Keywords: shape optimisation, unstructured remeshing, gradient based optimisation, discontinuities.

In this 'work-in-progress' paper, we show how gradient-only optimization algorithms can effectively be used for optimization problems characterized by noise and discontinuities; an example of such a function being the unstructured remeshing shape optimization problem. For this problem, gradient information is known to be (reasonably) reliable, as opposed to function value information, which may be deceptive.

In studying the unstructured remeshing shape optimization problem, we restrict ourselves to the displacement based shape optimisation problem. In order to solve the associated partial differential equations, we use the finite element method (FEM), and in particular, linear strain triangle (LST) elements. The finite element meshes are generated using the quadratically convergent mesh generator we have previously proposed [1]; it is based on a truss structure analogy, for which analytical gradients are available. The computational effort required in generating the analytical sensitivities is comparable to that of doing a single finite element analysis, hence the approach is highly efficient from a computational point of view.

Having opted for an unstructured remeshing strategy, a new mesh is generated in each consecutive iteration. This ensures that the mesh quality during each iteration is 'near-optimal' (mostly depending on the quality of the mesh generator used). This important advantage however comes at the cost of having the discontinuous function and gradients mentioned, due to remeshing and changes in finite element connectivity. (In addition, the preservation of continuity relationships may also be difficult to enforce in the general case.)

We therefore reflect on optimization strategies for the discontinuous unstructured remeshing shape optimization problem: conventional gradient based optimisers use both function value information and gradient information; hence both the function and gradient information are assumed to be reliable. However, as said, in unstructured remeshing shape optimisation it is known that the objective function is highly discontinuous due to remeshing, and changes in element connectivity [1,2]. As a consequence, the function value information is deceptive, which results in extremely poor performance for conventional gradient based optimisers.

The main aim of this paper therefore is to investigate whether gradient based optimisation can still effectively be used to optimise functions when the function value information is deceptive, but the gradient information is (reasonably) reliable. Consequently, we investigate whether gradient-only optimisation algorithms, like those proposed by Snyman [3], can effectively be used as optimisers.

To do this, we present two simple algorithms based on the well-known Broyden-Fletcher-Goldfarb-Shanno (BFGS) updating scheme. In particular, we modify the line searches used in the updating formulae. In the first implementation, we use only function values in the line search; in the second, only gradient information.

We then present two relatively simple example problems, being a discontinuous one-dimensional function, and the shape optimisation of a Michell-like structure. We show that gradient-only optimisation algorithms can indeed be effectively used for optimisation problems characterized by noise and discontinuities, although many difficulties remain. Gradient-only optimisation algorithms certainly by far outperform algorithms that also use deceptive function value information.

In on-going research, we are investigating a posteriori error estimates for the unstructured remeshing strategy we have developed, which may then be combined with an adaptive mesh refinement strategy to improve the quality of the finite element meshes as the shape iterations progress.

1

Wilke DN, Kok S, Groenwold AA. A quadratically convergent unstructured remeshing strategy for shape optimization. International Journal for Numerical Methods in Engineering 2006; 65:1-17. doi:10.1002/nme.1430

2

Brandstätter BR, Ring W, Magele C, Richter KR. Shape design with great geometrical deformations using continuously moving finite element nodes. IEEE Transactions on Magnetics 1998; 34:2877-2880. doi:10.1109/20.717670

3

Snyman JA. A gradient-only line search method for the conjugate gradient method applied to constrained optimization problems with severe noise in the objective function. International Journal for Numerical Methods in Engineering 2005; 62:72-82. doi:10.1002/nme.1189

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