Mathematical Determination of the Fréchet Derivative with Respect to the Domain for a Fluid-Structure Scattering Problem: Case of Polygonal-Shaped Domains

The characterization of the Frechet derivative of the elasto-acoustic scattered field with respect to Lipschitz continuous polygonal domains is established. The considered class of domains is of practical interest since two-dimensional scatterers are always transformed into polygonal-shaped domains when employing finite element methods for solving direct and inverse scattering problems. The obtained result indicates that the Frechet derivative with respect to the scatterer of the scattered field is the solution of the same elasto-acoustic scattering problem but with additional right-hand-side terms in the transmission conditions across the fluid-structure interface. This characterization has the potential to advance the state of the art of the solution of inverse obstacle problems.

[1]  Miguel C. Junger,et al.  Sound, Structures, and Their Interaction , 1972 .

[2]  N. Gol’dman,et al.  Regularization of Ill-Posed Problems by Iteration Methods , 1999 .

[3]  M. Costabel,et al.  Shape Derivatives of Boundary Integral Operators in Electromagnetic Scattering. Part II: Application to Scattering by a Homogeneous Dielectric Obstacle , 2010, 1002.1541.

[4]  D. S. Jones,et al.  LOW-FREQUENCY SCATTERING BY A BODY IN LUBRICATED CONTACT , 1983 .

[5]  Antoine Henrot,et al.  Variation et optimisation de formes : une analyse géométrique , 2005 .

[6]  F. Hettlich.  Frechet derivatives in inverse obstacle scattering , 1995 .

[7]  E. S. Palencia,et al.  Vibration and Coupling of Continuous Systems , 1989 .

[8]  J. Lions,et al.  Problèmes aux limites dans les équations aux dérivées partielles , 1965 .

[9]  Hyunjoong Kim,et al.  Functional Analysis I , 2017 .

[10]  Paul A. Martin,et al.  Fluid-Solid Interaction: Acoustic Scattering by a Smooth Elastic Obstacle , 1995, SIAM J. Appl. Math..

[11]  Necas Jindrich.  Les Méthodes directes en théorie des équations elliptiques , 2017 .

[12]  P. Grisvard.  Singularities in Boundary Value Problems , 1992 .

[13]  R. Potthast.  Domain Derivatives in Electromagnetic Scattering , 1996 .

[14]  David Jerison,et al.  The Neumann problem on Lipschitz domains , 1981 .

[15]  P. Ciarlet,et al.  A characterization of singular electromagnetic fields by an inductive approach , 2008 .

[16]  Houssem Haddar,et al.  On the Fréchet Derivative for Obstacle Scattering with an Impedance Boundary Condition , 2004, SIAM J. Appl. Math..

[17]  On the Characterization of the Fréchet Derivative with Respect to a Lipschitz Domain of the Acoustic Scattered Field , 1999 .

[18]  R. Kress,et al.  Inverse Acoustic and Electromagnetic Scattering Theory , 1992 .

[19]  A. Kirsch.  The domain derivative and two applications in inverse scattering theory , 1993 .