OptiStruct solves topological optimization problems using either the homogenization or density method.
Under topology optimization, the material density of each element should take a value of either 0 or 1, defining the element as being either void or solid, respectively. Unfortunately, optimization of a large number of discrete variables is computationally prohibitive. Therefore, representation of the material distribution problem in terms of continuous variables has to be used.
For the homogenization method, the material of the structure is represented as a porous continuum with certain periodic microstructure or layered composites of different ranks of densities. The homogenization method implemented in OptiStruct uses a material microstructure that contains periodic rectangular voids (hexahedral voids in 3-D). The design variables for each element are the breadth and depth of these rectangular voids and their orientations. These define the elasticity properties and the density of the material.
Using a normalized formulation, the density of an element may be determined by:
where (1.0 – a)(1.0 – b) represents the total volume of void in an element. It is easy to see that a=b=0 represents the state of void for this element, and a=1 or b=1 implies that the element is solid, i.e. filled with the 'real' material. Intermediate values of a and b represent fictitious material.
The void size variables are considered to be continuous variables varying between 0 and 1. The void orientation of each element is also a continuous variable, which is determined by the orientation of the principle strain. Note that while the real material is isotropic, the fictitious material of intermediate density is anisotropic.
With the density method, the material density of each element is directly used as the design variable, and varies continuously between 0 and 1; these represent the state of void and solid, respectively. As with the homogenization method, intermediate values of density represent fictitious material.
With this method, the stiffness of the material is assumed to be linearly dependent on the density. This material formulation is consistent with our understanding of common materials. For example, steel, which is denser than aluminum, is stronger than aluminum. Following this logic, the representation of fictitious material at intermediate densities is more realistic under the density approach. An anisotropic representation of the semi-dense material is not consistent with the behavior of the real isotropic material, although it is more 'efficient' due to optimal material orientation.
In general, the optimal solution of problems, using both formulations mentioned above, involves large gray areas of intermediate densities in the structural domain. Such solutions are not meaningful when we are looking for the topology of a given material, and not meaningful when considering the use of different materials within the design space. Therefore, techniques need to be introduced to penalize intermediate densities and to force the final design to be represented by densities of 0 or 1 for each element. The penalization technique used for the density approach is the "power law representation of elasticity properties," which can be expressed for any solid 3-D or 2-D element as follows:
where K and K represent the penalized and the real stiffness matrix of an element, respectively, r is the density and p the penalization factor which is always greater than 1.
In OptiStruct, the DISCRETE parameter corresponds to (p - 1). DISCRETE can be defined on the DOPTPRM bulk data entry. p usually takes a value between 2.0 and 4.0. For example, compared to the non-penalized formulation (which is equivalent to p=1) at r=0.3, p=2 reduces the stiffness of the element from 0.3 to 0.09 times the stiffness of the fully dense element. An additional parameter, DISCRT1D, can also be defined on the DOPTPRM bulk data entry. DISCRT1D allows 1-D elements to use a different penalization to 2-D or 3-D elements.
When minimum member size control is used, the penalty starts at 2 and is increased to 3 for the second and third iterative phases. This is done in order to achieve a more discrete solution. For other manufacturing constraints such as draw direction, extrustion, pattern repetition, and pattern grouping, the penalty starts at 2 and increases to 3 and 4 for the second and third iterative phases, respectively. Obviously, due to the existence of semi-dense elements, the analysis results may change dramatically when the design process enters a new phase using a different penalization factor.
Penalization formulations under the homogenization approach use the same principal. Because the relationships of elasticity properties to design variables differ for the homogenization and density methods, analysis results where intermediate densities are present will be different for the same volume fraction. However, when the volume fraction is 0 or 1, the results will be identical. At these values, the structural domain is void or solid, respectively, and the elastic property should be that of the real material for any formulation.
Each approach has its advantages and disadvantages. The homogenization approach has the advantage that the design can form rapidly along the lines of the force transmission path. This is due to the angular orthotropy of the intermediate dense elements. However, it has the disadvantages that there are more design variables and that it is difficult to uniquely determine the orientation angles of the voids for structures under multiple loading cases and for multiply constrained problems. The advantages of the density method are that it is more general and requires less design variables.
In OptiStruct, the homogenization method can only be used on homogeneous isotropic material, where the computed effective material property is an-isotropic for semi-dense elements. The density method can be used for both isotropic and an-isotropic materials (including composite material), where the computed effective material property is proportionate to its original material property.
Due to concerns over efficiency and general applicability, the density method is the default method for all problems except the simple case where a compliance minimization problem of isotropic material is defined through the use of the MATFRAC parameter. Also note that the density method is the only method implemented for manufacturing constraints (draw direction, extrusion, pattern repetition, and pattern grouping constraints).
Three types of finite elements can be defined as topology design elements in OptiStruct: Solid elements, shell elements, and 1-D elements (including ROD, BAR/BEAM, BUSH, and WELD elements). For the density method, each element has one variable representing its material density. For the homogenization method, which has limited availability for 2-D and 3-D elements, there are two void size variables for each shell element and three void size variables for each solid element. An additional variable for each element is the void orientation angle.
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