Shape Optimization of Shell Structure Acoustics
Harbir Antil
Department of Mathematical Sciencess
George Mason University
Sean Hardesty
Z-Terra Inc., Houston
Matthias Heinkenschloss
Department of Computational and Applied Mathematics
Rice University
Abstract
This paper provides a rigorous framework for the
numerical solution of shape optimization problems in shell structure acoustics.
The structure is modeled with Naghdi shell equations, fully coupled to
boundary integral equations on a minimally regular
surface, permitting the formulation of
three-dimensional radiation and scattering problems on a
two-dimensional set of reference coordinates.
We prove well-posedness of this model, and Fr\'{e}chet
differentiability of the state with respect to the surface shape.
For a class of shape optimization problems we prove existence of optimal solutions under slightly
stronger surface regularity assumptions. Finally, adjoint equations are used to
efficiently compute derivatives of the radiated field with respect to large numbers of shape
parameters, which allows consideration of a rich space of shapes, and thus,
of a broad range of design problems.
Keywords. Structural acoustics, Naghdi shell, boundary integral equation, shape derivative, shape optimization.