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27
An Implicit Finite Element Method for Elastic Solids in Contact
 In Comput. Anim
, 2001
"... 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 traditio ..."
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Cited by 27 (1 self)
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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 selfcontact such as skin folds.
Axioms and Variational Problems in Surface Parameterization
"... For a surface patch on a smooth, twodimensional surface in IR , lowdistortion 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 nonlinear el ..."
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Cited by 15 (3 self)
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For a surface patch on a smooth, twodimensional surface in IR , lowdistortion 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 nonlinear 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.
On Mathematical Models For Phase Separation In Elastically Stressed Solids
, 2000
"... 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. ..."
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Cited by 14 (6 self)
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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 \Gammalimit of the elastic GinzburgLandau energies 52 5.2. EulerLagrange equation for the sharp interface functional 60 6. The GibbsThomson 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
Hyperbolicity of the nonlinear models of Maxwell's equations
 Arch. Rat. Mech. Anal
, 2003
"... 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 2matrix, we show that polyconvexity of W implies the local wellposedness ..."
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Cited by 14 (2 self)
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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 2matrix, we show that polyconvexity of W implies the local wellposedness of the Cauchy problem within smooth functions of class H with s > 1 + d/2.
Simulation of Nonpenetrating Elastic Bodies Using Distance Fields
, 2000
"... : We present an efficient algorithm for simulation of nonpenetrating flexible bodies with nonlinear elasticity. We use finite element methods to discretize the continuum model of nonrigid objects and the fast marching level set method to precompute a distance field for each undeformed body. As the ..."
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Cited by 10 (2 self)
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: We present an efficient algorithm for simulation of nonpenetrating flexible bodies with nonlinear elasticity. We use finite element methods to discretize the continuum model of nonrigid 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 nonpenetration constraints between two colliding elastic bodies. This approach can automatically handle selfpenetration and interpenetration in a uniform manner. We combine quasiviscous Newton's iteration and adaptivestepsize 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, physicallybased animation, numerical analysis. 1 Introduction Due to recent advancements in physic...
Diffusion in PoroElastic Media
 Jour. Math. Anal. Appl
, 1998
"... . Existence, uniqueness and regularity theory is developed for a general initialboundaryvalue 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 se ..."
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Cited by 10 (7 self)
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. Existence, uniqueness and regularity theory is developed for a general initialboundaryvalue 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 quasistatic 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 quasistatic 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). ...
Parallel Adaptive Subspace Correction Schemes with Applications to Elasticity
 Comput. Methods Appl. Mech. Engrg
, 1999
"... : 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 featur ..."
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Cited by 8 (4 self)
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: 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 spacefilling curves. Keywords: subspace correction, iter...
A nonlinear elastic shape averaging approach
 SIAM Journal on Imaging Sciences
, 2008
"... 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 elast ..."
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Cited by 7 (6 self)
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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 multiscale 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.
A Continuum Mechanical Approach to Geodesics in Shape Space
"... In this paper concepts from continuum mechanics are used to define geodesic paths in the space of shapes, where shapes are implicitly described as boundary contours of objects. The proposed shape metric is derived from a continuum mechanical notion of viscous dissipation. A geodesic path is defined ..."
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Cited by 7 (3 self)
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In this paper concepts from continuum mechanics are used to define geodesic paths in the space of shapes, where shapes are implicitly described as boundary contours of objects. The proposed shape metric is derived from a continuum mechanical notion of viscous dissipation. A geodesic path is defined as the family of shapes such that the total amount of viscous dissipation caused by an optimal material transport along the path is minimized. The approach can easily be generalized to shapes given as segment contours of multilabeled images and to geodesic paths between partially occluded objects. The proposed computational framework for finding such a minimizer is based on the time discretization of a geodesic path as a sequence of pairwise matching problems, which is strictly invariant with respect to rigid body motions and ensures a 11 correspondence along the induced flow in shape space. When decreasing the time step size, the proposed model leads to the minimization of the actual geodesic length, where the Hessian of the pairwise matching energy reflects the chosen Riemannian metric on the underlying shape space. If the constraint of pairwise shape correspondence is replaced by the volume of the shape mismatch as a penalty functional, one obtains for decreasing time step size an optical flow term controlling the transport of the shape by the underlying motion field. The method is implemented via a level set representation of shapes, and a finite element approximation is employed as spatial discretization both for the pairwise matching deformations and for the level set representations. The numerical relaxation of the energy is performed via an efficient multiscale procedure in space and time. Various examples for 2D and 3D shapes underline the effectiveness and robustness of the proposed approach. 1
An Elasticity Approach to Principal Modes of Shape Variation
"... 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 s ..."
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Cited by 3 (3 self)
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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