Tensor product decomposed multi-material isogeometric topology optimization with explicit NURBS element stiffness computation

Based on the explicit stiffness computation formula of NURBS (non-uniform rational B-splines) element and tensor product decomposed explicit filter, a novel multi-material isogeometric topology optimization (MMITO) method is put forward to generate multi-material structures efficiently. The explicit NURBS element stiffness computation formula is: derived from the Bezier extraction operator under the combination of scaled and rotational geometric transformations between NURBS elements, and the stiffness matrices of NURBS elements can be computed by the stiffness matrix of a Bezier element without resorting to the time-consuming integral procedure. As the presented MMITO is a nested optimization framework, we apply the explicit filtering technique to update the multi-phase design variables efficiently in combination with the tensor product decomposition method. With the decomposed weight matrices of the filter and the explicit form of the NURBS stiffness matrix, we obtain the equivalent but much simplified preprocessing procedure and sensitivity analysis for the MMITO method. A comprehensive set of multi-material numerical examples are optimized to verify that, while improving the preprocessing efficiency and reducing the memory overhead, the present MMITO scheme is able to generate semblable two-dimensional and three-dimensional multi-material structures with identical structural performance to the traditional scheme. Therefore, the proposed MMITO method is a promising approach to optimizing multi-material structures.