José Carlos Bellido

Modal optimization and filtering in piezoelectric microplate resonators

A systematic design procedure to tailor the modal response of micro-resonators based on flexible plates with piezoelectric films is demonstrated. Sensors/actuators were designed by optimizing the surface electrode shapes in the plane dimensions. A numerical finite element procedure, which considers the effective surface electrode covering the piezoelectric film as a binary function on each element, has been implemented. Two design goals are considered: (i) optimized response (actuation or sensing) in a given mode; (ii) implementation of a modal transducer by filtering specific modes. For a given mode in a plate with arbitrary boundary conditions, our calculations allowed us to predict the top electrode layout reaching higher displacement in resonance than any other electrode design for the same structure. Microcantilevers and microbridges were fabricated and their modal response characterized by laser Doppler vibrometry. In comparison to a conventional square-shaped electrode, our experiments show that the implemented designs can increase the response in any desired resonant mode and simultaneously attenuate the contributions from other unwanted modes, by simply shaping the surface electrodes. Enhancement ratios as high as 42 dB, relative to a full-size electrode case, are demonstrated. The limitations imposed by the fabrication are also discussed.

Optimal design via variational principles: the one-dimensional case

By reformulating a typical optimal design problem in a variational format, we are able to prove existence results in a rather general framework allowing non-linear state equations and costs functional depending on derivatives of states. In this work, we restrict attention to the one-dimensional situation.

Optimal design via variational principles: the three-dimensional case

The analysis of some three-dimensional optimal design problems leads us to study variational principles under curl-free and divergence-free constraints simultaneously. We explicitly exploit the relationship between curl-free and div-free restrictions in order to take advantage of the accumulated experience in the classical curl case by the introduction of potentials. Our discussion takes place in the three-dimensional situation. This is a first contribution in the sense that we only deal with the most basic issues.

Piezoelectric modal sensors/actuators based on microplates applying surface electrode patterning

We demonstrate an advanced design procedure to fabricate resonators based on flexible plates with piezoelectric films. Two design strategies are considered: i) optimized response (actuation or sensing) in a given mode; ii) implementation of a modal transducer by filtering specific modes. Sensors/actuators were designed by optimizing the surface electrode shape in both dimensions. A numerical finite element procedure, which considers the effective surface electrode covering the piezoelectric film as a binary function on each element, has been developed. Microcantilevers and microbridges were fabricated and their modal response characterized by laser Doppler vibrometry. Our calculations allowed us to predict, for a given mode in a plate with arbitrary boundary condition, the top electrode layout with higher displacement in resonance than any other electrode design for the same structure. Our experiments show that the implemented designs can suppress the contributions of different modes simply by shaping the surface electrodes.

Relaxation of an Optimal Design Problem With an Integral-Type Constraint

We study a new relaxation for a two-dimensional optimal design problem in conductivity consisting of determining how to mix two given conducting materials in order to minimize the amount of one of them, subject to a constraint on the efficiency of the conducting properties of the mixture. Our approach here is different from that obtained in [10], and based on a local reformulation of the optimal design problem by means of the introduction of new potentials. The concept of constrained quasiconvexification is used in an important way.

Two-material optimal design for nonlinear elastica☆

The work considers an optimal design problem in the context of nonlinear elastica. More specifically, we deal with finding the best way of mixing fixed amounts of two different elastic materials, so as to minimize the tip deflection of a cantilever beam loaded on its free extreme under the assumption of large deflections. Applying an optimality criteria method to the relaxed problem, simulations give us numerical evidence that the original design problem admits classical solutions (i.e. there is no microstructure) and those are the same as the respective ones for the case of small deflections.

Optimal design of a microgripper-type actuator based on AlN/Si heterogeneous bimorph

This work presents a systematic procedure to design piezoelectrically actuated microgrippers. Topology optimization combined with optimal design of electrodes is used to maximize the displacement at the output port of the gripper. The fabrication at the microscale leads us to overcome an important issue: the difficulty of placing a piezoelectric film on both top and bottom of the host layer. Due to the non-symmetric lamination of the structure, an out-of-plane bending spoils the behaviour of the gripper. Suppression of this out-of-plane deformation is the main novelty introduced. In addition, a robust formulation approach is used in order to control the length scale in the whole domain and to reduce sensitivity of the designs to small manufacturing errors.

Optimal design of piezoelectric microtransducers for static response

In this work, we present a systematic procedure to design piezoelectric transducers by simultaneously optimizing the host structure and the electrode layout. The technique allows maximizing any electromechanical coupling of output efficiency of the transducer. Either the output current collected at the electrodes when a mechanical force is applied (sensors), or the in-plane displacement when a given voltage is applied to the electrodes (actuators) can be optimized. We introduce a new idea to avoid the typical problem in topology optimization of the appearance of gray areas, getting finally 0-1 designs, some of which have been manufactured. Also, mathematical demonstration of reciprocity of the piezoelectric effect is shown. Many MEMS-based actuators like microgrippers, surface probes, or micro-optical devices can be optimized following this procedure. A similar approach has been demonstrated previously in modal sensors/actuators, although restricted to the design of the electrode layout for a given structure. The novel method shown here allows the simultaneous optimization of both shapes, for electrode and structure in the static case.

Relaxation of an optimal design problem with an integral-type constraint

We study a new relaxation for a two-dimensional optimal design problem in conductivity consisting of determining how to mix two given conducting materials in order to minimize the amount of one of them, subject to a constraint on the efficiency of the conducting properties of the mixture. Our approach here is different from that obtained in [10], and based on a local reformulation of the optimal design problem by means of the introduction of new potentials. The concept of constrained quasiconvexification is used in an important way.