shape optimization

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displacement matching problem

Given an observed field u*, we want to determine the region (shape) where that field was originated. We assume that the observed field satisfies a partial differential equation (PDE) describing the physics of the problem. In this particular case, we consider the Poisson equation as the PDE. The field u* can be any physical quantity occurring in phenomena governed by the Poisson equation, for example, displacement of a membrane subject to external forces.

We model this inverse problem as a constrained nonlinear least squares problem where the objective function is the misfit between the current u and the target u* fields, and the constraint imposes u to satisfy the desired physical behavior. When u approaches u* we expect to find the solution shape.

As it is, the problem is ill posed. There is an infinitely large number of level set functions (phi in the equations) that can generate the same shape of interest. We regularize the problem by adding a regularization R term on the level set function so a unique solution is viable.

In order to solve the problem on a fictitious domain, we augment the Poisson equation with a penalization term that will enforce the homogeneous Dirichlet boundary conditions on the moving interface. Note that for interior points, i.e. points inside the domain of interest where X = 1 (X is the Heaviside functional), the augmented equation reduces to the original Poisson problem. For exterior points and a suitable choice of the penalization parameter we obtain a displacement field that satisfies the boundary conditions with an accuracy that depends on the mesh size. The finer the mesh, the more accurate is the solution.

Test cases for the displacement matching problem follow below. Click on image to play an animation in Quicktime format.

 

1D examples

quicktime animation 510K (128 elements)

More to come. Stay tuned...

 

2D examples

Copyright Alexandre Cunha
updated on jul/2004