Summary: We develop and test an algorithmic approach to the boundary design of elastic structures. The goal of our approach is two-fold: first, to develop a method which allows one to rapidly solve two-dimensional Lamé equations in arbitrary domains and compute, for example, stresses; second, to develop a systematic way of modifying the design to optimize chosen properties. At the core, our approach relies on two distinct steps. Given a design, we first apply an explicit jump immersed interface method to compute stresses for a given design shape. We then use a narrow band level set method to perturb this shape, and progress towards an improved design. The equations of two-dimensional linear elastostatics in the displacement formulation on arbitrary domains are solved quickly by domain embedding and by the use of fast elastostatic 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 at the second step to alter the shape, with velocities depending on stresses in the current design. We provide criteria for advancing the shape in an appropriate direction, and for correcting the evolving shape when given constraints are violated.