The focus of this work is on simultaneous shape, material and orientational design of plates. The shape optimization is related to the shape and position of a hole in the plate. The material design is the design of an orthotropic material that can be considered to be a fiber-net within each finite element. This fiber-net is then oriented in the individual elements of the finite element discretization. The object of the optimization is dynamics, specifically the eigenfrequencies. We maximize eigenfrequencies or gaps between eigenfrequencies to avoid resonance. Generally optimization of eigenfrequencies is also applied in case of minimizing the possibility of internal resonance or inverse dynamics where specific demands on eigenfrequencies are meet.
Using composite material we can tailor the behavior of a structure to meet specific demands. Many of the early papers dealing with the subject of orienting material to maximize eigenfrequencies considered only one orientation variable for each element. We assume that we are dealing with an orthotropic material and align part of the material in one direction and the rest of the element material in a direction perpendicular to the first direction. The integrated constitutive relation is then given as the sum of the two constitutive relations, so that a fiber-net is oriented individually in each finite element. The fibernet can be seen as a regularization of the formulation with unidirectional fibers, the basic advantage is the possibility to use fibers in both of the principal stress directions. That this is an advantage can be argued from the case of an isotropic stress state or a pure shear state where the choice of direction using unidirectional fibers is arbitrary.
The optimizations are performed using the finite element method for analysis and the optimization approach is a two-step method, where both a recursive optimization procedure based on an optimality criterion (OC) is used and mathematical programming (MP) with sensitivity analysis is applied to find the final optimized design.
The examples show that the two-step approach of using both OC and MP gives fast and robust convergence of the design. The paper shows that we can manipulate the eigenfrequencies of a plate by varying the material(fiber-net) and the orientation of the fiber-net together with the shape of a hole in the plate (within the physical limits of the problem).