# Topology Optimization Based Material Design for 3D Domains Using MATLAB
## 3 Model Order Reduction in Material Design Optimization  
The main objective of model order reduction (MOR) techniques is to reduce the dimensionality of a linear system of equations, which is solved multiple times. In the case of structural optimization procedure (STO) problems, including when the equilibrium equations system is solved multiple times, the use of such techniques can produce significant computational benefits for the entire process. In this part of the study, three MOR techniques are presented. We focus on their application to the material design optimization problem provided by the STO problem formulation. Although in this study three MOR techniques are integrated in the subsequently presented Matlab code, the code can easily be extended with the addition of new ones, i.e., the Proper Orthogonal Decomposition (POD), On-the-fly Reduced Order Model Construction, and the Approximate Reanalysis. These techniques have also been described in the work of the authors [17]; thus, their description will be presented very briefly herein.  
Among the three MOR techniques, the main difference is the construction of the reduced basis matrix \(\Phi\), which contains the reduced basis vectors. This matrix is utilized by the MOR to reduce the dimensionality of the equilibrium system of the equations. Thus, instead of solving the equilibrium system of the equations, a system with smaller size is solved following the expression of Equation (8):  
\[F=K\cdot U\rightleftharpoons F_{eff}\approx K_{eff}\cdot y \tag{8}\]  
where \(y\) is a reduced approximation of the displacement field and \(F_{eff}\) and \(K_{eff}\) are computed following the expressions of Equations (9) and (10), respectively:  
\[F_{eff}=\Phi\cdot F \tag{9}\]and  
\[K_{eff}=\Phi^{\mathsf{T}}\cdot K\cdot\Phi \tag{10}\]  
The dimensions of the reduced basis matrix \(\Phi\) are \(Nb\times dofs\). \(Nb\) is the number of the reduced basis vectors and dofs are the number of degrees of freedom for the system of equilibrium equations. To assess the accuracy of the resulting displacement field, the following expression of Equation (11) can be used:  
\[e^{2}=\frac{\|K\cdot\Phi\cdot y-F\|^{2}}{\|F\|^{2}} \tag{11}\]  
In this expression, the approximate solution \(\Phi\cdot y\) is utilized to compute the residual force resulting from the use of MOR. If the residual force increases, then the MOR reduces the basis matrix \(\Phi\), which needs to be updated to continue obtaining results of acceptable accuracy. In the case of STO problem formulations due to the use of the approximate solution \(\Phi\cdot y\) in the sensitivities calculation part, an additional term can be used to the expression of Equation (2) to account for the inaccuracy of the approximation, as provided in the following expression of Equation (12):  
\[\frac{\partial C}{\partial x_{e}}=-U_{e}^{T}\cdot\frac{\partial K_{e}}{ \partial x_{e}}\cdot U_{e}=-y^{T}\cdot\Phi^{T}\cdot\frac{\partial K}{\partial x _{e}}\cdot\Phi\cdot y-\sum_{i=1}^{N_{b}}\lambda_{i}^{T}\cdot\frac{\partial K_ {i}}{\partial x_{e}}\cdot U_{i} \tag{12}\]  
where \(\lambda_{i}\) is computed using the following expression of Equation (13) for each reduced basis \(Nb\).  
\[K_{i}\cdot\lambda_{i}=2\cdot y_{i}\cdot(F-K\cdot\Phi\cdot y_{i}) \tag{13}\]  
where \(K_{i}\) refers to the stiffness matrix corresponding to the reduced basis \(i\), \(y_{i}\) refers to the reduced displacement field of the reduced basis \(i\) and \(K\), \(\Phi\), and \(y\) refer to the current stiffness matrix, the reduced basis matrix, and the current reduced displacement field, respectively. In the following sections the construction of the reduced basis matrix for the three MOR techniques are presented.