# Design Elements

## Solid Elements

The SIMP method (Solid Isotropic Material with Penalty) is used in OptiStruct. In the SIMP method, a pseudo material density is the design variable, and hence it is often called density method as well. The material density varies continuously between 0 and 1, with 0 representing void state and 1 solid state. The SIMP method applies a power-law penalization for stiffness-density relationship in order to push density toward 0/1 (void/solid) distribution:(1)
$\stackrel{˜}{K}\left(\rho \right)={\rho }^{p}K$

Where, $\stackrel{˜}{K}$ is the penalized stiffness matrix of an element, $K$ is the real stiffness matrix of an element, $\rho$ is the density, and $p$ is the penalization factor (always greater than 1).

## Shell Elements

The SIMP method (Solid Isotropic Material with Penalty) is used in OptiStruct. In the SIMP method, a pseudo material density is the design variable, and hence it is often called density method as well. The material density varies continuously between 0 and 1., with 0 representing void state and 1 solid state. The SIMP method applies a power-law penalization for stiffness-density relationship in order to push density toward 0/1 (void/solid) distribution.(2)
$\stackrel{˜}{K}\left(\rho \right)={\rho }^{p}K$

Where, $\stackrel{˜}{K}$ is the penalized stiffness matrix of an element, $K$ is the real stiffness matrix of an element, $\rho$ is the density, and $p$ is the penalization factor (always greater than 1).

For isotropic material a non-zero base plate thickness can be defined. For a composite plate or a plate with anisotropic material, 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 Composite Topology and Free-size Optimization.

## 1D Elements

Only the density method is implemented for topology optimization of 1D elements. Currently available elements include ROD, BAR/BEAM, BUSH, and WELD elements. Each element is controlled by a single design variable that is the material density $\rho$ of this element that varies between 0 (numerically a small value is used) and 1.0. In essence, 0 represents nonexistence and 1.0 represents full existence of the corresponding element. The following power law representation of elastic properties is used to penalize intermediate density:(3)
$\stackrel{˜}{K}\left(\rho \right)={\rho }^{p}K$

Where, $\stackrel{˜}{K}$ and $K$ represent the penalized and the real stiffness matrix of an element, respectively, $p$ is the penalization factor which is always greater than 1. The penalty is controlled by the DISCRETE or DISCRT1D parameters, the value of these parameters correspond to ( $p$ - 1).