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

Stable finite element algorithms for analysing the vertebral artery

The research described in this thesis began with a single long-term objective: modelling of the vertebral artery during chiropractic manipulation of the cervical spine. Although chiropractic treatment has become prevalent, the possible correlation between neck manipulation and subsequent stroke in patients has been the subject of debate without resolution. Past research has been qualitative or statistical, whereas resolution demands a fundamental understanding of the associated mechanics. Analysis in the thesis begins with a study of the anatomy and properties pertinent to the chiropractic problem. This indicates that the complexity of the problem will necessitate a long-term multidisciplinary effort including a nonlinear finite element formulation effective in analysing image data for soft tissue modelled as nearly incompressible. This leads to an assessment of existing finite element methods and the conclusion that new equation solving techniques are needed to ensure numerical stability. Three techniques for effectively eliminating the source of numerical instability are developed and demonstrated with the aid of original finite element codes. Two of the methods are derived as modifications of matrix decomposition algorithms, while the third method constitutes a new finite element formulation. In addition, the understanding gained in developing these methods is used to produce a theorem for assessing a different but related problem: deformation of a nearly incompressible material subjected to a single concentrated force. Throughout the thesis, an interdisciplinary path from chiropractic problem to numerical algorithms is outlined, and results are in the form of mathematical proofs and derivations of both existing and new methods

The Hybrid High-Order Method for Polytopal Meshes: Design, Analysis, and Applications

International audienceHybrid High-Order (HHO) methods are new generation numerical methods for models based on Partial Differential Equations with features that set them apart from traditional ones. These include: the support of polytopal meshes including non star-shaped elements and hanging nodes; the possibility to have arbitrary approximation orders in any space dimension; an enhanced compliance with the physics; a reduced computational cost thanks to compact stencil and static condensation. This monograph provides an introduction to the design and analysis of HHO methods for diffusive problems on general meshes, along with a panel of applications to advanced models in computational mechanics. The first part of the monograph lays the foundation of the method considering linear scalar second-order models, including scalar diffusion, possibly heterogeneous and anisotropic, and diffusion-advection-reaction. The second part addresses applications to more complex models from the engineering sciences: non-linear Leray-Lions problems, elasticity and incompressible fluid flows

Spectral and High Order Methods for Partial Differential Equations ICOSAHOM 2018

This open access book features a selection of high-quality papers from the presentations at the International Conference on Spectral and High-Order Methods 2018, offering an overview of the depth and breadth of the activities within this important research area. The carefully reviewed papers provide a snapshot of the state of the art, while the extensive bibliography helps initiate new research directions

Potential-based Formulations of the Navier-Stokes Equations and their Application

Based on a Clebsch-like velocity representation and a combination of classical variational principles for the special cases of ideal and Stokes flow a novel discontinuous Lagrangian is constructed; it bypasses the known problems associated with non-physical solutions and recovers the classical Navier-Stokes equations together with the balance of inner energy in the limit when an emerging characteristic frequency parameter tends to infinity. Additionally, a generalized Clebsch transformation for viscous flow is established for the first time. Next, an exact first integral of the unsteady, three-dimensional, incompressible Navier-Stokes equations is derived; following which gauge freedoms are explored leading to favourable reductions in the complexity of the equation set and number of unknowns, enabling a self-adjoint variational principle for steady viscous flow to be constructed. Concurrently, appropriate commonly occurring physical and auxiliary boundary conditions are prescribed, including establishment of a first integral for the dynamic boundary condition at a free surface. Starting from this new formulation, three classical flow problems are considered, the results obtained being in total agreement with solutions in the open literature. A new least-squares finite element method based on the first integral of the steady two-dimensional, incompressible, Navier-Stokes equations is developed, with optimal convergence rates established theoretically. The method is analysed comprehensively, thoroughly validated and shown to be competitive when compared to a corresponding, standard, primitive-variable, finite element formulation. Implementation details are provided, and the well-known problem of mass conservation addressed and resolved via selective weighting. The attractive positive definiteness of the resulting linear systems enables employment of a customized scalable algebraic multigrid method for efficient error reduction. The solution of several engineering related problems from the fields of lubrication and film flow demonstrate the flexibility and efficiency of the proposed method, including the case of unsteady flow, while revealing new physical insights of interest in their own right

Estimation d’erreur a posteriori pour l’approximation de problèmes Laplaciens fractionnaires et applications en poro-élasticité

This manuscript is concerned with a posteriori error estimation for the finite element discretization of standard and fractional partial differential equations as well as an application of fractional calculus to the modeling of the human meniscus by poro-elasticity equations. In the introduction, we give an overview of the literature of a posteriori error estimation in finite element methods and of adaptive refine- ment methods. We emphasize the state–of–the–art of the Bank–Weiser a posteriori error estimation method and of the adaptive refinement methods convergence results. Then, we move to fractional partial differential equations. We give some of the most common discretization methods of fractional Laplacian operator based equations. We review some results of a priori error estimation for the finite element discretization of these equations and give the state–of–the–art of a posteriori error estimation. Finally, we review the literature on the use of the Caputo’s fractional derivative in applications, focusing on anomalous diffusion and poro-elasticity applications. The rest of the manuscript is organized as follow. Chapter 1 is concerned with a proof of the reliability of the Bank–Weiser estimator for three–dimensional problems, extending a result from the literature. In Chapter 2 we present a numerical study of the Bank–Weiser estimator, provide a novel implementation of the estimator in the FEniCS finite element software and apply it to a variety of elliptic equations as well as goal-oriented error estimation. In Chapter 3 we derive a novel a posteriori estimator for the L2 error induced by the finite element discretization of fractional Laplacian operator based equations. In Chapter 4 we present new theoretical results on the convergence of a rational approximation method with consequences on the approximation of fractional norms as well as a priori error estimation results for the finite element discretization of fractional equations. Finally, in Chapter 5 we provide an application of fractional calculus to the study of the human meniscus via poro-elasticity equations.Ce manuscrit traite d’estimation d’erreur a posteriori pour la discrétisation d’équations aux dérivées partielles standard et fractionnaires par les méthodes éléments finis ainsi que de l’application de l’analyse fractionnaire à la modélisation du ménisque humain par les équations de poro-élasticité. Dans l’introduction, nous donnons un aperçu de la littérature sur l’estimation d’erreur a posteriori pour les méth- odes éléments finis et des méthodes de raffinement adaptatif. Nous insistons particulièrement sur l’état de l’art de la méthode d’estimation d’erreur a posteriori de Bank-Weiser et sur les résultats de convergence des méthodes adaptatives. Ensuite, nous nous intéressons aux équations aux dérivées partielles fractionnaires. Nous présentons certaines méthodes de discrétisation d’équations basées sur l’opérateur Laplacien fractionnaire et donnons l’état de l’art sur l’estimation d’erreur a posteriori. Finalement, nous donnons un aperçu de la littérature concernant les applications de la dérivée fractionnaire au sens de Caputo en nous concentrant sur le phénomène de diffusion anormale et les applications en poro-élasticité

Numerical Methods for Partial Differential Equations

These lecture notes are devoted to the numerical solution of partial differential equations (PDEs). PDEs arise in many fields and are extremely important in modeling of technical processes with applications in physics, biology, chemisty, economics, mechanical engineering, and so forth. In these notes, not only classical topics for linear PDEs such as finite differences, finite elements, error estimation, and numerical solution schemes are addressed, but also schemes for nonlinear PDEs and coupled problems up to current state-of-the-art techniques are covered. In the Winter 2020/2021 an International Class with additional funding from DAAD (German Academic Exchange Service) and local funding from the Leibniz University Hannover, has led to additional online materials such as links to youtube videos, which complement these lecture notes. This is the updated and extended Version 2. The first version was published under the DOI: https://doi.org/10.15488/9248