Layout optimization with h‐adaptivity of structures

In this paper, we propose a new procedure for the layout optimization of structures making use of h‐adaptive methods. The method combines the topology optimization and the existing h‐adaptive finite element methods in order to: (i) improve the definition of the material boundary, i.e. the contour between the material and void regions; (ii) reduce the effective number of design variables; and (iii) bound the relative solution error. The refinement strategy is applied to a given element if: (a) the measure of the quality of the element is below a given lower bound; (b) the element is a ‘material element’ or has a side which forms the ‘material boundary’ of the given optimum layout; and (c) the element average error estimate is larger than a multiple of the average error of the mesh. After the h‐adaptive refinement, the mesh quality is improved with the application of a conditional Laplacian smoothing process. The formulation of the optimization problem is defined by the minimization of the compliance of the structure subjected to a volume and side constraints. The design variable is then the average density of the material, which is considered to be constant within each finite element.