Optimal Design of Piezoelectric Modal Transducers

Transcription

1 Arch Computat Methods Eng DOI /s ORIGINAL PAPER Optimal Design of Piezoelectric Modal Transducers David Ruiz 1 José Carlos Bellido 1 Alberto Donoso 1 Received: 25 October 2016 / Accepted: 4 November 2016 CIMNE, Barcelona, Spain 2016 Abstract Piezoelectric materials are those having the ability to convert electrical energy into mechanical one, and vice versa. Often surface bonded to structures, they are commonly used for sensing, acting and even for reducing noise and structural vibrations as part of active control systems. And, further, they can isolate specific mode shapes of structures when working as spatial filters in the frequency domain (i.e. modal transducers) by shaping properly the piezoelectric layers. This article is intended to revise that concept, initially conceived for beam-type structures only, and explain how it has been extended to plates and shells by means of optimization techniques. 1 Introduction Topology optimization is considered nowadays a quite consolidated conceptual tool for structural design, not only for pure academic purposes but also for industrial applications. This technique has succeeded in many physical situations to date, such as the design of metamaterials, compliant mechanisms or piezoelectric actuators, among others. Early works in piezoelectricity by using topology optimization date back to the end of the nineties. Since then, the underlying optimization tool has been applied & David Ruiz David.Ruiz@uclm.es José Carlos Bellido JoseCarlos.Bellido@uclm.es Alberto Donoso Alberto.Donoso@uclm.es 1 Departamento de Matemáticas, ETSII, Universidad de Castilla-La Mancha, Ciudad Real, Spain successfully, improving physical responses of piezoelectric devices in different contexts, especially in the field of modal transducers, where its contribution has been crucial. Piezoelectric modal transducers (the word transducer comprises both sensor and actuator terms) are those piezoelectric sensors or actuators which measure or excite a single mode of a structure, but remain insensitive to the rest; that is, they behave as ideal spatial filters in the frequency domain. The concept of modal transducer is due to [38]. In that outstanding work, the authors derived theoretically expressions for one-dimensional modal transducers, and then tested experimentally the devices manufactured. What makes the former possible it is precisely the orthogonality principle among the mode shapes of beam-type structures. Without going into details so far, that approach itself cannot be extended to plates and shells for arbitrary boundary conditions. However, it is possible to design modal transducers for such situations just changing the perspective and treating that issue as an optimization problem. The purpose of this article is precisely to review the more relevant advances in this field in the last ten years, and explain how optimization has let design such devices. Two situations are addressed: first when the host structure is fixed (the piezo is optimized only), and second when it is not (i.e. both structure and piezo are optimized). The latter case is more delicate and for that reason, an optimal design problem for statics is dealt with as a previous case in order to highlight the difficulties derived from that approach. The layout of this paper is the following. Section 2 is concerned with an introduction of piezoelectric materials and their properties. In Sect. 3, the philosophy of the topology optimization method is briefly introduced. Section 4 starts citing some of the more relevant works in the optimal design of piezoelectric transducers for statics,

2 D. Ruiz et al. making special emphasis on the contributions done by the authors of this paper. The purpose to do that has already been mentioned above, to point out some of the drawbacks that appear on designing modal transducers whenever the host structure is no longer fixed. After that, the emphasis in placed on Sect. 5, where it is described in detail how the problem of designing modal transducers can be regarded as an optimization problem. Both Sects. 6 and 7 address the problem when the host structure is fixed and when it is not, respectively. Finally, Sect. 8 is devoted to discuss about open problems and work currently in progress. Hereafter, boldface is used for vectors, matrices and tensors. 2 The Piezoelectric Effect The term piezo is a Greek word which means to press. Therefore, it is not surprising that piezoelectricity was first understood as the ability of certain crystals and ceramic materials, among others, to generate electrical energy from pressure [44]. This physical phenomenon, often called direct piezoelectric effect, was discovered in 1880 by the French physicists Jacques and Pierre Curie. But it was not until the following year that the converse (or reciprocal) effect was mathematically deduced and consequently, that was possible to deform such materials under an external electric field. Whenever this kind of materials generate a voltage proportional to their deformation, it is said that they work as sensors, and conversely, when they convert electrical energy into mechanical one, they do as actuators. The physics of the piezoelectric effect is somehow related to the electric dipole moments in solids. A dipole is made of two electrical charges of different polarity, one nearby the other. These dipoles form regions with local alignment called domains, typically randomly oriented, as depicted in Fig. 1 (left). However, they can be aligned as a whole by using a strong electric field (or a poling voltage) at elevated temperatures, as shown in Fig. 1 (right). After this poling process, the polarization remains and the material will expand or contract depending on the polarity of the external applied voltage. That is to say, if this polarity is the same as the one of the underlying poling voltage, then the piezo material will lengthen in this direction and its diameter will become shorter [see Fig. 2 (left)]. Conversely, the piezo will shorten in that vertical direction and become longer in any radial one [see Fig. 2 (right)]. Finally, if an alternating voltage is applied, the piezoelectric material will expand and contract cyclically at that particular target frequency. Although the magnitudes of piezoelectric displacements are small (they have been magnified in Fig. 2 for the sake of clarity), piezoelectric materials have been adapted to an impressive range of applications. They go from the typical quartz watches to scientific instrumental techniques with atomic resolution in the field of microscopy, apart form the typical ones, sensing and acting. An interesting application is the active control of structural vibrations. Active damping means adding piezoelectric actuators and sensors to the structure to be controlled. In such a case, a typical configuration of a piezoelectric transducer is shown in Fig. 3. It is formed by the host structure with a piezoelectric layer bonded to its upper surface. As the piezo material is dielectric, two electrodes on both sides of the piezo are required to collect the charge generated as sensor to an external measurement device. Another common architecture consists of two piezoelectric layers bonded to both the top and bottom of the structure to be controlled. By shaping the electrode in an appropriate way as Fig. 3 (right) shows, then the transducer can isolate a single mode of a structure from the rest. In such cases, the device behaves as a spatial filter in the frequency domain, what it is typically called modal transducer [38]. Once the piezoelectric effect has been explained, the equations that model this phenomenon are introduced. Hereafter, linear piezoelectricity is considered, that is, piezoelectric materials which have a linear response under changes in the electric field, electric displacement, mechanical stress and strain. The relationships among all these tensors can be fully described by a single pair of electromechanical equations. There are many equivalent ways to write them and the best choice will depend on the problem studied. Based on the IEEE standard for piezoelectricity [23], such equations are usually written as poling voltage V V - Fig. 1 Side view of the poling process in a cylinder-type probe; (left) before: domains randomly oriented; (right) after: overall polarization under the poling voltage Fig. 2 Piezoelectric material working as actuator: (left) in tension and (right) in compression

3 Optimal Design of Piezoelectric Modal Transducers C PIEZO STRUCTURE PIEZO ELECTRODE polyvinylidene fluoride (PVDF) [31]. The stiffness of PZT makes it well suited to be used as actuator, while the compliance of PVDF makes it a better candidate for sensing applications. Fig. 3 Schematic diagram of a piezoelectric transducer [17]); (left) cross section of a structure with a piezo layer surface bonded; (right) top view of the surface electrode e i ¼ s E ij r j þ d ki E k D k ¼ d ki r i þ r kl E ð1þ l; where i, j take on values 1; 2;...; 6 and k; l ¼ 1; 2; 3 (or x, y, z). e i is the strain tensor, r j is the stress tensor, s E ij is the compliance tensor, d ki are piezoelectric constants, E k is the electric field, D k is the electric displacement, and r il is the permittivity. The superscripts E and r indicate that the values of the tensors are obtained at constant electric field and constant stress, respectively. Assuming a transversely isotropic piezoelectric material poled along the axis 3 (or z), we can rewrite (1) in a matricial way as e 1 s E 11 s E 12 s E r 1 e 2 s E 21 s E 22 s E r 2 e 3 s E 31 s E 32 s E r e ¼ s E r e s E r 5 5 e s E 66 r ð2þ 0 0 d d E d 33 þ d E E 4 d r r 2 D d r 3 4 D 2 5 ¼ d r 4 D 3 d 31 d 32 d ð3þ 2 r þ 0 r r 33 Equation (2) stands for the direct piezoelectric effect, while Eq. (3) refers to the converse one. Those tensors involved might be slightly altered depending on the piezo material [44]. In practice, the most common piezoelectric materials used are lead zirconate titanate (PZT) [24] and E 1 E 2 E r 5 r 6 3 The Topology Optimization Method A basic engineering goal is to optimize properties and physical responses in structures and mechanisms. Most times, an efficient use of the amount of material is crucial in the following sense: only some optimized layouts would be the best in each case and therefore, better than any other distribution considered. The layout or topology of a structure is described by the connectivity of its structural elements, or equivalently by the number, shapes and position of its holes. Nowadays, the term topology optimization is conceived as a major conceptual tool for structural design. Basically, it solves the problem of distributing a limited amount of material in a design space to optimize a certain objective function while some required constraints are fulfilled. Roughly speaking, the point is to determine where to put the holes in the design domain occupied by the structure. Today, the topology optimization theory is well established and has been successfully applied to numerous situations to date. The number of references on this topic as well as the examples of industrial applications is really overwhelming. Interested readers are referred to the quintessential reference book [3]. Also, it is worthwhile to remark some surveys recently published on the topic itself and applications derived from the method [11, 26, 36, 64]. The general setting of a topology optimization problem over a fixed design domain X may be described as max v2f0;1g subject to: Gðv; u v Þ¼0 Wðv; u v Þ ð4þ f i ðv; u v Þ 0; i ¼ 1;...; m ð5þ where W is the cost function, (4) is the state equation, u v is the state, v is the control (also called design) and (5) are the constraints of the problem. The most classical problem in structural optimization is the minimum compliance one. It can be stated as follows: given the loads and the boundary conditions over a structure occupying a design domain, the goal is to find the stiffest structure among all with a prescribed volume fraction, say V 0. A generic design problem in 2d is sketched in Fig. 4 (left). In Fig. 4 (right), a possible (but invented) optimized solution is shown.

4 D. Ruiz et al. Having in mind that maximizing the stiffness of a structure is equivalent to minimize the elastic energy stored in the system, the formulation for the compliance is usually written as Z min v2f0;1g gðuþ ¼ X fu dx subject to: divðeðvþeðuþþ ¼ f; in X boundary conditions on ox Z 1 v dx V 0 jxj X? Ω ð6þ where f are the external forces applied to the structure, u is the displacement field, eðuþ is the symmetric strain gradient, and EðvÞ is the stiffness (fourth-order tensor). The design variable v can take two values only, v ¼ 1 and v ¼ 0, meaning that there is either material or void, respectively. That variable appears in the state equation through the properties of the material as follows EðvÞ ¼E A v þ E B ð1 vþ; P v 2f0; 1g χ = 0 χ = 1 Fig. 4 Left design domain with arbitrary boundary and a prescribed hole; right optimized layout where E A and E B are the stiffness tensor of the two different materials. Typically, the second material is void, usually modelled by a weak material with a very low stiffness value. Boundary conditions imposed on the boundary ox include points where either the displacements or the forces applied are known. The last constraint in (6) is a limit over the maximum amount of material allowed. Although this upper bound is not always totally needed, here it is, otherwise the solution for the compliance would be all the design domain filled with material, that is, the most stiffest structure. In general, this problem lacks 0 1 solutions due to the non-convexity of the set of feasible designs. This fact is reflected in the discretized problem as the following numerical instability: a larger number of holes appear on refining the mesh, what it is often called mesh-dependence. On the other hand, even for a coarse mesh, the resulting integer-type optimization problem is virtually impossible to solve because all possible configurations satisfying the P volume constraint should be evaluated. Indeed, this process does not guarantee convergence for finer meshes, as mentioned above. The usual way to proceed consists first in using the finite element method (FEM) to discretize the problem and second, replacing the integer variables v by density ones q. Now, these new variables, apart from taking the values 0 and 1, can take any value in the continuous interval q 2½0; 1. Whenever they do the latter, small gray transitions are generated between solid and void regions. In a specific finite element, an intermediate value of the density would indicate the local volume fraction of an hypothetical composite material. In principle, this is not desirable and make it difficult the manufacturing process. An efficient way to penalize these gray areas is the SIMP method [3, 4]. It is done through the following interpolation function EðqÞ ¼q p E A þð1 q p ÞE B ; q 2½0; 1 where p is the power that usually takes the value p ¼ 3 due to physical reasons [4]. However, the SIMP method does not alleviate by itself the mesh-dependence issue above mentioned. To this purpose, it is necessary to incorporate regularization techniques to the design problem in order to provide optimized solutions. These techniques are typically a combination of a mesh-independent filtering technique and a projection. Essentially, the former bounds the maxima oscillations on the design variables, and the latter together with SIMP help to reduce intermediate density values, forcing to obtain 0 1 solutions. This lets in turn control the minimum length scale in the structural elements at the end of the optimization process [20]. At this point, it is worthwhile to say something about the most two used filtering techniques: sensitivity- and densitybased filtering methods. Sensitivity filter [60] modifies the element sensitivities values (i.e. the derivatives of the objective function with respect to the element densities) by a weighted average of the element sensitivities in their neighborhood of radius r min (a parameter whose value is fixed beforehand, typically called filter radius). For regular meshes, the filtered sensitivity is given by og fog Pi2N e wðx i Þq i oq ¼ P i oq e q e i2n e wðx i Þ ; where x i is the spatial (center) location of element i, and the weighting function wðx i Þ is given by the cone-shape function wðx i Þ¼r min kx i x e k Despite it is has not yet proved that the filter ensures existence of solutions, the fact is that it has been successfully used in numerous applications to date, producing

5 Optimal Design of Piezoelectric Modal Transducers mesh-independent designs. Maybe, motivated by the fact that on modifying the sensitivities in that heuristic way, it is not clear at all what objective function is actually being minimized, now the tendency is to use the density filter instead [6]. Following the same philosophy it is more intuitive because the densities are filtered rather than the sensitivities P i2n ~q e ¼ e wðx i Þq i Pi2N e wðx i Þ ; Like the sensitivity filter, it also produces mesh-independent design, but in this case this is a fact which was mathematically proved by Bourdin [5]. Finally, to ensure both black-and-white designs and that the smallest solid feature in the final design will be equal or larger than 2r min, a projection is performed over the filtered density. A possible scheme to do that is a smoothed Heaviside function proposed in [20] q e ¼ 1 e b ~q e þ ~q e e b ; ð7þ where b determines the sharpness of the projection. Using FE-notation the (regularized) discrete compliance problem may be written as min q2½0;1 f T u subject to: q ¼ SðqÞ Kð qþu ¼ f v T q V 0 where f is the load vector, u is the displacement vector, K is the global stiffness matrix, v is the volumes vector, and q is the regularized (filtered and projected) density field after the regularization process expressed by S. As an illustrative example, Fig. 5 shows the optimal cantilevered structure Ω Fig. 5 Example of a cantilever beam: (top) design domain; (bottom) optimized layout for V 0 ¼ 0:5. The black bar of 2r min width represents the minimum allowable feature size for structural elements P composed of bars of 2r min width at least and V 0 ¼ 0:5 (50% of material). Concerning the optimizer to solve the discrete problem, the typical choice is a gradient-based method, an algorithm that needs in each iteration step the value of the objective function and constraints, and more importantly their derivatives with respect to the densities. A very good candidate is MMA [72], a mathematical programming algorithm able to cope with many variables and a moderate number of constraints. MMA is quite stable and has been used efficiently in many situations so far. Currently, the tendency is to control feature sizes of both material and void phases, not only to avoid typical hinges or corners in the final structures, but also to obtain design robust to possible errors in the manufacturing process. That is performed by using the so-called robust approach [21, 58, 63, 75]. Finally, the extension to the three-dimensional case is straightforward, but the computation time increases considerably. At this point, a parallel computing framework is necessary for conducting very large scale topology optimization problems (see [1]). 4 Optimal Design in Piezoelectricity As we have underlined above, piezoelectricity is clearly a promising field nowadays and at the cutting edge of researching, so it is not surprising that the number of papers on optimization together with piezoelectric materials have increased considerably in the last years. The purpose of this section is not to make an extensive literature review on that topic, but rather cite some of the more relevant works, finishing and making special emphasis on some the contributions done for statics by the authors of this paper. We honestly think that this can help to better understand how to overcome some of the drawbacks that appear in the next section on designing modal transducers, especially in the case that the host structure is no longer fixed. The first studies on piezoelectricity and topology optimization were about designing optimal periodic piezocomposite, that is, materials with prescribed piezoelectric properties. Unit cell and homogenization based topology optimization studies can be found in [65] for the two dimensional case as well as in [61, 66 68] for the threedimensional case. Closer to our purpose, a first attempt on the design of inplane piezoelectric resonators using topology optimization is the work of [69], where the material layout of a host structure is optimized, but keeping fixed the piezoelectric material. That was a very interesting work, but on imposing a constraint in the position of the piezoelectric material, the search of optimized solutions is rather limited. Since then,

6 D. Ruiz et al. some authors began to perform simultaneous design of both host structure and piezoelectric layers by using topology optimization techniques. We mention here some of them. [34] considered the optimization of the piezoelectric layers together with the polarization profile for statics. [7] and [41], among others, optimized both the host structure and the piezo layers to design in-plane multi-phase actuators. More recently, [29] have gone further, on including as a third design variable the spatial distribution of the control voltage in the optimization problem, in some way connected with the polarity of the piezoelectric layers. In contrast to the works commented above, the problem studied by the authors of this paper for static response [52] assumes that the piezoelectric material is surface bonded to the structure wherever it is, so the design variables are essentially two, the material layout of the whole structurepiezo and the electrode profile. Indeed, we deal with two optimization problems, somehow equivalents, as we report later on. From the sensor point of view, the problem consists in maximizing the output charge generated by an external mechanical actuation. From the actuator perspective, the problem is about maximizing the output displacement generated by an external input voltage. Next, we will present and study both problems in detail. 4.1 The Sensor Problem We start studying the first of the two aforementioned problems, that is, the sensor problem. Here, we will focus on the in-plane case for the sake of clarity, and the reader is referred to [54] for the out-of-plane case Continuous Formulation We consider a two-dimensional structure as a design domain, more specifically, a L x L y rectangular plate clamped at its left side C u (see Fig. 6). Unless otherwise stated, two piezoelectric layers of negligible stiffness and mass compared to the plate are surface bonded to the host C q out Γ u SIDE VIEW PIEZO STRUCTURE PIEZO TOP VIEW? F in Ω STRUCTURE AND PIEZO k in χ s = 1 χ s = 0 VOID Fig. 6 Design domain for the sensor problem ELECTRODE PROFILE χ p = 1 χ p = -1 structure. This assumption is very common, especially when working at the micro scale as we do here. In this case, the whole structure is subjected to an external mechanical actuation (the input), typically modelled with a in-plane force F in and a spring with stiffness k in applied in C t, a very small area around the midpoint of its right edge. Two design variables are used in our approach. The first one v s represents the topology of the structure-piezo as a whole, and it can take on two values in every point of the domain, v s 2f0; 1g meaning that there is void or material (both structure and piezo). The second variable v p represents the polarity of the electrode that covers the piezoelectric layers. This variable can take two values v p 2f 1; 1g, meaning negative or positive polarity. Something common as well as advisable is to include a constraint over the maximum allowed displacement Uin max at the input port of the device. The usual way to do that consists in adding the previously mentioned spring at the input port with stiffness k in and the force F in such that U max in ¼ F in k in. That is expressed through a Robin boundary condition. The design problem consists in determining simultaneously the electrode profile and the material layout of the whole structure-piezo for a fixed volume fraction V 0 in order to maximize the current q out. This optimization problem may be formulated as Z ou max : q out ¼ v p e 31 v s ;v p X ox þ e ov 32 dx oy subject to divðe s v s eðu; vþþ ¼ 0 in X ðe s v s eðu; vþþ n þðk in ; 0Þ ðu; vþ ¼F in u; v ¼ 0 on C u Z 1 v j X j s dx V 0 X v s 2f0; 1g v p 2f 1; 0; 1g on C t ð8þ where n is the normal outer vector to the boundary of X. Concerning the physical parameters, E s is the stiffness tensor of the host structure, and e 31 and e 32 are the piezoelectric stress/charge constants in x and y directions respectively. In the following, we consider same piezoelectric constants in both spatial directions, and piezo axes coincident with the geometric ones of the plate, as in e.g. hexagonal class 6 mm crystals. Based on these assumptions, it is important to notice that the piezoelectric properties are not needed in the problem formulation and just appear as a scaling factor that we will omit in the discrete problem for the sake of simplicity.

7 Optimal Design of Piezoelectric Modal Transducers Discrete Formulation As usual, this kind of optimal design problems do not admit optimal solutions in the sense that we usually understand for the reasons reported in the previous section. The standard approach consists in discretizing the design domain in finite elements and letting each element has two variables, the first one concerning the host structure and the piezo material, and the second one is related to the polarization profile of the electrode. On the one hand, SIMP method replaces the term E s v s in (8) by the interpolation function E min q p s þ 1 qp s Es ; 0 q s 1 where E min is a very small stiffness used in void regions in order to prevent singularities in the stiffness matrix, p is the power and q s is the material density. On the other hand, the polarization variable v p is directly replaced by the polarization density with the continuous expression ð2q p 1Þ, being now 0 q p 1. In [30] an efficient way to relax this variable is proposed, but in this work, in line with [34], we have checked that a penalty factor in q p has no significant effects. This fact has perfect sense since the variable in the optimum always take the values 1 and 1 (and not 0), otherwise the output charge would be lower. The discrete formulation of the problem in FE-notation may be written as max q s ;q p : q out ¼ Gðq p Þ T U subject to ðkðq s Þþk in 1 in ÞU ¼ F in L V v T q s V 0 q s 2½0; 1 q p 2½0; 1 where K and U are the global stiffness matrix and vector of displacements respectively, 1 in is a zero matrix that takes the value 1 at the input port, L is a zero vector which takes the value 1 again at the input port, q s and q p are the vectors of the material and polarization densities, respectively, and v is a vector containing the volume of the elements. The vector G is expressed as G ¼ð2q p 1ÞB; where B is the usual FE strain-displacement matrix Computation of Sensitivities ð9þ At this point, we need the computation of the sensitivities of the objective function and the constraints with respect to the design variables in each element, denoted by qs ðeþ and qp ðeþ, in which we have discretized the design domain. The usual way to do that is by means of the adjoint method. It consists in adding the state equation (as a null term) to the cost through a multiplier q out ¼ G T U þ k T ððk þ k in 1 in ÞU F in LÞ; ð10þ where k T acts as Lagrange s multiplier. Differentiating and rearranging terms, we arrive at (10) oq out ¼ k T oqs ðeþ ok oq ðeþ s U; where k T is the solution to the adjoint problem ðk þ k in 1 in Þk ¼ G: Notice that we can take advantage of the matrix decomposition made when we solved both linear systems of equations since the matrix is the same. On the other hand, the derivative of the cost with respect to the electrode density qp ðeþ is direct, due to the linear dependence of the cost with respect to this variable oq out oq ðeþ p ¼ ogt U ¼ 2B T U: oq ðeþ p Finally, the corresponding derivatives of the volume constraint are given by ov oq ðeþ s ¼ v ðeþ ; and ov oq ðeþ p ¼ 0: Numerical Approach and Examples On performing numerical simulations in the previous problem, it is observed that gray areas still appear concerning variable q s in the final designs, though the sensitivity filter is used. That makes perfect sense and it can be explained just remembering the physics of the problem. Not only those gray areas are providing (artificial) output charge to the device as being covered by an electrode, but also their contribution is bigger than the one corresponding to solid areas since areas with intermediate values of densities are more flexible than solid ones. A way to avoid this is to modify the cost functional by multiplying it by a new term Rðq s Þ that penalizes in a progressive way these gray areas [52] and [54]). The heuristic scheme proposed R, shown in Fig. 7 is a piecewise linear function, with very small slope but positive in the first part, and much steeper in the second one. The parameters g and f are tunable, but we have kept them fixed in all the simulations because it was very effective in removing gray areas (we start taking g ¼ 0:8 and f ¼ 0:01,

8 D. Ruiz et al. and after some iterations these values eventually change to g ¼ 0:9 and f ¼ 0). It is important to emphasize that the lack of differentiability in R does not affect the numerical results at all. However, if needed, it can be smoothed as we will see in the next section. Hence, maximizing now q out R instead of q out the problem is successfully solved because the gray areas almost completely disappear and the designs obtained are really close to 0 1 designs. This interpolation scheme is now introduced in the expression of the output charge in (9) as follows Gðq s ; q p Þ¼Rðq s Þð2q p 1ÞB ð11þ Therefore, a new term appears in the derivative of the cost with respect to the material density oq out ¼ k T oqs ðeþ oq ðeþ s 1 R ζ 0 oq ðeþ s ok oq ðeþ s U þ ogt U oq ðeþ s where og T ¼ or ð2q p 1ÞB (1 ζ) ρs e + ζ η 1 η ζ η ρ e s Fig. 7 Penalization function Rðq s Þ Although all this analysis is valid for structures at the macro scale, the focus on the examples is done at the micro scale. Next, our optimization approach is illustrated through several numerical examples of interest in the area of MEMS (microelectromechanical systems) because of the sizes used. Regarding applications, many MEMS-based actuators like surface probes, micro-grippers or micro-optical devices can be optimized following the procedure shown in this work. We consider a square plate clamped at its left side of length L ¼ 1000 lm and thickness t ¼ 20lm. The material ρ s η 1 properties for the host structure are those corresponding to silicon, that is, Young s modulus E ¼ 130 GPa and Poisson s ratio m ¼ 0:28. The piezoelectric constant only appears multiplying the cost as a scale factor, so it does not affect the optimized designs. Due to symmetry only a half of the design domain is discretized using a mesh of elements. Figure 8 shows both material layout and polarity density for two different volume fractions and k in ¼ N/m. By means of this new interpolation function the material distribution is almost black and white (0 1 design). Same happens with the electrode layout, the whole structure is being covered by red and blue colors, positive and negative polarity, respectively. Another meaning of the colors of the electrode is the stress, red areas are in tension while blue areas are in compression. Figure 9 shows a preliminary test device recently fabricated [54]. A passive area is included in the design in order to study its operation when moving into fluids. That part has been done in collaboration with the Department of Mathematics and Microsystems, Actuators and Sensor Group of the University of Castilla-la Mancha. The microsensor has been manufactured in Vienna by the Institute of Sensor and Actuators Systems. The results of its performance will be published elsewhere. 4.2 The Actuator Problem In this section, we will study the optimal design problem from the actuator view. This case study is very especially Fig. 8 Optimized design for the sensor for two different volume fractions; (left) material density and (right) electrode density. This example has been extracted from [52]. a V 0 ¼ 0:3 andb V 0 ¼ 0:5. (Color figure online)

9 Optimal Design of Piezoelectric Modal Transducers Fig. 9 Optimized microsensor with a passive area important on designing MEMS. Contrary to what happened when the piezo layer worked as sensor, the input is an electrical signal (voltage V in ) and the parameter to be maximized is the output displacement at the output port. The goal now is to find both structure layout and electrode profile that maximize an output displacement U out for a given output stiffness spring k out (see Fig. 10). This parameter is fixed when we choose an application, meaning that the output stiffness that the structure has to beat is not the same in every case, e.g. this value is different in a micro-gripper used in medicine than in an electrical switch. Imposing in-phase identical transducers in Fig. 10, we can restrict ourselves to in-plane motion when working as actuators. Under this consideration the piezoelectric force is modelled as a force due to an initial strain, produced by the applied external voltage. The reader is referred to [54] for more details about it. The discrete formulation for the actuator case may be expressed as max : U out ¼ L T U q s ;q p subject to ðkðq s Þþk out 1 out ÞU ¼ jgðq s ; q p Þ v T q s V 0 q s 2½0; 1 q p 2½0; 1 where j is another constant comprising the thickness of the piezoelectric layers, a piezoelectric constant and the voltage applied, among others. Again L is a zero vector which takes the value 1 now at the input port, and 1 out is a zero matrix taking the value 1 at the element corresponding to the output port. Using k out ¼ N/m and DV in ¼ 1000 V, the displacement of the output port of the optimized design in Fig. 11 (left) is U out ¼ 6:38 lm. However, using as starting point the corresponding optimized sensor for the same value of spring stiffness k in ¼ N/m, the layout does not change but the displacement is much bigger, U out ¼ 25:43 lm, as Fig. 11 (right) shows. We can get two conclusions of this example, the first one is that this problem is very prone to local optima, and the second one is that for k in ¼ k out, both optimized designs, sensor and actuator, are exactly the same. The fact that both designs are exactly the same comes from the nature of the piezoelectric effect and this is rigorously justified in the next section. Same conclusions are valid for both, in-plane and out-of-plane cases, as reported in [54]. 4.3 Reciprocity of the Piezoelectric Effect In this section, we will demonstrate that the optimized designs for the sensor and actuator problem are the same, provided the input and output springs are equal, k in ¼ k out. As we mentioned above, this is a consequence of the reciprocity of the piezoelectric effect. To check that the optimized solutions for the sensor and the actuator problems are the same, we have just to notice that writing the solutions of the state equation for the sensor problem as U ¼ F in ðkðq s Þþk in 1 in Þ 1 L; being the matrix ðkðq s Þþk in 1 in Þ symmetric and regular, we can rewrite the sensor problem as max : F in Gðq s ; q p Þ T ðkðq s Þþk in 1 in Þ 1 L q s ;q p + V - in SIDE VIEW PIEZO STRUCTURE PIEZO TOP VIEW? U out Ω STRUCTURE AND PIEZO k out χ s = 1 χ s = 0 VOID Fig. 10 Design domain for the actuator problem ELECTRODE PROFILE χ p = 1 χ p = -1 subject to Fig. 11 Optimized designs for the actuator case; (left) using as starting point the optimized design for the sensor problem; (right) using another starting point. This example has been extracted from [54]

10 D. Ruiz et al. v T q s V 0 q s 2½0; 1 q p 2½0; 1 Operating in the same way, the actuator problem can be rewritten as max : jgðq s ; q p Þ T ðkðq s Þþk out 1 out Þ 1 L q s ;q p subject to v T q s V 0 q s 2½0; 1 q p 2½0; 1 being the matrix ðkðq s Þþk out 1 out Þ symmetric and regular. Finally, both optimization problems are the same (up to a positive factor in the cost) when the parameters k in and k out coincide. 4.4 Optimal Design of Piezo Unimorph Microgrippers As it has been mentioned above, our model considers two piezo layers perfectly bonded to both sides of the host structure. Unfortunately, things change at the micro-scale. Owing to the current manufacturing techniques, it is not possible to get this configuration and only one piezo layer can be placed over the structure. That makes the structure moves also out-of-plane, which, in general, it could spoil and dramatically reduce the performance of the device. When fabricating a sensor this fact is not truly a problem, since the deformation is produced by an external force. However, the main problem appears when working as actuator, making it impossible to design, for instance, a genuine microgripper-type actuator. The only way to give a systematic answer to this problem is through an appropriate formulation that maximize the in-plane response of the device at the same time as reduce bending at certain points as much as possible. More specifically, as Fig. 12 (left) shows, u 1 is the in-plane displacement to be maximized, and both u 2 and u 3 are the out-of-plane displacements to be suppressed. Concerning geometrical constants, the length and the height of the passive area are set to L p ¼ 1000 lm and h p ¼ 50 lm (it has been magnified in the design domain for the sake of clarity), respectively. Finally, the separation between the two points, where the bending is cancelled, is 25 lm. Figure 12 (right) shows the optimal actuator corresponding to k out ¼ 20 N/m. The structure layout is a pure black and white design, and just orange and blue colors represent parts of the structure with different polarity. Fig. 12 Optimized designs of a piezo microgripper-type actuator; (left) design domain; (right) electrode profile [55]. (Color figure online) In that Fig. 12 also have been implemented the so-called robust approach which lets minimize the sensitivity of the target of the microactuator under fabrication errors as well. Since the displacements obtained are small compared with the size of the actuator, a linear elasticity model has been used. Nonlinear modelling is being considered [56] in order to model piezo MEMS exhibiting large displacements. 5 Modal Filtering by Piezo Transducers That concept of modal transducers was introduced by [38]. In that remarkable work [38], the authors first derived theoretically expressions for beam-type structures and onedimensional plates (basically treated as beams with a slight modification in the flexural rigidity term), and then tested experimentally the devices manufactured. Modal transducers are intended to isolate a particular mode of a structure (eigenmode) from the rest, that is to say, they become as ideal spatial filters for a driving frequency different to the target one. Furthermore, due to the reciprocity of the piezoelectric effect, it is also proved that the pair modal sensor/actuator which measures/excites the same mode present the same pattern.

11 Optimal Design of Piezoelectric Modal Transducers Having in mind that only the portion of piezo covered by electrode will be electrically active, the problem of designing a modal sensor for 1d structures is confined to determining the surface electrode width. The charge collected by the effective surface electrode may be expressed (up to a scaling factor) by Z L qðtþ ¼ FðxÞ o2 w dx ð12þ 0 ox2 where L is the length of the beam, w is the vertical displacement, and FðxÞ plays the role of the effective electrode. By using modal expansion, the displacement w can be written as wðx; tþ ¼ X1 / j ðxþg j ðtþ; ð13þ j¼1 where g j ðtþ and / j ðxþ are the jth modal coordinate and mode shape (eigenmode), respectively. Inserting eq. (13) into (12), we arrive at qðtþ ¼ X1 where B j ¼ Z L 0 j¼1 B j g j ðtþ; FðxÞ / 00 j ðxþ dx: As the eigenmodes are orthogonal to each other with respect to the unit weight function, modal transducers are found by tailoring the surface electrode with areas of positive, null or negative polarity, according to the curvature of the mode of interest. In other words, if FðxÞ is choosen as a constant times the second derivative of a particular mode shape, say the kth, then that mode is isolated from the rest; that is to say, the coefficient corresponding to that mode, B k, is maximized and the rest are cancelled at all. That FðxÞ function contains all the necessary information to shape the modal transducer: on the one hand, its absolute value indicates the gain distribution of the transducer along the x-direction in accordance with its profile, and on the other hand, its sign means whether the polarity is positive or negative. As an illustrative example, the electrode profile (in gray) isolating the second mode shape for a cantilever beam is shown in Fig. 13. White areas mean parts of the structure not covered by electrode. When considering plates with arbitrary boundary conditions, the expression for the charge becomes Z qðtþ ¼ Pðx; yþ Gðx; y; tþ dx dy S where S is the area covered by electrode, G is a function depending on both piezoelectric constants and strains that we will detail later, and P refers to the polarity of the effective surface electrode. Basically, it can take three possible values only:?1 (electrode with positive polarity), 0 (no electrode) and -1 (electrode with negative polarity). Unfortunately, the way to proceed in 1d cannot be extended to the 2d case mainly due to the non-validity of the aforementioned orthogonality principle for plate-type structures with arbitrary boundary conditions, that is, the identities Z / i / j dx dy ¼ 0; for i 6¼ j; S WIDTH (+) φ (x) 2 F(x) (-) Fig. 13 (Top) second mode shape of a cantilevered beam; (bottom) electrode profile that isolates that mode shape do not hold, in general. However, even in the cases for it can (for example, plates with pinned boundary conditions), an intermediate-values polarity distribution P is required. It would imply that P 2½ 1; 1, which could be really difficult to achieve in practice, as pointed out in [8]. Since then, many authors have studied the underlined problem in detail to date and the more relevant are mentioned here. In [32], though the results obtained from genetic algorithms over rough meshes are satisfactory, the implementation requires extra interface circuits. [71] proposed structures composed of many small piezo patches of different and uniform thickness. [49] introduced a new porous distributed electrode concept. More recently, electrode-shaping techniques have been performed in [50] to detect modes only, but not to cancel others. Also [77] reduce the sound radiation in shells under harmonic excitations. An interesting alternative would be to regard the design of modal transducers as an optimization problem, where the design variable (polarity) that controls the electrode profile takes exclusively 1 or 1 values. A first try in this direction was the work of [27], where the continuous shape of a sensor is optimized taking some points of the parameterized boundary as design variables. Although the L x x

12 D. Ruiz et al. polarization profile is also initially considered in the model, it is no longer used in the optimization process as a design variable. Following the ideas described in such works [27] and using the topology optimization method, we point up the technique developed by some of the authors of this paper in [28] to systematically design modal transducers for plates moving out-of-plane. That is explained in detail in the next section. 6 Design of Modal Transducers via Optimization: Host Structure Fixed The aim of this section is to discuss how the problem of designing modal transducers can be elegantly regarded as an optimization problem. In all the analysis, the host structure is supposed to be fixed and therefore, is not included in the optimization process so far. As before, a L x L y rectangular plate of arbitrary boundary conditions is considered as design domain (see Fig. 14). It is worth mentioning again that piezoelectric properties are not needed in the problem formulation and just appear as a scaling factor that we will omit for the sake of simplicity. Concerning the physics of the problem, on deforming the whole structure, the charge q collected by the piezoelectric material working as sensor can be expressed, up to a scaling factor, by [38] qðtþ ¼ Z Ly Z Lx v p 0 0 ðh s þ h p Þ 2 ou ox þ ov oy o 2 w ox 2 þ o2 w oy 2 dx dy; ð14þ where (u, v, w) is the displacement vector, h s is the thickness of the plate and h p is the one of the piezoelectric layers. Notice that here v p plays the role of the previous function P. C q(t) SIDE VIEW PIEZO STRUCTURE PIEZO TOP VIEW? Ω STRUCTURE AND PIEZO χ s = 1 χ s = 0 VOID Fig. 14 Design domain for modal filtering [53] ELECTRODE PROFILE χ p = 1 χ p = -1 By using modal expansion, the displacements u, v, w are written as uðx; yþ ¼ X1 / j ðx; yþg j ðtþ j¼1 vðx; yþ ¼ X1 j¼1 wðx; yþ ¼ X1 j¼1 w j ðx; yþg j ðtþ u j ðx; yþg j ðtþ ð15þ where U j ðx; yþ ¼ð/ j ðx; yþ; w j ðx; yþ; u j ðx; yþþ is the jth mode shape, and g j ðtþ the jth modal coordinate. Inserting (15) in(14) we arrive at qðtþ ¼ X1 j¼1 F j g j ðtþ; that is, the response of the sensor can be rewritten in such a way that the spatial terms appear separated from the temporal ones, and therefore the coefficient F j ¼ Z Ly Z Lx v p 0 0 ðh s þ h p Þ 2 o/ j ox þ ow j oy o 2 u j ox 2 þ o2 u j oy 2!) dx dy ð16þ depends on the jth mode shape only. It is important to remark that such coefficients F j depend explicitly on v p and the eigenmodes U j. These are computed through the eigenmode equation, AðU j Þ¼0; ð17þ just once, since the host structure is fixed and therefore it does not come into play in the optimization problem. This is radically different to the case considered in the next section. As we work with a plate-type model, the curvature is null and then in-plane modes appear decoupled from outof-plane modes. That means that Eq. (16) will not be considered, and shorter expressions will be used instead. In this way, when considering in-plane modes only (both piezo layers are in-phase polarized), then U j ¼ð/ j ; w j ; 0Þ and Z Ly Z Lx o/ j F j ¼ v p 0 0 ox þ ow j dx dy: oy Oppositely, when considering out-of-plane modes only (both piezo layers are out-of-phase polarized), then U j ¼ ð0; 0; u j Þ and F j ¼ Z Ly Z Lx v p 0 0 o 2 u j ox 2 þ o2 u j oy 2! dx dy

13 Optimal Design of Piezoelectric Modal Transducers where the term ðh s þ h p Þ=2, being a constant multiplicative factor, is removed. The simplest problem coming to our mind would be to find the electrode profile that maximizes the mode sensitivity of a specific mode (say the kth), max : F kðv p Þ v p 2f 1;1g In fact, this problem can be solved analytically by saying that 8! >< v p ¼ 1; if o 2 u j ox 2 þ o2 u j oy 2 0 >: 1; otherwise However, these designs do not cancel the influence of the rest of modes, as the coefficients F j are not forced to be null in the problem formulation. Rather to this, the really interesting problem indeed is to find a modal sensor that filters the kth mode among the first M modes, and simultaneously cancels the rest of them, that is, max v p 2f 1;1g : F kðv p Þ subject to F j ðv p Þ¼0; j ¼ 1;...; M; j 6¼ k As reported in [15], since the problem is linear in v p in both the objective function and the constraints, it can be proved that an optimal solution taking values on 1 or 1 always exists, and therefore neither relaxation nor regularization are needed. The reader is referred to that work [15] for a more detailed discussion on the mathematical issues of that problem as well as all information concerning the proof. Anyhow, for numerical reasons is practical to relax the design variable by replacing in the model v p 2f 1; 1g to q p 2½ 1; 1, and once the problem has been relaxed, the discrete problem can efficiently be solved by using the simplex method, without requiring filters nor projection or penalization methods, and corroborating the existence of classical solutions. Even there is no need, the problem can be solved by more difficult and sophisticated techniques like level sets, as in [48]. The discrete problem in FE-notation may be expressed by max : F k ðq p Þ G T q k q p p subject to G T j q p ¼ 0; j ¼ 1;...; M; j 6¼ k q p 2½ 1; 1 where G j is the term concerning the strains in the jth mode. As it is mentioned above, a similar formulation is obtained in the in-plane case [16]. Figure 15 (left) shows the electrode profiles that maximize the sixth mode shape for a plate fixed at its left edge, first without filtering and second when considering the first 20 out-of-plane modes are the rest are canceling. Same when the structure is moving in-plane only and the first 12 modes are taking into account appears in Fig. 15 (right). Another example for different boundary conditions is depicted in Fig. 16. In that case, a plate simply-supported at all four edges for which the first mode is isolated among the first M ¼ 10, 15, 20 and 26 modes. That approach was also successfully generalized to other geometries [14], and even to shells [13]. As an example, Figs. 17 and 18 show electrode profiles for a circular plate and a cylindrically curved panel, respectively. Later, the investigation followed by designing micro transducers that were manufactured and tested showing quite good performance in general [57]. However, we could check that part of the discrepancies between the tested filters and the ideal ones come from the fabrication. To illustrate this, now we focus on Fig. 19 (b) that shows Fig. 15 (Top) sixth out-of-plane mode (left) and sixth in-plane one (right) for a plate fixed at its left edge. (Middle) electrode profiles that maximize those mode sensitivities. (Bottom) electrode profile that isolate those modes when considering the first 20 out-of-plane modes only, and the first 12 in-plane modes only, respectively. This example has been partly extracted from [53]

14 D. Ruiz et al. Fig. 16 Polarization profiles toisolate the first mode for a plate simply-supported at all four edges when considering M ¼ 10 (a), M ¼ 15 (b), M ¼ 20 (c) and M ¼ 26 (d). This example has been extracted from [15] S R 0 0 L X Z Y X Fig. 18 (Top) second mode for a cylindrical shell cantilevered at its left curved side. (Middle) eletrode profile in the parametric domain (x, s) that isolates that mode among the first 20 modes. (Bottom): same in the real domain (x, y, z). This example has been partly extracted from [13] Fig. 17 Polarization profiles that measure the second mode (left) and the twelfth mode (right) for a clamped circular plate when considering 16 modes. This example has been extracted from [14] the optimal electrode that isolates the fundamental bending mode (Fig. 19a) of a micro bridge from the first 14 out-ofplane modes. As it is noticed from the manufactured device (Fig. 19c), in the lithography process [35] a uniform nullpolarity phase (gap-phase) corresponding to v p ¼ 0ofa few microns was required between areas of opposite polarity in order to avoid short-circuiting. Clearly, the sensitivity to that gap-phase strongly depends on the scale of the device, and at the micro scale, as in our case, the width of the gap is much more critical than for a macro structure, even before having manufactured them. This fact is highlighted in Fig. 20, where the (theoretical) optimized designs obtained through our approach are represented for different width values of the gap phase. It is observed how some mode coefficients rise with the introduction of a gap-phase. Clearly, that makes us conclude that the gap-phase must be somehow included in the problem formulation rather than imposing it over the optimal structure at a later stage. Basically, the idea is to enforce a new-material phase (the gap) of prescribed width just in the interfaces of areas of opposite polarity. As this new phase is imposed

15 Optimal Design of Piezoelectric Modal Transducers 0 x Fig. 21 Electrode pattern that isolates the 1st mode among the first 14 modes in a plate clamped at its both extreme sides; (top) the gap is imposed a posteriori; (below) the gap in orange is considered in the formulation. This example has been extracted from [17] 50 μm GAP Fig. 19 (Top) first bending mode for a micro plate of 680 lm in length and 200 lm in width clamped at its both extreme sides. (Middle) electrode profile that isolates the first mode from the first 14 out-of-plane modes. (Bottom) manufactured design with a gap phase of width w gap ¼ 5 lm. This example has been extracted from [57] Normalized coefficient, F k Mode number w gap= 0 w gap= 2 w gap= 5 w = 10 Fig. 20 Mode sensitivity in Fig. 18 (middle) when a gap-phase of width 0, 2, 5 and 10 lm is imposed a posteriori. This picture has been extracted from [17] precisely where the gradient norm of the design field is high (theoretically infinite), the new interpolation function has to incorporate somehow a filtered underlying gradient norm [17]. gap Motivated also for manufacturing reasons, it is also advisable to impose an outside frame of gap-phase in the designs, typically of the same width as the one imposed inside. Figure 21 shows the optimal structure with the gapphase (now in gray) imposed at a later stage (top) and the one obtained with the formulation that includes both the inside and outside gap-phase in orange (below). Despite both designs showing quite similar layouts, it is pointed out in Fig. 22 that small variations in the geometry imply really significative changes in the response of the other coefficients. Fortunately, again, the optimization process has succeeded in canceling practically the influence of the rest of the modes. It is important to notice that the gap issue should be taken into account not only for designing piezo modal transducers, but also piezo transducers in general presenting regions with opposite polarities in contact. In principle, these results could be improved if we do not restrict our attention to predetermined host structures, and contemplate this as an additional optimization variable. In this way, the host structure is no longer fixed, and therefore both the material layout of the host structure and the electrode profile are simultaneously optimized. This new problem is studied in the next section. 7 Design of Modal Transducers via Simultaneous Optimization of Structure and Electrode As we have commented above, motivated by many applications, particularly in microelectronics, finding the best as possible transducer is a must. In the previous subsection we have seen a general systematic method, adapted to many useful situations, for obtaining optimal transducers of a prescribed shape. We could go even further if the host

16 D. Ruiz et al. Normalized coefficient, F k Mode number zero gap phase gap phase not optimized gap phase optimized Fig. 22 Normalized amplitude for the first 14 modes (refer to Fig. 21). This picture has been extracted from [17] structure comes into the optimization play together with the polarization profile. This is the propose of this subsection. We will study the problem of designing modal transducers that maximize the charge associated to a mode shape while filtering the rest. The motivation to study this problem is twofold: first, because it is of interest in applications, specially in microelectronics, and second because there are serious difficulties present in it that are worst dealing with. The main novelty, and difficulty, here is that both objective function and constraints will depend on the eigenmodes. As their shape and order will change during the optimization process because of the host structure is no longer fixed, they will appear the classical difficulties involved when dealing with eigenfrequencies and eigenmodes such as spurious modes, mode tracking and switching or repeated eigenfrequencies. Beyond this remarkable difficulties, we need to compute derivatives of eigenmodes corresponding to multiple eigenfrequences, and, as far as the authors may claim, this has not been done before for practical problems. This problem was studied in [53] and [51] and this section relies very much of those references. We start by reviewing related literature. Just few papers dealing with eigenmodes optimization appear in the literature. Next those are briefly discussed. In [22], a specific (single) mode shape of a fiber laser package is designed in order to minimize the elongation of the fiber under dynamic excitation. In [43], a multi-objective function is formulated in order to find optimal configurations that simultaneously satisfy (simple) eigenfrequency, eigenmode, and stiffness requirements at certain points of a vibrating structure. A similar problem is treated in [45]. The novelty there is to include the electromechanical coupling coefficient in the objective function so that the energy conversion is maximized for a specific mode. In [74], eigenmodes appear in the constraints only. One of the objectives of that work is to determine the material distribution of a structure that maximizes the fundamental frequency and at the same time synthesize the first two modes. However, the list is more extensive when talking about eigenfrequencies optimization. The readers are referred to [12, 18, 25, 42], among many others. In our new model, we are interested in mode shapes rather than frequencies in the spectrum. For that reason we are not numbering modes according to the position in the spectrum of the corresponding frequencies. As it is very well known, this may be unpractical when working with this kind of problems due to mode switching [42]. Indeed, what we do is, for a given design structure v s, and among the modes for this structure, to select the closest to a prescribed mode shape of the homogeneous square plate. Of course the term closest needs to be made precise and concrete. We will do this in the next section when dealing with the discrete problem. As it is mentioned above, apart from the classical difficulties of the topology optimization problems, we have to deal with the well-known ones of working with eigenproblems. The first one is concerning the modelling of the stiffness and mass of the structure. A suitable interpolation is required in order to avoid spurious modes. Mode switching, that is, change in the order of the vibration modes, is another issue that will happen in all examples. Also we will find repeated eigenvalues during the optimization process. In this problem we are interested in specific mode shapes, but the basis of eigenvectors associated to repeated eigenvalues is not unique. We have to look for the basis of eigenvectors that are more similar to the reference (eigenvectors of the previous iteration). Also a method for the computation of the derivatives of eigenvectors associated to a repeated eigenvalue is needed. All these issues and the way to overcome them are widely commented below. Other new issues are the appearance of gray areas and the normalization of eigenvectors. Concerning the first one, the interpolation scheme introduced in the static case is not the best choice because this one cannot stop the appearance of these areas. With respect to the normalization of eigenvectors, the cost depends explicitly of an eigenvector, then the normalization chosen has a big influence in the optimization process. 7.1 Continuous Formulation We consider a fixed domain in which the host structure will be contained. The formulation of the problem is similar to the fixed host structure case analyzed in Sect. 6, but, since we consider the host structure as varying and part of the optimization problem, there are essential differences. We start by define a new optimization variable v s 2f0; 1g, in the usual way: v s ðxþ ¼1 if there is structure in the point x 2 X, and v s ðxþ ¼0 if there is void. As the host structure

17 Optimal Design of Piezoelectric Modal Transducers are no longer fixed modes will change during the optimization problem depending on v s, so that eigenmode equation (17) becomes Aðv s ; U j Þ¼0; ð18þ and modes depend on v s, and so coefficients F j do, F j ¼ F j v p ; U j ðv s Þ : The physics of the problem is given through the eigenmode equation (18), and it could be an in-plane or out-of-plane plate problem, or even a shell problem, as mentioned in the previous section. We will made it precise in the examples below. In order to state the optimization problem we start by selecting J mode shapes, and for the sake of simplicity we number these eigenmodes and their corresponding eigenfrequencies increasingly from 1 to J, dropping the rest in (15) (i.e., we truncate the sum (15) atj ¼ J). Then the design problem aims to find a modal sensor that filters the kth mode (i.e. the coefficient F k is maximized) among the set of J modes. This is mathematically formulated as: max : F k v p ; U k ðv s Þ v s ;v p subject to Aðv s ; U j Þ¼0 jf j ðv p ; U j ðv s ÞÞj a; j ¼ 1;...; J; j 6¼ k v s 2f0; 1g v p 2f 1; 1g We include a small parameter a in order to avoid the too restrictive constraint of requiring F j ¼ 0. Since the election of this parameter is a delicate issue, and clearly a bad choice of a could imply a lack of solutions for the problem, we have opted here for using a bound-type formulation to reformulate the optimization problem. This basically implies to consider a now as an extra non-negative variable, rather than an input parameter. In this way, we maximize F k a so that the response to the kth mode is maximized whereas a is decreased, minimizing then the response of the rest of modes. Taking this into account and dropping the design variables in the coefficients for the sake of clarity, the new (bound) formulation is the following: max v s ;v p ;a : F k a subject to Aðv s ; U j Þ¼0 jf j j a; j ¼ 1;...; J; j 6¼ k v s 2f0; 1g v p 2f 1; 1g a 0 The variable a in our problem takes a very small value in all examples, as we will see below. Notice that this variable is introduced to suppress as much as possible the amplitudes of the rest of the modes. Of course, in case it could take bigger values, a very good idea would be normalize it as the rest of design variables. It is also important to note the absence of a volume constraint on v s. This constraint in not necessary mathematically for the problem to be non trivial, and further it does not add anything meaningful in the model from a physical point of view. 7.2 Discrete Formulation As usual, we replace the design variables v s and v p, taking values on f0; 1g, by their continuous versions, q s and q p, taking values on [0, 1]. Once the design variables have been relaxed, we include in the model a penalization according to the SIMP methodology. However, the way that q e s appears in the stiffness term is a delicate issue when working in eigenproblems optimization. Often a straightforward use of the SIMP method to penalize it leads to the appearance of the so-called localized or spurious modes. The point is that stiffness term basically depends on the power p on the element density ðq e s Þp (typically p ¼ 3), and the mass term is just linearly dependent on the element density q e s.asqe s tends to zero, the ratio of the stiffness and the mass goes towards zero too, which means that very low eigenfrequencies may appear in low density regions, spoiling the frequency analysis. To eliminate these artificial modes, the use of tailored interpolation functions for both stiffness and mass terms are required. The first idea to overcome this problem was proposed in [47]. It consists first, in linearising the stiffness part for low densities (below a threshold value) and second, no taking into account the nodes surrounded by these elements, when computing the eigenvectors. Other options are the approach of [73], which consists in setting the mass term to 0 in these low density regions, or the RAMP interpolation [70] which has a non-zero gradient for null density values. This is precisely the approach used here to alleviate problems concerning spurious modes. In such a case, the material stiffness, E s v s, is replaced wherever appears by the interpolation function q e s Eðq e s Þ¼E min þ 1 þ qð1 q e s Þ E s; 0 q e s 1 where E min is a very small Young s modulus used to avoid singularities, E s is the Young s modulus of the host structure and q is a penalization factor. Concerning the electrode polarity, for convenience first we change the term

18 D. Ruiz et al. q p by 2q p 1, thus the polarity variable is now taking values in [0, 1]. Next difficulty to deal with is the appearance of large gray areas. The nature of the problem is prone to these, since more flexible areas generate a bigger electric charge than the solid ones. In order to avoid this problem, the interpolation scheme introduced for the static problem through function R (Eq. (11) and Fig. 7), 8 f >< Rðq e s Þ¼ g qe s q e s 2½0; g ð1 fþq >: e s þ f g q e s 2ðg; 1 1 g where g and f are tuning parameters, penalizes in a progressive way the charge generated in gray areas. This function worked correctly in the static case, removing these gray areas, but in this case, its lack of derivability at g has a negative effect in the numerical results, spoiling symmetry. Because of this reason, we have replaced this scheme by a smoothed version of it, namely Rðq e s Þ¼e cð1 qe s Þ ð1 q e s Þe c ð19þ where c is another tuning parameter. This expression clearly reminds us the erode operator introduced in [62] used to control the feature sizes in void regions. We can see R with different values of c in Fig. 23. This interpolation is introduced in the expression of the charge collected for a vibration mode F j ¼ Y T U j where Y ¼ Rðq s Þð2q p 1ÞB and B is the usual FE strain-displacement matrix that will adopt an expression or another one depending on each case (in-plane or out-of-plane plate model). U j is the jth global eigenvector obtained from the equations ðk l j MÞU j ¼ 0; j ¼ 1;...; J and U T j MU j ¼ 1; j ¼ 1;...; J where l j is the jth eigenvalue, (i.e. the square of the jth eigenfrequency), K and M are respectively, the global stiffness and mass matrix. Both of them are symmetric and positive definite, then the eigenvalues of our problem are real and positive. The last step before writing the complete problem formulation is the choice of the vibration modes. We want to optimize the charge collected in one of them while minimizing the rest. We are interested in mode shapes, and 1 R γ =0 γ =1 γ =3 γ =5 0 ρ e 1 s Fig. 23 Interpolation scheme R proposed in (19) for different values of c during the optimization process, the shape of the structure (q s ) will change, then the modes shapes could change its order, so we need a method to ensure that we are always maximizing and canceling the same mode shapes during the optimization. For reference propose, we start by choosing J desired mode shapes of the homogeneous square plate, that we number increasingly from 1 to J; U 1 ;...; U J. The method chosen for finding the mode shapes closest to those reference ones, U 1 ;...; U J, is the MAC ( Modal Assurance Criterion ) [33, 74]. MAC uses the next correlation criterion for vectors: ðu T WÞ 2 MACðU; WÞ ¼ ðu T UÞðW T WÞ where the value of MAC is always between 0 and 1 and equals to 1 if only if U ¼ W. Given a design q s and once we have computed the modes of the structure given by fu l g L l¼1 (with L [ J large enough), we select U j ; j ¼ 1;...; J as the solution of the discrete optimization problem max : MAC U j ; U l 1 l L The eigenvectors corresponding to the L lowest eigenvalues are computed by using the subspace iteration method [2]. The eigenvectors calculated by this method are orthonormal with respect to the mass matrix, but in our problem this is a difficulty, due to the fact that this matrix, since it depends on the design, and furthermore the coefficients F j depends linearly on the length of the eigenvectors. Due to this, comparison of cost and constraints of one iteration with the next in which the design has changed, and so the mass matrix, may cause convergence troubles of the numerical algorithm. We have fixed this issue by forcing eigenvectors to have unit norm, simply dividing each one by its norm, so that all eigenvectors belong to the

19 Optimal Design of Piezoelectric Modal Transducers unit sphere. Then, by doing this to the set of eigenvectors obtained by the subspace iteration method, we have a set of unitary (with respect to the identity matrix) eigenvectors, but orthogonal with respect to the mass matrix. From a mathematical point of view, we have continuous and bounded cost and constraints, and so, existence of solution for the discrete problem is guaranteed. This normalization has another advantage, in the case of using MAC as a constraint in the problem (e.g. for mode tracking [33]), the computation of its sensitivity with respect to q s is much easier than using the normalization with the mass matrix [74]. With the unitary normalization the expression for the MAC would just be MACðU; WÞ ¼ðU T WÞ 2 since U T U ¼ W T W ¼ 1. The discrete problem formulation written in the usual topology optimization format would be max : F k a q s ;q p ;a subject to ðk l j MÞU j ¼ 0; j ¼ 1;...; J U T j MU l ¼ 0; j; l ¼ 1;...; J; j 6¼ l U T j U j ¼ 1; j ¼ 1...; J jf j j a; j ¼ 1;...; J; j 6¼ k q s 2½0; 1 q p 2½0; 1 a 0 It is important to remark that working with the normalization respect to the mass matrix is also possible, but our numerical algorithm works better with the normalization respect to the identity matrix. The normalization used has to be taken into account when computing the sensitivities. 7.3 Computation of Sensitivities The computations of eigenmodes derivatives might be a difficult issue that have been dealt with in the literature before [19, 37, 40, 76] as an academic problem, but, as far as the authors may claim, never for a real problem as complex as the one considered here. In this section is devoted to the detailed computation of derivatives of the coefficients F j with respect to the design variables. These derivatives will be necessary for the numerical method that we are going to implement to simulate the problem. For a given element of the discretization e, we have to compute the derivative of F j with respect to q e s and qe p. Derivative with respect to q e p is trivial since this variable only appears in the coefficient F j, not in the eigenproblem equations, and it does linearly. We remind the expression of F j : F j ¼ Y T U j ¼ Rðq s Þð2q p 1ÞBU j ð20þ The derivative of the expression (20) with respect to q e p is of j ¼ 2Rðq s ÞBU j oq e p As we commented above, the computation of the sensitivity with respect to the density variable q e s is more difficult, since the eigenproblem depends on this variable. Differentiating the expression (20) with respect to the density variable q e s we arrive at of j oq e ¼ oyt s oq e U j þ Y T ou j s oq e s The computation of this sensitivity leads us to calculate the derivative of the associated eigenvectors. Recall that vector Y depends on both variables q e s and qe p. Indeed oy oq e ¼ or s oq e ðq s Þð2q p 1ÞB s It is very well-known that in order to compute derivatives of eigenvectors, multiplicity of the corresponding eigenvalues is crucial, and so we will distinguish two cases according to whether this eigenvalue is simple or multiple Simple eigenfrequencies Before starting to calculate sensitivities of the eigenvectors, we recall that we are considering a basis of eigenvectors, i.e. solutions of the eigenproblem ðk l j MÞU j ¼ 0; j; l ¼ 1;...; J; j 6¼ l ð21þ which is M-orthogonal and unitary with respect to the identity matrix, that is to say, eigenvectors satisfy U T j MU l ¼ 0; j; l ¼ 1;...; J; j 6¼ l U T j U j ¼ 1; j ¼ 1;...; J ð22þ From now on, prime stands for derivatives with respect to material densities ðþ 0 ¼ oðþ oq Differentiating (21) and the s. e equation in (22) we arrive at the system ( ðk l j MÞU 0 j ¼ ðk0 l j M 0 ÞU j þ l 0 j MU j U T j U0 j ¼ 0 ð23þ The derivative of its eigenvector, U 0 j, with respect to an element material density is the solution of this problem. Equation (23) is a linear system of N unknowns and N þ 1 equations. Matrix ðk l j MÞ is singular as it has a redundant row that has to be substituted by last line in (23)

20 D. Ruiz et al. in order to compute U 0 j. The best choice to solve this system seems to be Nelson s method (see [37, 46] and the references therein). This method is powerful for computing the eigenvector derivatives with single eigenvalues, since it requires only the knowledge of the pair ðl j ; U j Þ. The disadvantage is the computation time because we have to solve the system in (23) once for each design variable q e s. One solution is proposed in [73]. If we look carefully, the derivative that we really need is the derivative of the coefficients F j, that is a scalar, so now we will calculate the derivative of this scalar instead of the derivative of the eigenvector. We consider the augmented function c ¼ Y T U j P T j ðk l jmþu j k j ðu T j U j 1Þ ð24þ where P T j and k j are the vector and scalar Lagrange s multipliers associated to the jth mode. Indeed c ¼ Y T U j. Now we differentiate (24) with respect to the material density c 0 ¼ðY 0 Þ T U j þ Y T U 0 j ðp 0 jþ T ðk l j MÞU j P T j ðk0 l j M 0 ÞU j P T j ðk l jmþu 0 j þ l 0 j PT j MU j k 0 j ðut j U j 1Þþ2k j U T j U0 j taking into account that ðk l j MÞU j ¼ 0 and U T j U j 1 ¼ 0 and rearranging terms c 0 ¼ðY 0 Þ T U j P T j ðk0 l j M 0 ÞU j þ Y T P T j ðk l jmþ 2k j U T j þ l 0 j PT j MU j U 0 j Now, choosing the pair ðp j ; k j Þ, if possible, such that Y T P T j ðk l jmþ 2k j U T j ¼ 0 P T j MU j ¼ 0 we arrive at c 0 ¼ðY 0 Þ T U j P T j ðk0 l j M 0 ÞU j ð25þ Now we have to calculate the value of the scalar and vector multipliers. We multiply the first line in (25) byu j Y T U j P T j ðk l jmþu j 2k j U T j U j ¼ 0 ð26þ taking into account that the first term in (26) is the expression for the coefficient F j, the second is the eigenproblem equation and the last one is the eigenvector normalization, then k j ¼ 1 2 F j Introducing this last equation in (25) we get the equation system to calculate the adjoint state P j ðk l j MÞP j ¼ Y F j U j U T j MP j ¼ 0 ð27þ Now we have to justify that this last system admits a solution P j, and therefore there exists a pair ðp j ; k j Þ satisfying (25). Equation (27) is a system of N unknowns and N þ 1 equations, and last equation is used to replace one of the redundant equations of the first row of the system, and this justifies existence of solution for this system. For solving it we use Nelson s method. The advantage now is that the aforementioned technique is applied just once, rather than N times, for a given eigenvector, since the independent term in system (27) does not depend on the design variable. Following Nelson s method, the complete solution P j in (23) can be expressed in terms of a particular solution Q j and a homogeneous solution cu j P j ¼ Q j þ cu j ð28þ the particular solution Q j is found by solving the system by replacing the redundant equation (that we call ith), G j Q j ¼ f j ; where G j is the regular matrix obtained from ðk l j MÞ when zeroing out the ith row and column (that corresponds to the largest value of the components of U j ) and placing 1 in the ith position in the diagonal, and f j is the vector column Y F j U j with the ith element replaced by 0. The unknown coefficient c is obtained by substituting (28) in the second line of (27), obtaining c ¼ QT j MU j U T j MU j In summary, the complete process for the computation of the derivatives of the coefficient F j with respect to the material density is the following: 1. Solve the eigenproblem ðk l j MÞU j ¼ Ensure U T j U j ¼ 1 (by dividing the obtained eigenvectors by its norm). 3. Compute G j ðk l j MÞ and f j Y F j U j. 4. Find the largest element of U j. We call its position the ith. 5. Construct G j by zeroing out the ith column and row, and placing 1 in the ith diagonal element of G j. 6. Construct f j by zeroing out the ith element of f j. 7. Solve the system G j Q j ¼ f j. 8. Compute c ¼ QT j MU j U T j MU j 9. Compute P j ¼ Q j þ cu j

21 Optimal Design of Piezoelectric Modal Transducers 10. Compute for each element of the discretization c 0 ¼ðY 0 Þ T U j P T j ðk0 l j M 0 ÞU j Repeated eigenfrequencies Multiple eigenvalues may appear in different ways. The most obvious one is due to certain symmetry in the boundary conditions, like in a simply-supported plate (the displacements of edges in any direction are impeded, only rotation is allowed in these points). We can see in Fig. 24 the second and third out-ofplane mode shape for the boundary conditions commented above. As we can see in this example, the mode shapes are exactly the same, but turned on themselves 90, and consequently both corresponds to the same eigenfrequency. Another situation that could origin repeated eigenvalues is whenever mode switching happens and then two eigenvalues cross in the spectrum diagram, and in this case, at least during a few iterations, two eigenvalues can be considered as identical until they definitely separate into simple ones again or not. We will see examples of this pathology in the next section. As we will see in the next section, multiple eigenvalue issue happens in the most of the examples, so it is important the right computation of the sensitivities for this case. Nelson s method is only valid if rankðk l j MÞ¼N 1, and this only happens when l j is simple. Another difficulty in computing derivatives of eigenvectors corresponding to multiple eigenvalues is that there are infinitely many M- orthogonal basis associated to a multiple eigenvalue, and only for one of those we can compute the derivatives of the eigenvector forming that basis. As pointed out in [10], when a design variable is perturbed, the eigenvalue splits into m (being m the eigenvalue multiplicity) distinct eigenvalues. In order that eigenvector derivatives exist, the eigenvector basis has to be adjacent to the m distinct eigenvectors that appear when we perturb such a design variable (adjacent means here closest, as we will see below). If we do not restrict ourselves to that adjacent basis, eigenvectors of any other basis are discontinuous with respect to the design variable, and consequently nondifferentiable. It is important to remark that similar to the single eigenvalue problem, we do not need to calculate the derivatives of the eigenvalues previously, they will be calculated during the process. If these values are needed (for example in the case of the maximization of the first eigenvalue problem) there exist some directs methods that compute these values straightforward without computing eigenvectors derivatives (see for instance [59]). The method used in this work is the one developed in [10]. We assume that the multiplicity of the multiple eigenvalues is two, in fact, in our examples whenever multiple eigenfrequencies occurs they are double. We call Fig. 24 Out-of-plane mode shapes with the same frequency for a simply-supported plate. a 2nd mode shape and b 3rd mode shape the double eigenvalue l 1 ¼ l 2, and the M-orthogonal basis associated to l 1 (and provided by the subspace iteration method) U 1 ; U 2. For convenience we store this eigenvectors by columns in the matrix U ¼ðU 1 ; U 2 Þ. The first step is to compute the adjacent eigenvectors for which the derivatives can be calculated. We recall our eigenproblem, written now in matrix form, ðk KMÞU ¼ 0 U T U ¼ I where K is the eigenvalue matrix K ¼ l 1 0 ; l 1 ¼ l 2 : 0 l 2 The procedure to find the adjacent basis for which derivatives can be computed is the following. We consider two new vectors obtained from the eigenvector basis associated to l 1 through the expression. Any other M-orthogonal basis of eigenvectors associated to l 1 can be written as Z ¼ UC; where the columns of Z are the new eigenvectors and C is a orthonormal matrix ðc T C ¼ IÞ. Clearly the columns of Z are M-orthogonal. We impose now that Z satisfies the eigenvalue problem KZ ¼ MZK ð29þ In order to obtain the matrix C. we start differentiating (29) with respect to the material density variable K 0 Z þ KZ 0 ¼ M 0 ZK þ MZ 0 K þ MZK 0 ð30þ rearranging terms ðk l 1 MÞZ 0 ¼ðl 1 M 0 K 0 ÞZ þ MZK 0 ð31þ

22 D. Ruiz et al. and multiplying by U T U T ðk l 1 MÞZ 0 ¼ U T ðl 1 M 0 K 0 ÞZ þ U T MZK 0 taking into account that U T ðk l 1 MÞ¼0 and Z ¼ UC U T ðk 0 l 1 M 0 ÞUC ¼ U T MUCK 0 ð32þ we can rewrite (32) as DC ¼ CK 0 where ð33þ D ¼ UT ðk 0 l 1 M 0 ÞU U T MU where we understand that a matrix dividing is equivalent to multiplying (by the left) by its inverse and K is the diagonal matrix of the eigenvalue derivatives, that is K 0 ¼ l l 0 : 2 The problem in (33) is a small eigenproblem, whose dimension is equal to the multiplicity of the repeated eigenvalue, dimension two in our case. Eigenvalues for this this problem are the derivatives l 0 1 ; l0 2. It is important to remark that the value of the adjacent eigenvectors Z does not depend on the initial eigenvectors U. A word must be said now on eigenvalues derivatives for multiple eigenfrequencies. In the case we are dealing with a double eigenvalue, l 1 ¼ l 2, we have two values for their derivatives as a result of the double multiplicity. For the double eigenvalue l 1, the subgradient, i.e. the set of slopes of any tangent straight line to the graph of l 1, is a closed interval [9], the extreme points of such an interval are just l 0 1 and l0 2. In Fig. 25 we can see the evolution of both eigenvalues with respect to the element density. As pointed out in [19], Dailey s method breaks down when l 0 1 ¼ l0 2, and in that paper, a method was introduced for computing eigenvectors derivatives in such a case. If l 0 1 ¼ l0 2, then the subgradient of l 1 is a singleton, and that means that l 1 is differentiable [9], what can only happens if the equality l 1 ¼ l 2 occurs independently on the chosen variable. This does not happen in our case, not even when we consider symmetric boundary conditions, and thus Dailey s method is enough for our purposes. Once the adjacent eigenvectors have been calculated, next step is the computation of their derivatives. For this particular case of multiplicity two, rankðk l 1 MÞ¼ N 2, and Nelson s method does not work, we use Dailey s method instead. First, as we did in the single eigenvalue case, we write the derivative of the eigenvector in terms of a particular and a homogeneous solution, Z 0 ¼ V þ ZC; being V the solution of the system GV ¼ f; ð34þ where G is the matrix ðk l 1 MÞ obtained by zeroing out the two rows and columns (since the rank is N 2 we need to replace two equations instead of one) containing the largest elements, and setting both diagonal elements equal to 1. The vector f is the column vector obtained from ðl 1 M KÞZ þ MZK by zeroing out the same rows. Now we have to calculate the matrix C. We start differentiating the normalization equation Z T Z ¼ I ðz 0 Þ T Z þ Z T Z 0 ¼ 0 We introduce (34) in(35) ðv þ ZCÞ T Z þ Z T ðv þ ZCÞ ¼0 Operating we arrive at C þ C T ¼ V T Z Z T V Q ð35þ Next step is the computation of the values of the elements of matrix C. We start differentiating (30) K 00 Z þ K 0 Z 0 þ K 0 Z 0 þ KZ 00 M 00 ZK M 0 Z 0 K M 0 ZK 0 M 0 Z 0 K MZ 00 K MZ 0 K 0 M 0 ZK 0 MZ 0 K 0 MZK 00 ¼ 0 rearranging terms we get ðk 00 l 1 M 00 ÞZ þ 2ðK 0 l 1 M 0 ÞZ 0 þðk l 1 MÞZ 00 2M 0 ZK 0 2MZ 0 K 0 MZK 00 ¼ 0 Multiplying by Z T and taking into account that Z T ðk l 1 MÞ¼0 Fig. 25 Evolution of the eigenvalue versus a density variable

23 Optimal Design of Piezoelectric Modal Transducers Z T ðk 00 l 1 M 00 ÞZ þ 2Z T ðk 0 l 1 M 0 ÞZ 0 2Z T M 0 ZK 0 2Z T MZ 0 K 0 Z T MZK 00 ¼ 0 We introduce (34) into this last expression Z T ðk 00 l 1 M 00 ÞZ þ 2Z T ðk 0 l 1 M 0 ÞðV þ ZCÞ 2Z T M 0 ZK 0 2Z T MðV þ ZCÞK 0 Z T MZK 00 ¼ 0 Now multiplying (31) byz T we get Z T ðk 0 l j M 0 ÞZ ¼ Z T MZK 0 ð36þ Introducing this last expression in (36) and rearranging terms ðz T MZÞCK 0 ðz T MZÞK 0 C þ 1 2 ðzt MZÞK 00 ¼ Z T ðk 0 l 1 M 0 ÞV Z T ðm 0 Z þ MVÞK 0 þ 1 2 ZT ðk 00 l 1 M 00 ÞZ Last step is premultiplying by ðz T MZÞ 1 on both sides of the equation CK 0 K 0 C þ 0:5K 00 ¼ðZ T MZÞ 1 Z T ðk 0 l j M 0 ÞV Z T ðm 0 Z þ MVÞK 0 þ 0:5Z T ðk 00 l j M 00 ÞZ R It is important to remark that K 00 is diagonal, whereas CK 0 K 0 C always has zeros on the diagonal. This provides a neat separation of C and K 00. Finally C can be built as q 11 =2 r 12 =ðl 0 2 C ¼ l0 1 Þ r 21 =ðl 0 1 l0 2 Þ q 22=2 where Q ¼½q ij and R ¼½r ij. For more details in these calculations for computing the derivatives of Z 0 we refer the interested readers to [10]. With Dailey s method we can obtain the derivatives of the adjacent eigenvectors, Z 0, but we are interested in following mode shapes, and therefore we cannot replace U by Z in the problem formulation. We need an expression for U 0 in order to update design variables in our numerical algorithm. As pointed out by [10] there is only one basis of eigenvectors associated to repeated eigenvalues, that can be differentiated, we call this basis Z. For the computation of U 0, we proceed heuristically in the following way. First, we notice that for our fixed design variable q e s, the matrix C is constant and U ¼ ZC T the derivative of the initial eigenvector U is computed as U 0 ¼ Z 0 C T where U 0 is a matrix that stores in its columns the derivatives of the eigenvectors. This is the way to construct the derivatives of the desired mode shapes with respect to the design variable, understanding that it is not its gradient with respect to design variables but the partial derivatives matrix due to lack of differentiability in general [19]. In summary, we will show the complete process for the computation of the derivatives with respect to the material density for eigenvectors associated to repeated eigenvalues: Computation of mode shapes U 1. Solve the eigenproblem ðk l j MÞU j ¼ Ensure U T U ¼ I. Computation of adjacent eigenvectors 1. Compute D ¼ UT ðk 0 l j M 0 ÞUÞ. U T MU 2. Solve the eigenproblem DC ¼ CK Compute Z ¼ UC. Computation of the derivatives of Z 1. Compute G ¼ðK lj MÞ and f ¼ MZK 0 ðk 0 l 1 M 0 ÞZ. 2. Find the two largest elements of the matrix U, we call this positions ith and kth. 3. Construct G by zeroing out the ith and kth columns and rows, and placing 1 in the ith and kth diagonal elements of G. 4. Construct f by zeroing out the ith and the kth elements of f. 5. Solve the system GV ¼ f. 6. Compute Q ¼ V T Z Z T V. 7. Compute R ¼ðZ T MZÞ 1 Z T ðk 0 l 1 M 0 ÞV Z T ðm 0 Z þ MVÞK 0 þ 0:5Z T ðk 00 l 1 M 00 ÞZÞ. q 8. Construct C ¼ 11 =2 r 12 =ðl 0 2 l0 1 Þ r 21 =ðl 0 1 l0 2 Þ q. 22=2 9. Compute Z 0 ¼ V þ ZC. Computation of the derivatives of eigenvectors U 1. Compute U 0 ¼ Z T C Compute Fj 0 ¼ðY0 Þ T U j þ Y T U 0 j: This method allows us to compute the (partial) derivative of the eigenvectors with respect to a design variable q e s. We have to repeat the complete process once per

24 D. Ruiz et al. discretization element. In this case, we cannot save computation time as we did in the previous one because all parameters depend on the design variable. This fact makes the process very slow. 7.4 Numerical Approach In this section we sketch the numerical algorithm implemented in order to simulate our optimization problem. As we commented in Sect. 7.2, we have chosen a square-shape plate and we discretize it by finite elements. We have considered plane rectangular bilinear elements (8 degrees of freedom per element) for the in-plane case, and rectangular Kirchhoff plate elements (12 degrees of freedom per element) for the out-of-plane case. The optimization algorithm flow is the following: 1. Choose J mode shapes of the homogeneous square plate with the same boundary conditions than the one of our problem. The eigenvectors chosen fu j g J j¼1 will be our reference in the first iteration step. 2. Initialize the design variable q ¼ðq s ; q p Þ. The initial value of this variables are the same for all elements, q e s ¼ 1 and qe p ¼ 0:5. We initialize our iteration counter i ¼ Compute L modal shapes (L [ J, large enough) for the plate to be optimized. n U ðiþ j o L l¼1 4. By means of MAC, identify the J closest modes to the ones of reference, among the set of previously computed L modes. Relabel the sequence again from 1toJ, n U ðiþ j o J j¼1 5. Check the multiplicity of the eigenvalues. In order to do this, we check whether the averaged distance between two consecutive eigenvalues is greater or not than a certain tolerance, k l jþ1 l j l k s. In our examples j we choose s ¼ for such a value, and it is updated dividing its value by 1.5 each 10 iterations, until it reaches the minimal value of For the jth eigenvalue we have two options: it is simple: then compute the coefficient F ðiþ j and its derivatives with respect to both variables. The derivative with respect to the material density is computed by the Nelson s method, explained in Sect it is multiple: we assume the jth eigenvalue multiplicity to be two (in all simulations with repeated eigenvalues this is the multiplicity that we have found) and for simplicity in the exposition that it coincides with the ðj þ 1Þth eigenvalue. Then we proceed in the following way: Given a M-orthogonal basis of eigenvectors (in our case, the one given by the subspace iteration method), fw ðiþ j ; W ðiþ jþ1g, we find a new basis of eigenvectors, f ~U ðiþ j ; ~U ðiþ jþ1g that follows the previous mode shapes as ~U ðiþ j ; ~U ðiþ jþ1 ¼ W ðiþ j ; W ðiþ H jþ1 where H is a orthogonal matrix solution of the finite dimensional optimization problem U ði 1Þ j ; U ði 1Þ jþ1 M ðiþ ~U ðiþ j ; ~U ðiþ jþ1 max fh:hh T ¼Ig Remark that ~U ðiþ j Normalize U ~ ðiþ j ; U ~ ðiþ jþ1 as U ðiþ j ¼ 1 k ~U ðiþ k j and ~U ðiþ jþ1 are M-orthogonal. ~U ðiþ j ; U ðiþ jþ1 ¼ 1 ~U ðiþ k ~U ðiþ jþ1 k jþ1 Compute the coefficients F ðiþ j and F ðiþ jþ1 using these eigenvectors. Calculate the new eigenvectors Z ðiþ ¼ U ðiþ C as commented in Sect. 7.3, being U ðiþ ¼ U ðiþ j ; U ðiþ jþ1 Get their derivatives Z ðiþ 0 by using the Dailey s method. This is we have been called the adjacent basis of the jth eigenvector, as commented in Sect Find the required derivatives as 0C ðu ðiþ Þ 0 ¼ Z ðiþ T Compute the derivative of the coefficients 0 and F ðiþ 0 F ðiþ j jþ1 6. Update design variables by using MMA. 7. Until convergence, go back to step 3, taking fu ðiþ j g J j¼1 as the new reference. The part of the algorithm related with multiple eigenvalues is delicate since there are an infinite number of basis and only for one of them the derivatives can be computed. As we commented above, the part concerning the derivatives

25 Optimal Design of Piezoelectric Modal Transducers of eigenvectors associated to single eigenvalues is much faster than the one related with repeated eigenvalues, since we cannot apply adjoint method in the current case. The values of the tunable parameters work out in almost all cases, nevertheless, as we commented above, this problem in prone to local maxima issue. In some specific cases these values must be changed in order to be adapted to the particular situation. It is important to remark that in order to avoid checkerboard and mesh-dependence issues, a standard density filter has been used. Further, we have used a continuation approach over the radius filter. Once the derivatives have been computed, and the algorithm has been presented, we can see the examples corresponding to different boundary conditions and for inplane and out-of-plane cases. 7.5 Numerical Examples Next we illustrate our approach with several examples corresponding to different boundary conditions. Common to all of them is the design domain, a square plate, and the number of modes to be considered, that is, J ¼ 4, for both in-plane and out-of-plane cases. Due to possible mode switching we have to compute more than 4 modes since it might happen that any of the modes from the 5th on could switch to the forth, for instance. In all examples that we show here we compute the first 8 modes, and that is enough. Then, according to the previous notation we fix L ¼ 8. The mesh used is 50 by 50, which means N e ¼ 2500 elements. From now on, wherever appears the expression jth mode shape really means mode shape similar to the initial jth mode shape. We say that because when tracking a specific mode, it is more than likely to change its order in the spectrum due to mode switching. Figure 26 shows a flowchart with the complete algorithm process Plate Clamped at Its Left Edge For the first example, the plate is clamped at its left side. The vibration modes considered first are in-plane. In these examples we want to maximize the charge generated by the first mode shape of the homogeneous square plate, while the charge collected for the 2nd, 3rd and 4th modes shapes is as small as possible. The mode shapes of the square homogeneous plate is our reference for the first iteration. We show in Fig. 27 the optimized design for this example. In Fig. 27a we can see the material density, black means solid material, white means void. Almost all the design is black and white, and only insignificant gray pixels appear. In Fig. 27b we can see the electrode polarity, red and blue mean positive and negative polarity, in fact, the color that represents each polarity is not important because the polarity changes with the frequency of the alternating current. It is important to point out that the whole structure is being covered by electrode, and then, the whole structure is electrically affected. This variable always takes extreme values as we commented above. Actually, the electrode profile contains all the information needed to understand results. Hence, hereafter this will be the figure to be shown instead of both, in which we only show the polarity density q p. We remember that there is no structure neither piezo where there is no electrode. When working with vibration modes, symmetry must be present in the designs. We can see that our first example is practically symmetric. Mode switching appears in this example. The first mode shape does not change its position, and it is always the first vibration mode, nevertheless, the second and the third mode shapes change its order one with the other, in fact, during a few iterations the frequencies are almost equal. Attention should be paid also to the fourth mode since at the end of the process its frequency is very near to the one of the fifth mode, although they do not change the order. We can see mode switching also in higher modes, but it does not affect since we are interested only in the first four. In Fig. 28 we can see the evolution of the frequencies with the iterations. This example justifies the use of our method (that relies on Dailey s method) to correctly compute the derivatives in the points where the frequencies are repeated. In Fig. 29 we can see the evolution of the order of the four chosen mode shapes with the iterations. Mode switching is represented in Figs. 28 and 29 with small circles. This means that our third and second mode shapes suppressed are the most similar to the second and the third ones of the homogeneous plate, respectively. This fact can also be corroborated just checking that the MAC between these modes and the reference are really close to 1. At this point we want to remark the importance of the value of the tolerance s. Numerically two repeated eigenvalues will never be exactly the same (with the exception of those due to symmetric boundary conditions), then we have to allow a small tolerance. If this value is very small, the algorithm will consider single two eigenvalues when they are similar enough to be considered repeated; on the other hand, if this value is big, we are making a mistake considering two eigenvalues repeated when they are different enough. Both cases may cause the algorithm to break down, so this tolerance might be a critical tuning parameter, although we have used it here in a fixed manner (explained in the previous section). In Fig. 30 we can see four designs. It is noticed that the void region (white color) is in general small, and it will be empty in some examples. For each example we add the percentage gain (defined as the ratio of the normalized

26 D. Ruiz et al. Fig. 26 Flowchart of the process coefficient maximized under simultaneous optimization to the one under single optimization minus 1, and then altogether multiply by 100). We illustrate in Fig. 31 the correspondence between our four mode shapes optimized and the first four mode shapes of the homogeneous plate. The same analysis can be done whether the plate is moving now out-of-plate. Electrodes profiles for such situations are depicted in Fig Plate clamped at its left and right edges Now the plate is clamped at its left and right edges. In Figs. 33 and 34 we show the optimized designs for in-plane and out-of-plane cases. Again, the multiple eigenvalues issue is present in most of the examples. When the void region is null (Fig. 34d), there is no gain, because the optimized design is exactly the

27 Optimal Design of Piezoelectric Modal Transducers Fig. 27 Optimized designs that isolate the first mode shape when considering the first 4 modes in-plane. a Material density and b electrode polarity. (Color figure online) Fig. 29 Evolution of the order of the vibration modes with the iterations homogeneous plate with the polarity variable (q p ) optimized. For the sake of brevity only the optimized designs have been included, omitting graphs with the evolution of cost, frequencies or switching of the modes Plate Clamped at All Four Edges For the last example the plate is clamped at all four edges. Owing to these boundary conditions, it is well-known that the first eigenvalue for a homogeneous square plate moving in-plane is double, so the basis of eigenvectors that is our reference for the first iteration is not unique. We have taken as reference the mode shapes with horizontal and vertical nodal lines for the first and the second mode shapes, respectively, that are the mode shapes of an almost square plate (whose first and second eigenfrequencies are single). Once we have the reference we can start to maximize the charge collected, and after a few iterations both frequencies become simple due to small void regions that favors the movement of the mode shape that we are optimizing. In Fig. 35 we can see the optimized designs for in-plane movement. The electrode profiles that isolate first and second mode shape, make perfect sense. We have finished with optimized designs whose nodal lines have the same orientation than the ones of the reference. As we expected, the design for the second mode shape is the same as the first one but rotated 90. In both cases the gain is the same. If we look carefully the optimized designs for the third and fourth mode shapes, is very easy to see that the first two frequencies are always the same during all the optimization process, in other words, the first two modes are always double. In Fig. 36 the evolution of the frequencies with respect to the iterations is shown. We can see in Fig. 37 the optimized designs for the outof-plane case. The first couple of repeated eigenvalues is the one formed by the second and the third modes. As happened with the in-plane case, the couple is always double when we isolate one of the single modes (in this case the first or the fourth). This is shown in Fig. 38. The optimized electrodes of the second and third modes are the same but again rotated 90, as expected. 7.6 Results and Conclusions Fig. 28 Evolution of the frequencies with the iterations In this section, we will show the results that have been obtained with the method proposed. The objective has been

28 D. Ruiz et al. Fig. 30 Electrode profile that isolate the first mode shape (a), the second (b), the third (c) and the fourth (d) when considering the first 4 modes in-plane for a left-side clamped plate Fig. 31 Left the first (a), the second (b), the third (c) and the fourth (d) in-plane mode shapes for a homogeneous square plate clamped at its left side; right the fourth in-plane mode shape optimized closest to the previous ones the design of piezoelectric modal transducers by optimizing simultaneously the structure and the electrode layout. In order to show the results, we represent the optimized collected charge for the first example (left-clamped plate with in-plane motion) with a bar graph (Fig. 39). It corresponds with Fig. 30. It is important to remark that each coefficient F j is normalized with the same coefficient obtained for the homogeneous plate. In all cases we show the percentage gain. Black bar is the normalized F j coefficient for the homogeneous plate, gray bar is the same coefficient obtained with the method proposed. As we can see, in these four cases we are able to improve the charge that we collect when we optimize at the same time both variables, structure and electrode. We check that the larger the void the better optimization wins. As we expected, if this void area is small, the designs do not change enough with respect to the homogeneous plate, so this charge will be practically the same. In the four cases presented (isolating the modes one by one) the charge of the suppressed modes is almost zero. Since the case of optimizing the polarity of the electrode for the homogeneous plate is a linear problem, we can get exactly 0 in the suppressed modes, then we will not see any black bar in these modes. Our problem is non-linear, then these values, in general, will not be exactly 0.

29 Optimal Design of Piezoelectric Modal Transducers Fig. 32 Electrode profile that isolate the first mode shape (a), the second (b), the third (c) and the fourth (d) when considering the first 4 modes out-of-plane for a left-side clamped plate Fig. 33 Electrode profile that isolate the first mode shape (a), the second (b), the third (c) and the fourth (d) when considering the first 4 modes in-plane for a left and right-side clamped plate We have shown the first example with a bar graph, but for sake the of simplicity, the rest of the results are shown in Tables 1, 2 and 3 for the different boundary conditions. The results are shown as the gain in percent with respect to the optimization of the electrode polarity only for the homogeneous plate, as a consequence make better the performance of such piezo devices. As we can see we can conclude that we have improved the charge collected in the desired vibration mode, while suppressing the rest. 8 Final Comments and Future Research Lines At the present time piezoelectricity continues being a vibrating research area without a doubt and, in our opinion, is one the most cross-cutting research fields. In particular, now that tiny is at the cutting edge of technology, it is more and more common to find piezoelectric devices in applications taking place at the micro scale. A good example of this occurs in bio-engineering, where modal transducers have shown quite good performance as biosensors. The

30 D. Ruiz et al. Fig. 34 Electrode profile that isolate the first mode shape (a), the second (b), the third (c) and the fourth (d) when considering the first 4 modes out-of-plane for a left and right-side clamped plate Fig. 35 Electrode profile that isolate the first mode shape (a), the second (b), the third (c) and the fourth (d) when considering the first 4 modes in-plane for a four-side clamped plate objective of a biosensor is to determine with high accuracy the amount of important substances for biological reasons such as proteins and glucose. The way in that these kind of devices work is as simply as following changes in the natural frequency of the sensor. A summary of the process is shown in Fig. 40 for a possible biosensor with a cantilever-shape. First, we need to cover the sensor with a specific polymer, in the sense that it performs the function of glue with the substance that will be detected. In the case of a change in the substance we need to change the polymer too. After covering the sensor with the polymer layer (represented with the green layer), we measure the natural frequency (f r0 ) of a desired vibration mode [Fig. 40 (top)]. The selection of this vibration mode is made according to the application and the electronic circuit that adapts the electrical signal to be measured. Once the substance that we want to detect make appearance [blue dots in Fig. 40 (middle)], gets stuck to the sensor [Fig. 40 (bottom)] increasing the mass of the system. This change in the mass produces a change in the natural frequency measured (f r1 ) that is directly related with the amount of substance. A few words must be said about this method of detecting