Design Variables

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.

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. Intermediate values of density represent fictitious material. 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 stiffer than aluminum. Following this logic, the representation of fictitious material at intermediate densities does reflect engineering intuitions.

Penalization

In general, the optimal solution of problems involves large gray areas of intermediate densities in the structural domain. Such solutions are not meaningful when you 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. OptiStruct uses different penalization techniques in different settings of topology optimizations. These are SIMP, RAMP or Polynomial.

Where, $\tilde{K}$ and $K$ represent the penalized and the real stiffness matrix of an element, respectively, $\rho$ is the density and $p$ is the penalization factor which is always greater than 1. SIMP is used for the vast majority of setups.

Discreteness

As described above, for the sake of interpretability it is desirable to achieve a discrete design, i.e. a design where the vast majority of elements is either 0 or 1. In addition to interpretability, the performance of the structure, reported at the end of the optimization, becomes inaccurate when large amounts of medium dense elements exist. This is because the medium dense elements, despite their low penalized stiffness, can still have a significant influence on the structural behavior. A reanalysis after interpretation of the structure is the only reliable way to verify the performance of the structure. This becomes even more important when comparing structures obtained with different penalty schemes, for instance comparing a structure obtained with DRAW and a structure with OVERHANG, i.e. a topology obtained with SIMP and one obtained with RAMP. The different penalty schemes can further amplify differences between the reported performances as medium dense elements are penalized differently.

Estimating Discreteness

In practice, it is very difficult to obtain a design that is comprised of densities exclusively equal to 0.0 or 1.0. In reality a transition zone usually exists between a solid member and voids. This transition zone is acceptable in most cases. It should be avoided however, to have large areas containing mainly medium dense elements. To quantify this, OptiStruct reports a discreteness index in the .out file after each iteration (since version 2017.2.3). This indicates the amount of medium dense elements in the structure by calculating the ratio between the volume from elements with densities of at least 0.9 and that of the entire design space. For a perfectly discrete model this value would be 1.0 but for structures where a transition zone exists, this parameter should be 0.5 or higher. When values are smaller than that at the end of the optimization, the topology should at least be interpreted with caution and possibly measures should be taken to improve discreteness.

Density                      %
------------------------------
0.0-0.1                   64.6
0.1-0.2                    0.9
0.2-0.3                    0.8
0.3-0.4                    1.3
0.4-0.5                    1.2
0.5-0.6                    2.1
0.6-0.7                    1.2
0.7-0.8                    4.1
0.8-0.9                    5.6
0.9-1.0                   18.2

Improving Discreteness

Lattice Optimization

While it is desirable to achieve purely solid and void designs in regular topology optimization, Lattice structure opens up the freedom of realizing porous zones (medium dense regions) into lattice regions. The penalty used for this type of optimization is usually reduced to increase porous zones in the design. The penalty and thus the amount of lattice structure is controlled via LATPRM, POROSITY. Refer to Lattice Structure Optimization for additional information.

Penalty for SIMP

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 $\rho$ =0.3, $p$ =2 reduces the stiffness of the element from 0.3 to 0.09 times the stiffness of the fully dense element. The default DISCRETE is 1.0 for shell dominant structures, and 2.0 for solids dominant structures with member size control and no manufacturing constraints (the dominance is defined by the proportion of number of elements). An additional parameter, DISCRT1D, can also be defined on the DOPTPRM Bulk Data Entry. DISCRT1D allows 1D elements to use a different penalization to 2D or 3D elements.

Three types of finite elements can be defined as topology Design Elements in OptiStruct: Solid elements, shell elements, and 1D elements (including ROD, BAR/BEAM, BUSH, and WELD elements).