Hierarchical MEMS synthesis and optimization - Semantic Scholar

Hierarchical MEMS Synthesis and Optimization Ying Zhang1, Raffi Kamalian2, Alice M. Agogino3, and Carlo H. Séquin4 1

Systems Engineering Program, University of California, Berkeley Department of Mechanical Engineering, University of California, Berkeley 4 Department of Electrical Engineering and Computer Sciences, University of California, Berkeley 2, 3

ABSTRACT A hierarchical MEMS synthesis and optimization architecture has been developed for MEMS design automation. The architecture integrates an object-oriented component library with a MEMS simulation tool and two levels of optimization: global genetic algorithms and local gradient-based refinement. An object-oriented data structure is used to represent hierarchical levels of elements in the design library and their connectivity. Additionally, all elements encapsulate instructions and restrictions for the genetic operations of mutation and crossover. The parameterized component library includes distinct low-level primitive elements and high-level clusters of primitive elements. Surface micro-machined suspended resonators are used as an example to introduce the hierarchical MEMS synthesis and optimization process. Keywords: MEMS Synthesis, Multi-Optimization, Genetic Algorithms, Evolutionary Design.

1. INTRODUCTION MEMS (Microelectromechanical Systems) is a rapidly growing technology for miniature devices using fabrication processes similar to those of integrated circuits (ICs). MEMS technologies are emerging for a diverse range of applications in bioengineering, automotive engineering, telecommunications, environmental monitoring and space exploration. Similar to the early days of IC design, hand calculations are often the norm in MEMS design practice as there are no powerful or widely accepted computer-aided design tools. Different pathways have been used to approach automated synthesis of MEMS devices. Muhkerjee, Fedder, et al.[1,2] used numerical parameter optimization to optimize the geometry of a given MEMS configuration and applied it to a system level design of band pass filters. Since the configuration of the design was fixed, the method only allowed the local optimization of a parameterized design. Ma, Li and Antonsson[3,4] applied genetic algorithms (GAs) to automated mask-layout and process synthesis for MEMS. Their focus was on evolving the optimal mask and process sequence given a user defined final shape and not on the early stages of MEMS design synthesis. Zhou, et al.[5,6] were the first to demonstrate MEMS

Figure 1. "Winning" MOGA design[7].

1

Ying Zhang: [email protected], Tel: (510)6438146. Raffi Kamalian: [email protected], Tel: (510)6438146. 3 Alice M. Agogino: [email protected], Tel: (510)6426450. 4 Carlo H. Séquin: [email protected], Tel: (510)6425103 2

96

Smart Structures and Materials 2005: Smart Electronics, MEMS, BioMEMS, and Nanotechnology, edited by Vijay K. Varadan, Proceedings of SPIE Vol. 5763 (SPIE, Bellingham, WA, 2005) 0277-786X/05/$15 · doi: 10.1117/12.600376

component synthesis using a multi-objective genetic algorithm (MOGA) on a simple MEMS device, a ‘meandering resonator’ for three objectives. Zhou[7] was also the first to run a successful "human against machine" competition in which students in a graduate MEMS design class at the University of California at Berkeley competed with her MOGA algorithm in the design of a comb drive and a serpentine spring. The best design was judged to be one that met the spring deflection requirement and other constraints with the minimum envelope area. Thirty-five homework solutions were collected and approximately twenty of them had complete designs. The best MOGA design was smaller than all of the feasible student designs; only one student had a design that was even close. Kamalian, et al.[8,9] extended Zhou’s work to more advanced MEMS synthesis problems and explored the role of geometric constraints and human interaction in the MEMS resonator synthesis case study. Using a "test-feedback" strategy (the feedback loop in Figure 2) the best MOGA designs generated were fabricated, tested and evaluated. The results were used to improve the design rules so as to guide the synthesis algorithms to reach Pareto-optimal designs faster, taking expected simulation and fabrication errors into account. The work of Zhou and Kamalian required recoding the MOGA formulation for each new problem, motivating Graf's[10] development of an object-oriented data structure that could be used to easily assemble and configure new design problems. In this paper, we present a general hierarchical MEMS synthesis and optimization architecture using object-oriented programming that brings our MEMS synthesis tool a step closer to practical MEMS design automation. The evolutionary design starts from a set of randomly mutated populations based on an initial design. Each individual of the population is composed of primitive elements or clusters of primitive elements in an object-oriented component library. This population goes through GA operations (crossover and mutation) following the instructions and restrictions encapsulated within each component element. This process drives the evolutionary algorithm towards designs that satisfy the performance specifications and design constraints. Automated synthesis of a suspended MEMS resonator is used as an example to explain and test the hierarchical synthesis process in this paper. The advantages of integrating a conventional numerical optimization algorithm with a global stochastic optimization method – MOGA – are explored.

2. MEMS DESIGN SYNTHESIS AND OPTIMIZATION In this section, our hierarchical design synthesis process and optimization strategy are described (shaded box in Figure 2). An object-oriented component library was developed to enable rapid prototyping and reuse of successful MEMS design components. 2.1. Design Building Blocks for Component Library Each MEMS object component is represented in the data structure as an entity with its geometrical parameters, connectivity, operation instructions and restrictions. The connectivity of the component is determined by the properties of its external nodes, which not only represent the connection with the other components, but also are associated with instructions concerning any symmetry properties. Each component also has the following flags: mutation, crossover, and symmetry. These flags determine the type of mutation and crossover operations allowed by the component during the GA evolutionary processes and whether it has a symmetric counterpart in the synthesized MEMS device. The component library consists of two levels of GA building blocks: low-level primitive elements and high-level clusters, which are composed of primitive elements. Each building block provides an implementation for the optimization routines to manipulate the block. The following functions are common to each block: The mutate(•) function determines what mutation operation are allowed in each building block. The rotate_z(α) function rotates the block around it’s center in the x-y plane by the specified angle α. The mirror_x(•) and mirror_y(•) functions are used to invert the x- and y-coordinates of the block, respectively. All the building blocks offer an interface to interact with the MOGA optimization process. In order to synthesize the design that can be built using the existing MUMPs (Multi-User MEMS Processes) fabrication technology[11], every primitive building block has an invisible rectangular, axis-aligned bounding box used for layout intersection checks[10]. With a focus on the design of MEMS suspensions, the key primitive building blocks currently implemented in the component library include anchors, beams, plates, and comb drives. Clusters – aggregations of primitive building blocks, such as a dog-bone (I-shaped) center mass, polyline, serpentine and compound springs – have also been added

Proc. of SPIE Vol. 5763

97

into the component library. Following the same data structure, high-level clusters, such as examples of commonly used MEMS devices, will be added into the design library in order to provide a case library.

Design Objectives Constraints, Stop Criteria Fabricated & Validated Designs Knowledge Base and Component Library Casebased Indexed Design Library

Fabrication & Simulation Constraints

Evaluation, Validation & Refinement of Design Rules

Design Encoding Scheme: object elements & clusters

Initial Designs

Evolutionary Synthesis Algorithm: genetic selection, mutation, Pareto-ranking based on performance

Local Optimization, & Human Refinement

Simulation Engine

Fabrication, Testing & Characterization Finite Element Analyses (FEM)

Synthesized Designs

Design Synthesis Module

Figure 2. MEMS design synthesis flow chart.

2.2. MEMS Design Synthesis Process To start the synthesis process, the designer needs to specify the design objectives, constraints and stopping criteria. Drawing from the MEMS design component library, an initial valid design or a set of designs is loaded into the design synthesis module, instantiating the building block data structure in the component library (see Figure 2). In the design synthesis module, the MOGA optimization algorithm mutates the initial design(s) to generate the population for the first generation. All of the designs in a generation are evaluated by a MEMS simulator – called SUGAR[12,13] – to determine their performance attributes. If the MOGA stopping criteria are not satisfied, a fitness value

98


is assigned to each design based on its performance attributes. For problems with multiple objectives, a Pareto ranking of the designs is performed. The genetic operations: selection, mutation and crossover are then applied to designs in the current generation to generate the designs of the next generation. The new generation goes through the same process again. The MOGA optimization process stops when the stopping criteria are met. Our research shows that the GA stochastic searching method is best suited for deriving the general shape and topology at the conceptual design stage. However, finding a global optimal design in a practical time period can be difficult due to the expensive computational costs on the simulation for large populations in each generation. A conventional gradient-descent optimization algorithm has been implemented to further refine the best designs resulting from MOGA synthesis. The flow chart for the MEMS design synthesis process is shown in shaded box in Figure 2.

3. CASE STUDY -- MEMS RESONATOR DESIGN 3.1. Resonator Topology and Building Blocks The automated synthesis of a MEMS suspension – a surface micro-machined resonator – is used as an example to illustrate the design synthesis process in Figure 2. The resonator design is composed of an I-shaped center mass, comb drives, anchors and springs as shown in Figure 3 for an example with x- and y-symmetry. The I-shaped center mass is a cluster composed of a rigid plate and four short, wide beams, that can be treated as rigid elements. The suspension is a I-shaped center mass Comb-drive

anchors spring

Figure 3. Composition of a resonator example.

polyline spring, a cluster of three-beam primitives. In this example, the dimensions of center mass and comb drives are fixed and the springs are evolved to achieve the design requirement. Each building block in the design is represented as an entity with its geometrical variables, connectivity, operation instructions and restrictions. For example, the I-shaped center mass is shown in Figure 4 with geometrical dimensions in micrometers. Since the dimensions of the center mass are fixed for this example, the crossover and mutation flags for this component are turned off. The connectivity of the center mass is represented by the six nodes shown in Figure 4. Besides the location parameter, each node also offers optional parameters to define an x- or y-axis symmetry options. For the symmetric example in Figure 3, the symmetry flags for the spring and for the anchor connected to it are set to be true. 200 100 100

40

Connecting node

Figure 4. I-shaped center mass component for the resonator (unit: µm).


99

3.2. Experiment Setup To test the object-based MOGA for MEMS design synthesis, the resonator design specifications used in Zhou, et al.[6,8] were used to synthesize designs through the new hierarchical MEMS design process. For the fixed I-shaped center mass the geometry shown in Figure 4 was used. Each comb drive attached to the center mass had 11 fingers, 50µm long by 4µm wide with 3µm gap, plus a 4µm wide spine. To explore the flexibility of the new data structure, the spring used for the resonator not only included polyline springs (called ‘meandering springs’ in Zhou, et al.[6,8]), but also a new ‘serpentine spring’ clusters. The parameter constraints for the springs are listed in Table 1. There are three objectives in this example: (1) minimize the deviation from the target resonant frequency of 10,000Hz; (2) guarantee that the lateral stiffness, Kx, is at least 10 times larger than the vertical (direction of resonance) stiffness, Ky; and (3) minimize the device area, defined by a rectangular bounding box around the device. Several cases with different symmetry and angle constraints were subjected to the resonator synthesis program. Each test case was subjected to five independent MOGA runs, each running for 50 generations with a population of 400 for each generation. The design with the minimum area, subject to the constraint that Kx/Ky ≥ 10, and that its resonant frequency lies within 5% of the target resonant frequency, was recorded as the best design after MOGA optimization. The best design in each case was then further subjected to gradient-descent optimization with the following goals: (1) Bring the resonant frequency to within 0.01% (1Hz) of the Target of 10,000 Hz, and (2) minimize the area of the bounding rectangle. Table 1. Design parameter constraints for MEMS resonator case study.

Polyline Spring Parameter Max number of beams per leg Min number of beams per leg Max beam length Min beam length Max beam width Min beam width

Value 7 1 100µm 10 µm 10 µm 2 µm

Serpentine Spring Parameter Max number of loops Min number of loops Max beam length Min beam length Max beam width Min beam width

Value 5 1 100 µm 10 µm 10 µm 2 µm

Table 2. Resonator synthesis testing cases.

Case 1 Case 2 Case 3 Case 4 Case 5 Case 6

Non-symmetric polyline spring legs with unconstrained angle x-symmetric polyline spring legs with unconstrained angle xy-symmetric polyline spring legs with unconstrained angle xy-symmetric polyline spring legs with 90° angles xy-symmetric serpentine springs xy-symmetric resonator with mixed springs

3.3. Resonator Design Results after MOGA Optimization The best designs resulting from the MOGA optimization process for different test cases are shown in Figure 5. The designs in cases 1, 2, 3 and 4 are resonator designs with polyline springs, with different symmetry and angle constraints. We note that the least constrained case 1 did not initially result in a feasible design after five MOGA runs. In this case, the MOGA optimization was repeated using the design with the best performance from the previous run as an initial design. Twelve good feasible designs were generated and the best one is shown in Figure 5a. In general, one would expect an optimal design problem with fewer constraints on the search space to yield designs with equal or better performance than those that are more highly constrained. However, this example demonstrates clearly that the more design knowledge one can put into the problem specification the better results one will typically get. For this resonator example, engineering intuition and experience tell us that an asymmetrical suspension geometry is not likely to improve the desired behavior. Thus the introduction of x or xy-symmetry simplifies the problem. It reduces the dimension of the search space by a factor of four and thus allows for an exponentially deeper search in a given amount of time. Thus the likelihood of finding a more optimal design is very high.

100


(a) Case 1*

(b) Case 2

ωr=10,407Hz Area=181E-9m2 Kx/Ky=12 (c) Case 3

ωr=9,958Hz Area=168E-9m2 Kx/Ky=17 (d) Case 4**

ωr=10,151Hz Area=171E-9m2 Kx/Ky=13



101

(e) Case 5

(f) Case 6


ωr=9,999.5Hz Area=149E-9m2 Kx/Ky=13

*special initial design with performance close to the design specification **best result after discarding degenerate design with straight legs Figure 5. The best MOGA-generated designs for all cases.

Case 7

ωr=10,005Hz; Area=139E-9m2; Kx/Ky=11 Figure 6. The best MOGA result for case 7.

102


Case 6, which incorporates the higher-level cluster of the serpentine spring, yields the best result among the six cases. Although this result may be a fluke, it does indicate that such higher-level primitives play an important role in finding the best designs, since they also incorporate some engineering design knowledge. Thus stimulated by the success of the serpentine suspension, we started a new test case number 7, in which each corner of the center mass is suspended by a simple serpentine spring. The anchor points are offset in the y-direction from the connections on the center mass, and the maximum length of beams are extended to 300µm from the original 100µm. The width of the serpentine loops is 12µm. The width of the long beams in the serpentine is 2µm and the width of the short beams is 6µm. This design was subjected to MOGA optimization with the length of the loops and the number of loops as mutable parameters. The resulting design shown in Figure 6 has the smallest area of all of the cases – approximately 7 % lower than case 6. 3.4. Design Refinement with Gradient-Descent Optimization A gradient-descent local optimization algorithm was used to further refine the resonator designs with the goal of reducing both the area and frequency deviation from the target of 10,000 Hz. The best designs from all cases were used as the skeletal configurations, allowing only the spring beam lengths and widths as variables for optimization. The objective function for the gradient optimization was:

f ( x ) = A( x) + λ ( F − 10000) 2

Eq. (1) Here, A is the device area with unit E-10m2, F is the resonant frequency of the design, and λ is a penalty factor for any deviation from the resonant frequency, which starts from a low value and is increased by three times every iteration until the convergence criteria are met. The gradient of the objective function was calculated using finite differences. We first ran some optimizations with a low starting value of λ, (typically λ =1E-05), thus treating this problem as an evenly balanced multi-objective optimization problems. Typically, both the area and the frequency deviation could be reduced (Table 3). However, for practical engineering applications of these resonators, the resonant frequency has to lie rather close to the desired target, or the design is worthless. One way to give the frequency target priority over area concerns is to use a much larger value for λ. With a starting value of 0.001, we obtain the results presented in Table 4. Now the frequency deviations are no larger than 1Hz in all cases. Of course the areas are now slightly larger than those in Table 3. Note that after gradient optimization the designs with the smallest areas are Case 6 and 7; the difference between the two being negligible in spite of their substantially differing structures. Table 3. Results of gradient-decent optimization formulated to improve the performance of both resonant frequency and layout area.

Case 1 ωr=10,368Hz Area=175E-9m2 Kx/Ky=15

Case 2* ωr=10,036Hz Area=171E-9m2 Kx/Ky=21



Case 5 Case 6 Case 7* ωr=10,158Hz ωr=10,000Hz ωr=10,000Hz Area=172E-9m2 Area=145E-9m2 Area=143E-9m2 Kx/Ky=19 Kx/Ky=11 Kx/Ky=11 * No design was found that improved both performance objectives. Table 4. The results of gradient optimization formulated to drive optimization to the exact target frequency (large λ) .

Case 1 ωr=10,000Hz Area=181E-9m2 Kx/Ky=13 Case 5 ωr=9,999.9Hz Area=181E-9m2 Kx/Ky=21

Case 2 ωr=9,999.9Hz Area=209E-9m2 Kx/Ky=16 Case 6 ωr=10,000Hz Area=145E-9m2 Kx/Ky=11

Case 3 ωr=9,999.9Hz Area=178E-9m2 Kx/Ky=14 Case 7 ωr=10,000Hz Area=143E-9m2 Kx/Ky=11



103

4. CONCLUSIONS AND FUTURE RESEARCH Our investigations of genetic algorithms over the last couple of years has shown that MOGA alone rarely can lead to practically useful engineering designs. Such design problems often involve one or more explicitly specified target parameters, such as the resonance frequency in our example designs. The sparse sampling of the solution space by MOGA will rarely hit that desired value. Gradient-descent optimization, on the other hand, can easily fine tune one or two parameters to their precise target values, while corrupting other design objectives as little as possible. However, it cannot make discrete changes, such as adding an extra beam segment to the suspension, and thus cannot produce new “innovative designs.” The combination of the two algorithms is needed to produce useful engineering solutions. A hierarchical MEMS design synthesis and optimization architecture has been developed for the design of micromachined suspensions. The synthesis process integrates an object-oriented component library with a multi-objective evolutionary algorithm to find promising conceptual designs and topologies. Refinement of the best designs are then performed through gradient-descent optimization of the free variables associated with the most promising topology. The design synthesis architecture was applied to a surface micro-machined MEMS suspension with different design constraints and symmetry options. Gradient-descent optimization at the end of the design synthesis process is an effective way to refine promising design candidates with computational efficiency. The average time for each MOGA run for the polyline spring resonator example was about 3.5 hours on a 2.40GHz Pentium 4 processor. The average time for the subsequent gradient-descent optimization to refine the MOGA output was 0.8 hours. In the resonator design example, gradient-descent optimization was an easy way to “nail” the resonant frequency to the target value with minimal cost in area. (a) Starting design

(b) Gradient Optimized design


ωr=10,001Hz Area=198E-9m2 Kx/Ky=1042

Figure 7. Results of gradient optimization for a given configuration (design specifications: ωr=10,000Hz, Kx/Ky ≥ 10, and minimum area)

The evolutionary algorithm was able to find good structural starting points for the local gradient optimizer. As a point of comparison, gradient optimization was applied to the same starting design (Figure 7a) that was used for the symmetric cases 3 and 4 in the above MOGA runs. Figure 7b shows that the gradient-descent method by itself reached a local optimum within 30 min., achieving a design that was very close to the target frequency, but with an area of around

104


200E-9m2, which is significantly larger than that achieved by the demonstrated hierarchical method. In addition, optimization drove the design to a layout that is nearly degenerate, with a high stiffness prone to residual stress problems. Clearly, the initial configuration does not have enough degrees of freedom to allow a reasonable design to emerge with just adjustments of continuous parameters; it is also devoid of any elements that an experienced engineer might use as a starting point, as for instance exemplified by Figure 6. The building blocks in the component library were developed so that they could be used for rapid prototyping of a wide range of MEMS suspensions. The results of this study and others [10] show the advantage of drawing from a suitable set of building blocks. They also confirm that any constraints on the search space based on engineering domain knowledge are highly advantageous to focus the evolution on the more promising configurations. The unconstrained case 1, was not able to find an acceptable design after 50 generations when using a randomly generated starting design. However, it was able to find a competitive design when seeded with a reasonably good starting design. The more constrained test cases did much better. This clearly demonstrates how important it is to introduce as much engineering domain knowledge as possible right from the start in order to keep the MOGA search effective. Towards this goal we plan to expand the object library with a range of historically successful initial designs. The development of an indexed case library is one of our directions of future research. Along these lines, we also intend to extend the data structures for the component library to be able to model more complex MEMS designs for the case library.

ACKNOWLEDGMENTS This research was conducted in part through NSF grant CCR-DES/CC-0306557. The authors gratefully acknowledge the work of Sebastian Graf in developing the object-oriented data structure during his internship with the Berkeley Sensors and Actuator Center at UC Berkeley [10].

REFERENCES 1.

T. Mukherjee, Y. Zhou, G. K. Fedder, “Automated Optimal Synthesis of Microaccelerometers”, Technical Digest of the 12th IEEE International Conference on Micro Electro Mechanical Systems (MEMS '99), pp. 326-331, 1999. 2. Q. Jing, H. Luo, T. Mukherjee, L. R. Carley, and G. K. Fedder, “CMOS Micromechanical Bandpass Filter Design Using a Hierarchical MEMS Circuit Library”, in Proceedings of the 13th IEEE International Conference on Micro Electro Mechanical Systems (MEMS 2000), pp. 187-192, 2000. 3. Ma, L. and E.K. Antonsson, “Robust Mask-Layout Synthesis for MEMS,” Technical Proceedings of the 2001 International Conferecne on Modeling and Simulation of Microsystems, Chapter 5: Optimization, Nanotech 2001, 1, pp. 128-131. 4. Li, H. and Antonsson, E.K., “Evolutionary Techniques in MEMS Synthesis’’, Proc. DETC'98, 1998 ASME Design Engineering Technical Conferences, 1998. 5. Zhou, N., Zhu, B., Agogino, A. M. and Pister, K. S. J., “Evolutionary Synthesis of MEMS MicroElectronicMechanical Systems Design,” Intelligent Engineering System through Artificial Neural Networks Proceedings of the Artificial Neural Networks in Engineering (ANNIE2001), 11, ASME Press, pp.197-202, 2001. 6. Zhou, N., Agogino, A. M., and Pister, K. S., “Automated Design Synthesis for Micro-Electro-Mechanical Systems (MEMS),” CD-ROM Proc. of ASME Design Automation Conference, 2002. 7. Zhou, N., “Simulation and Synthesis of MicroElectronicMechanical Systems” PhD Thesis, UC Berkeley, Berkeley, CA, 2002. 8. Kamalian, R., Agogino, A.M., and Takagi, H., “The Role of Constraints and Human Interaction in Evolving MEMS Designs: Microresonator Case Study,” Proceedings of DETC/DAC, 2004 Design Engineering Technical Conference, Design Automation Track, Paper # DETC2004-57462, CD ROM, ISBN # I710CD, 2004. 9. Kamalian, R., "Evolutionary Synthesis of MEMS Devices," Doctoral Dissertation, Department of Mechanical Engineering, University of California at Berkeley, Fall 2004. 10. Graf, S., “GA Building Blocks and Data Structures for MEMS/NEMS Design Automation and Synthesis,” Diploma Thesis, Department of Computer Science, RWTH Aachen (in cooperation with the Department of Mechanical Engineering, University of California, Berkeley), Oct. 2004.


105

11. MUMPs manufacturing services, MEMSCAP: http://www.memscap.com/memsrus/crmumps.html 12. Clark, J.V., Bindel, D., Zhou, N., Nie, J., Kao, W., Zhu, E., Kuo A., Pister, K.S.J., Demmel, J., Govindjee, S., Bai, Z., Gu, M. and Agogino, A.M., “Addressing the Needs of Complex MEMS Design,” Proceedings of the 15th IEEE International MEMS Conference, pp. 204-209, 2002. 13. http://www-bsac.eecs.berkeley.edu/cadtools/sugar/sugar/

106
