# Analysis Problem Size Reduction Technique

In continuum topology optimization of sparse structures with limited structural material usage and realistically large design loads, geometrical instabilities become a definite possibility. If modeled, geometrical instabilities in structural systems can result in finite deformations. The modeling of finite deformations in mixed solid-void grid-like meshes used in continuum structural topology optimization can result in excessive distortion of void or low-density elements that can in turn lead to numerical difficulties solving the structural analysis problem. Since the optimization process in continuum topology optimization typically removes structural material from low stress, or low-sensitivity areas, fairly substantial regions of low-density elements are common. As these elements are highly compliant, they contribute very little to structural stability, while being subject to excessive deformation that creates numerical difficulties. Therefore, it is sometimes advantageous to identify these large regions of void and low-density elements and to remove them, at least temporarily, from the structural analysis problem. An automated algorithm for identifying such regions and removing them from the structural analysis problem is presented and discussed below. It is worth noting that the procedure proposed and investigated here is reversible in that it permits low-density regions of the structure to return as high-density structural regions even after they have previously been removed from consideration during structural analysis.

The essence of the proposed analysis problem reduction technique can be captured in the three steps listed below:

1. All finite elements in the structural analysis model that are devoid of solid material, or nearly so, are identified as “void” elements. (Typically, in the examples presented below, if an element’s volume fraction of solid material is less than or equal to.002, it is identified as “void”.)

2. All nodes that are members only of “void” elements are identified as “prime” nodes. The degrees of freedom of such “prime” nodes are restrained, reducing the size of the analysis problem.

3. If only “prime” nodes comprise an element, that element is then denoted as a “prime” element. Such “prime” elements are then neglected in the structural analysis problem so that if they undergo excessive distortion it does not create any singularities in the system of finite element equations. It is worth noting, that “prime” elements are those that are surrounded by “void” elements.

A graphical description supporting the explanation of this technique for reducing the analysis problem is presented in Figure 1.

The matter of reducing the analysis problem by neglecting significant regions of void elements has been addressed in preceding works (9,10). The current reduction techniques have proven to be both robust and efficient in all of the example problems presented in the section below. The techniques are especially powerful and effective when applied in design problems involving extremely sparse structures, since highly refined meshes are needed when very stringent material resource constraints are imposed. When a fine mesh is employed with a very limited amount of structural material, the proposed reduction techniques will allow for dramatic savings in computing effort.

Figure 1. Schematic of partial mesh to illustrate analysis problem reduction technique. Nodes having vanishing design variable values are denoted with open circles; filled circles denote nodes associated with nonzero design values; nodes represented by open circles with X’s are “prime” nodes whose degrees of freedom are restrained in the size reduction method. Elements designated with “S” are at least partially solid and those with “V” are devoid of material. Those designated with “P” are prime and need not be considered during structural analysis since all of their degrees of freedom are restrained.

Examples

The validity of the proposed methodology is demonstrated in this article on three problems, where the first two are benchmark linear stability design problems with extreme sparsity and the third involves design of a very sparse, hinge-free gripper compliant mechanism. In all problems presented, the void material is assumed to have a stiffness equal to that of structural material scaled down by a factor of 10-6.