Optimal design of piezoelectric microtransducers

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(1)UNIVERSITY OF CASTILLA-LA MANCHA Escuela Técnica Superior de Ingenieros Industriales Departamento de Matemáticas. OPTIMAL DESIGN OF PIEZOELECTRIC MICROTRANSDUCERS. Author: David Ruiz Gracia Supervisors: José Carlos Bellido Guerrero Alberto Donoso Bellón. Doctoral Thesis Ciudad Real, 2015.

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(3) Acknowledgments First of all, I would like to thank my parents Ma de Gracia and Vicente and my brother Alberto. It is difficult to express with words all the support that the family can give to you all these years. Thanks Marı́a. You have lived very close all the way up here. Without your support it would have been impossible to finish this project. Of course, I want to thank my supervisors José Carlos and Alberto. The effort that you have done is huge. Thanks for your advices that have allowed me to grow as research scientist. I want to thank also everyone that has surrounded me all these years: Javi T., Ángel, Rober, Mora, Israel, Lucı́a, Javi B., Victoria, Noemi, Alicia, Julián, Mirella, Paco, etc. The list is endless and probably I have forgotten many names. This doctoral thesis has been carried out thanks to a fellowship associated to the project of the Ministerio de Ciencia e Innovación entitled “Una perspectiva variacional en EDP’s: control y diseño” (MTM2010-19739), that also has financially supported the researching. Also I want to thank the project “Técnicas variacionales en control y diseño óptimos: análisis, simulación y aplicaciones” (MTM2013-47053-P) its financial support..

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(5) Contents Acknowledgments. iii. Contents. v. List of Figures. vii. List of Tables. xi. Resumen de la tesis. 1. 1 Introduction 1.1 Motivation of the thesis . . . . . . . . . . . . . . . . . . . . 1.2 Piezoelectric effect . . . . . . . . . . . . . . . . . . . . . . . 1.2.1 Classical applications of piezoelectric materials . . . 1.3 Introduction to topology optimization . . . . . . . . . . . . 1.3.1 Other physical contexts where topology optimization 1.4 Optimal design of modal piezoelectric transducers . . . . . 1.4.1 Modal sensor/actuators . . . . . . . . . . . . . . . . 1.4.2 Statement of the problems analysed in this thesis . . 1.5 Structure of the thesis and results of our research . . . . . . 1.6 Possible applications derived from our designs . . . . . . . . 1.6.1 Actuators . . . . . . . . . . . . . . . . . . . . . . . . 1.6.2 Sensors . . . . . . . . . . . . . . . . . . . . . . . . . 1.7 Review of the related scientific literature . . . . . . . . . . .. . . . . . . . . . . . . . . . . helps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . .. . . . . . . . . . . . . .. 11 11 13 17 17 25 28 28 32 35 38 38 40 43. 2 Design of piezoelectric transducers for static 2.1 The sensor problem . . . . . . . . . . . . . . . 2.1.1 Continuous formulation . . . . . . . . 2.1.2 Discrete formulation . . . . . . . . . . 2.1.3 Computation of sensitivities . . . . . . 2.1.4 Numerical approach and examples . . 2.2 The actuator problem . . . . . . . . . . . . . 2.2.1 Discrete formulation . . . . . . . . . . 2.2.2 Computation of sensitivities . . . . . . 2.2.3 Numerical approach and examples . .. . . . . . . . . .. . . . . . . . . .. . . . . . . . . .. 45 46 46 49 51 52 62 64 65 66. response . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . .. . . . . . . . . .. . . . . . . . . .. . . . . . . . . .. . . . . . . . . ..

(6) vi. Contents 2.3 2.4 2.5. Reciprocity of the piezoelectric effect . . . . . . . . . . . . . . . . . . . Manufacturability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 3 Mode filtering by piezoelectric transducers 3.1 Continuous formulation . . . . . . . . . . . . . 3.2 Discrete formulation . . . . . . . . . . . . . . . 3.3 Computation of sensitivities . . . . . . . . . . . 3.3.1 Simple eigenfrequencies . . . . . . . . . 3.3.2 Repeated eigenfrequencies . . . . . . . . 3.4 Numerical approach . . . . . . . . . . . . . . . 3.5 Examples . . . . . . . . . . . . . . . . . . . . . 3.5.1 Plate clamped at its left edge . . . . . . 3.5.2 Plate clamped at its left and right edges 3.5.3 Plate clamped at all four edges . . . . . 3.6 Results and conclusions . . . . . . . . . . . . .. . . . . . . . . . . .. . . . . . . . . . . .. . . . . . . . . . . .. . . . . . . . . . . .. . . . . . . . . . . .. . . . . . . . . . . .. . . . . . . . . . . .. . . . . . . . . . . .. . . . . . . . . . . .. . . . . . . . . . . .. . . . . . . . . . . .. . . . . . . . . . . .. . . . . . . . . . . .. 67 68 69 71 72 75 79 80 83 89 91 91 95 97 98. 4 Conclusions of the thesis and future research 105 4.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 4.2 Future research . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108 Bibliography. 113.

(7) List of Figures 1 2 3 4. 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 1.10 1.11 1.12 1.13 1.14 1.15 1.16 1.17 1.18 1.19 1.20 1.21 1.22 1.23 1.24 1.25 1.26 1.27. Diversas capas que forman un transductor piezoeléctrico. . . . . . . . Condiciones de contorno para el sensor (a) y el actuador (b). . . . . . Resumen del proceso. . . . . . . . . . . . . . . . . . . . . . . . . . . . Configuración óptima para filtrar: (a) primera, (b) segunda, (c), tercera y (d) cuarta forma modal, cancelando las otras tres considerando las cuatro primeras. . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 2 3 6. Examples of MEMS. . . . . . . . . . . . . . . . . . . . . . . . . . . . . Piezoelectric converse (top) and direct (bottom) effect. . . . . . . . . . Molecule of water. Mills (2015). . . . . . . . . . . . . . . . . . . . . . . Macro and microcrystalline quartz. . . . . . . . . . . . . . . . . . . . . Deformation in a crystal of quartz. . . . . . . . . . . . . . . . . . . . . Example of piezoelectric sensors. . . . . . . . . . . . . . . . . . . . . . Examples of piezoelectric actuators. . . . . . . . . . . . . . . . . . . . Quartz crystal oscillator. Duino (2012). . . . . . . . . . . . . . . . . . Optimized design of a compressor bracket. Altair (2013). . . . . . . . (a) Sizing, (b) shape and (c) topology optimization. . . . . . . . . . . Design domain with boundary and load conditions. . . . . . . . . . . . Design domain (left) and optimized design (right). . . . . . . . . . . . Optimized designs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Synthesis of compliant mechanisms. . . . . . . . . . . . . . . . . . . . Optimal design in dynamics. . . . . . . . . . . . . . . . . . . . . . . . Optimal design of materials. . . . . . . . . . . . . . . . . . . . . . . . . Numerical simulations for band-gap problems. . . . . . . . . . . . . . . Scheme of a MSA. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Mode shapes (top) and top views of optimized sensors (bottom). . . . Different steps of the process. . . . . . . . . . . . . . . . . . . . . . . . Design domain and boundary conditions for the static sensor problem. Optimization of electrode (left). Optimization of structure (center). Simultaneous optimization (right). . . . . . . . . . . . . . . . . . . . . Normalized output charge. . . . . . . . . . . . . . . . . . . . . . . . . . Manufactured optimized design for the sensor case. . . . . . . . . . . . Microgripper. Chronis (2004). . . . . . . . . . . . . . . . . . . . . . . . Microinjector of DNA. Aten (2014). . . . . . . . . . . . . . . . . . . . Energy harvesters. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 11 14 14 15 16 17 18 18 19 19 21 23 25 26 27 27 28 30 30 32 34. 8. 36 37 38 39 39 40.

(8) viii. List of Figures 1.28 Non-invasive blood pressure sensor. Concardiac (2014). . . . . . . . . 1.29 Array of cantilevers. Jensen (2013). . . . . . . . . . . . . . . . . . . . .. 41 42. 2.1 2.2. 46. 2.24 2.25 2.26. Side view of a piezoelectric transducer. . . . . . . . . . . . . . . . . . . Design domain, boundary conditions and design variables for the sensor case. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Design domain for the sensor problem. . . . . . . . . . . . . . . . . . . Flowchart of the process. . . . . . . . . . . . . . . . . . . . . . . . . . . Optimized designs for sensor case. Material density (left) and electrode profile (right). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Optimized designs for the sensor case with R = ρ3s material layout (left) and polarity layout (right). . . . . . . . . . . . . . . . . . . . . . Penalization function R(ρs ). . . . . . . . . . . . . . . . . . . . . . . . Optimized designs for sensor case with interpolation showed in Figure 2.7 material (left) and electrode density (right). . . . . . . . . . . . . . Convergence of the cost. . . . . . . . . . . . . . . . . . . . . . . . . . . Optimized design for sensor case in-plane, material (left) and electrode density (right). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Sensor output qout versus volume fraction V0 for kin = 2 × 105 N/m. . Optimized design for sensor case out-of-plane, material (left) and electrode density (right). . . . . . . . . . . . . . . . . . . . . . . . . . . . . Optimized design for sensor case in-plane, material (left) and electrode density (right). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Optimized design for sensor case out-of-plane, material (left) and electrode density (right). . . . . . . . . . . . . . . . . . . . . . . . . . . . . Rotary in-plane sensor. . . . . . . . . . . . . . . . . . . . . . . . . . . . Optimized design for out of plane sensor. Green means points where only rotation is allowed. . . . . . . . . . . . . . . . . . . . . . . . . . . Design domain, boundary conditions and variables for actuator case. . Cross section of a structure. Piezo working in tension (a) and in compression (b). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Cross section of a structure for out-of-phase excitation. . . . . . . . . . Design domain for the actuator problem. . . . . . . . . . . . . . . . . . Optimized design for the actuator case in-plane (left) and out-of-plane (right) cases. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Optimized design for the actuator case in-plane (left) and out-of-plane (right) cases using as starting point the optimized design for the sensor problem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Optimized sensor design (left). Optimized actuator design (center). Optimized actuator using as starting point the optimized design of the sensor (right). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Optimized in-plane sensor. . . . . . . . . . . . . . . . . . . . . . . . . . Optimized in-plane sensor with passive areas. . . . . . . . . . . . . . . Optimized designs for different optimization processes. . . . . . . . . .. 3.1. Design domain and boundary conditions.. 72. 2.3 2.4 2.5 2.6 2.7 2.8 2.9 2.10 2.11 2.12 2.13 2.14 2.15 2.16 2.17 2.18 2.19 2.20 2.21 2.22. 2.23. . . . . . . . . . . . . . . . .. 47 50 53 54 55 56 57 57 58 59 59 60 60 61 61 62 62 63 64 66. 66. 67 69 69 70.

(9) List of Figures 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9. 3.10. 3.11. 3.12. 3.13. 3.14. 3.15 3.16. 3.17 3.18. 4.1 4.2 4.3. ix. Interpolation scheme R proposed in (3.9) for different values of γ. . . . 77 Out-of-plane mode shapes with the same frequency for a simply-supported plate. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 Evolution of the eigenvalue vs. a density variable. . . . . . . . . . . . . 86 Flowchart of the process. . . . . . . . . . . . . . . . . . . . . . . . . . . 92 Optimized designs that isolate the first mode shape when considering the first 4 modes in-plane. . . . . . . . . . . . . . . . . . . . . . . . . . 93 Evolution of the frequencies with the iterations. . . . . . . . . . . . . . 94 Evolution of the order of the vibration modes with the iterations. . . . 94 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. . . . . . . . . . . . . . . . . . . . 95 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. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96 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. . . . . . . . . . . . . . . . . 97 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. . . . . . . . . . . . . . 98 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. . . . . . . . . . . . 99 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. . . . . . . . . . . . . . . . . . . 100 Evolution of the frequencies with the iteration when isolating fourth mode. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 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 four-side clamped plate. . . . . . . . . . . . . . . . . 101 Evolution of the frequencies with the iteration when isolating first mode.101 Comparison between the normalized coefficients Fj obtained under simultaneous optimization (gray) versus the ones got under single optimization (black) when the first mode shape (a), the second (b), the third (c) and the fourth (d) are isolated and considering the first 4 modes in-plane. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 Example of optimized design where a gap-phase is required. . . . . . . 108 Example of optimized design with hinges. . . . . . . . . . . . . . . . . 109 Optimized design fabricated. . . . . . . . . . . . . . . . . . . . . . . . 110.

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(11) List of Tables 1.1. Comparison of output charge for different optimization processes . . .. 37. 2.1. Comparison of output charge for different optimization process . . . .. 70. 3.1 3.2 3.3. Gain for the first boundary conditions . . . . . . . . . . . . . . . . . . 102 Gain for the second boundary conditions . . . . . . . . . . . . . . . . . 102 Gain for the third boundary conditions . . . . . . . . . . . . . . . . . . 103.

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(13) Resumen de la tesis El objetivo de esta tesis doctoral es el diseño óptimo de microtransductores piezoeléctricos. Comencemos definiendo el concepto de transductor. Entendemos como tal a aquel dispositivo que tiene la capacidad de trabajar como sensor, como actuador o como ambos. En nuestro caso, la actuación sobre el dispositivo es piezoeléctrica, aunque existen otros tipos: magnética, térmica, mecánica, etc. Un transductor piezoeléctrico puede convertir energı́a mecánica en eléctrica mediante el efecto piezoeléctrico directo o viceversa, mediante el efecto inverso. El dispositivo está formado por diversas capas como podemos observar en la Figura 1. La capa más gruesa es la capa base, que es la encargada de hacer de soporte de las capas piezoeléctricas. Dichas capas están formadas por un material dieléctrico, el cual no conduce la corriente pero es capaz de generar un campo eléctrico interno cuando le aplicamos un campo eléctrico externo (a diferencia de un material aislante que no puede generarlo). Para conseguir generar este campo eléctrico interno añadimos dos electrodos a cada capa piezoeléctrica, los cuales son los encargados de generar el campo eléctrico externo. A la hora de entender el funcionamiento tenemos que tener en cuenta si las capas de material piezoeléctrico están trabajando como sensor o como actuador. Si están trabajando como sensor, una fuerza mecánica externa deformará la estructura. Debido al efecto piezoeléctrico, esta deformación generará un voltaje en el material piezoeléctrico, que será recogido por los electrodos. En el caso de que esté trabajando como actuador, aplicamos un voltaje a los electrodos y estos generarán un campo eléctrico, polarizando el material piezoeléctrico que a su vez generará una deformación debido al efecto piezoeléctrico inverso. Podemos restringir el desplazamiento de la estructura al plano de la misma si polarizamos ambos piezoeléctricos en fase, es decir, ambos están polarizados en la misma dirección o si la fuerza mecánica está aplicada dentro del mismo plano de la estructura. Si por el contrario estamos interesados en el movimiento fuera del plano debemos polarizar ambos piezoeléctricos con polaridad opuesta o aplicar una fuerza en el plano perpendicular al plano de la estructura. La reciprocidad del efecto piezoeléctrico permite a la estructura trabajar tanto como sensor como actuador, por contrapartida, debido al tamaño y a las múltiple capas que lo forman presentan una elevada dificultad de fabricación. Las técnicas clásicas de fabricación no sirven para fabricarlos y tenemos que buscar por lo tanto otras más recientes como “sputtering”, litografı́a, etc. Ya hemos definido qué es un transductor piezoeléctrico y cómo funciona, queda destacar el tamaño de los dispositivos que vamos a diseñar. Debido a su gran número.

(14) 2. Resumen de la tesis. Piezoeléctrico Base. Electrodos. Piezoeléctrico. Figura 1: Diversas capas que forman un transductor piezoeléctrico. de aplicaciones, nos hemos centrado en los MEMS (“Micro-ElectroMechanical Systems”). Este tipo de sistemas tiene una serie de ventajas, la principal es el tamaño, que oscila entre 1µm y 1mm, lo cual permite colocarlos en prácticamente cualquier lugar, desde automóviles hasta el interior del cuerpo humano. La segunda ventaja es que la superficie de estos dispositivos es muy grande en comparación con su volumen, este hecho permite que efectos como el piezoeléctrico o electrostático dominen a otros tales como la gravedad o inercias térmicas. La tercera gran ventaja es que la mayorı́a de los MEMS están formados únicamente por una pieza, y toda la deformación se consigue mediante la elasticidad de la estructura. Este hecho evita que haya rozamiento entre diversas partes, por lo tanto no necesitan mantenimiento (reemplazo de piezas por desgaste, lubricación, etc.). Llegados a este punto ya tenemos todos los ingredientes para entender el objetivo de la tesis el cual es, como ya comentamos anteriormente, el diseño óptimo de microtransductores piezoeléctricos. Entendemos como tal al diseño de la geometrı́a de la estructura dentro de un recinto al que llamamos dominio, con el objetivo de optimizar una cierta caracterı́stica del dispositivo, bajo restricciones de diseño (cantidad de material usado, desplazamientos máximos, etc.). Un ejemplo clásico de diseño óptimo es la maximización de la rigidez de una estructura para un peso máximo fijado. En nuestro caso, el objetivo es maximizar la carga recogida por los electrodos cuando el dispositivo está trabajando como sensor, o maximizar el desplazamiento de una parte de la estructura cuando aplicamos un voltaje determinado a los electrodos, es decir, cuando está trabajando como actuador. La herramienta matemática para conseguir llegar a los diseños optimizados es la optimización topológica. Este proceso consiste en determinar la geometrı́a óptima de una estructura con el objetivo de optimizar un parámetro (peso, rigidez, conductividad, etc.) bajo unas ciertas restricciones. En el proceso de optimización vamos a trabajar con dos variables: la primera es la variable de la estructura, que definirá la topologı́a (forma) del transductor. La segunda es la polarización del electrodo que puede ser negativa o positiva. No es necesario que todo el electrodo que cubre el material piezoeléctrico tenga la misma polaridad, la inclusión de zonas con distinta polaridad mejora el rendimiento del transductor, y por ello la utilizamos como variable. Durante la tesis vamos a analizar dos casos de estudio, el primero es el diseño óptimo bajo condiciones de trabajo estáticas, el segundo en un régimen dinámico en el dominio de la frecuencia, donde entran en juego modos de vibración..

(15) 3. Resumen de la tesis. ?. ?. Fin. +. V_in. C qout (a). (b). Figura 2: Condiciones de contorno para el sensor (a) y el actuador (b). 1. Diseño óptimo de transductores piezoeléctricos bajo régimen estático Dado que nuestros transductores pueden trabajar tanto como sensores como actuadores, vamos a estudiar por separado los dos modos de funcionamiento, ya que los objetivos son diferentes en cada caso. • Diseño óptimo del sensor: el objetivo es la maximización de la carga recogida por los electrodos debido al efecto piezoeléctrico directo para un volumen dado. En la Figura 2(a) podemos ver el dominio y las condiciones de contorno para un posible caso de estudio: una estructura empotrada en su lado izquierdo, con desplazamiento vertical y giro impedido en el caso de trabajar fuera del plano y con desplazamiento impedido en ambas direcciones para el caso de trabajar dentro del plano de la estructura. A la hora de buscar el diseño óptimo debemos definir dos entradas: la primera es la fuerza mecánica externa que aplicamos, la segunda es la rigidez que debe tener la estructura en el punto de aplicación de la fuerza. Estas dos entradas son las que modelan la actuación sobre el dispositivo. Una vez discretizado mediante elementos finitos, el problema discreto escrito en la formulación clásica de la optimización topológica quedarı́a: max : qout = G(ρp )T U. ρs ,ρp. sujeto a: K(ρs ) + kin 1in U v T ρs ρs ρp. = ≤ ∈ ∈. fin L V0 [0, 1] [0, 1]. donde G es un vector que contiene el tensor de deformaciones, U es el vector de desplazamientos, K es la matriz de rigidez, 1in y L son una.

(16) 4. Resumen de la tesis matriz y un vector de ceros con un 1 en el grado de libertad de la entrada, v es el volumen de un elemento, V0 es la máxima fracción de volumen permitida. kin y fin son la rigidez y fuerza que modelan la entrada. ρs y ρp son las variables de estructura y polaridad del electrodo respectivamente. Es importante destacar que estamos trabajando con una placa sin curvatura, por lo que podemos estudiar de forma independiente cuando trabajamos en el plano de la estructura y cuando no. En la Figura 2(a) hemos tratado de reflejar ambos casos mediante dos muelles, uno colocado de forma paralela a la estructura que representa el primer caso, y otro colocado perpendicular a la misma que representa el segundo. • Diseño óptimo del actuador: el objetivo en este caso es la maximización del desplazamiento de una parte de la estructura, bajo una restricción sobre su volumen máximo. En la Figura 2(b) mostramos el dominio y las condiciones de contorno para este caso. Las entradas van a ser el voltaje aplicado a los electrodos y la rigidez que debe vencer la parte de la estructura para la cual queremos maximizar el desplazamiento. De nuevo, el problema discreto (discretizado mediante elementos finitos) escrito en la formulación clásica de la optimización topológica quedarı́a: max : Uout (ρs , ρp ). ρs ,ρp. sujeto a: K(ρs ) + kout 1out U vT ρs ρs ρp. = ≤ ∈ ∈. F(ρs , ρp ) V0 [0, 1] [0, 1]. donde Uout es el desplazamiento que queremos maximizar, kout es la rigidez que queremos vencer, 1out es una matriz de ceros con un 1 en el grado de libertad de la salida y F es el vector de fuerza generado por el efecto piezoeléctrico. Es importante destacar que la fuerza en este caso no es constante y depende tanto de la topologı́a como de la polaridad del electrodo. Al igual que sucede con el sensor, podemos separar los desplazamientos dentro del plano de la estructura y fuera del mismo, lo cual está representado en la Figura 2(b) con dos muelles al igual que hicimos en el caso del sensor. La resolución de ambos problemas es muy interesante, tanto desde el punto de vista matemático como de las aplicaciones. La primera dificultad aparece una vez discretizado el problema. Ambas variables de diseño usadas en el modelado del problema son continuas, sin embargo, buscamos que dichas variables tomen valores discretos en el óptimo. La variable estructural ρs toma el valor 1 cuando en ese punto colocamos material y 0 cuando no. La variable de polarización.

(17) Resumen de la tesis. 5. ρp puede tomar el valor 0 cuando la polaridad de ese punto es negativa, y 1 cuando es positiva. No existen algoritmos eficientes para resolver el problema con variables discretas, y la evaluación de todas y cada una de las combinaciones posibles conllevarı́a un tiempo de computación inadmisible. La solución consiste en reemplazar estas variables discretas por variables continuas, que pueden tomar cualquier valor entre 0 y 1 y a las que llamaremos variables de densidad. Mediante métodos de penalización intentaremos que dichas variables únicamente tomen los valores extremos 0 y 1, que son para los que el problema tiene sentido fı́sico. La naturaleza del problema va a favorecer la aparición de zonas con valores en la variable estructural entre 0 y 1. En estas zonas se produce una mezcla de materiales a escala microscópica (llamada microestructura) imposible de fabricar, por lo que su aparición debe ser evitada a toda costa. El método propuesto para conseguirlo es una de las novedades presentadas en esta tesis. Consiste en la introducción de un esquema de interpolación en la expresión de la carga recogida para el caso del sensor, y en la expresión de la fuerza piezoeléctrica en el caso del actuador. Otros problemas tales como la dependencia del mallado y el problema del “tablero de ajedrez” son resueltos a través de ciertas técnicas (llamadas de filtrado) que veremos en la memoria. Finalmente, el último gran problema son los óptimos locales. En este trabajo hemos intentado evitarlos a través de la utilización de métodos de continuación y de diferentes inicializaciones de las variables en el método iterativo del optimizador usado. Desde el punto de vista de las aplicaciones es muy importante mejorar las prestaciones de estos dispositivos. Debido a su diminuto tamaño la carga eléctrica generada por el sensor es muy pequeña; en el caso del actuador el desplazamiento conseguido por el voltaje aplicado es muy bajo debido a la rigidez del material con el que está fabricado. El incremento de dichos parámetros es crucial a la hora de poder seguir reduciendo su tamaño y facilitar su introducción en lı́quidos, donde las pérdidas son mucho más elevadas que en aire. Durante la tesis también se ha demostrado matemáticamente que tanto el problema del sensor como el del actuador son equivalentes, es decir, los diseños óptimos de ambos problemas son los mismos siempre que la rigidez de entrada y de salida sean la misma (kin = kout ). Este hecho se produce debido a la reciprocidad del efecto piezoeléctrico, y es por lo tanto una demostración de que realmente se cumple la dicha propiedad. Una vez obtenido el diseño óptimo para unos datos de entrada determinados el siguiente objetivo es la fabricación. Para ello debemos convertir las variables de la estructura y electrodo que el algoritmo nos devuelve en forma de matrices, en curvas parametrizadas para poder fabricarlo. Algunos de los diseños obtenidos han sido fabricados y están a la espera de poder ser testeados. Podemos ver un resumen simplificado del proceso en la Figura 3. Para este caso hemos resuelto el problema del sensor para un dominio cuadrado con desplazamientos impedidos en el lateral izquierdo. La fuerza aplicada está dentro del plano de la estructura y la fracción máxima de volumen permitido es del 40%..

(18) 6. Resumen de la tesis. Figura 3: Resumen del proceso. 2. Diseño óptimo de sensores/actuadores modales piezoeléctricos El objetivo de este caso de estudio es el diseño de transductores modales. Este tipo de dispositivos son eléctricamente sensibles a un modo de vibración, es decir, son capaces de aislar un modo de vibración del resto. De ahora en adelante vamos a llamar filtrado al proceso de quedarnos únicamente con la forma modal deseada, cancelando el resto. Dado que sustituimos los desplazamientos causados en la estructura por una fuerza estática (ya sea de origen mecánico o piezoeléctrico) por formas modales, pasamos a trabajar en un régimen dinámico. En este caso resulta más claro estudiar el problema desde el punto de vista del sensor. Como ya hemos comentado anteriormente, el objetivo del problema es filtrar un modo de vibración al mismo tiempo que maximizamos su carga. El número de modos de vibración de una estructura continua es infinito y obviamente no tiene sentido trabajar con tal número de modos. Cuando discretizamos, dicho número queda truncado y es igual al número de grados de libertad de la estructura. Por lo general, aún truncando dicho número, éste suele ser demasiado grande por lo que únicamente tenemos en cuenta un cierta cantidad de modos que corresponden a los de más baja frecuencia, ya que la influencia del resto no es importante. Los modos de vibración utilizados en el problema no tienen por qué ser consecutivos, sin embargo, a la hora de formular el problema los vamos a numerar desde 1 hasta J por simplicidad. El problema discreto escrito en la formulación clásica de la optimización topológica quedarı́a:.

(19) Resumen de la tesis. 7. max : Fk (ρs , ρp ). ρs ,ρp. sujeto a: (K − µj M)Φj = 0, j = 1, . . . , J ΦTj MΦl = 0, j, l = 1, . . . , J, j 6= l ΦTj Φj = 1, j = 1...,J |Fj | ≤ Fk , j = 1, . . . , J, j 6= k ρs ∈ [0, 1] ρp ∈ [0, 1] donde Fi es la expresión de la carga recogida por los electrodos para el modo i-ésimo, K y M son las matrices globales de rigidez y masa respectivamente, Φi es la forma modal del modo i-ésimo, es un valor muy pequeño y J es el número de modos de vibración elegidos. Las variables de diseño son las mismas que en el caso estático. Al igual que ocurre con el anterior caso de estudio, este problema es matemáticamente muy interesante. Aparte de todas las dificultades comentadas anteriormente, que también aparecen de nuevo en este problema, tenemos que sumar la dificultad de trabajar con modos de vibración. Los tres principales problemas que surgen al ahora de trabajar con modos son: • Modos espúreos: este tipo de modos de vibración aparece debido a un incorrecto modelado de los materiales que forman la estructura. Aparecen a frecuencias muy bajas y distorsionan el análisis del espectro. En el modelo matemático los huecos son modelados como zonas con muy baja densidad, y por lo tanto tienen masa y rigidez, aunque muy pequeñas. Si la relación entre dichos parámetros es muy grande (lo cual significa que su masa es grande en comparación con su rigidez) dichas zonas comenzarán a moverse antes que las zonas sólidas haciendo que aparezcan estos modos indeseados. Eligiendo una relación adecuada entre rigidez y masa para las zonas de baja densidad se pueden evitar estos modos que no existen en la realidad. • Intercambio de modos: durante el proceso de optimización la estructura va variando su forma, y este hecho va a hacer que los modos de vibración que querı́amos optimizar y cancelar cambien de orden, de modo que necesitamos una forma de asegurarnos que siempre estamos optimizando o suprimiendo las mismas formas modales. • Frecuencias repetidas: debido principalmente a simetrı́a en las condiciones de contorno y al intercambio del orden de los modos, en ocasiones aparecen dos modos de vibración con la misma frecuencia. Matemáticamente es un problema muy complejo, debido a que la base de autovectores que son solución del problema ya no es única. Además, de las infinitas bases que.

(20) 8. Resumen de la tesis existen, sólo una de ellas es derivable, lo cual genera severas complicaciones a la hora de calcular las sensibilidades. En esta tesis doctoral se ha conseguido solventar todos estas dificultades, siendo la principal aportación de la misma la resolución de un problema de optimización con autovectores incluidos tanto en el coste como en las restricciones. En la Figura 4 podemos observar algunos de los diseños óptimos obtenidos. Para este ejemplo hemos elegido un dominio cuadrado donde los desplazamientos de los cuatro bordes están impedidos. Los modos de vibración elegidos son los correspondientes a desplazamientos fuera del plano de la placa. El número de modos de vibración usados es cuatro, en cada diseño optimizamos la carga recogida en uno y cancelamos la de los tres restantes.. (a) Modo primero optimizado.. (b) Modo segundo optimizado.. (c) Modo tercero optimizado.. (d) Modo cuarto optimizado.. Figura 4: Configuración óptima para filtrar: (a) primera, (b) segunda, (c), tercera y (d) cuarta forma modal, cancelando las otras tres considerando las cuatro primeras. Estos ejemplos tienen un importante interés práctico, ya que esperamos que la mejora de la carga recogida en los electrodos sea suficiente para facilitar su introducción en lı́quidos, ya que los diseños sin optimizar, debido a la viscosidad y densidad del fluido, generalmente generan una carga eléctrica muy pequeña la cual no nos permite usarlo como sensores de estos parámetros. Para que el lector se pueda hacer una idea del tamaño de los dispositivos que han sido fabricados vamos a comentar brevemente sus dimensiones. Es importante remarcar que para el caso de los transductores modales no se ha fabricado ninguno.

(21) Resumen de la tesis. 9. de los diseños obtenidos hasta la fecha, únicamente se han fabricado para el caso estático. La capa base tiene un espesor de 20µm y está formada por silicio (Si). Esta capa sirve de soporte para las dos capas de material piezoeléctrico, de un espesor de 620nm y formada por nitruro de aluminio (AlN). Existen materiales piezoeléctricos con mejores propiedades que el AlN, como por ejemplo el PZT (titanato de plomo con zirconato de plomo), sin embargo, no todos los materiales son compatibles con el complejo proceso de fabricación. Finalmente las capas de electrodo tienen un espesor de entre 50 y 100nm y los materiales usados son cromo (Cr) y oro (Au). En todos los ejemplos el dominio es un cuadrado de 1000µm de lado. Para poder hacernos una idea de estas dimensiones daremos dos datos: el tamaño promedio de una célula es de 10µm y el grosor medio del cabello de una persona adulta es de 100µm. Como resumen final podemos concluir que hemos cumplido con éxito los objetivos propuestos en la tesis. Hemos resuelto dos casos de estudio satisfactoriamente habiendo solventado todas las dificultades que han aparecido, aportando nuevas ideas cuando ha sido necesario. Incluso algunos de los dispositivos diseñados han sido fabricados en Austria con las colaboraciones de Institute of Sensor and Actuator Systems y Austrian Center of Competence for Tribology, y están a la espera de ser testeados en el laboratorio de tecnologı́a electrónica de la universidad de Castilla-la Mancha por Microsystems, Actuators and Sensors Group para verificar su correcto funcionamiento..

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(23) Chapter 1. Introduction 1.1. Motivation of the thesis. This doctoral thesis is focused on the optimal design of MEMS (“MicroElectroMechanical Systems”). Nowadays this kind of devices is a very important part of our lives. The applications of MEMS cover a large number of fields: medicine, entertainment, automotive industry, aeronautic industry, energy harvesting, biotechnology, electronics, etc. The main characteristic of these devices is their size, MEMS are at the millimeter scale. We can see a pair of examples in Figure 1.1. The size is an advantage, since they can be placed everywhere: cars, planes, shoes, and even inside the human body. Of course, it may also be a disadvantage: classical techniques of fabrication cannot be used to manufacture these devices, what makes this process more expensive. Some techniques used to fabricate MEMS are deposition, patterning, or etching processes. Depending on the application of the device, the actuation over it comes from different nature: • Mechanical actuation: We apply one or more mechanical forces directly over. 200 m. (a) Microgripper. Lee (2006).. (b) Microresonator. Schmid (2015).. Figure 1.1: Examples of MEMS..

(24) 12. Chapter 1. Introduction the device. Some examples are microgrippers, cantilevers or energy harvesters. • Inertial actuation: This kind of actuation could be included inside the previous group. In this case inertial forces cause the deformation of the structure. A perfect example is the accelerometer used to measure movements in one, two or three dimensions. • Thermal actuation: Heat is applied to the whole structure or only in a part of it, getting a thermal deformation in the structure. Thermostats or temperature sensors are the typical examples of this kind of actuation. • Electrostatic actuation: The actuation over the device is an electric field. The problem with this type of actuation is that the electric force decreases with the square of the distance between the charged bodies, but in MEMS the aspect ratio is really low (one dimension is smaller than the rest, then this force is bigger enough to be relevant). Microswitches are examples of this actuation. • Magnetic actuation: The deformation or the movement is produced by a magnetic field. We can use the Lorentz force (an electrical current produces a magnetic force) or we can apply directly a magnetic field. Micromotors use this kind of actuation. • Piezoelectric actuation: In this last case, we apply a voltage over a piezoelectric material, and thanks to the piezoelectric effect, it is deformed. A very common use of this actuation takes place in crystal quartz watches.. In this thesis, we focus on piezoelectric actuation. We can find very important features in this kind of actuation, the first one, mentioned above, is that the application of a voltage produces a deformation in the structure, and the application of the same voltage but with different polarity, produces an equivalent deformation but in the opposite direction. This allows us to fabricate resonators working in traction and compression, that is, an oscillating behavior. Oscillators are very interesting for some applications as we will see below. Another interesting advantage is the reciprocity of the piezoelectric effect, meaning that a deformation in a structure induces an electric signal, whose charge can be collected and measured. The piezoelectric effect allows the device to work as sensor and as actuator. In the first case the deformation produced in the structure as a consequence of an external actuation, generates an electrical charge that can be measured. In the second case we control the deformation produced in the structure by applying an electrical voltage. This effect is produced in the piezoelectric layer, that is bonded to the top and bottom of the structure. Over this layer we can find another thin layer of conductive material connected to a electric source, the electrode. Its aim is the collection of the electric charge in the sensor case and the polarization of the piezoelectric layer with different polarities in the actuator case. At this point, it is important to remark that only the part of the structure that is covered by the electrode is electrically affected, meaning that we only collect charge or make a deformation in these parts of the structure. Vibrations of a structure can be controlled by using the piezoelectric effect. Although a continuous structure has infinite vibration modes, generally the ten or.

(25) 1.2. Piezoelectric effect. 13. twenty first ones define its dynamic behavior. The excitation or measurement of one specific vibration mode in some fields like electronics, is very important. This leads us to consider a new problem optimizing the response of one desired mode, while the response of other modes is suppressed as much as possible. This problem is named as mode filtering and the devices that achieve this goal are called “modal sensors/actuators”, hereafter MSA. The electrode of MSA is optimized in order to reach the filtering of a particular vibration mode. As we will see below, its development at the microscale is an important emerging field of study. The characteristics of MEMS not only depend on the properties of the material of fabrication, but also depend on the shape of the host structure. At this point mathematics are the key, and more specifically, the topology optimization method. This is an important conceptual tool that let us optimize a specific characteristic of a structure as its weight, stiffness, displacement, etc. Structural optimization is not only related with micromechanisms, it is also an useful tool at the macro-scale. Two examples where structural optimization has been widely used are in the automotive and aeronautic industry. In both cases, it is very interesting the reduction of the weight of the structural components that form a car or a plane, but keeping the stiffness as bigger as possible. The reduction of weight means that we make an efficient use of material and then the fuel consumption is also reduced. In the next two sections we will see an introduction to the piezoelectric effect and the topology optimization method, as well as some examples of applications of each one of them.. 1.2. Piezoelectric effect. The word “piezo” is derived from the Greek word which means pressure. This effect was discovered in 1880 by the French physicists Jacques and Pierre Curie. Piezoelectricity is the ability of certain kind of materials to generate an electric polarization in its mass under the action of a mechanical stress on it, that creates a potential difference and an electrical charge in its surface. This effect is reciprocal, that is, we can get a deformation in the material when we apply an external electric field. The first case is called direct piezoelectric effect, the second one, converse piezoelectric effect. We can see both cases in Figure 1.2. We can find in nature materials with this ability: quartz, topaz, tourmaline-group minerals, wood, silk, dry bone or lead titanate (PbTiO3 ). Also we can synthesize materials with this effect: barium titanate (BaTiO3 ), potassium niobate (KNbO3 ), lithium tantalate (LiTaO3 ), bismuth ferrite (BiFeO3 ), bismuth titanate (Bi4 Ti3 O12 ), zinc oxide (ZnO) or lead zirconate titanate (Pb[Zrx Ti1−x ]O3 0 ≤ x ≤ 1) (the most common piezoelectric ceramic in use nowadays). The nature of this effect is related to the electric dipole moments in solids. We can understand a dipole as two electrical charges of different polarity, one nearby the other. The electric dipole moment measures the separation between both charges. In nature, most materials have a neutral global charge, since they have the same number of positive and negative charges. However, we can find materials with an asymmetric.

(26) 14. Chapter 1. Introduction. - 0 + -. -. - 0 +. 0 V. 0. - 0 +. 0 V. -. V. Converse effect F. F. - 0 +. - 0 +. - 0 +. V. V. F. V. F. Direct effect. Figure 1.2: Piezoelectric converse (top) and direct (bottom) effect. configuration of the atoms that produces a small polarization in the molecule and therefore a dipole. They are called polar materials and a good example of them is the water, whose molecule is showed in Figure 1.3. Once we have understood the nature of the effect, we will see what physically happens. We choose quartz as example. Quartz is a ceramic material, formed by crystals that are groups of molecules ordered in a particular way, in this case, a hexagonal crystal (Figure 1.4(a)). As we can see in Figure 1.4(b), the crystal of quartz is formed by silicon (pink) and oxygen (red). The atoms of oxygen tend to attract electrons of the atoms of silicon, so the first one is charged with negative polarity, and the second one is charged with positive polarity. If we apply a deformation over one axis, positive and negative charges get closer and an electric field appears, and. Figure 1.3: Molecule of water. Mills (2015)..

(27) 15. 1.2. Piezoelectric effect. (a) Macrocrystalline Gemtec (2012).. quartz.. (b) Crystalline structure. Genchem (2014).. Figure 1.4: Macro and microcrystalline quartz. consequently a difference of voltage is generated. In the converse case, we apply a difference of voltage instead of a mechanical force, generating then an electric field that produces a force in the opposite direction (depending on the polarity of the atoms). This force produces a deformation in the material. In Figure 1.5 we can see a deformed crystal of quartz. This could be produced by applying either a mechanical force (direct effect) or a voltage (converse effect). Once the piezoelectric effect has been explained, we will introduce the equations that model this phenomenon. Piezoelectricity is a combination of the mechanical and electric behavior of the material, then equations that model the electric and mechanical part are coupled. The mechanical part is modelled with the well known Hooke’s law using tensorial notation as S = sT → Sij = sijkl Tkl , where S is the strain field tensor, s is the inverse of the stiffness tensor and T is the stress tensor. The electric behavior of the material is modelled as D = εE → Di = εij Ej , where D is the electric displacement, ε is the permittivity and E is the electric field strength. We can combine both laws into the so-called coupled equations. These equations model the direct and converse piezoelectric effect, respectively. In both we can find a mechanical term with the stress and an electric one with the electric field S = sT + δ t E → D = δT + εE →. Sij = sijkl Tkl + dkij Ek Di = dijk Tjk + εij Ej ,. and δ is the tensor of dielectric constants for the direct piezoelectric effect.. (1.1).

(28) 16. Chapter 1. Introduction. = Si =O Figure 1.5: Deformation in a crystal of quartz..        . We can rewrite   S1 s11  s21 S2     S3   =  s31  S4    0 S5   0 0 S6     +   . (1.1) in a matricial way as  s12 s13 0 0 0  s22 s23 0 0 0    s32 s33 0 0 0    0 0 s44 0 0   0 0 0 s55 0   0 0 0 0 s66  0 0 d31   0 0 d32   E1  0 0 d33   E2  0 d24 0   E3 d15 0 0  0 0 0 . .   D1 0  D2  =  0 D3 d31 . 0 0 d32. ε11 + 0 0. 0 ε22 0. 0 0 d33. 0 d24 0. d15 0 0.  0   0   0  . T1 T2 T3 T4 T5 T6. T1 T2 T3 T4 T5 T6.         (1.2).        . (1.3).   0 E1 0   E2  ε33 E3. Equation (1.2) models the direct piezoelectric effect and eq. (1.3) the converse piezoelectric effect..

(29) 17. 1.3. Introduction to topology optimization. 5mm. (a) Piezoelectric microphone. (2015).. Korg. (b) Micropiezoelectric Dtech (2010).. accelerometer.. Figure 1.6: Example of piezoelectric sensors.. 1.2.1. Classical applications of piezoelectric materials. We can find applications of the piezoelectric effect in many fields. In this section we will comment some of the most classical ones. We can divide the applications into three groups: • Sensors: Thanks to the ability of generating an electric charge from an external excitation, we can use this kind of materials as sensors. Some of the most common applications of this group are microphones and accelerometers. The deformation is produced by a wave sound in the first case, and by inertial forces in the second. Examples of both applications are shown in Figure 1.6. • Actuators: In this case, we take advantage of the converse piezoelectric effect to control changes in the shape of the material produced by an electric field. A big change in the electric field causes a small change in the shape, so the sensibility of this kind of devices is very high. Two of the classical applications are fuel injectors in car engines and loud speakers (see Fig.1.7). In both cases the deformations are produced by an electric signal. • Frequency standard: Several piezoelectric materials (e.g. quartz) can be used as frequency standards, as the vibration frequency of them is practically constant with almost no variations when they work as oscillators. The most known application is the use in quartz clocks, where the vibration of a crystal of quartz is used to measure the time. Another important application of this group is radio transmitters. In Figure 1.8 we can see a quartz crystal oscillator.. 1.3. Introduction to topology optimization. Industry is always looking for an efficient use of material in order to reduce costs and improve its features at the same time. We can define topology optimization as a process to find the optimal material distribution in a given domain that optimizes a.

(30) 18. Chapter 1. Introduction. (a) Piezoelectric injector diesel. Johan (2015).. (b) Piezoelectric loudspeaker. (2014).. Kunst. Figure 1.7: Examples of piezoelectric actuators.. Figure 1.8: Quartz crystal oscillator. Duino (2012). certain objective function (stiffness, deflection, vibration frequency, etc.) while some design constraints (holes, mode filtering, weight, etc.) are satisfied. It is easy to convince oneself that the improvement of materials is interesting from the point of view of the industry. The development of this optimization process, together with the use of computational tools like CAD (“Computer Aided Design”) and CAM (“Computer Aided Manufacturing”), allow us to simulate in the computer the design obtained with the optimization process instead of fabricating it. This is a very important advance, since we can save time and money. An example of the advantage of this method in the automotive industry is shown in Figure 1.9, where the weight of a compressor bracket has been reduced in a 20%. This kind of optimization belongs to a group called structural optimization, that is divided into three categories: • Sizing optimization: This process starts with a pre-established structure and the objective is to find its optimal dimension. In Figure 1.10(a) we can see a truss structure where the design variable is the thickness of the bars..

(31) 19. 1.3. Introduction to topology optimization. Design before optimizing. Optimized design. Figure 1.9: Optimized design of a compressor bracket. Altair (2013).. • Shape optimization: In this case the domain of the structure does not change its topology. This process consists in optimizing the shape of the holes (internal and external). Shape optimization does not change the number of holes placed inside the structure, only their shape. We can see an example in Figure 1.10(b). • Topology optimization: This is the most general case of structural optimization. As we commented above, the objective is to find the optimal distribution of a limited amount of material. In this case, the process can introduce or remove any number of holes. An example is shown in Figure 1.10(c).. (a). (b). (c). Figure 1.10: (a) Sizing, (b) shape and (c) topology optimization. In our case we use topology optimization. Let us consider a fixed domain Ω. We can define the general setting of a topology optimization problem as.

(32) 20. Chapter 1. Introduction. max Φ(χ, uχ ) χ. subject to: G(χ, uχ ) = 0 fi (χ, uχ ) ≤ 0,. (1.4) i = 1, ..., m. (1.5). χ ∈ {0, 1}. where Φ is the cost function, (1.4) is the state equation, uχ is the state, χ is the control (also called design) and (1.5) are the constraints of the problem. In order to know the cost of a certain design, we have to compute it through its associate state equation (1.4). There has been a great deal of work in the subject during the last ten years, and consequently the amount of examples of applications of topology optimization and references is really overwhelming. Well known contributions on topology optimization are: Bendsøe and Sigmund (2003) is the quintessential reference book on this topic, where we can find a unified presentation of methods for the optimal design of topology, shape and material for continuum and discrete structures. This book is an updated and expanded version of the previous book Bendsøe (1995). Another recent account is the book of Christensen and Klarbring (2009), where an introduction to structural optimization is presented, as well as the treatment of different problems and methods. This book is focused on numerical solutions methods for discrete and discretized linear elastic structures. Methods for structural optimization discretized with finite elements and heavily focused in computing are presented in Spillers and MacBain (2009), where we can also find a discussion between classical methods (as heuristic schemes) and new methods based on mathematical programming. A state of-the-art report on optimality criteria methods is introduced in Rozvany (1989), where we can also find a discussion about the optimality criteria and their applications in the context of continua and direct analytical solutions. In Kirsch (1993) an introduction to the fundamentals and applications of optimum structural design is presented. In this book it is shown a collection of selected topics presented in a unified approach. In Hafka and Gürdal (1992) are presented basic details to understand optimality conditions and optimization methods, as well as the dealing with the two issues of constructing the approximate problem and obtaining sensitivity derivatives. In Deaton and Grandhi (2013) is presented a survey with the new developments, improvements and applications of finite element-based topology optimization from 2000 to 2012. In Sigmund and Maute (2013) we can find an overview, comparison and critical review of the different approaches, their strengths, weaknesses, similarities and dissimilarities in topology optimization. Of course, this is only a very small part of the vast amount of literature about this subject. The interested reader can find a large amount of development and applications in the proceedings of the 10-th and 11-th World Congress on Structural and Multidisciplinary Optimization (WCSMO). At this point we will introduce the most classical problem in structural optimization: the minimum compliance problem. We will show the formulation of the problem.

(33) 21. 1.3. Introduction to topology optimization. and how to tackle it. Also, fundamental issues that appear in this kind of problems will be discussed and overcome. The minimum compliance problem could 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 V0 . The maximization of the stiffness of a structure is equivalent to minimize the elastic energy stored in the system. We can see a general design domain with the boundary conditions and the design variables in Figure 1.11. For the sake of simplicity in the exposition, let us assume that we are in the 2D situation. Ω. f. Ω χ=0. χ=1. Figure 1.11: Design domain with boundary and load conditions. The formulation of the problem would be Z min f u dx. χ∈{0,1}. (1.6). Ω. subject to: ( div E(χ)ε(u) = f,. in Ω. boundary conditions in ∂Ω Z 1 χ dx ≤ V0 |Ω| Ω. (1.7) (1.8). where f are the external forces applied to the structure, u is the displacement field, ε(u) is the symmetric strain gradient, E(χ) is the stiffness (fourth-order tensor), Ω is the design domain and χ is the design variable. This variable appears in the state equation through the properties of the material. We consider the objective function (1.6) as the energy of deformation produced by the external applied force, then in order to maximize the stiffness of the structure, we minimize this energy. The state equation in (1.7) models the elasticity equation under small displacements hypothesis. The Hooke’s law for two materials can be expressed as follows E(χ) = EA χ + EB (1 − χ),. χ ∈ {0, 1}. (1.9).

(34) 22. Chapter 1. Introduction. where EA and EB are the stiffnesses of the two different materials. In the examples dealt in this work the second material is void, typically modelled by a very weak material with a very low stiffness EB . Boundary conditions include points where we know the displacements and points where we know the forces applied. The last constraint in (1.8) is a limit over the maximum amount of material allowed. We need this limit, otherwise the solution of the compliance would be all the design domain filled with material. It is easy to understand the trivial solution since the stiffest structure is the one with the most amount of material. The design variable can only take two values, χ = 1 and χ = 0, meaning that there is material or void, respectively. 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: when we increase the number of elements in which the structure is discretized, new holes appear in the optimized design. In order to find optimized designs, we use a numerical approach, otherwise we have to evaluate all possible combinations and the computation time will be huge. In addition, the evaluation of all these combinations does not ensure convergence due to the instability commented below. We discretize the design domain by using the finite element method (FEM). The discrete form of the problem would be min f T u χ. subject to: K(Ee )u = V (χ) ≤. f V0. where f is the load vector, u is the displacement vector, K is the stiffness matrix, that depends on the Young’s modulus Ee in each element e, numbered as e = 1, . . . , N , V (χ) and V0 is the amount of material and the maximum amount allowed. We can write K as K=. N X. Ke (Ee ). e=1. It is important to remark that the global stiffness P of the structure K is obtained by assembling the stiffness of each element, then does not represent a summation in the strict sense of the word. The usual way to solve this discrete problem is replacing our integer variables χ by density ones ρ. These new variables can take the values in the continuous interval ρ ∈ [0, 1]. The process of replacing the integer variables by continuous ones is called relaxation. Although this step is completely required, a new issue appears: gray areas. In these areas the variable ρ takes any value between 0 and 1. Intermediate values of density means microstructure: a mix of material and void at a microscale that is impossible to manufacture in principle. The method used to avoid this problem.

(35) 1.3. Introduction to topology optimization. 23. consists in penalizing this intermediate values of the variable. An efficient way to perform it is the SIMP (“Solid Isotropic Material with Penalization”) method. If we replace our integer variable by the density one, and introduce the SIMP method in (1.9) we get the next expression E(ρ) = ρp EA + (1 − ρp )EB where p is the power that usually takes the value p = 3 due to physical reasons (Bendsøe and Sigmund 1999). We can see the first example in Figure 1.12 for a mesh of 50 × 25 elements. The design domain and the boundary conditions are shown in Figure 1.12(a) and the optimized design in Figure 1.12(b).. Figure 1.12: Design domain (left) and optimized design (right). As we can see, this SIMP interpolation itself does not ensure the existence of solution. If we refine now the mesh, we will find new designs again and the alternance of materials will be more usual when the number of elements of the mesh is bigger. This effect is called checkerboard problem. The usual way to overcome it is the use of a filter (Sigmund 1994 and 1997). We distinguish between two filtering techniques (Sigmund 2007), density- and sensitivity-based methods (Bruns and Tortorelli 2001 and Bourdin 2001). Before commenting both techniques, the concept of neighbourhood will be introduced. We define the neighbourhood of an element Ne as the elements whose center is closer to the element center than the filter radius R Ne = {i | kxi − xe k ≤ R} where xi is the spatial center of the element i. • Density filtering: it was introduced by Bruns and Tortorelli (2001) and mathematically proved as a viable approach by Bourdin (2001). The objective is avoiding abrupt variations of the density variable. Two remarkable and recent references are Svanberg and Svärd (10-th WCSMO Proceedings, 2013) where the authors present four different density filters to be used within a relaxation/penalization method like SIMP, and Lazarov and Sigmund (2011) where.

(36) 24. Chapter 1. Introduction a Helmholtz-type partial differential equation is applied as an alternative to standard density filter in topology optimization problems. We define the modified element density as a function of the densities of the elements in the neighbourhood. The expression for the new density is X w(xi )vi ρi ρ̃e =. i∈Ne. X. w(xi )vi. i∈Ne. where vi is the volume of the i-th element and the weight function w(xi ) is the linearly decaying function w(xi ) = R − kxi − xe k. (1.10). With the new densities, we use FEM to solve the state equation and we change the sensitivities (derivatives of cost and constraints with respect to the design variable) in a consistent way. It is important to remark the physical sense of each variable. The initial design variable ρ is only an intermediate variable used to calculate ρ̃, that is the real variable that represents the physical density of the element. • Sensitivity filtering: it was introduced by Sigmund (1994,1997) and has become very popular in academic and commercial software. In this case we use the initial densities to solve the state equation, and then we change the real sensitivities, that is, the derivatives of the cost with respect to the design variables, by filtered sensitivities. The expression for the new sensitivities is X f ∂f = ∂ρe. w(xi )ρi. i∈Ne. ρe /ve. X. ∂f /vi ∂ρi. w(xi ). i∈Ne. where the weight function w(xi ) is the linearly decaying function given in (1.10). We take into account that the minimum value of ρe is not 0 but a small value ρmin = 10−3 . Although it has been applied successfully in many situations to date, the existence issue for the mesh-independence has not yet proved. However, it has recently showed in Sigmund and Maute (2012) that the sensitivities computed correspond to the compliance of a problem modelled by non-local elasticity. With the combination of the SIMP method and the filters the issue of the lack of classical solutions has been successfully solved. Concerning the optimizers, there exist two kind of methods for solving the optimization problem: optimality criteria methods and gradient-based methods. Optimality criteria methods are heuristic algorithms based on the necessary optimality condition. This is a classical method to.

(37) 25. 1.3. Introduction to topology optimization. solve optimization process, but in some problems is quite difficult the obtainment of the optimality criteria. Gradient-type methods are algorithms that need in each iteration step the value of the objective function and constraints and their derivatives with respect to the densities. This kind of methods, in general, converge slower to the solution than the former, however they are easier to implement, since we only have to compute the sensitivities of both the cost and the constraints. In this work we have used a gradient-based method called MMA (“Method of Moving Asymptotes”) introduced by Svanberg (1987). This method replaces our non-convex problem by a convex quadratic one, assuring that if there is a solution the method will find it. MMA has shown to be an useful tool in many problems of structural optimization to date. For the next example, we keep the design domain and boundary conditions of the first one, refining the mesh to 180 × 90 elements. We choose a radius filter R = 1.2, the power p = 3 and the maximum, volume fraction V0 = 0.4. In Figure 1.13(a) and (b) we can see the optimal design when we use a sensitivity and a density filter, respectively. In both cases the variable takes the values 0 or 1, meaning that there is no microstructure, we call them “0-1” or “black and white” designs. The value of the objective function for both optimized designs is very similar, but we can see that even though the topologies are almost the same, new bars appear in the second case. We can conclude that the topology of the optimized designs depends on the filtering technique.. (a) Sensitivity filter.. (b) Density filter.. Figure 1.13: Optimized designs. In all cases the design domain is two-dimensional, however the extension to the three-dimensional case is straightforward, but the computation time increases considerably (see Andreassen et al. 2014).. 1.3.1. Other physical contexts where topology optimization helps. The problem of minimizing the compliance has been chosen to show how the topology optimization method works. The typical issues that appear in this optimization problem have been commented and solved, but also there are other fields where we.

(38) 26. Chapter 1. Introduction. can improve the performance of the structures involved. We will comment some of the most representative ones. • Synthesis of compliant mechanisms. The displacements obtained in this kind of mechanisms come from their flexibility, meaning that they are always working with elastic deformation. They are usually designed at the microscale. They are formed by only one piece or a very few parts, then do not need lubrication and there is not wearing between parts either. In Figure 1.14 we can see the design domain, boundary conditions and optimized design for a force inverter. In this mechanism we apply a mechanical force at the middle point of its left edge, and we obtain a displacement at the middle point of its right edge, but in the opposite direction.. uout. Fin. (a) Design domain and boundary conditions.. (b) Optimized design.. Figure 1.14: Synthesis of compliant mechanisms. We can improve the performance of these devices by introducing the supports as a new variable, then we do not restrict ourselves to the supports given by the problem. The simultaneous design of structure and supports let us obtain gains up to 50%. • Dynamics problems. We can distinguish two kind of problems inside this section, free vibrations and forced vibration problems. The first one is commonly used to increase the first natural frequency of a structure. This is interesting when we want to avoid vibrations that can collapse it. We can see an example of a structure with the first eigenfrequency maximized in Figure 1.15(a). The second case is very similar to the minimum compliance problem, we want to maximize the stiffness of a structure. In fact, the only difference is that the mechanical force depends on time, then we have to take into account inertial forces. An example is shown in Figure 1.15(b).. • Metamaterial design. The response of a structure depends on the mechanical properties of the material which is made of. We can improve or create materials with characteristics.

(39) 1.3. Introduction to topology optimization. 27. (a) Maximization of the first eigenfre- (b) Minimization of compliance for a dyquency. namic load.. Figure 1.15: Optimal design in dynamics. that are not usual in nature. This problem is solved in a domain that we call unit cell. The structure with the new characteristics is formed by repeating these cells periodically. In Figure 1.16(a) we can see a material with negative thermal expansion coefficient, this means that it contracts when we apply heat. In Figure 1.16(b) a material with negative Poisson’s ratio is shown, that means that when we stretch the structure, it gets bigger in the perpendicular axis.. (a) Microstructure with negative Poisson’s coefficient. Dapogny (2014a).. (b) Microstructure with negative thermal expansion coefficient. Dapogny (2014b).. Figure 1.16: Optimal design of materials. Both examples are very interesting in industry. For instance, we can create a structure with parts with different thermal expansion coefficients that do not change its size with temperature changes. The same idea can be used with the Poisson’s coefficient. • Band gap. Propagation of waves in the structure might be interesting in certain situations. In optoelectronics we are interested in guiding the waves from one point to.

(40) 28. Chapter 1. Introduction another by a designed way. An example of wave-guiding is shown is Figure 1.17(a). A very interesting example where we want to avoid the propagation of waves is in the case of earthquakes. In this particular case we try to avoid the propagation of the wave in order to minimize the damage produced. In Figure 1.17(b) we show how the propagation of the seismic wave is suppressed in the right part of the structure.. (a) Phononic waveguide. Femto (2008).. (b) Pressure wave with seismic crystals. Alagoz (2009).. Figure 1.17: Numerical simulations for band-gap problems. The problem of band gap covers from microscale in electronics to macro-scale with the design of buildings.. 1.4. Aim of the thesis: optimal design of modal piezoelectric transducers. In this section, we will make a brief introduction to modal transducers. We will comment the mathematical process to design them in one and two dimensions. Once this concept has been introduced, we will set the aim of this doctoral thesis and the mathematical formulation of the problems that will be addressed.. 1.4.1. Modal sensor/actuators. In Section 1.1, we introduced the concept of modal sensors/actuators (MSA). A modal sensor/actuator is a device that can excites/measures a specific mode of a structure. Thanks to the reciprocity of the piezoelectric effect, the shape that observes a vibration mode is the same that the one which excites it. This concept was introduced by Lee and Moon (1990). The first point to be taken into account is that only the area of the structure that is covered by the electrode can be electrically affected, in order to collect charge in the sensor case, or to generate the induced force by the piezoelectric effect in the actuator case. The objective is the design of the polarization profile of the piezoelectric layer in order to obtain the MSA. Let us consider a laminated plate with electrodes bonded to both, top and bottom surfaces. Subscript L = 1, 2, 3.

(41) 1.4. Optimal design of modal piezoelectric transducers. 29. denotes respectively first, second and third layer. In our case the charge collected in the electrodes is the same, then q1 = q3 ; q1 is the charge collected in the first layer and q3 in the third. The expression for the collected charge in one layer is Z ∂2w ∂2w ∂2w ql (t) = −z̃l P e31 2 + e32 2 + 2e36 dx dy, l = 1, 3 (1.11) ∂x ∂ y ∂x∂y S where e31 , e32 and e36 are piezoelectric constants, z̃l is the z-coordinate of the midplane of the l-th layer, z̃1 = z̃3 = −(hp + hs )/2, hp and hs are the thicknesses of the structure and piezoelectric layers, respectively, S is the area covered by the electrode and w = w(x, y, t) is the out-of-plane displacement. P = P (x, y) is a distributed function that represents the polarity of the electrode, meaning P = −1, P = 1 and P = 0 negative, positive or null-polarity (which means that there is no electrode), respectively. A scheme with the geometric parameters is shown in Figure 1.18. From (1.11) we can obtain the one-dimensional sensor equation, assuming that there is no y-dependence in the displacement, w = w(x, t). The expression for the charge collected, up to a scaling factor, is Z Lx ∂2w (1.12) ql (t) = F(x) 2 dx ∂x 0 where Lx is the length of the plate and F(x) is the normalized surface electrode width, which is given by the expression F(x) =. Ly /2. Z. P (x, y) dy −Ly /2. The objective of a MSA is to isolate the response of one vibration mode. The out-of-plane displacement of a one-dimensional plate can be decomposed into the modal summation of the vibration modes ∞ X. w(x, t) =. Am (t)φm (x),. (1.13). m=1. where Am (t) and φm (x) are the modal coordinate and the mode shape of mode m, respectively. Now, inserting expression (1.13) into (1.12), we arrive at ql (t) =. ∞ X. Am (t)Bm. m=1. where Bm = −. Z 0. a. F(x). d2 φm (x) dx dx2. (1.14). From expression eq.(1.14) we can see choose F equal to a scaling factor times the second derivative of the mode shape. In order to get negative values of F, we have to vary the polarization profile P (x, y). With this concept we can create a modal.

(42) 30. Chapter 1. Introduction Lx Electrode. Ly. P(x,y). Top view. hp. 1 Piezoelectric layer 2. hs. Host structure. hp. 3 Piezoelectric layer. Side view. Figure 1.18: Scheme of a MSA. sensor that measures directly the mode chosen. In Figure 1.19 an example of MSA for the first and second vibration modes is shown. The figures on the top represent the mode shapes and the figures on the bottom illustrate the surface electrode. For the first eigenmode the polarity of the whole electrode is positive, whereas for the second eigenmode the electrode shows both polarities.. (+). (+). (-). (a) First eigenfrequency.. (b) Second eigenfrequency.. Figure 1.19: Mode shapes (top) and top views of optimized sensors (bottom). For beams and one-dimensional plates (basically treated as beams with a slight modification in the flexural rigidity term), the problem of finding MSA reduces to computing the normalized surface electrode width F(x), where this function contains all the necessary information to construct the device: on the one hand, its absolute value indicates the gain distribution of the transducer and on the other hand, it forces the polarization profile (positive or negative) of the piezoelectric layers to vary along the x-direction in accordance with its profile. The condition which lets us construct MSA is the orthogonality principle among the vibration mode shapes of the structure to be controlled. This orthogonality.