that nevertheless covers the most widespread demands of academic FEM software. ...
We present an easily accessible, object oriented code (written exclusively in Matlab) for adaptive finite element simulations in 2D. It features various refinement routines for triangular meshes as well as fully vectorized FEM ansatz spaces of arbitrary polynomial order and allows for problems with very general coefficients. In particular, our code can handle problems typically arising from iterative linearization methods used to solve nonlinear PDEs. Due to the object oriented programming paradigm, the code can be used easily and is readily extensible. We explain the basic principles of our code and give numerical experiments that underline its flexibility as well as its efficiency.
... Our method leads to a fully local and parallelizable solution technique for the spectral fractional Laplacian with computable L 2 error. Our method is valid for any finite element degree (however, for the sake of brevity we do not show results with higher degree finite elements) and for one, two and three dimensional problems [39]. ...
... In addition, its computational stencil is highly local which is particularly appealing for three-dimensional problems see e.g. [39]. Finally, our choice of the Bank-Weiser estimator is also justified in section 6.2.1. ...
... Our method is based on the Bank-Weiser finite element error estimator introduced in [21] and its implementation in the FEniCSx software described in [39]. ...
This chapter is about practical situations where the user is interested to estimate the error due to finite element approximation.
