A parametric level set method for the design of distributed piezoelectric modal sensors

Distributed modal filters based on piezoelectric polymer have especially become popular in the field of active vibration control to reduce the problem of spillover. While distributed modal filters for one-dimensional structures can be found analytically based on the orthogonality between the mode shapes, the design for two-dimensional structures is not straightforward. It requires a continuous gain variation in two dimensions, which is not realizable from the current manufacturing point of view. In this thesis, a structural optimization problem is considered to approximate distributed modal sensors for two-dimensional plate structures, where the thickness is constant but the polarization can switch between positive and negative. The problem is solved through an explicit parametric level set method. In this framework, the boundary of a domain is represented implicitly by the zero isoline of a level set function. This allows simultaneous shape and topology changes. The level set function is approximated by a linear combination of Gaussian radial basis functions. As a result, the structural optimization problem can be directly posed in terms of the parameters of the approximation. This allows to apply standard optimization methods and bypasses the numerical drawbacks, such as reinitialization, velocity extension and regularization, which are associated with the numerical solution of the Hamilton-Jacobi equation in conventional methods.Since the level set method based on the shape derivative formally only allows shape but not topology transformation, the optimization problem is firstly tackled with a derivative-free optimization algorithm. It is shown that the approach is able to find approximate modal sensor designs with only few design variables. However, this approach becomes unsuitable as soon as the number of optimization variables is growing. Therefore, a sensitivity-based optimization approach is being applied, based on the parametric shape derivative which is with respect to the parameters of the radial basis functions. Although the shape derivatives does not exist at points where the topology changes, it is demonstrated that an optimization routine based on a SQP solver is able to perform topological changes during the optimization and finds optimal designs even from poor initial designs. In order to include the sensors' distribution as design variable, the parametric level set approach is extended to multiple level sets. It turns out that, despite the increased design space, optimal solutions always converge to full-material polarization designs. Numerical examples are provided for a simply supported as well as a cantilever square plate.Doctorat en Sciences de l'ingénieur et technologie