Numerical Solution of an Inverse Obstacle Scattering Problem for Elastic Waves via the Helmholtz Decomposition (Communications in Computational Physics, 2019)
Consider an inverse obstacle scattering problem in an open space whichis filled with a homogeneous and isotropic elastic medium. The inverse problem isto determine the obstacle's surface from the measurement of the displacement on anartificial boundary enclosing the obstacle. In this paper, a new approach is proposedfor numerical solution of the inverse problem. By introducing two scalar potentialfunctions, the method uses the Helmholtz decomposition to split the displacement ofthe elastic wave equation into the compressional and shear waves, which satisfy acoupled boundary value problem of the Helmholtz equations. The domain derivativeis studied for the coupled Helmholtz system. In particular, we show that the domainderivative of the potentials is the Helmholtz decomposition of the domain derivativeof the displacement for the elastic wave equation. Numerical results are presented todemonstrate the effectiveness of the proposed method.
Keywords: The elastic wave equation; inverse obstacle scattering; domain derivative; the Helmholtz decomposition.