Adaptive unstructured remeshing using gradient-only optimisation algorithms for shape optimisation
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 performing the shape optimization, we use the quadratically convergent unstructured remeshing strategy we have previously proposed; it is based on a truss structure analogy, for which computationally efficient analytical gradients are available. While the unstructured remeshing strategy per se allows for increased flexibility (e.g. large shape changes per iteration), this comes at the cost of the discontinuities mentioned, due to remeshing and changes in finite element connectivity. We therefore reflect on optimization strategies for the discontinuous unstructured remeshing shape optimization problem. To do this, we present two simple algorithms based on the well-known Broyden- Fletcher-Goldfarb-Shanno (BFGS) updating scheme, in which we modify the line searches used in the updating formulae. We then present two relatively simple example problems, being a discontinuous one-dimensional function, and the shape optimisation of a Michell-like structure. © 2006 Civil-Comp Press.