Simultaneous Optimization of Shape and Anisotropy

In this chapter, the example for the exactly solvable simultaneous shape and anisotropy optimization is presented. The following optimization problem is solved. The task is to determine the optimal anisotropy together with the a priori unknown boundary of a single hole win the plane made of an orthotropic elastic material. The orientation of principal axes of material orthotropy is arbitrary and is principally different from point to point over the surface of the infinite flat plate. Both the optimal shape of the hole and orientation of principal axes in the plate are searched from the condition of fixed elastic energy for a given area of the hole. Based on the acknowledged solution of optimization problem for the isotropic material, the optimal orientation of orthotropy axes is found. The principal result demonstrates the optimality of the elliptical shape of the hole. The hole is surrounded by the orthotopic material with the orientation of the orthotropy axes in the directions of isolines of the elliptic coordinates.