5. On a new phase field model for the approximation of interfacial energies of multiphase systems
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Elie Bretin
Abstract
This chapter is devoted to a new multiphase field approximation model for a category of interfacial energies of multiphase systems which appear in material sciences or image processing. Our model has several advantages when the surface tensions satisfy a suitable embedding property (namely the ℓ1-embeddability): (i) it can be explicitly derived from the surface tensions; (ii) the Γ-convergence to the multiphase perimeter can be proven; (iii) it is convenient for robust numerical approximation of the associated gradient flow. Several applications are presented, in particular to droplets dynamics.
Abstract
This chapter is devoted to a new multiphase field approximation model for a category of interfacial energies of multiphase systems which appear in material sciences or image processing. Our model has several advantages when the surface tensions satisfy a suitable embedding property (namely the ℓ1-embeddability): (i) it can be explicitly derived from the surface tensions; (ii) the Γ-convergence to the multiphase perimeter can be proven; (iii) it is convenient for robust numerical approximation of the associated gradient flow. Several applications are presented, in particular to droplets dynamics.
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
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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
