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We consider a vibrating system with piezoelectric patches whose minimum vibration frequency is to be minimized subject to some constraint on the patch geometry. A numerical scheme is constructed using the level set method devised in [12] and the projected gradient method devised in [14] to optimize the patch geometry. An integral equation approach introduced in [4, 17, 20, 21] is also presented for the computation of the corresponding eigenvalue problem. Introduction. A piezoelectric material can respond to mechanical forces/pressures and generate an electric charge/voltage. This piezoelectric phenomenon is called the direct piezoelectric effect. On the other hand, an electric charge/field applied to the material induces mechanical stresses or strains, and this phenomenon is called the converse piezoelectric effect. In active piezoelectric structures, the direct effect is used for structural measurements while the converse effect is used for active vibration controls of the continua. In recent years, piezoelectric materials are being used increasingly in noise control [5, 6], vibration control [3, 8] and shape control [9, 10]. They have been effectively employed in areas such as acoustics for noise cancellations with applications to reduce interior noise in aircraft, aerodynamics to adjust wing surfaces and electronics where they are used in the reading heads in videocassette recorders and in compact discs as positioning devices. Another application is in adaptive structures for shape control by piezo-actuation. Also adaptive materials and structures are presently being used in a variety of applications involving static control such as robotic and space structures. One of the important issues in the use of piezo actuators is their optimal deployment to minimize their weight and enhance system performance. In many applications, the piezo materials are used in the form of several patches in order to provides flexibility in choosing their locations which can be optimized to improve the effective of the control [2]. This work concerns a closed-loop displacement feedback control of a thin rectangular plate reinforced with a sensor patch and an actuator patch. The sensor senses the bending strains of the plate and generates a signal which is amplified and sent to the actuator. The actuator then generates a corresponding signal which causes the plate to bend in the opposite direction. The optimal shapes of the patches (under some constraint) are to be determined to minimize the minimum vibration frequency. We consider the equation of motion, a fourth order hyperbolic equation derived in [19], with simply supported boundary conditions. In the classical approach presented in [20], the modeling equation is converted into a certain integral equation to which a kernel can be determined explicitly. Consequently, the kernel is expressed in terms of the patch shapes by converting the domain integrals over the patches into the corresponding line integrals over their boundaries. Then optimizing the shapes of the patches amounts to optimizing the parameterizations of their boundaries with the admissible set composed of all the reasonable parameters. ∗Received December 1, 2002; accepted for publication March 22, 2004. †Department of Mathematics, 121-1984 Mathematics Road, University of British Columbia, Vancouver, BC, Canada V6T 1Z2 (jyzhang@math.ubc.ca).

Level Set Method for Shape Optimization of Plate Piezoelectric Patches