of the void is determined by orienting the void in the direction which is most effective in reducing strain energy density. The orientation angle is not actually a variable; its optimum value is determined at each iteration. In the cell of shell elements, each void is rectangular-shaped, with its length and width determined by design variables a and b. The orientation angle of the void is determined by orienting the void in the direction which is most effective in reducing strain energy density. The orientation angle is not actually a variable; its optimum value is determined at each iteration.
|
Note: |
The void orientation is not a physical representation but a mathematical approximation. |
Design Domain Representative cell in a shell element
Design variables a and b for a shell design element.
Each shell element has a base thickness (T0) and a total thickness (T). The void is not present in the base thickness layer. The base thickness (T0) is always present regardless of void size and is located at the center of the element. Total thickness (T) is always full height regardless of the size of the void. This material distribution is quite different from a thickness design variable when determining the bending stiffness of the element.
In the homogenization method, the effective two-dimensional material property (thickness is not involved) for a given void size (a and b) is calculated first for the perforated material, assuming that there are an infinite number of such voided cells. The element stiffness matrix is calculated similar to a sandwich plate. The plate property is calculated with the center, T0 plate, having the original material property and the top and bottom void plates having the equivalent an-isotropic material property. This plate property is rotated to the optimal void orientation and then used for creating the element stiffness matrix.
Design properties T0 and T for shell design element.
The material density of the shell element is equal to one minus the area 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 means that only the base thickness of the element is present. A material density of 1 means that void size is zero.
|
Note: |
A material density of zero does not mean that the element has no mass if T0 is non-zero. To ensure that the stiffness matrix is not ill-conditioned, mindens and T0 should not both be set to zero simultaneously. |
In the density method, the rho-mat is used directly as the density design variable. The effective material property is equal to a scalar (function of the rho-mat) times the original material property. The procedure used in the homogenization method is the same as that used to calculate the element stiffness matrix. In this case, there is no angle design variable.
For a composite plate or a plate with an-isotropic material, only the density method can be used, and the base plate thickness must be zero (the limitation of the current development).
Topology optimization of composites has certain unique characteristics and is discussed in a separate section, Composite Topology and Free-size Optimization.
Go To
Design Variables for Topology Optimization