Shape Optimization of an Arterial Bypass by Using Embedding Method

In this paper a new numerical technique in the study of aorto-coronaric bypass anastomoses configurations using unsteady Stokes equations is considered. The theory of optimal control based on notations of the measure theory, functional analysis and linear programming (see [1]), is applied in order to optimize the shape of the zone of the incoming branch of the bypass (the toe) into the coronary (Figure 1). Agoshkov et.al [2], applied optimal control by perturbation theory and provided a new approach to the problem, with the goal of improving arterial bypass graft on the basis of a better understanding of fluid dynamics aspects involved in the bypass studying. The aim of this article is to provide design indications in the perspective of future development for prosthetic bypasses. First the problem is expressed as an shape optimization problem by adding a suitable optimality criteria, where the problem has governed by variational forms on a fixed domain. Then by using an embedding method, the class of admissible shapes is replaced by a class of positive Radon measures. The optimization problem in measure space is then approximated by a linear programming problem. The optimal measure representing optimal shape is approximated by the solution of this linear programming problem. Numerical example is also given