Solid Elements

In the homogenization method, the material is perforated by an infinite number of periodically distributed voids. In each of the solid cells containing a void, the void is shaped like a rectangular prism with a length, width, and height determined by variables a, b, and c. The orientation of the void is determined by angles , , and . These angles are determined by orienting the void in the direction that is most effective in reducing strain energy density.

A cell in a solid element:

Design variables a, b, and c for a solid design element.

The effective material property for a given void size (a, b, and c) is calculated for the perforated material, assuming that there are an infinite number of such voided cells. The equivalent an-isotropic material property is rotated to the optimal void direction and is then used in the calculation of the elemental stiffness matrix. Therefore, each solid element has three void size design variables and three void angle design variables.

The material density for a solid element is equal to 1, minus the volume of the void:

This term is dimensionless during optimization and may vary within the range of the mindens parameter (default = 0.01) to 1. A material density of zero indicates there is no material in the element and results in an ill-conditioned stiffness matrix. The mindens should never be set to zero for models with solid design elements. A material density of 1 indicates the void size is zero.

In the density method, the material’s density is directly used as the density design variable. The effective elasticity property is equal to a scalar function of the density, times the original material property. In this case, there are no angle design variables.

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Design Variables for Topology Optimization