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©2012 Civil-Comp Ltd |
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A. Pospíšilová, M. Lepš, D. Rypl and B. Patzák
Department of Mechanics, Faculty of Civil Engineering, Czech Technical University in Prague, Czech Republic
Keywords: shape optimization, particle swarm, optimization, NURBS, isogeometric analysis, Distmesh tool.
full paper (pdf) -
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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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