Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Frequency-explicit stability estimates for time-harmonic elastodynamic problems in nearly incompressible materials

T. Chaumont-Frelet, S. Nicaise

Abstract

We consider time-harmonic elastodynamic problems in heterogeneous media.cWe focus on scattering problems in the high-frequency regime and incnearly incompressible media, where the the angular frequency $ω$ and ratio of the Lamé parameters $λ/μ$ may both be large. We derive stability estimates controlling the norm of the solution by the norm of the right-hand side up to a fully-explicit constant. Crucially, under natural assumptions on the domain and coefficients, this constant increases linearly with $ω$ and is uniform in the ratio $λ/μ$.

Frequency-explicit stability estimates for time-harmonic elastodynamic problems in nearly incompressible materials

Abstract

We consider time-harmonic elastodynamic problems in heterogeneous media.cWe focus on scattering problems in the high-frequency regime and incnearly incompressible media, where the the angular frequency and ratio of the Lamé parameters may both be large. We derive stability estimates controlling the norm of the solution by the norm of the right-hand side up to a fully-explicit constant. Crucially, under natural assumptions on the domain and coefficients, this constant increases linearly with and is uniform in the ratio .
Paper Structure (33 sections, 28 theorems, 199 equations)

This paper contains 33 sections, 28 theorems, 199 equations.

Table of Contents

  1. Introduction
  2. Settings
  3. Domain
  4. Tensor notations
  5. Functional setting
  6. Notations related to the boundary
  7. Material coefficients
  8. Frequency
  9. Dissipative boundary
  10. Stress tensor
  11. Model problem
  12. Standard key identities
  13. Gå rding identites
  14. Integration by parts identities
  15. Dirichlet boundary
  16. ...and 18 more sections

Key Result

Lemma 3.1

For all $\boldsymbol u \in \boldsymbol H^1_{\Gamma_{\rm Dir}}(\Omega)$ solution to eq_helmholtz_weak, we have

Theorems & Definitions (50)

  • Remark 2.1: Relaxed assumptions on $\lambda$
  • Lemma 3.1: Gå rding-type identities
  • Lemma 3.2: Rellich identity
  • Corollary 3.3: Rellich identity with lower regularity
  • proof
  • Lemma 3.4: Zero-order term identity
  • Lemma 3.5: Dirichlet boundary
  • Theorem 3.6: Morawetz identity
  • proof
  • Proposition 4.1: Korn inequality
  • ...and 40 more