439 research outputs found
A Discretize-Then-Optimize Approach to PDE-Constrained Shape Optimization
We consider discretized two-dimensional PDE-constrained shape optimization
problems, in which shapes are represented by triangular meshes. Given the
connectivity, the space of admissible vertex positions was recently identified
to be a smooth manifold, termed the manifold of planar triangular meshes. The
latter can be endowed with a complete Riemannian metric, which allows large
mesh deformations without jeopardizing mesh quality; see arXiv:2012.05624.
Nonetheless, the discrete shape optimization problem of finding optimal vertex
positions does not, in general, possess a globally optimal solution. To
overcome this ill-possedness, we propose to add a mesh quality penalization
term to the objective function. This allows us to simultaneously render the
shape optimization problem solvable, and keep track of the mesh quality. We
prove the existence of a globally optimal solution for the penalized problem
and establish first-order necessary optimality conditions independently of the
chosen Riemannian metric.
Because of the independence of the existence results of the choice of the
Riemannian metric, we can numerically study the impact of different Riemannian
metrics on the steepest descent method. We compare the Euclidean, elasticity,
and a novel complete metric, combined with Euclidean and geodesic retractions
to perform the mesh deformation
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