Quasi-Monte Carlo sparse grid Galerkin finite element methods for linear elasticity equations with uncertainties

We explore a linear inhomogeneous elasticity equation with random Lam\'e parameters. The latter are parameterized by a countably infinite number of terms in separated expansions. The main aim of this work is to estimate expected values (considered as an infinite dimensional integral on the parametric space corresponding to the random coefficients) of linear functionals acting on the solution of the elasticity equation. To achieve this, the expansions of the random parameters are truncated, a high-order quasi-Monte Carlo (QMC) is combined with a sparse grid approach to approximate the high dimensional integral, and a Galerkin finite element method (FEM) is introduced to approximate the solution of the elasticity equation over the physical domain. The error estimates from (1) truncating the infinite expansion, (2) the Galerkin FEM, and (3) the QMC sparse grid quadrature rule are all studied. For this purpose, we show certain required regularity properties of the continuous solution with respect to both the parametric and physical variables. To achieve our theoretical regularity and convergence results, some reasonable assumptions on the expansions of the random coefficients are imposed. Finally, some numerical results are delivered

Phase-field fracture modeling, numerical solution, and simulations for compressible and incompressible solids

In this thesis, we develop phase-field fracture models for simulating fractures in compressible and incompressible solids. Classical (primal) phase-field fracture models fail due to locking effects. Hence, we formulate the elasticity part of the phase-field fracture problem in mixed form, avoiding locking. For the elasticity part in mixed form, we prove inf-sup stability, which allows a stable discretization with Taylor-Hood elements. We solve the resulting (3x3) phase-field fracture problem - a coupled variational inequality system - with a primal-dual active set method. In addition, we develop a physics-based Schur-type preconditioner for the linear solver to reduce the computational workload. We confirm the robustness of the new solver for five benchmark tests. Finally, we compare numerical simulations to experimental data analyzing fractures in punctured strips of ethylene propylene diene monomer rubber (EPDM) stretched until total failure to check the applicability on a real-world problem in nearly incompressible solids. Similar behavior of measurement data and the numerically computed quantities of interest validate the newly developed quasi-static phase-field fracture model in mixed form.DFG/SPP 1748/392587580/E

SOLID-SHELL FINITE ELEMENT MODELS FOR EXPLICIT SIMULATIONS OF CRACK PROPAGATION IN THIN STRUCTURES

Crack propagation in thin shell structures due to cutting is conveniently simulated using explicit finite element approaches, in view of the high nonlinearity of the problem. Solidshell elements are usually preferred for the discretization in the presence of complex material behavior and degradation phenomena such as delamination, since they allow for a correct representation of the thickness geometry. However, in solid-shell elements the small thickness leads to a very high maximum eigenfrequency, which imply very small stable time-steps. A new selective mass scaling technique is proposed to increase the time-step size without affecting accuracy. New ”directional” cohesive interface elements are used in conjunction with selective mass scaling to account for the interaction with a sharp blade in cutting processes of thin ductile shells

Proceedings of the FEniCS Conference 2017

Proceedings of the FEniCS Conference 2017 that took place 12-14 June 2017 at the University of Luxembourg, Luxembourg

Advanced Computational Engineering

The finite element method is the established simulation tool for the numerical solution of partial differential equations in many engineering problems with many mathematical developments such as mixed finite element methods (FEMs) and other nonstandard FEMs like least-squares, nonconforming, and discontinuous Galerkin (dG) FEMs. Various aspects on this plus related topics ranging from order-reduction methods to isogeometric analysis has been discussed amongst the pariticpants form mathematics and engineering for a large range of applications