Shape calculus in differential forms

Description

Differential forms play an increasingly important role in the theoretical analysis and numerical solution of a variaty of partial differential equations (PDEs) [1,2]. In this project, we investigate an intrinsic perspective in differential forms of shape derivatives of the solutions to PDEs, which is of immense significance in the field of PDE-constrained shape design, optimization and sensitivity analysis. Fundamental Hadamard structure theorems will be addressed for shape derivatives of boundary and domain integrals by the exterior calculus of differential forms. Higher order shape derivatives can be systematically derived in a recursive way. With resort to such an approach, one can express the abstract framework for variational formuations of concrete PDEs and incarnate the differential forms in terms of avatars like scalar functions and vector fields to derive explicit formulae by straightforward calculus for shape derivatives of solutions to certian PDE under concern. As a model problem, we illustrate the power of the theory by deriving the shape derivatives of solutions to elliptic PDEs with Dirichlet, Neumann and Robin boundary conditions. In particular, the Dirichlet boundary condition will be rigorously derived in the weak sense via dual variational formulation for the first time.

Publications / Preprints or SAM-Reports

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References

[1] Douglas N. Arnold, Richard S. Falk, and Ragnar Winther, Finite element exterior calculus, homological techniques, and applications, Acta Numerica, (2006), pp. 1-155.
[2] Ralf Hiptmair, Finite elements in computational electromagnetism, Acta Numerica, 11 (2002), pp. 237-339.

Funding

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Contacts

Prof. Ralf Hiptmair
Dr. Jingzhi Li