ABSTRACT
Goong Chen* and Jianxin Zhou* Texas A&M University, College Station, Texas
Robert C. McLean Oklahoma State University, Stillwater, Oklahoma
A B ST R A C T
The Hadamard formula for shape gradient involves only boundary integrals of func
tionals. This makes the application of the boundary element method extremely attractive
in the shape design and control of systems governed by partial differential equations. In
this paper, we develop the boundary element method for shape optimization problems with
a quadratic cost governed by the potential equation. This approach is based upon the the
ory of boundary integral equations derived from simple layer potential representations of
the state and the adjoint state. Boundary element computations are carried out for two
concrete examples involving expanding circular and elliptical domains, effectively yield
ing numerical solutions for the optimal shapes and control variables. Piecewise constant
boundary elements are used in the discretization, and numerical results are illustrated by
computer graphics.
