This work focuses on the simulation of mechanical contact between nonlinearly elastic objects such as the components of the human body. The computation of the reaction forces that act on the contact surfaces (contact forces) is the key for designing a reliable contact handling algorithm. In traditional methods, contact forces are often defined as discontinuous functions of deformation, which leads to poor convergence characteristics. This problem becomes especially serious in areas with complicated self-contact such as skin folds.

For a surface patch on a smooth, two-dimensional surface in IR , low-distortion parameterizations are described in terms of minimizers of suitable energy functionals. Appropriate distortion measures are derived from principles of rational mechanics, closely related to the theory of non-linear elasticity. The parameterization can be optimized with respect to the varying importance of conformality, length preservation and area preservation. A finite element discretization is introduced and a constrained Newton method is used to minimize a corresponding discrete energy. Results of the new approach are compared with other recent parameterization methods.

Contents 1. Introduction 2 2. The diffuse interface model 7 3. Existence for the diffuse interface system 12 3.1. The gradient flow structure 12 3.2. Assumptions 15 3.3. Weak solutions 16 3.4. The implicit time discretisation 17 3.5. Uniform estimates 21 3.6. Proof of the existence theorem 25 3.7. Uniqueness for homogeneous linear elasticity 26 4. Logarithmic free energy 29 4.1. A regularised problem 32 4.2. Higher integrability for the strain tensor 36 4.3. Higher integrability for the logarithmic free energy 42 4.4. Proof of the existence theorem 45 5. The sharp interface limit 46 5.1. The \Gamma--limit of the elastic Ginzburg--Landau energies 52 5.2. Euler--Lagrange equation for the sharp interface functional 60 6. The Gibbs--Thomson equation as a singular limit in the scalar case 70 7. Discussion 79 8. Appendix 81 9. Notation 86 References 90 1 1. Introduction We study a mathematical model describing phase separation in multi-- component alloy

: We present an efficient algorithm for simulation of non-penetrating flexible bodies with nonlinear elasticity. We use finite element methods to discretize the continuum model of non-rigid objects and the fast marching level set method to precompute a distance field for each undeformed body. As the objects deform, the distance fields are deformed accordingly to estimate penetration depth, allowing enforcement of non-penetration constraints between two colliding elastic bodies. This approach can automatically handle self-penetration and inter-penetration in a uniform manner. We combine quasi-viscous Newton's iteration and adaptive-stepsize incremental loading with a predictorcorrector scheme. Our numerical method is able to achieve both numerical stability and efficiency for our simulation. We demonstrate its effectiveness on a moderately complex animated scene. Keywords: Deformation, physically-based animation, numerical analysis. 1 Introduction Due to recent advancements in physic...

: In this paper, we give a survey on the three main aspects of the efficient treatment of PDEs, i.e. adaptive discretization, multilevel solution and parallelization. We emphasize the abstract approach of subspace correction schemes and summarize its convergence theory. Then, we give the main features of each of the three distinct topics and treat the historical background and modern developments. Furthermore, we demonstrate how all three ingredients can be put together to give an adaptive and parallel multilevel approach for the solution of elliptic PDEs and especially of linear elasticity problems. We report on numerical experiments for the adaptive parallel multilevel solution of some test problems, namely the Poisson equation and Lam'e's equation. Here, we emphasize the parallel efficiency of the adaptive code even for simple test problems with little work to distribute, which is achieved through hash storage techniques and space-filling curves. Keywords: subspace correction, iter...

. Existence, uniqueness and regularity theory is developed for a general initial-boundary-value problem for a system of partial differential equations which describes the Biot consolidation model in poroelasticity as well as a coupled quasistatic problem in thermoelasticity. Additional effects of secondary consolidation and pore fluid exposure on the boundary are included. This quasi-static system is resolved as an application of the theory of linear degenerate evolution equations in Hilbert space, and this leads to a precise description of the dynamics of the system. 1. Introduction We shall consider a system modeling diffusion in an elastic medium in the case for which the inertia effects are negligible. This quasi-static assumption arises naturally in the classical Biot model of consolidation for a linearly elastic and porous solid which is saturated by a slightly compressible viscous fluid. The fluid pressure is denoted by p(x; t) and the displacement of the structure by u(x; t). ...

We consider the class of nonlinear models of electromagnetism that has been described by Coleman & Dill [7]. A model is completely determined by its energy density W (B, D). Viewing the electromagnetic field (B, D) as a 3 2-matrix, we show that polyconvexity of W implies the local well-posedness of the Cauchy problem within smooth functions of class H with s > 1 + d/2.

Abstract. A physically motivated approach is presented to compute a shape average of a given number of shapes. An elastic deformation is assigned to each shape. The shape average is then described as the common image under all elastic deformations of the given shapes, which minimizes the total elastic energy stored in these deformations. The underlying nonlinear elastic energy measures the local change of length, area, and volume. It is invariant under rigid body motions, and isometries are local minimizers. The model is relaxed involving a further energy which measures how well the elastic deformation image of a particular shape matches the average shape, and a suitable shape prior can be considered for the shape average. Shapes are represented via their edge sets, which also allows for an application to averaging image morphologies described via ensembles of edge sets. To make the approach computationally tractable, sharp edges are approximated via phase fields, and a corresponding variational phase field model is derived. Finite elements are applied for the spatial discretization, and a multi-scale alternating minimization approach allows the efficient computation of shape averages in 2D and 3D. Various applications, e. g. averaging the shape of feet or human organs, underline the qualitative properties of the presented approach.

Abstract. Concepts from elasticity are applied to analyze modes of variation on shapes in two and three dimensions. This approach represents a physically motivated alternative to shape statistics on a Riemannian shape space, and it robustly treats strong nonlinear geometric variations of the input shapes. To compute a shape average, all input shapes are elastically deformed into the same configuration. That configuration which minimizes the total elastic deformation energy is defined as the average shape. Each of the deformations from one of the shapes onto the shape average induces a boundary stress. Small amplitude stimulation of these stresses leads to displacements which reflect the impact of every single input shape on the average. To extract the dominant modes of variation, a PCA is performed on this set of displacements. To make the approach computationally tractable, a relaxed formulation is proposed, and sharp contours are approximated via phase fields. For the spatial discretization of the resulting model, piecewise multilinear finite elements are applied. Applications in 2D and in 3D demonstrate the qualitative properties of the presented approach. 1

Abstract. The conformal formulation of the Einstein constraint equations is first reviewed, and we then consider the design, analysis, and implementation of adaptive multilevel finite element-type numerical methods for the resulting coupled nonlinear elliptic system. We derive weak formulations of the coupled constraints, and review some new developments in the solution theory for the constraints in the cases of constant mean extrinsic curvature (CMC) data, near-CMC data, and arbitrarily prescribed mean extrinsic curvature data. We then outline some recent results on a priori and a posteriori error estimates for a broad class of Galerkin-type approximation methods for this system which includes techniques such as finite element, wavelet, and spectral methods. We then use these estimates to construct an adaptive finite element method (AFEM) for solving this system numerically, and outline some new convergence and optimality results. We then describe in some detail an implementation of the methods using the FETK software package, which is an adaptive multilevel finite element code designed to solve nonlinear elliptic and parabolic systems on Riemannian