Abstract
Shape gradients of shape differentiable shape functionals constrained to an interface problem (IP) can be formulated in two equivalent ways. Both formulations rely on the solution of two IPs, and their equivalence breaks down when these IPs are solved approximatively. We establish which expression for the shape gradient offers better accuracy for approximations by means of finite elements. Great effort is devoted to provide numerical evidence of the theoretical considerations.
The work of A. Paganini was partly supported by ETH Grant CH1-02 11-1.
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Notes
- 1.
We tacitly assume that the vector field \({\mathcal V}\) vanishes on \(\partial \Omega \) because we are mostly interested in the contribution of the interface.
- 2.
We denote by C a generic constant, which may depend on \(\Omega \), its discretization, the source function f, and the coefficient \(\sigma \). Its value may differ between different occurrences.
- 3.
Repeating the experiments for \(m_i+n_i\le 3\) produces results in agreement with the observations made for \(m_i+n_i\le 5\). Therefore, the arbitrary choice of restricting the sum of the indices to 5 does not seem to compromise our observations.
- 4.
In experiment 1 new meshes are always adjusted to fit the curved interface.
- 5.
The experiments are performed in MATLAB and are based on the library LehrFEM developed at ETHZ. Mesh generation and uniform refinement are performed with the functions initmesh and refinemesh of the MATLAB PDE Toolbox.
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Paganini, A. (2015). Approximate Shape Gradients for Interface Problems. In: Pratelli, A., Leugering, G. (eds) New Trends in Shape Optimization. International Series of Numerical Mathematics, vol 166. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-17563-8_9
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DOI: https://doi.org/10.1007/978-3-319-17563-8_9
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