Design of piezoelectric microtransducers based on the topology optimization method

Microsyst Technol (2016) 22:1733 1740 DOI 10.1007/s00542-016-2828-0 TECHNICAL PAPER Design of piezoelectric microtransducers based on the topology optimization method David Ruiz 1 Alberto Donoso 1 José Carlos Bellido 1 Martin Kucera 2,3 Ulrich Schmid 2 José Luis Sánchez Rojas 4 Received: 21 July 2015 / Accepted: 8 January 2016 / Published online: 23 January 2016 Springer-Verlag Berlin Heidelberg 2016 Abstract In this work, a systematic procedure based on the topology optimization method to design piezoelectric transducers in a static in-plane and out-of-plane framework is presented. The layout of the host structure and the electrode profile of the piezoelectric material are optimized simultaneously. Our numerical approach shows a significant improvement by using simultaneous optimization rather than optimizing separately the host structure and the electrode profile. Two important issues in this problem are the appearance of large gray areas (a mixture of solid and void material) in the optimized designs for the in-plane case and branches (disconnected areas) for the out-of-plane one, that actually distort the real performance of such transducers. To solve these problems and in order to obtain efficient and really close to 0 1 designs, a new interpolation function, appearing in the objective function for the sensor and in the source term for the actuator, is required. Thanks to the reciprocity of the piezoelectric effect, we get to prove analytically and to corroborate numerically how such a fact leads to the same optimized designs for sensors and actuators. A similar approach has shown quite good performance in the design of modal sensors/actuators, although restricted to the design of the electrode layout for a given * David Ruiz david.ruiz@uclm.es 1 Departamento de Matemáticas, ETSII, University of Castillala Mancha, Ciudad Real, Spain 2 3 4 Institute of Sensor and Actuator Systems, Vienna University of Technology, Vienna, Austria Austrian Center of Competence for Tribology, AC2T research GmbH, Wiener Neustadt, Austria Microsystems, Actuators and Sensors Group, ETSII, University of Castilla-la Mancha, Ciudad Real, Spain structure. Finally, some of the optimized designs for the in-plane case have been fabricated. In several cases passive pieces are included to study movement into fluids. 1 Introduction: state of the art In Donoso and Bellido (2009) a systematic procedure for designing piezoelectric modal sensors/actuators by optimizing the polarization profile of the piezoelectric material was introduced. In Sánchez-Rojas et al. (2010), the previous method was used for designing microtransducers that were manufactured and tested showing quite good performance. In those works, the host structure to which the piezoelectric material is bonded is fixed, namely, a rectangular plate structure. Motivated by many applications, particularly in microelectronics, it is interesting to investigate whether making this ground structure free as a design variable, together with the polarization profile, it could improve the performance of the designed transducers. These ideas were reported in Ruiz et al. (2013), where the authors introduced a method for the design of in-plane sensors for static response by simultaneously optimizing the host structure and the polarization profile. In this paper, we extend our previous investigation, including further the actuator problem in the in-plane and out-of-plane situations. In the current work the aim is the maximization of the output of the transducers. The optimization of this feature allows us to manufacture smaller devices or to increase the range of applications. A first attempt to efficiently design in-plane piezoelectric resonators using topology optimization was presented in Silva and Kikuchi (1999), where the material layout of a host structure is optimized, but keeping fixed the piezoelectric material. That was a very interesting work, but

1734 Microsyst Technol (2016) 22:1733 1740 on imposing a constraint in the position of the piezoelectric material, the search of optimized solutions is rather limited. Therefore, in the last decade, some authors have performed simultaneous design of both host structure and piezoelectric layers by using topology optimization techniques. In Kögl and Silva (2005), the optimization of the piezoelectric part together with the polarization distribution is considered. The authors took a three-layer plate and two piezo layers attached to the top and bottom surfaces of the base layer which was fixed. They proposed an artificial piezoelectric material model with penalization on both piezoelectric constants and polarization. In Carbonari et al. (2007) and Luo et al. (2010), among others, the optimized layout of both the host structure and the piezoelectric part is simultaneously addressed, obtaining interesting designs of in-plane multi-phase actuators. Other authors have gone further, as reported in Kang and Tong (2008a), where not only the topologies of both host and piezoelectric layers are simultaneously optimized, but also a spatial distribution of the control voltage (somehow connected with the polarity of the piezoelectric layers) is included in the optimization problem. Later, Kang and Tong (2008b) introduced an interpolation scheme in the tri-level actuation voltage term improving their approach, as the new optimized designs shown in their recent works Kang et al. (2011, 2012) for designing in-plane and out-of-plane piezoelectric transducers, respectively. In contrast to the works commented above and in order to simplify the equilibrium equation, the problem studied in this paper assumes that the piezoelectric material is surface bonded to the structure wherever it is. Both layers, the host structure and the piezoelectric one, are included in the same variable that is called structure-piezo variable. The second variable used in this problem models the electrode profile that covers the piezoelectric layer. The optimization problem consists in maximizing the output charge generated by a given input force in the case of sensors, and maximizing the output displacement generated by an external input voltage through the piezoelectric layer in the case of actuators. From the point of view of topology optimization there is one issue to be solved: appearance of large gray areas (microstructure) in the optimized designs, a mixture of solid and void material that cannot be manufactured. These parts are represented in the designs with the gray color. What really happens is that the strain in these low density areas is larger than in the rest of the design, and therefore they produce a higher charge for the sensor problem, or higher displacement for the actuator one, increasing then the objective function and being favored during the optimization process. It was introduced in Ruiz et al. (2013) a new interpolation scheme that avoids the generation of these gray areas. Such an interpolation function penalizes intermediate values of the structure variable, which only appears in the objective function for the sensor problem, and it will appear in the source term of the equilibrium equation for the actuator problem. The well-known reciprocal piezoelectric effect (Lee and Moon 1990) also analyzed in this optimization context, makes us to obtain a rigorous mathematical justification of the interesting fact that the actuator and sensor problems are equivalent, in the sense that they admit the same optimized solutions. Thus, we conclude that the reciprocity of the piezoelectric effect also holds in optimization: the optimized sensors and actuators coincide. The layout of the paper is as follows. In Sect. 2 we deal with the sensor problem. The interpolation scheme to avoid the gray areas in the optimized designs mentioned above is introduced, and several examples are shown. Section 3 is devoted to the actuator problem, and we also discuss the local optima issue since these problems seem to be very sensitive to it. In Sect. 4 we analyze the equivalence of the sensor and actuator problems owing the reciprocity of the piezoelectric effect. In Sect. 5 a discussion on the benefits of the simultaneous optimization is given together with an example. In Sect. 6 we comment some aspects related to the fabrication process of the optimized designs. Finally, in Sect. 7 the main conclusions of the paper are remarked. 2 The sensor problem We consider a plate-type structure clamped at its left edge and subject to either an in-plane or out-of-plane force F in in the midpoint of its right edge, as shown in Fig. 1. The host structure is bonded to both the top and the bottom surfaces with two piezoelectric layers (of negligible stiffness compared to the plate) working as sensors or actuators. The electrical wiring of these piezoelectric films depends on each particular case. In-phase excitation is required to restrict ourselves to the in-plane charge detection and outof-phase excitation is needed for the out-of-plane one. The Fig. 1 Design domain for the sensor problem

Microsyst Technol (2016) 22:1733 1740 method is also valid for the case of one single piezoelectric film placed on one of the surfaces of the plate, but in this case the response is reduced to half, and out-of-plane displacement may occur in the actuator even if we are working with in-plane regime. As explained in the introduction, two design variables χ s and χ p are used. χ s is a characteristic function that represents the material layout such that χ s {0, 1} meaning void and structure (and piezo as well), respectively. χ p is another characteristic function that represents the polarity of the electrode, that is χ p { 1, 0, 1}, negative, null or positive polarity, respectively. The standard approach consists in discretizing the design domain in finite elements and letting each one of them has two variable densities as design variables, that is, ρ s is the usual spatial material density and ρ p is the polarization density. The material densities appear when we relax the design variables by using the modified SIMP method (Bendsøe and Sigmund 1999): E = E min + ρ s p (E s E min ), 0 ρ s 1, where E min is a very small stiffness assigned to void regions in order to prevent the stiffness matrix from becoming singular, and p is the power. Concerning the polarization profile, the electrode function χ p is replaced by the continuous expression (2ρ p 1), being now 0 ρ p 1. In Kang and Tong (2008b) an efficient way to relax this variable is proposed. In our work, in line with Kögl and Silva (2005), we have checked that a penalty factor in ρ p showed no significant effects and this is mathematically justified by the fact mentioned above, namely, for a given host structure there exists a classical polarity profile that covers the whole structure (Donoso and Bellido 2009). It is worth mentioning here the self-penalization effect of piezoelectric materials pointed out in Wein et al. (2011). On deforming the whole structure-piezo under the input force, an electric signal is induced owing to the piezoelectric effect, so the objective of the problem is to find the one that stores the maximum strain energy, and consequently maximizes the output charge. This output can be expressed as follows q out = R(ρ s )(2ρ p 1)B T U, where B is the usual FE strain-displacement matrix and R(ρ s ) is a heuristic interpolation function proposed to avoid gray areas for this particular problem. This new interpolation scheme was introduced by first time in Ruiz et al. (2013). The mathematical expression is given by ζ η ρe s ρs e [0, η] R(ρs e ) = (1 ζ )ρ e s + ζ η 1 η ρ e s [η,1] (1) (2) (3) 1735 where η and ζ are tuning parameters. U is the global displacement vector. Since we work with a plate-type model, the curvature is null, then the in-plane and the out-of-plane displacements can be decoupled. The spatial expression for the in-plane case would be U = (u, v, 0) and the one for the out-plane situation U = (0, 0, w). Notice that the role of the electrode is here crucial because only the area of the piezoelectric sensor covered by an electrode will be electrically affected. That linear dependence of q out on ρ p leads us to expect that such a variable will take on only extreme values in order to maximize the sensor response. Indeed, this is so, and it is rigorously proved in Donoso and Bellido (2009), that given a host structure the optimal electrode profile, ρ p, takes on values 1 or 1, so that, at the end of the optimization process the whole optimal structure will be covered by electrode with areas of positive or negative polarity. After the discretization and rewriting the objective function, the topology optimization problem would be max : q out = G(ρ s, ρ p ) T U ρ s,ρ p subject to (K(ρ s ) + k in 1 out )U = F in L v T ρ s V 0 ρ s [0, 1] ρ p [0, 1] where K is the global stiffness matrix and L and 1 out are a zero vector and a zero matrix which take the value 1 at the element corresponding to the input port. ρ s and ρ p are the vectors of both design variables (spatial material and polarization density, respectively) and v is a vector containing the volume of the elements. The vector G is composed of different terms G = R(ρ s )(2ρ p 1)B We add a typical constraint, the maximum structural volume fraction V 0. In Ruiz et al. (2013) is also discussed the role of the volume constraint for the sensor problem, showing that the maximum value of the objective function is not a monotone function of the volume fraction, so that there exists an optimal volume constraint (in the examples shown in that paper about 80 % of material). The input of the problem can be modelled by a force F in and by a spring with stiffness k in (Sigmund 1997). We illustrate our optimization approach through several numerical examples of interest in the area of MEMS because of the sizes used. We consider a square plate of length L = 1000 µm and thickness h s = 50 µm. The material properties for the host structure are those corresponding to silicon, that is, Young s modulus E = 130 GPa and Poisson s ratio ν = 0.28. As the piezoelectric properties (4) (5) (6)

1736 Microsyst Technol (2016) 22:1733 1740 only appear as a scale factor in the objective function, they do not affect the optimized designs. The discrete problem is numerically solved by using MMA (Svanberg 1987). Due to symmetry, only a half of the design domain is discretized by using a mesh of 50 100 elements. A classical issue in topology optimization problems is the dependence of the designs with the size of the mesh. In order to avoid this issue a filtering technique has been used. This technique consists in computing the element sensitivity as a weighted average over elements within a fixed neighborhood, whose size is defined by the filter radius. We have used a continuation approach over the value of this parameter, starting with a filter radius equal to 15 % of the smallest dimension and decreasing in a linear way to 1.2 times the element size. This continuation approach has also been used over the power p in (1), varying from 1 to 3 at the same time as the filter size. In this case, the sensitivity analysis has not been included here because the computations are straightforward and they do not provide anything new. In order to justify the use of the interpolation function given by (3), we first perform numerical simulations for the in-plane situation with no penalization function in the objective function, that is, R 1. In this case a horizontal in-plane force F in = 2N is applied in the mid-point of the right edge, the volume fraction V 0 = 0.4 and the stiffness of the spring is set to 20,000 N/m. For the out-of-plane case this stiffness is changed to 500 N/m while the force applied and the maximum volume fraction keep constant. As observed in Fig. 2a, b, a lot of gray areas concerning variable ρ s appear in the final designs, though the well-known mesh-independent filter is used. This kind of structure tends to appear because physically gray areas are more flexible than the solid ones, which in turn contribute to the sensor output as being covered by electrode. A significant improvement is obtained by using the scheme given by (3), as clearly suggested in Fig. 3a, b, where the material layout obtained is really close to black and white, and the electrode profile is nearly to blue and red, meaning areas in tension and compression, respectively. Although the parameters η and ζ in (3) are tunable, we have kept them fixed in all simulations (we start taking η = 0.8 and ζ = 0.01, and after some iterations these values eventually Fig. 3 Material layout (left) and electrode profile (right) with R corresponding to Eq. (3) Fig. 2 Material layout (left) and electrode profile (right) with R 1 Fig. 4 Design domain (left) and optimized design (right)

Microsyst Technol (2016) 22:1733 1740 1737 for the first case and another placed perpendicular to the plate in the second one. The discrete formulation in FE-notation would be max : u out = L T U ρ s,ρ p (7) subject to (K(ρ s ) + k out 1 out )U = F v T ρ s V 0 ρ s [0, 1] ρ p [0, 1] (8) where 1 out is a zero matrix taking the value 1 at the element corresponding to the output port. The load vector is Fig. 5 Material layout (left) and electrode profile (right) change to η = 0.9 and ζ = 0). The problem is successfully solved because gray areas have disappeared and the designs obtained are really close to 0 1 designs. In Fig. 4a, b are shown two examples of rotary sensor for in-plane case. In the second example a circular passive are has been included. We fix the maximum volume fraction to V 0 = 0.5 (this value includes the passive area). For the sake of brevity only the electrode profile is included, since this variable contains all the information about the optimal design. Concerning the out-of-plane case, in Fig. 5a, b are shown two optimized designs. In both examples the force is applied in the center of the structure. Regarding the boundary conditions, in the first case the corners are simply supported and in the second one the edges are simply supported. In both cases the maximum volume fraction is V 0 = 0.5. To end up this section we would like to remark that the optimized sensor does not change by keeping fixed the ratio F in k in, which can be interpreted as a maximum displacement at the input port when the force F in is applied. F = κg where κ = E s d 31 V in (1 ν)t p in-plane w e E p E s t 2 d 31 V in 6E p t p + E s t w e is the width of the square finite element, d 31 is another piezoelectric constant, G is the vector presented before, E s and t are the stiffness and thickness of the host structure respectively, and E p and t p are the stiffness and thickness of the piezoelectric layer. The expression has been simplified since t p t. The voltage applied to the piezoelectric layer is V in = 1000 V. Such an unrealistic voltage is chosen in order to minimize errors in the numerical process. The objective function can be rewritten including the state equation where the input appears premultiplying the Fig. 6 Actuator problem out-of-plane (9) (10) 3 The actuator problem Contrary to what happened above, whenever an input voltage V in is applied to the whole structure, the piezoelectric layers work as actuators. In such a case, the objective of the actuator problem is to find the design of volume V 0 that maximizes the output displacement for a given output stiffness spring k out (see Fig. 6). The in-plane and out-of-plane situations are represented with a spring placed in parallel Fig. 7 Optimized designs for actuator in-plane (left) and out-of-plane (right)

1738 Microsyst Technol (2016) 22:1733 1740 a physical point of view we understand it as a consequence of the reciprocity of the piezoelectric effect, and further to this we are showing that this reciprocal effect commutes with optimization, which is a mathematical fact. To check that optimized solutions of both problems are the same, we just have to notice that writing the solutions of the state equation for the sensor problem as Fig. 8 Optimized designs for actuator in-plane (left) and out-of-plane (right) using as starting point the optimized design for the sensor case expression. Since the input voltage is a constant in our process, the optimal designs do not depend on this parameter, but from now on all the displacements shown correspond to this V in. Taking V 0 = 0.4, k out = 2000 N/m, the thickness and the stiffness of the layer of AlN are t p = 1 µm and E p = 2 10 12 N/m, respectively, and the piezoelectric constant d 31 = 2 10 12 m/v corresponding to AlN. The optimized actuator in Fig. 7(left) is obtained when the usual homogeneous distribution (in the first iteration, the value of the structure variable is the same in all the elements) is used as starting guess. The optimized designs are shown in Fig. 7. In such a case, the value of the output displacement for the in-plane example is u out = 6.38 µ m. Now, we repeat the same simulation, but changing the starting point. Instead of using an homogeneous distribution, we use the optimized design of the sensor case. The optimized designs for this new starting point are shown in Fig. 8. The value of the output displacement for the in-plane case is now u out = 25.34 µm. It is remarkable that the design obtained is exactly the same than for the sensor case, and the displacement in the output is larger than the one obtained with the previous starting point. We can get two conclusions of this example, the first is that this problem is prone to local optima and the second one is that for k in = k out, both optimized designs, sensor and actuator, are exactly the same. We rigorously justify this fact in the next section. From now on, the examples will be focused on the in-plane regime, but all the conclusions can be extrapolated to the out-of-plane one. 4 Reciprocity of the piezoelectric effect As we have mentioned above, we show that the sensor and the actuator problems are equivalent, in the sense that they have exactly the same optimized solutions, structure and polarization profile, provided the input and output springs are equal, k in = k out. Indeed, this fact is very easy and almost direct to check and it is just an observation, but we think that it is worth mentioning. On the other hand, from U = F in (K(ρ s ) + k in 1 in ) 1 L, being the matrix (K(ρ s ) + k in 1 in ) symmetric and regular, we can rewrite this problem as max : F in G(ρ s, ρ p ) T (K(ρ s ) + k in 1 in ) 1 L ρ s,ρ p subject to v T ρ s V 0 ρ s [0, 1] ρ p [0, 1] and operating in the same way, the actuator problem can be rewritten as max : λg(ρ s, ρ p ) T (K(ρ s ) + k out 1 out ) 1 L ρ s,ρ p subject to v T ρ s V 0 ρ s [0, 1] ρ p [0, 1] being the matrix (K(ρ s ) + k out 1 out ) symmetric and regular. Finally, both optimization problems are the same (up to a positive factor in the objective function) when the parameter k in and k out coincide. 5 Advantages of simultaneous optimization (11) (12) (13) (14) (15) In this section, we justify why we perform simultaneous optimization. To this purpose, we show an example in which we compare: 1. Optimization of the electrode profile while keeping fixed the host structure. 2. Optimization of the structure while keeping fixed the electrode profile. 3. Simultaneous optimization of the host structure and the electrode profile. The example shown is focused on the sensor situation, but the same conclusions are also valid for the actuator case. The first (Fig. 9a), the second (Fig. 9b) and the third (Fig. 9c) pairs correspond to the first, second and third cases

Microsyst Technol (2016) 22:1733 1740 1739 Fig. 10 Manufactured sensor with passive areas Fig. 9 Material layout (left) and electrode profile (right) when optimizing: a the electrode profile only; b the host structure only; c both of them respectively. The left column stands for the host structure, and the right one stands for the electrode profile. Optimized value of the objective function q out is pointed out in the figures, showing the great improvement obtained by using simultaneous optimization (a gain of 2.3173 0.9261 2.5 for this example). 6 Manufacturability of the devices Preliminary test devices designed with our method have been fabricated as in Kucera et al. (2013). In this section we comment on certain considerations that have to be taken into account for the fabrication of the devices designed with this optimization process. A first step towards fabrication is the conversion of the vectors ρ s and ρ p resulting from the numerical procedure outlined above, into curves representing the layout of the structure and the electrodes, respectively. This parametrization and the conversion to a CAD/ CAM model are presented in Chacón et al. (2014). Besides, each of these layouts has to be implemented satisfying technology-related restrictions during fabrication. The narrowest feature allowed in a fabricated micro-structure, to make it reliable for actuation and/or vibrations, certainly depends on the total mass and dimensions of the device. For a practical application of the optimized layouts shown in the paper, a post-processing might be required so that features narrower than a certain threshold are not allowed. It is important to remark that this processing has not been included in the optimization procedure. In Wang et al. (2011) it is presented a robust formulation based on projection methods that allows to control the maximum and minimum size of the different parts of the structure. Regarding the electrodes, a minimum spacing between adjacent metal areas of different sign, which is determined by lithographical resolution and limitations from the etching process, has to be considered. This issue has been recently solved for the problem of mode filtering in Donoso and Sigmund (2015). Consequently, making the final design compatible with the design rules of the foundry may reduce its sensor/actuator performance, with respect to the optimum performance obtained with the model presented here. In the current work both issues have been alleviated in the post-processing. In order to get better optimal designs both difficulties should be taken into account in the optimization process. Some of the devices shown in this paper have been fabricated and in several cases passive pieces have been included in the designs to study the improvement when moving into fluids (Fig. 10). The results of their performance will be published elsewhere. 7 Conclusions In this work, we have proposed a procedure to design piezoelectric transducers by simultaneously optimizing the

1740 Microsyst Technol (2016) 22:1733 1740 host structure and the electrode profile. We point out here the tendency of appearing large gray areas due to the physics of the problem, and that is the reason why it is necessary to introduce a penalization scheme that avoid this issue and let us obtain efficient designs. The proposed interpolation scheme works out, obtaining black and white designs in any situation. Owing to the reciprocal piezoelectric effect, we prove that the optimized design for a specific transducer does not change independently whether it is working as sensor or as actuator. It is important to remark all the difficulties appeared during the whole process. On one hand, gray areas have to be avoided in order to get 0 1 designs during the design process, on the other hand, during the manufacturing process, all the designs to be fabricated have to be parametrized and imported to CAD files. Notice that the manufacturing of this kind of structures is quite complicated due to the size ( µm) and the multiple layers that form a piezoelectric microtransducer. Acknowledgments This work has been supported by MINECO MTM2010-19739 and DPI2012-31203 (Spain), Consejería de Educación, Cultura y Deportes de la Junta de Comunidades de Castilla- La Mancha, European Fund for Regional Development through grant PEII-2014-010-P and the Austrian Research Promotion Agency within the COMET-K2 Project XTribology (Project-no. 824187). References Bendsøe MP, Sigmund O (1999) Material interpolation schemes in topology optimization. Arch Appl Mech 69(9 10):635 654 Carbonari R, Silva E, Nishiwaki S (2007) Optimum placement of piezoelectric material in piezoactuator design. Smart Mater Struct 16(1):207 220 Chacón JM, Bellido JC, Donoso A (2014) Integration of topology optimized designs into CAD/CAM via an IGES translator. Struct Multidiscip Optim 50(6):1115 1125 Donoso A, Bellido JC (2009) Systematic design of distributed piezoelectric modal sensors/actuators for rectangular plates by optimizing the polarization profile. Struct Multidiscip Optim 38(4):347 356 Donoso A, Sigmund O (2015) Topology optimization of piezo modal transducers with null-polarity phases. Struct Multidiscip Optim. doi:10.1007/s00158-015-1330-0 Kang Z, Tong L (2008a) Integrated optimization of material layout and control voltage for piezoelectric laminated plates. J Intell Mater Syst Struct 19:889 904 Kang Z, Tong L (2008b) Topology optimization-based distribution design of actuation voltage in static shape control of plates. Comput Struct 86(19 20):1885 1893 Kang Z, Wang X, Tong L (2011) Combined optimization of bimaterial structural layout and voltage distribution for in-plane piezoelectric actuation. Comput Methods Appl Mech Eng 200(13 16):1478 1497 Kang Z, Wang X, Luo Z (2012) Topology optimization for static shape control of piezoelectric plates with penalization on intermediate actuation voltage. J Mech Des 135(5):051006 Kögl M, Silva E (2005) Topology optimization of smart structures: design of piezoelectric plate and shell actuators. Smart Mater Struct 14(2):387 399 Kucera M, Manzaneque T, Sánchez-Rojas JL, Bittner A, Schmid U (2013) Q-factor enhancement for self-actuated self-sensing piezoelectric MEMS resonators applying a lock-in driven feedback loop. J Micromech Microeng 23(8):085009 Lee CK, Moon FC (1990) Modal sensors/actuators. J Appl Mech 195(1):17 32 Luo Z, Gao W, Song C (2010) Design of multi-phase piezoelectric actuators. J Intell Mater Syst Struct 21(18):1851 1865 Ruiz D, Bellido JC, Donoso A, Sánchez-Rojas JL (2013) Design of in-plane piezoelectric sensors for static response by simultaneously optimizing the host structure and the electrode profile. Struct Multidiscip Optim 48(5):1023 1026 Sánchez-Rojas JL, Hernando J, Donoso A, Bellido JC, Manzaneque T, Ababneh A, Seidel H, Schmid U (2010) Modal optimization and filtering in piezoelectric microplate resonators. J Micromech Microeng 20(5):055027 Sigmund O (1997) On the design of compliant mechanisms using topology optimization. Mech Struct Mach 25(4):493 524 Silva E, Kikuchi N (1999) Design of piezoelectric transducers using topology optimization. Smart Mater Struct 8(3):350 364 Svanberg K (1987) The method of moving asymptotes: a new method for structural optimization. Int J Numer Methods Eng 24(2):359 373 Wang F, Lazarov B, Sigmund O (2011) On projection methods, convergence and robust formulations in topology optimization. Struct Multidiscip Optim 43(6):767 784 Wein F, Kaltenbacher M, Kaltenbacher B, Leugering G, Bänsh E, Schury F (2011) On the effect of self-penalization of piezoelectric composites in topology optimization. Struct Multidiscip Optim 43:405 417