4. High-order topological expansions for Helmholtz problems in 2D
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Victor A. Kovtunenko
Abstract
Methods of topological analysis are inherently related to singular perturbations. For topology variation, a trial geometric object put in a test domain is examined by reducing the object size from a finite to an infinitesimal one. Based on the singular perturbation of the forward Helmholtz problem, a topology optimization approach, which is a direct one, is described for the inverse problem of object identification from boundary measurements. Relying on the 2d setting in a bounded domain, the high-order asymptotic result is proved rigorously for the Neumann, Dirichlet, and Robin-type conditions stated at the object boundary. In particular, this implies the first-order asymptotic term called a topological derivative. For identifying arbitrary test objects, a variable parameter of the surface impedance is successful. The necessary optimality condition of minimum of the objective function with respect to trial geometric variables is discussed and realized for finding the center of the test object.
Abstract
Methods of topological analysis are inherently related to singular perturbations. For topology variation, a trial geometric object put in a test domain is examined by reducing the object size from a finite to an infinitesimal one. Based on the singular perturbation of the forward Helmholtz problem, a topology optimization approach, which is a direct one, is described for the inverse problem of object identification from boundary measurements. Relying on the 2d setting in a bounded domain, the high-order asymptotic result is proved rigorously for the Neumann, Dirichlet, and Robin-type conditions stated at the object boundary. In particular, this implies the first-order asymptotic term called a topological derivative. For identifying arbitrary test objects, a variable parameter of the surface impedance is successful. The necessary optimality condition of minimum of the objective function with respect to trial geometric variables is discussed and realized for finding the center of the test object.
Kapitel in diesem Buch
- 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
Kapitel in diesem Buch
- 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
