A simultaneous approach for compliance minimization and piezoelectric actuator design considering the polarization profile

This article addresses the compliance problem along with the piezoelectric actuator design for active vibration control. The topology structural design is obtained by solving a compliance minimization problem with volume constraint, whereas the actuator design is carried out by the maximization of a control performance index written in terms of the controllability Gramian. This measure describes the ability of the actuator to move the structure from an initial condition to a desired final state, at rest for instance, in a finite time interval. The actuator design is also characterized by the polarization profile, which is defined according to the distribution of an additional design variable. Therefore, the actuators can yield both tensile and compressing fields at different points of the structure using the same applied control voltage. To achieve this goal, a material interpolation scheme based on the solid isotropic material with penalization and the piezoelectric material with penalization and polarization (PEMAP-P) models is employed, and both the optimum structure/actuator layout and polarization profile are obtained simultaneously. The sensitivities with respect to the polarization and design variables are calculated analytically. Numerical examples are presented considering the design and vibration control for a cantilever beam, a beam fixed at both ends, and an L-bracket structure to show the efficiency of the proposed formulation. The control performance of the designed structures are analyzed employing a linear-quadratic regulator simulation, and these results are compared to verify the influence of the optimized polarization profiles.