Topology Optimization
The general objective of compliance-based topology optimization is to seek for a material distribution $\mathbf{\rho}$, a scalar field representing the material density at each point on a design domain $\Omega$, to obtain the minimal structural compliance $c(\mathbf{\rho}, \mathbf{u})$, or equivalently, the least strain energy $e(\mathbf{\rho}, \mathbf{u})$, under force equilibrium between external force load $\mathbf{f}$ and internal elasticity force $-\frac{\partial e}{\partial \mathbf{u}}$ with displacement $\mathbf{u}$: \begin{equation} \underset{\mathbf{\rho}}{\text{min}} \quad c(\mathbf{\rho}, \mathbf{u}) = e(\mathbf{\rho}, \mathbf{u}) \quad \text{s.t.} \quad \begin{cases} \frac{\partial e}{\partial \mathbf{u}}(\mathbf{\rho}, \mathbf{u}) = \mathbf{f} \ \mathbf{Du} = \mathbf{0}\ V(\mathbf{\rho}) \le \hat{V}. \end{cases} \label{eq:formulation} \end{equation} Here $\mathbf{Du} = \mathbf{0}$ is the discretized Dirichlet boundary condition where $\mathbf{D}$ selects the zero displacement nodes, $V(\mathbf{\rho}) = \int_{\Omega_0} \rho d\mathbf{X}$ is the total volume of the structure, and $\hat{V}$ is an upper bound specified by users to avoid trivial solutions.
The conventional Solid Isotropic Material with Penalization Method (SIMP) descritizes the density field on a fixed grid, which has limited capability in capturing intricate structures. On the other hand, Lagrangian representations suffer from a lower computational performance. We propose a hybrid Lagrangian–Eulerian topology optimization (LETO) method. LETO transfers density information from freely movable Lagrangian carrier particles to a fixed set of Eulerian quadrature points. The quadrature points act as MPM particles embedded in a lower-resolution grid and enable a subcell multidensity resolution of intricate structures with a reduced computational cost (See Fig.1).
