Structural boundary design via level set and immersed interface methods

We develop and test an algorithmic approach to the boundary design of elast ic structures. The goal of our approach is two-fold: first, to develop a me thod which allows one to rapidly solve the two-dimensional Lame equations i n arbitrary domains and compute, for example, the stresses, and second, to develop a systematic way of modifying the design to optimize chosen propert ies. At the core, our approach relies on two distinct steps. Given a design , we first apply an explicit jump immersed interface method to compute the stresses for a given design shape. We then use a narrow band level set meth od to perturb this shape and progress towards an improved design. The equat ions of 2D linear elastostatics in the displacement formulation on arbitrar y domains are solved quickly by domain embedding and the use of fast elasto static solvers. This effectively reduces the dimensionality of the problem by one. Once the stresses are found, the level set method, which represents the design structure through an embedded implicit function, is used in the second step to alter the shape, with velocities depending on the stresses in the current design, Criteria are provided for advancing the shape in an appropriate direction and fur correcting the evolving shape when given cons traints are violated. (C) 2000 Academic Press.