Reconstructing unknown inclusions for the biharmonic equation

TL;DR

This paper develops a method to detect and reconstruct unknown obstacles with complex shapes in a biharmonic equation setting using the enclosure method and complex geometrical optics solutions.

Contribution

It extends the enclosure method to penetrable obstacles governed by the biharmonic equation, including non-convex parts, using complex geometrical optics solutions.

Findings

Successfully reconstructs non-convex obstacle parts.

Validates the method for penetrable obstacles with biharmonic equations.

Provides theoretical justification for the reconstruction approach.

Abstract

Herein, we study an inverse problem for detecting unknown obstacles by the enclosure method using the Dirichlet--to--Neumann map for measurements. We justify the method for an penetrable obstacle case involving a biharmonic equation. We use complex geometrical optics solutions with a logarithmic phase to reconstruct some non--convex parts of the obstacle.