# Harbir Antil - Research

## Shape Optimization of Shell Structure Acoustics

For a description of PDE constrained optimization problems we refer to:
link
## Goal

We provide 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.
## Collaborators

## Achievements

- We provide well-posedness of a new coupled model; shell and acoustics.
- We prove the Fr\'{e}chet differentiability of the state with respect to the surface shape.
- Invoking adjoint equations we efficiently compute derivatives of the radiated field (cost functional)
with respect to large numbers of shape parameters.

## Applications

This approach can be applied to efficiently design:
- musical instruments
- loud speakers

## Related Publications

**H. Antil**, M. Heinkenschloss, and S. Hardesty.

* Shape Optimization of Shell Structure Acoustics. *

SIAM J. Control Optim. 55 (3), 1347--1376, 2017.

**H. Antil**, M. Heinkenschloss, and S. Hardesty.

* Supplementary Materials: Shape Optimization of Shell Structure Acoustics. *

Department of Computational and Applied Mathematics, Rice University (2017). Technical Report