Multi-physics interpolation for the topology optimization of piezoelectric systems

Unlike single-physics static structural problems, the topology optimization of piezoelectric systems involves three different material coefficient groups, such as the mechanical, electrical, and electromechanical coefficient groups, which correspondingly necessitate three values of penalty exponents. Earlier investigations have shown that stable convergence cannot be ensured if the exponents are selected arbitrarily or based simply on typical rules used for static structural problems. Here, stable convergence implies that the use of more materials can yield better system performance and a distinct void–solid distribution is favored over an intermediate material distribution during the optimization process. However, no rule for choosing the values of penalty exponents has yet been studied for piezoelectric systems and in fact, they have been chosen mainly by trial and error. In this work, two conditions that the three penalty exponents must satisfy for stable convergence are derived for one-dimensional problems and their effectiveness for two-dimensional problems is investigated. The first condition is an intrinsic condition ensuring better energy conversion efficiency between mechanical and electric energy for more piezoelectric material usage and the second one is an objective-dependent condition favoring a distinct void–solid distribution over an intermediate material distribution for the same amount of piezoelectric material used. In this study, we consider the design of piezoelectric actuators, sensors and energy harvesters that can be analyzed by static analysis. Several numerical examples are solved to check the validity of the proposed conditions.