Convergence in the incompressible limit of new discontinuous Galerkin methods with general quadrilateral and hexahedral elements

Standard low-order finite elements, which perform well for problems involving compressible elastic materials, are known to under-perform when nearly incompressible materials are involved, commonly exhibiting the locking phenomenon. Interior penalty (IP) discontinuous Galerkin methods have been shown to circumvent locking when simplicial elements are used. The same IP methods, however, result in locking on meshes of quadrilaterals. The authors have shown in earlier work that under-integration of specified terms in the IP formulation eliminates the locking problem for rectangular elements. Here it is demonstrated through an extensive numerical investigation that the effect of using under-integration carries over successfully to meshes of more general quadrilateral elements, as would likely be used in practical applications, and results in accurate displacement approximations. Uniform convergence with respect to the compressibility parameter is shown numerically. Additionally, a stress approximation obtained here by postprocessing shows good convergence in the incompressible limit

Variational Methods in Shape Space

This dissertation deals with the application of variational methods in spaces of geometric shapes. In particular, the treated topics include shape averaging, principal component analysis in shape space, computation of geodesic paths in shape space, as well as shape optimisation. Chapter 1 provides a brief overview over the employed models of shape space. Geometric shapes are identified with two- or three-dimensional, deformable objects. Deformations will be described via physical models; in particular, the objects will be interpreted as consisting of either a hyperelastic solid or a viscous liquid material. Furthermore, the description of shapes via phase fields or level sets is briefly introduced. Chapter 2 reviews different and related approaches to shape space modelling. References to related topics in image segmentation and registration are also provided. Finally, the relevant shape optimisation literature is introduced. Chapter 3 recapitulates the employed concepts from continuum mechanics and phase field modelling and states basic theoretical results needed for the later analysis. Chapter 4 addresses the computation of shape averages, based on a hyperelastic notion of shape dissimilarity: The dissimilarity between two shapes is measured as the minimum deformation energy required to deform the first into the second shape. A corresponding phase-field model is introduced, analysed, and finally implemented numerically via finite elements. A principal component analysis of shapes, which is consistent with the previously introduced average, is considered in Chapter 5. Elastic boundary stresses on the average shape are used as representatives of the input shapes in a linear vector space. On these linear representatives, a standard principal component analysis can be performed, where the employed covariance metric should be properly chosen to depend on the input shapes. Chapter 6 interprets shapes as belonging to objects made of a viscous liquid and correspondingly defines geodesic paths between shapes. The energy of a path is given as the total physical dissipation during the deformation of an object along the path. A rigid body motion invariant time discretisation is achieved by approximating the dissipation along a path segment by the deformation energy of a small solid deformation. The numerical implementation is based on level sets. Chapter 7 is concerned with the optimisation of the geometry and topology of solid structures that are subject to a mechanical load. Given the load configuration, the structure rigidity, its volume, and its surface area shall be optimally balanced. A phase field model is devised and analysed for this purpose. In this context, the use of nonlinear elasticity allows to detect buckling phenomena which would be ignored in linearised elasticity

A virtual element method for transversely isotropic elasticity

This work studies the approximation of plane problems concerning transversely isotropic elasticity, using a low-order virtual element method (VEM). The VEM is an alternative finite element method characterised by complete freedom in determining element geometries that are otherwise polygonal in two dimensions, or polyhedral in three. Transversely isotropic materials are characterised by an axis of symmetry perpendicular to a plane of isotropy, and have applications ranging from fibre reinforcement to biological materials. The governing equations of the transversely isotropic elasticity problem are derived and a virtual element formulation of the problem is presented along with a sample implementation of the method. This work focuses on the treatment of near-incompressibility and near-inextensibility. These are explored both for homogeneous problems, in which the plane of isotropy is fixed; and non-homogeneous problems, in which the fibre directions defining the plane of isotropy vary with position. In the latter case various options are explored for approximating the non-homogeneous terms at an element level. The VEM approximations are shown through a range of numerical examples to be robust and locking-free, for a selection of element geometries, and fibre directions corresponding to mild and strong inhomogeneity

A survey of Trefftz methods for the Helmholtz equation

Trefftz methods are finite element-type schemes whose test and trial functions are (locally) solutions of the targeted differential equation. They are particularly popular for time-harmonic wave problems, as their trial spaces contain oscillating basis functions and may achieve better approximation properties than classical piecewise-polynomial spaces. We review the construction and properties of several Trefftz variational formulations developed for the Helmholtz equation, including least squares, discontinuous Galerkin, ultra weak variational formulation, variational theory of complex rays and wave based methods. The most common discrete Trefftz spaces used for this equation employ generalised harmonic polynomials (circular and spherical waves), plane and evanescent waves, fundamental solutions and multipoles as basis functions; we describe theoretical and computational aspects of these spaces, focusing in particular on their approximation properties. One of the most promising, but not yet well developed, features of Trefftz methods is the use of adaptivity in the choice of the propagation directions for the basis functions. The main difficulties encountered in the implementation are the assembly and the ill-conditioning of linear systems, we briefly survey some strategies that have been proposed to cope with these problems.Comment: 41 pages, 2 figures, to appear as a chapter in Springer Lecture Notes in Computational Science and Engineering. Differences from v1: added a few sentences in Sections 2.1, 2.2.2 and 2.3.1; inserted small correction

Variational Discretization of Higher Order Geometric Gradient Flows Based on Phase Field Models

In this thesis a phase field based nested variational time discretization for Willmore flow is presented. The basic idea of our model is to approximate the mean curvature by a time-discrete, approximate speed of the mean curvature motion. This speed is computed by a fully implicit time step of mean curvature motion, which forms the inner problem of our model. It is set up as a minimization problem taking into account the concept of natural time discretization. The outer problem is a variational problem balancing between the L2-distance of the surface at two consecutive time steps and the decay of the Willmore energy. This is a typical ansatz in case of natural time discretization as it is used in the inner problem. Within the Willmore energy the mean curvature is approximated as mentioned above. Consequently our model is a nested variational and leads to a PDE constraint optimization problem to compute a single time step. It allows time steps up to the size of the spatial grid width. A corresponding parametric version of this model based on finite elements on a triangulation of the evolving geometry was investigated by Olischläger and Rumpf. In this work we derive the corresponding phase field version and prove the existence of a solution. Since biharmonic heat flow is a linear model problem for our nested time discretization of Willmore flow we transfer our model to the linear case. Moreover we present error estimates for the fully discrete biharmonic heat flow and validate them numerically. In addition we compare our model with the semi-implicit phase field scheme for Willmore flow introduced by Du et al. which leads to the result that our nested variational method is significantly more robust. An application of our nested time discretized Willmore model consists in reconstructing a hypersurface corresponding to a given lower-dimensional apparent contour or Huffman labeling. The apparent contour separates the regions where the number of intersections between the hypersurface and the projection ray is constant and the labeling which specifies these intersection numbers is called Huffman labeling. For reconstructing the hypersurface we minimize a regularization energy consisting of the scaled area and Willmore energy subject to the constraint that the Huffman labeling of the minimizing surface equals the given Huffman labeling almost everywhere. To solve the corresponding phase field problem we use an algorithm alternating the minimizes of the regularization and mismatch energy. Moreover we use a multigrid ansatz. In most parts of this work our nested variational problem is solved by setting up the corresponding Lagrange equation and solving the resulting saddle point problem. An alternative is presented in the last part of this work. It deals with the problem of solving the linear model problem as well as our nested variational problem with an Augmented Lagrange method