We develop a new optimisation technique that combines multiresolutionsubdivision surfaces for boundary description with immersed finite elements forthe discretisation of the primal and adjoint problems of optimisation. Similarto wavelets multiresolution surfaces represent the domain boundary using acoarse control mesh and a sequence of detail vectors. Based on themultiresolution decomposition efficient and fast algorithms are available forreconstructing control meshes of varying fineness. During shape optimisationthe vertex coordinates of control meshes are updated using the computed shapegradient information. By virtue of the multiresolution editing semantics,updating the coarse control mesh vertex coordinates leads to large-scalegeometry changes and, conversely, updating the fine control mesh coordinatesleads to small-scale geometry changes. In our computations we start byoptimising the coarsest control mesh and refine it each time the cost functionreaches a minimum. This approach effectively prevents the appearance ofnon-physical boundary geometry oscillations and control mesh pathologies, likeinverted elements. Independent of the fineness of the control mesh used foroptimisation, on the immersed finite element grid the domain boundary is alwaysrepresented with a relatively fine control mesh of fixed resolution. With theimmersed finite element method there is no need to maintain an analysissuitable domain mesh. In some of the presented two- and three-dimensionalelasticity examples the topology derivative is used for creating new holesinside the domain.
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