CAD inspired hierarchical partition of unity constructions for NURBS-based, meshless design, analysis and optimization

In this paper we develop a unified representational paradigm for design and analysis that is inspired by the set theoretic Boolean operations of the constructive solid geometry (CSG) procedure of computer‐aided design (CAD). We develop the notion of a primitive design state (corresponding to a CSG primitive region), which is characterized by a description of the geometrical shape of the primitive and by a description of the material distribution within the primitive. We analogously define a primitive behaviour state that is associated with a primitive design state and that is determined through a global analysis problem. We define a global multi‐level design problem to determine the primitive design states. We define Boolean operations on the fields belonging to primitive design and behaviour spaces and show that the compositions of these fields amount to a hierarchical partition of unity construction. We propose to use non‐uniform rational B‐splines (NURBS) to discretize the geometry, material, and behavioural fields (local approximations) defined on the primitive regions. We show that this leads to recursive partition of unity constructions on the knot spaces (sub‐local approximations) due to the partition of unity property of NURBS basis functions. Since the primitive shapes are parametrically defined using NURBS, the numerical quadrature is carried out on the parametric sub‐regions of the NURBS entities. This eliminates the need for either a background mesh or a pseudo‐mesh for quadrature. The developed methodology is implemented in a symbolic, meshless computational framework written using the java language (jNURBS) and was demonstrated on two classes of problems. The first class is the simultaneous optimal design of shape and multi‐material distributions. The procedure is argued as being a generalization of classical topology optimization to designing objects made of functionally graded materials with pre‐defined geometrical constraints, such as holes or fixed boundary shape. The second class of problems demonstrated in the paper is optimal design to mitigate the effects of cracks. Here, we identify the optimal locations of holes introduced to reduce the stress intensity factor of an edge crack. Copyright © 2007 John Wiley & Sons, Ltd.