Simultaneous isogeometrical shape and material design of functionally graded structures for optimal eigenfrequencies

• A concurrent isogeometric shape and material design technique for FGMs is introduced.
• Applicability of the method for eigenfrequency optimization of FGM structures is investigated.
• Volume fraction variations are captured using the IGA NURBS-based parameterization.

A new optimization strategy for eigenfrequency optimization of functionally graded (FG) structures within the framework of isogeometric analysis (IGA) approach is introduced. The proposed methodology which utilizes a concurrent procedure by combining the shape and material composition optimization of these structures employs an extended form of the standard IGA method by allowing for gradation of material properties through patches. The distributions of the graded material properties are considered as imaginary surfaces over the computational domain and captured in a fully isogeometric formulation using the same NURBS-based parameterization which is employed for the geometry modeling as well as the solution approximation. Considering the in-plane coordinates of the control points defining the design boundary surfaces as well as the applicates of all the control points describing the variations of material properties as design variables, we subsequently adopt a mathematical programming algorithm to simultaneously find the optimum shape and material composition of FG structures. A couple of illustrative numerical examples in 2D elasticity with eigenfrequencies as their either constraints or objective functions are presented to demonstrate the high performance of the proposed methodology. It will be seen that the obtained results by this concurrent optimization procedure have much better dynamic performance compared to the optimal results of the simple shape or material composition design.