7. Discrete varifolds and surface approximation
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Blanche Buet
Abstract
In this chapter, we give a brief account of the notion of discrete varifolds, which are general and flexible tools to represent in a common framework regular surfaces and a large category of discrete representations of surfaces, e.g. point clouds, triangulated surfaces or volumetric representations. In this setting, a new notion of discrete mean curvature can be defined, relying only on the varifold structure and not on any specific feature of the underlying discretization type. This notion of discrete mean curvature is obtained thanks to a regularization of the so-called first variation of the varifold, it is easy to compute, and we prove that it has nice convergence properties. We illustrate this notion on 2D and 3D examples.
Abstract
In this chapter, we give a brief account of the notion of discrete varifolds, which are general and flexible tools to represent in a common framework regular surfaces and a large category of discrete representations of surfaces, e.g. point clouds, triangulated surfaces or volumetric representations. In this setting, a new notion of discrete mean curvature can be defined, relying only on the varifold structure and not on any specific feature of the underlying discretization type. This notion of discrete mean curvature is obtained thanks to a regularization of the so-called first variation of the varifold, it is easy to compute, and we prove that it has nice convergence properties. We illustrate this notion on 2D and 3D examples.
Chapters in this book
- Frontmatter I
- Contents V
-
Part I
- 1. Geometric issues in PDE problems related to the infinity Laplace operator 3
- 2. Solution of free boundary problems in the presence of geometric uncertainties 20
- 3. Distributed and boundary control problems for the semidiscrete Cahn–Hilliard/Navier–Stokes system with nonsmooth Ginzburg–Landau energies 40
- 4. High-order topological expansions for Helmholtz problems in 2D 64
- 5. On a new phase field model for the approximation of interfacial energies of multiphase systems 123
- 6. Optimization of eigenvalues and eigenmodes by using the adjoint method 142
- 7. Discrete varifolds and surface approximation 159
-
Part II
- Preface 173
- 8. Weak Monge–Ampère solutions of the semi-discrete optimal transportation problem 175
- 9. Optimal transportation theory with repulsive costs 204
- 10. Wardrop equilibria: long-term variant, degenerate anisotropic PDEs and numerical approximations 257
- 11. On the Lagrangian branched transport model and the equivalence with its Eulerian formulation 281
- 12. On some nonlinear evolution systems which are perturbations of Wasserstein gradient flows 304
- 13. Pressureless Euler equations with maximal density constraint: a time-splitting scheme 333
- 14. Convergence of a fully discrete variational scheme for a thin-film equation 356
- 15. Interpretation of finite volume discretization schemes for the Fokker–Planck equation as gradient flows for the discrete Wasserstein distance 400
- Index 417
Chapters in this book
- Frontmatter I
- Contents V
-
Part I
- 1. Geometric issues in PDE problems related to the infinity Laplace operator 3
- 2. Solution of free boundary problems in the presence of geometric uncertainties 20
- 3. Distributed and boundary control problems for the semidiscrete Cahn–Hilliard/Navier–Stokes system with nonsmooth Ginzburg–Landau energies 40
- 4. High-order topological expansions for Helmholtz problems in 2D 64
- 5. On a new phase field model for the approximation of interfacial energies of multiphase systems 123
- 6. Optimization of eigenvalues and eigenmodes by using the adjoint method 142
- 7. Discrete varifolds and surface approximation 159
-
Part II
- Preface 173
- 8. Weak Monge–Ampère solutions of the semi-discrete optimal transportation problem 175
- 9. Optimal transportation theory with repulsive costs 204
- 10. Wardrop equilibria: long-term variant, degenerate anisotropic PDEs and numerical approximations 257
- 11. On the Lagrangian branched transport model and the equivalence with its Eulerian formulation 281
- 12. On some nonlinear evolution systems which are perturbations of Wasserstein gradient flows 304
- 13. Pressureless Euler equations with maximal density constraint: a time-splitting scheme 333
- 14. Convergence of a fully discrete variational scheme for a thin-film equation 356
- 15. Interpretation of finite volume discretization schemes for the Fokker–Planck equation as gradient flows for the discrete Wasserstein distance 400
- Index 417
