Optimization of holes in plates for control of eigenfrequencies

In this paper we study the optimization of a hole of given area which is placed in the interior of a plate with an arbitrary external boundary. To avoid stress concentrations the shape of the hole must be smooth (continuous curvature). The objectives of the optimization are the eigenfrequencies of the plate with the hole. The optimization is performed in relation to maximizing the first eigenfrequency or maximizing the gap between the first and second eigenfrequency. An inverse solution is also shown, i.e. finding the shape and position of a hole in the plate that result in a specified eigenfrequency. To obtain a smooth boundary of the hole we use an analytical description of the hole. A rather general parameterization with only seven design parameters is applied, including the possibility of going from an ellipse to a rectangle or even to a triangle. Optimal designs are obtained iteratively using mathematical programming, where each of the redesigns are based on finite element (FE) analysis and sensitivity analysis. Mindlin plate theory is the basis for the FE analysis and the semi-analytical sensitivity analysis includes only the elements on the boundary of the hole.