Keywords: shape optimization, particle swarm, optimization, NURBS, isogeometric analysis, Distmesh tool.

Isogeometric analysis (IGA) is a recently introduced method which builds upon the concept of isoparametric elements and upgrades it to the geometry level. Although the original intention was to span the gap between the computer aided design (CAD) and the finite element method (FEM), the various advantages and range of applicability make the IGA an interesting alternative to the widely used FEM. It has been shown that the IGA outperforms the classical FEM in various aspects (accuracy, robustness, system condition number, etc.). Another distinct advantage of the IGA over the FEM consists in the conciseness of the parametrization of the design variable space, which makes the IGA attractive for shape optimization problems.

Particle swarm optimization (PSO) is a nature-inspired method for simulation of social behavior of several particles by mimicking for example bird flocks or fish schools. Its main advantage is the simplicity of updating rules for a particle's position and velocity terms. The geometrical meaning of the moving particles predetermines this method for solving geometrical problems as well as constrained problems where for example the boundary problem can be tackled as elastic collisions of particles against a wall.

This paper describes the application of the PSO to the shape optimization of two-dimensional domains described by NURBS (non-uniform rational B-splines) and analyzed using the NURBS-based IGA. The regularization of the optimization problem, preventing undesirable clustering of control points of the underlying geometry leading to invalid geometry or parametrization, is achieved by controlling the magnitude of perturbation of the design variables within the PSO using a background mesh. This mesh, however, does not have to comply with the requirements on a standard (for example finit element) computational mesh, as it does not have to follow the exact geometry. Meshes on individual NURBS patches of the geometry need not match in a compatible way, the mesh does not have to respect small features on those parts of the domain, which is not optimized etc. Thus construction of such a mesh (using the MATLAB Distmesh tool is utilized) which is simple and does not introduce a bottleneck to the whole process. Although the proposed approach can be applied for the optimization of both location of control points of the geometry and weight of those control points (since the mesh can be generally of higher spatial dimension than the optimized domain), only the location of control points is currently considered in the optimization.

The paper shows a combination of the above methods. The IGA is a step towards a CAD which, as an addendum, has several advantages over the classical FE analysis in obtaining the mechanical response of a structure. The precise description of the geometry predetermines the IGA as a solution to the shape optimization problem. The particle swarm optimization algorithm is then characterized by a physical meaning of a group of flying particles which can utilize the inner properties of the dynamics of the particles. The shape optimization problem is difficult from the regularity point of view. Therefore, not only limitations within the PSO have been used in this paper, but also the second, background mesh produced by the Distmesh tool has been utilized. The proposed approach has been applied to the benchmark problems. The solution obtained is in reasonable agreement with the theoretical results only for the most coarse resolution.

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