Automated Design of a MEMS Resonator - Semantic Scholar

Automated Design of a MEMS Resonator Jason D. Lohn, William F. Kraus, and Gregory S. Hornby

Abstract—As the application and complexity of microelectromechanical (MEMS) devices increases, there is a corresponding need for automated design and optimization tools to augment engineers’ design skills. Evolutionary computation provides a set of tools that may prove very effective in this application domain. Here we present a novel evolutionary computation encoding scheme for specifying MEMS designs, and use this scheme to evolve a MEMS meandering resonator. Using fewer evaluations, we are able to evolve designs that perform as well as those previous reported.

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I. INTRODUCTION

icroelectromechanical systems (MEMS) have been successfully applied in a number of areas, including accelerometers, gyroscopes, pressure sensors, and optical switches. As the desire for improved performance and increased complexity of MEMS devices progresses, there is a corresponding need for automated design and optimization methodologies that allow a designer to explore more of the potential design space. Genetic algorithms have been applied to the design of simulated MEMS devices [3][7][8][9], as well as MEMS automated mask-layout and process synthesis [4][5][6]. This has been facilitated by the availability of MEMS simulation software. Nevertheless, simulating all but the most trivial MEMS device remains computationally expensive, and there is a premium on developing encoding schemes that can efficiently explore more design space using fewer total evaluations. Both Zhou et. al. [8][9] and Kamalian et. al. [3] employed a genetic algorithm to optimize the size, orientation and number of segments that made up the legs of a meandering resonator. While the simulated performance of the evolved designs was impressive, the designs themselves were characterized by acute angles between the beams comprising the resonator legs, making them unsuitable for fabrication. In addition, constraints placed on intersections between beams meant that not all genotypes produced valid designs, reducing the overall effective population size. Our goal was to develop a concise and extendable genetic programming language that was flexible enough to support the construction of a wide a variety of valid topologies,

Manuscript received March 15, 2007. This work was supported in part by NASA Ames Research Center grant NNX07AE45A. Jason D. Lohn is with Carnegie Mellon University, West Campus/Bldg 23, 5000 Forbes Ave, Pittsburgh, PA, 15213, USA (phone: 650-335-2802; fax: 650-603-7032; e-mail: [email protected]). William F. Kraus and Gregory S. Hornby are with the University of California at Santa Cruz, NASA Ames Research Center, Mountain View, CA 94035 USA (e-mail: bkraus, [email protected]).

while at the same time facilitating the evolution of devices with desirable performance characteristics, and using as few simulator evaluations as possible. While previous work used genetic algorithms as a means of parameter optimization within a fairly constrained design space, our goal was to develop an encoding scheme that facilitated a wide range of valid structures. In this paper, we present the results of experiments designed to evolve a MEMS meandering resonator using a genetic programming language, and we compare these results to similarly evolved devices [3][8][9]. II. EXPERIMENTAL SETUP A genetic algorithm was used to evolve the legs of a 2-D meandering resonator as described in [3]. A meandering resonator consists of a central mass suspended by up to four ‘legs,’ each leg comprised of a series of beams terminated

center mass

Fig. 1. A meandering resonator consists of a center mass suspended by four legs, each leg comprised of a series of beams terminated by an anchor.

by an anchor. We attempted to evolve a meandering resonator using a both a single objective genetic algorithm (SOGA) as well as a multi-objective genetic algorithm (MOGA), and compared our results to those in [3] and [8][9], respectively. A. Encoding Scheme The central mass of the resonator was fixed for all generated designs, and consisted of four 100 um by 20 um by 2 um beams laid out in a square. This was the same dimensions used in [3] and [8][9] [R. Kamalian, pers. comm., N. Zhou, pers. comm.], although they differ from those published in [8][9] as a result of typographical errors [N. Zhou, pers. comm.]. Each leg of the resonator was encoded in a separate,

variable length chromosome, consisting of a linear byte array. Each individual in the population had four such chromosomes, and each chromosome had an initial length of 30 instructions. The length of an individual chromosome could increase or decrease over time due to crossover, and was not constrained. The encoding scheme used a “growth” metaphor, and contained eight instructions which were encoded into the lower 4 bits of each byte: go_straight, go_straight_w, go_left, go_right, branch, terminate, begin_repeat, and end_repeat. The initial frequency of each instruction in the population, as well as the likelihood that any given byte would mutate into a given instruction, was arbitrarily set at 0.5625 for go_straight instruction, and 0.0625 for all other instructions. Note that the higher combined probability of the go_straight and go_straight_w instructions meant that any given section of a leg was more likely to go straight than to turn. Because the instruction set constrained the orientation of beams to right angles, all evolved devices exhibited Manhattan geometry. This not only simplified the beam intersection tests, but also produced designs that were more amenable to fabrication, avoiding the potential kinks and sharp acute angles found in previous evolved designs [3][7][8][9]. Each leg was constructed by reading the chromosome from beginning to end, extending the leg outward from a predetermined corner of the center mass. Construction would end when the reading head encountered either the end of the chromosome, or a top-level terminate instruction. The go_straight, go_straight_w, go_left, and go_right instructions extended the leg by fixed 20 um segment in the prescribed direction. The branch instruction caused the reading head to effectively split – the reading head would continue reading the chromosome and building the leg until either a terminate instruction or the end of the chromosome was reached. If the former, the reading head would begin implementing subsequent instructions from the point at which the branch initially occurred. Branches could be recursive. reading head SSBLSRSR . S

reading head SSBLSRSR . S

reading head SSBLSRSR . S Fig. 2. Each leg of the resonator was constructed by sequentially reading the corresponding chromosome and growing the leg outward from a corner of the central mass.

The high four bits of the go_straight_w, go_left, and go_right instructions contained a beam width specification, which was allowed to vary between 2 and 10 um. A distinction was made between the go_straight_w and the much higher frequency go_straight instruction in order to avoid excessive changes in beam width, which would have made any evolved design less amenable to fabrication. Beam height for all sections of a leg was fixed at 2 um. A repeat was started when a begin_repeat instruction was encountered. The end of a repeat was signified by either an end_repeat or a terminate instruction. Nested repeats were not allowed (i.e. repeating sections within repeating sections were ignored), since preliminary test runs had indicated that nested repeats could quickly produce complex fractal-like device topologies that overwhelmed the MEMS simulator. The high four bits of the begin_repeat instruction encoded the number of times a section would be repeated, anywhere from one to three times. Instructions that resulted in a collision with the central mass were simply ignored. Instructions that traced over an existing path were also ignored. Loops resulting from the intersection of two beams at right angles were allowed. After all legs were constructed, an anchor was placed at the terminal end of any free standing beam. An anchor’s width was set to the beam width of the immediately preceding leg segment. An anchor’s length was arbitrary fixed at 10 um. In order to prevent the evolution of designs with excessively long legs, the total size of the resonator was constrained to a 500 um by 500 um square surrounding the resonating mass. Instructions that caused a leg segment to extend beyond this boundary were ignored. This encoding scheme allowed for resonators with less than four legs, and because problematic instructions were ignored, all chromosomes produced valid designs. Finally, some of the trials had an additional “symmetry” constraint. In these cases, only one chromosome was evolved, and the instructions from that chromosome were used to build all four resonator legs. Note that because the encoding scheme ignored invalid instructions, this constraint could still result in designs that were not exactly symmetrical. B. Genetic Algorithm Two experiments were conducted using a SOGA and a MOGA, respectively. In both cases, a generational genetic algorithm with roulette wheel selection was used. The point mutation probability per gene was set to 1%, and the crossover rate was set to 80%. One point crossover occurred between respective pairs of chromosomes. In the SOGA experiment, the objective was to evolve a resonator that had a lowest modal frequency of 14.916 KHz (93723 ω), where fitness was inversely proportional to the absolute deviation of a device’s simulated modal frequency from the target frequency [3]. Elitism was implemented by copying the top 5% from each generation into the next.

The objective of the MOGA experiment was to evolve a resonator that, in addition to having a lowest modal frequency of 14.916 KHz (93723 ω), also had x and y stiffness values of 1.90 N/m and 0.56 N/m, respectively [8][9][N. Zhou, pers. comm.]. (In [3],[8], and [9], the units for the target frequency were incorrectly reported in Hertz, instead of radians per second.) Fitness values were assigned according to pareto ranking, where individuals on the pareto front had the highest fitness. Elitism was implemented by copying all individuals on the pareto front from each generation into the next, limited to a maximum of one third the population size. All designs was evaluated using SUGAR (version 3.0), a software package developed at the University of California at Berkeley which utilizes nodal analysis to simulate the behavior of MEMS devices [1][2]. Modal frequency and stiffness were calculated using a modified version of the Matlab script used by Zhou [pers. comm.]. Stiffness values were calculated by applying a small force at each of the four corners of the center mass and finding the deflection of the device [N. Zhou, pers. comm.]. Forty trials were conducted for each of the SOGA and MOGA experiments. In each case, ten asymmetrical design trials were run using a population size of 25, ten asymmetrical design trials were run using a population size of 100, ten symmetrical design trials were run using a population size of 25, and ten symmetrical design trials were run using a population size of 100. Each trial was initialized with a different random seed, and all trials were terminated after 100 generations since we consistently observed the algorithm converging well before.

evolved designs with a deviation of less than a single Hz, and all trials produced designs that were within 14 Hz of the target frequency. For population sizes of 100, the asymmetrical SOGA trials were significantly better at finding designs near the target frequency than were the symmetrical SOGA trials (Wilcoxon-Mann-Whitney test, p = 0.043). However, for population sizes of 25, the asymmetrical and symmetrical SOGA trials were not significantly different (WilcoxonMann-Whitney test, p = 0.315). While the asymmetrical SOGA trials with a population size of 100 performed significantly better than asymmetrical SOGA trials with a population size of 25 (Wilcoxon-MannWhitney test, p = 0.015), there was no significant difference between the symmetrical SOGA trials of different population sizes (Wilcoxon-Mann-Whitney test, p = 0.094). B. MOGA Results The 520 pareto solutions found across all asymmetrical MOGA trials (both Npop = 100 and Npop = 25) were pooled in order to find the pareto front across all asymmetrical trials. Of the 218 unique pareto solutions that remained, only 25 were on the pareto front - 6 of these were from 4 trials using a population size of 25, and remaining 19 were from 6 trials using a population size of 100. The fact that there were so many duplicate solutions was due to the fact that different phenotypes generated devices that had identical frequency and stiffness values as a consequence of

III. RESULTS A. SOGA Results Of the ten asymmetrical SOGA trials with a population size of 100, five generated designs matched the target frequency, after 47, 58, 81, 94, and 97 generations respectively. Nine of the ten trials produced designs that were within a single Hertz of the target frequency, and all trials produced designs that were within 4 Hz. For the asymmetrical SOGA trials with a population size of 25, the best design appeared after 90 generations, deviating from the target frequency by only 0.01 Hz. Six of the ten trials evolved designs with a deviation of less than a single Hz, and all trials produced designs that were within 5 Hz of the target frequency. Of the ten symmetrical SOGA trials with a population size of 100, five generated designs were within a single Hertz the target frequency, after 20, 28, 39, 80, and 96 generations respectively. All ten trials produced designs that were within 4 Hz of the target frequency. For the symmetrical SOGA trials with a population size of 25, the best design appeared after 39 generations, deviating from the target frequency by 0.11 Hz. Four of the ten trials

Fig. 3. Evolved asymmetrical SOGA designs from one of the best Npop = 100 and the best Npop = 25 trials, respectively.

Fig. 4. Evolved symmetrical SOGA designs from one of the best Npop = 100 and the best Npop = 25 trials, respectively. TABLE I DEVIATIONS FROM THE TARGET FREQUENCY (HZ) FOR THE BEST EVOLVED INDIVIDUAL FROM EACH SOGA TRIAL AT THE END OF 100 GENERATIONS FOR NPOP = 100 (10000 TOTAL EVALUATIONS) AND NPOP = 25 (2500 TOTAL EVALUATIONS). Trial 1 2 3 4 5 6 7 8 9 10 Avg

Asymmetrical Npop=100 0.05 0.84 0.01 0.00 0.00 0.71 0.00 0.00 3.60 0.00 0.52

Npop=25 0.28 0.01 0.95 0.81 4.63 1.13 0.15 0.60 4.70 2.43 1.57

Symmetrical a Npop=100 0.05 0.11 1.84 3.11 0.28 0.03 0.04 3.12 1.22 2.44 1.22

Npop=25 0.64 0.11 4.69 13.34 12.01 7.54 5.04 0.12 1.26 0.24 4.51

the presence of introns – non-coding sections of the genome. If we compare the remaining 25 unique solutions to the 6 solutions presented by Zhou et. al. [8][9], all but one are on the pareto front, whereas three of the solutions found by Zhou et. al. [8][9] are not. The 251 pareto solutions found across all symmetrical MOGA trials (both Npop = 100 and Npop = 25) were pooled in order to find the pareto front across all symmetrical trials. Of the 164 unique pareto solutions that remained, only 9 were on the pareto front – all of these were from 7 trials using a population size of 100.

Fig. 5. The top plot displays the average frequency deviation of the best SOGA individual for each generation across ten trials for each type of run. The bottom plot displays the average frequency deviation as a function of the number of evaluations.

If we compare these remaining 9 unique solutions to the 6 solutions presented by Zhou et. al. [8][9], 6 are on the pareto front, whereas 3 of solutions found by Zhou et. al. [8][9] are not. Finally, when we pooled the pareto solutions (N = 771) across all asymmetrical and symmetrical trials and removed the 399 redundant values, 28 solutions were on the pareto front. Of these, 18 were from asymmetrical trials. When we include Zhou’s 6 solutions, 26 of the 28 solutions evolved using our genetic programming scheme remain on the pareto front, whereas only two of Zhou’s solutions remain.

IV. DISCUSSION The use of a genetic programming language to construct MEMS device topology, rather than simply optimizing a set of parameters using a traditional genetic algorithm, allows the evolutionary process to explore much more of the design space. While using a genetic algorithm to optimize a set of parameters presupposes knowledge about the topology, genetic programming relaxes this constraint since the topology itself is evolved along with the parameter values. The implementation of a concise encoding scheme becomes increasing important as the computing resources required to simulate a MEMS device increases. In addition, our genetic programming representation allows the implicit

Fig. 7. A pareto solution from a symmetrical MOGA trial. This design was evolved using a population size of 100.

Fig. 6. Three examples pareto solutions from the asymmetrical MOGA trials. The top two designs were evolved using a population size of 100. The bottom design was evolved using a population size of 25.

construction of more complicated structures such as loops, branches, and even serpentine springs. And unlike previous work [3][8][9], all of the phenotypes generated by the genetic programming encoding were geometrically viable. The number of evaluations required to achieve a design goal is a better indicator of the efficiency of a genetic algorithm than total computer time used, since the latter can vary widely depending on system load and hardware used. Using a SOGA, Kamalian et. al. [3] were able to achieve convergence to a stated design objective of a modal frequency within 100 ω (15.9 Hz) of the target by 15 generations for a population of 400 individuals (6000 evaluations). The best individual was 11 ω (1.75 Hz) from

the target frequency [R. Kamalian, pers. comm.].). (Note that in [3], the units for the design objective were incorrectly reported as Hertz [R. Kamalian, pers. comm.].) Using a genetic programming language, we were able to achieve the same design goal using, on average, less than a third to one half of the number of evaluations - by generation 17 for an asymmetrical design population size of 100 (1700 evaluations), by generation 28 for a symmetrical design population size of 100 (2800 evaluations), by generation 43 for an asymmetrical design population size of 25 individuals (1075 evaluations), and by generation 66 for a symmetrical design population size of 25 individuals (1650 evaluations). Additionally, our encoding scheme was able to find several designs that were within a single Hertz of the target frequency, and all of our trials achieved the design goal of [3]. Interestingly, while Kamalian et. al. [3] reported that simulated annealing (SA) was in some cases faster than genetic algorithms in synthesizing valid designs, they nevertheless concluded that SAs were less robust than GAs for many MEMS synthesis problems. For the MOGA experiment, we were able to evolve designs that had similar or better performance to the designs presented by Zhou et. al. [8][9] while requiring fewer total evaluations (Zhou et. al. [8][9] used a population size of 400 for 30 generations). This was achieved despite the fact that we constrained our designs to Manhattan geometry and explicitly restricted their overall size. Although symmetrical designs are easier for a human to design, our results suggest that asymmetrical designs were as good if not better at producing topologies with the desired performance. Presumably, the use of asymmetry increases the potential design space available to the search algorithm. The potential for non-intuitive, asymmetrical design solutions favors the use of an automated design process such as evolutionary programming. One of the advantages of using smaller population sizes is that they require fewer evaluations for the same number of

generations. However, there is an increased risk that the smaller population will converge too quickly and become trapped on a local optimum. Choosing the appropriate population parameters is a tradeoff between reducing the number of evaluations and potentially reducing the effective search space. Nevertheless, the efficacy of the encoding scheme itself can play an important role in how well a genetic algorithm searches the design space. For example, a scheme which always produces valid topologies, and allows for the development of introns within the genome, can be very beneficial. For example, some studies have suggested that introns can facilitate the evolutionary process by protecting good building blocks from the destructive effects of crossover [10]. This particularly seems to be the case for our MOGA experiment, where identical phenotypes were created from different genotypes. The ease and reliability of fabrication is an important consideration in any MEMS design methodology. The compactness of the overall size and the explicit Manhattan geometry of the designs evolved using the genetic programming encoding described in this paper would appear to better candidates for fabrication than the designs presented by Zhou et. al. [8][9]. In fact, overall device size, as well as amenability to fabrication, would be ideal candidates for additional objectives in future MOGA experiments. V. CONCLUSION Evolving MEMS structures using evolutionary computation is a relatively new line of research. By employing a flexible genetic programming language, we were able to achieve similar or better results than previously reported using smaller population sizes and fewer total evaluations. Several areas conducive to future investigation include compensating for fabrication by taking into account parameter “drift” when evaluating a device design, coevolving the frequency distribution of the programming instruction set, applying the encoding scheme to more complex problems using an industrial strength MEMS simulator, and parallelizing the genetic algorithm to run on a cluster.

ACKNOWLEDGMENT We gratefully acknowledge helpful conversations with Ningning Zhou and Stephen Bart. REFERENCES [1]

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