Density-based topological design of structures subjected to water pressure using a parametric loading surface

Topological design of structures has enjoyed a period of steady growth since the publication of a seminal paper by Bendsøe and Kikuchi. Nowadays, topology design can be recognized as a discipline in structural and multidisciplinary optimization with its idiosyncrasies, specific formulations, and solution techniques. A class of problems, known as topological optimization with design-dependent loading (see Rozvany and Prager (1979) for first closed-form solutions and Fuchs and Moses (2000) for semianalytical results) present often some intrinsic obstacles. Such is the case, in particular, in the presence of pressure loads. Indeed, when water pressure is the applied load, the density-based technique seems to run into trouble because not until the structure has converged is there a clear water/material interface where the pressure can be applied. Several authors have proposed solutions, and this paper adds its contribution. We introduce, independent of the material distribution, a parametric loading surface, the shape of which is an additional unknown. Water pressure is applied at this surface as if it were the water/material interface. By doing so, one factually severs the physical link between the smooth pressure surface and the indistinct material distribution. In order to remove any lingering material from the water region, a small elastic modulus E is assumed there. Additional care was exercised for computing the sensitivities as the loading surface meanders through the fixed grid of the finite element model. For this purpose, a smooth transition for E from the water to the material zone is imposed to alleviate any numerical instabilities that may occur in computing the sensitivities of E over the abrupt transition at the loading surface. Several cases, primarily the design of optimal dams, were tested. The method proved very robust and produced crisp water/material interfaces, which coalesced with the loading surfaces.