Shape optimization for brake squeal
Brake squeal, which usually causes in the frequency range between 1 and 15 kHz, is a phenomenon which has been one of the most difficult concerns in design of brake systems. It causes customer dissatisfaction and increases warranty costs. The present paper proposes a cost function which is expected to be effective to prevent the brake squeal. The cost function is defined with the real part of a complex eigen frequency for a target natural vibration which causes the brake squeal. In the present paper, we assume that a brake system consists of a rotor and a pad between which the Coulomb friction occurs. The Coulomb friction makes an equation of motion asymmetric with respect to displacement and adjoint displacement in the weak form. Then, eigen values for natural vibrations become complex values. On the other hand, it has been known that brake squeal is an unstable vibration induced by friction. From this observation, we considered the real part of a complex eigen frequency for a natural vibration which causes the brake squeal becomes a candidate of a cost function to prevent the brake squeal. A shape optimization problem is formulated using the objective cost function of the real part of a complex eigen frequency and a constraint function of volume for the pad. The main problem consists of a weak form combining the weak forms of the liner elastic continua for the rotor and the pad including the terms of the Coulomb friction and the stiffness in normal direction on the sliding boundary. The shape derivative, which is defined as the Fréchet derivative with respect to domain variation, of the real part of a complex eigen frequency is evaluated with the solutions of the main problem and a adjoint problem which is derived theoretically by the adjoint variable method. To solve the shape optimization problem to minimize the real part of the complex eigen frequency with the volume constraint of the pad, we use an iterative algorithm based on the H1 gradient method. The H1 gradient method is used to keep the smoothness of the boundary. The volume constraint is satisfied using the KKT conditions in the shape optimization problem. We developed a computer program to solve the shape optimization problem based on the iterative algorithm. The boundary value problems are solved by the finite element method using an original code. Numerical example using a simple model is going to be shown in the paper.
Introduction to shape optimization : shape sensitivity analysis
This book presents modern functional analysis methods for the sensitivity analysis of some infinite-dimensional systems governed by partial differential equations. The main topics are treated in a general and systematic way. They include many classical applications such as the Signorini problem, the elastic-plastic torsion problem and the visco-elastic-plastic problem. The "material derivative" from which any kind of shape derivative of a cost functional can be derived is defined. New results about the wave equation and the unilateral problem are also included in this book, which is intended to serve as a basic reference work for the algorithmic approach to shape optimization problems.
Solution to boundary shape optimization problem of linear elastic continua with prescribed natural vibration mode shapes
This paper presents a practical method of numerical analysis for boundary shape optimization problems of linear elastic continua in which natural vibration modes approach prescribed modes on specified sub-boundaries. The shape gradient for the boundary shape optimization problem is evaluated with optimality conditions obtained by the adjoint variable method, the Lagrange multiplier method, and the formula for the material derivative. Reshaping is accomplished by the traction method, which has been proposed as a solution to boundary shape optimization problems of domains in which boundary value problems of partial differential equations are defined. The validity of the presented method is confirmed by numerical results of three-dimensional beam-like and plate-like continua.