Adaptive Topology Optimization Using a Continuous Approximation of Material Distribution
Authors: Lambe Andrew B. , Czekanski Aleksander 
Published: February 2017

Structural topology optimization seeks to distribute material in a design domain to produce the stiffest structure for a given mass or the lightest structure for a given strength. In the density-based approach to topology optimization, the design domain is divided into small elements and an optimization algorithm determines whether each element in the optimal design contains solid material or void. Solutions obtained using this method may suffer from a variety of issues, such as a checkerboard pattern of solid and void elements, large transition regions between solid and void parts of the structure, and dependence of the final solution on the initial mesh. Typically, these issues are mitigated using filters, projection functions, or a combination of the two. However, applying these techniques requires the user to select a few parameter values and the optimal design strongly depends on the selected parameters.
This work presents an alternative approach to addressing the aforementioned issues in density-based topology optimization. Rather than assigning a separate design variable to each element in the domain, a continuous approximation of the density field is used. This field is interpolated using finite element shape functions with the scaling coefficients of these shape functions acting as design variables in the optimization problem. Although this technique is known to produce an optimal design that is free of checkerboard patterns, it leads to a large transition region at the boundary of the structure whose size depends on the size of the finite elements used. To systematically reduce the size of this transition region, the finite element mesh is locally refined near the structural boundary and the design is optimized again. Because the mesh implicitly controls the size of the transition region, local refinement and optimization continue until the smallest cells in the mesh reach an acceptable resolution. A local refinement indicator is developed to identify and refine cells lying in the transition region. Local isotropic mesh refinement is used to maintain reasonable cell sizes over most of the design domain and, consequently, keep the computational cost of both the finite element analysis and the optimization down. Anisotropic mesh refinement may also be used with a suitable indicator, though it is not demonstrated here.
While both continuous density parametrization and adaptive mesh refinement have been applied independently to problems in topology optimization, this work applies them simultaneously for the first time. Structural designs produced by this method are shown to be free of checkerboard patterns and contain features whose size is largely controlled by the initial coarse mesh. In addition, the boundary can be sharply identified for additional processing, such as translation to a CAD file in preparation for fabrication and manufacturing. A disadvantage of the current method is that small features may emerge in the refined parts of the mesh after multiple refinements. Computations were carried out using open-source finite-element analysis and optimization tools. Results are presented for a pair of well-known two-dimensional topology optimization test problems. While not demonstrated in this work, the methodology can be extended easily to three-dimensional problems.