面向器件行为的MEMS宏建模方法研究
摘要
随着MEMS产品的多样化和功能结构的复杂化,微机电系统已成为多个学科相互融合的系统,其功能的实现是多个能量域相互耦合、综合作用的结果,这样,系统级建模与仿真越来越依赖于器件级宏模型的准确性。针对强耦合、非线性的MEMS器件,如何有效实现在多能量域耦合分析的基础上提取宏模型,完成系统级多信号仿真分析,成为当前MEMS设计领域的重要课题。本文针对典型的MEMS器件,研究了三种不同适应条件下的半解析宏建模方法,即在原MEMS器件(或子系统)数值计算基础上,通过一定的降阶算法得到解析宏模型。论文结合扭转微镜、微梁、电容式压力传感器等典型MEMS器件对三种宏建模方法的适应性和有效性进行了分析。
本文的主要研究内容包括:
1) 基于振动理论中的模态叠加原理,结合Lagrange动力学方程,研究了能量保守系统的宏建模方法。并用静电-结构耦合的扭转微镜为例,建立其宏模型,在ANSYS中对该宏模型进行分析。
2) 对于MEMS中经常遇到的有阻尼问题的能量耗散系统,针对线性问题研究了基于Arnoldi算法的宏建模方法,并在线性系统Arnoldi算法的基础上,总结其空间投影原理,结合Taylor展开技术,研究了弱非线性系统的宏建模方法。以一个双端固定微梁为例,综合考虑结构-静电-流体(空气阻尼)多域耦合关系,用一个二次系统描述其弱非线性特点,并对该二次系统进行了降阶宏建模,对其准确性进行分析。
3) 针对有阻尼、强非线性的能量耗散系统,研究了基于KL分解的宏建模方法,对MEMS器件在设计空间内进行一定数值计算,利用相应的计算结果作为最初的随机向量,通过KL分解生成基函数,将系统动态特性投影到少量的基函数上,结合Galerkin方法生成与时间相关的各基函数叠加系数,从而获得降阶模型。一个电容式压力传感器为例,用该方法进行宏模型的获取和分析。
Due to the nonlinear coupled energy domain behavior of MEMS (Micro Electro Mechanical System) devices, the development of MEMS CAD tools becomes a more challenging task. The coupled relevant field quantities (mechanical, thermal, electric, magnetic, optical, chemical, etc.) are consistently described by a set of time-dependent partial differential equations, direct simulation based on fully meshed structures involves thousands of degrees of freedom. In order to perform efficient prediction on system level, it is essential to build accurate macro models for MEMS devices on device level.In the thesis, three methods of semi-analytic macro modeling compatible with different terms are studied and realized to solve the questions of typical MEMS devices such as the torsion micro mirror, fixed-fixed micro beam, capacitive pressure sensor, etc. The main contents in this thesis are described as follows:1) Based on a modal representation of coupled domains, the behavior of a coupled system can be assembled in Lagrange equation. In the mode superposition method, the thousands of original system freedoms are reduced to several generalized coordinates (mode coordinates). With the example of a torsion micro mirror, the efficient and accuracy is verified. Because of the modularization of the energy domain, the method isn't available to the energy dissipation system.2) Arnoldi algorithm does not need to solve the original large scale ODEs but directly reduce it to a lower order model by computing an orthogonal subspace which spans the same Krylov subspace. With the Taylor series expansion and the Arnoldi process, weakly nonlinear MEMS devices model is efficiently reduced via a second order system. Unfortunately it can not be used to handle strong nonlinear system accurately.3) To solve the question of strongly nonlinear MEMS devices, the KL decomposition method is studied. In the method, the first and most important is the signals (data shots), which are required to represent the behaviors of the original system well. By the KL decomposition on the signals, the global basis functions are gained. With the Galerkin method, the time-dependent coefficients of each basis function are chosen. A capacitive pressure sensor is used to verify the method.
本文的主要研究内容包括:
1) 基于振动理论中的模态叠加原理,结合Lagrange动力学方程,研究了能量保守系统的宏建模方法。并用静电-结构耦合的扭转微镜为例,建立其宏模型,在ANSYS中对该宏模型进行分析。
2) 对于MEMS中经常遇到的有阻尼问题的能量耗散系统,针对线性问题研究了基于Arnoldi算法的宏建模方法,并在线性系统Arnoldi算法的基础上,总结其空间投影原理,结合Taylor展开技术,研究了弱非线性系统的宏建模方法。以一个双端固定微梁为例,综合考虑结构-静电-流体(空气阻尼)多域耦合关系,用一个二次系统描述其弱非线性特点,并对该二次系统进行了降阶宏建模,对其准确性进行分析。
3) 针对有阻尼、强非线性的能量耗散系统,研究了基于KL分解的宏建模方法,对MEMS器件在设计空间内进行一定数值计算,利用相应的计算结果作为最初的随机向量,通过KL分解生成基函数,将系统动态特性投影到少量的基函数上,结合Galerkin方法生成与时间相关的各基函数叠加系数,从而获得降阶模型。一个电容式压力传感器为例,用该方法进行宏模型的获取和分析。
Due to the nonlinear coupled energy domain behavior of MEMS (Micro Electro Mechanical System) devices, the development of MEMS CAD tools becomes a more challenging task. The coupled relevant field quantities (mechanical, thermal, electric, magnetic, optical, chemical, etc.) are consistently described by a set of time-dependent partial differential equations, direct simulation based on fully meshed structures involves thousands of degrees of freedom. In order to perform efficient prediction on system level, it is essential to build accurate macro models for MEMS devices on device level.In the thesis, three methods of semi-analytic macro modeling compatible with different terms are studied and realized to solve the questions of typical MEMS devices such as the torsion micro mirror, fixed-fixed micro beam, capacitive pressure sensor, etc. The main contents in this thesis are described as follows:1) Based on a modal representation of coupled domains, the behavior of a coupled system can be assembled in Lagrange equation. In the mode superposition method, the thousands of original system freedoms are reduced to several generalized coordinates (mode coordinates). With the example of a torsion micro mirror, the efficient and accuracy is verified. Because of the modularization of the energy domain, the method isn't available to the energy dissipation system.2) Arnoldi algorithm does not need to solve the original large scale ODEs but directly reduce it to a lower order model by computing an orthogonal subspace which spans the same Krylov subspace. With the Taylor series expansion and the Arnoldi process, weakly nonlinear MEMS devices model is efficiently reduced via a second order system. Unfortunately it can not be used to handle strong nonlinear system accurately.3) To solve the question of strongly nonlinear MEMS devices, the KL decomposition method is studied. In the method, the first and most important is the signals (data shots), which are required to represent the behaviors of the original system well. By the KL decomposition on the signals, the global basis functions are gained. With the Galerkin method, the time-dependent coefficients of each basis function are chosen. A capacitive pressure sensor is used to verify the method.
引文
[1] 苑伟政,马炳和.微机械与微细加工技术.西安:西北工业大学出版社,2000
[2] S.D. Senturia, R.M. Harris, et al. A Computer-Aided Design System for Microelectromechanical Systems. Journal of Microelectromechanical Systems. 1992, 1(1): 3-13.
[3] F. Maseeh, "IntelliCAD: the CAD for MEMS", in WESCON/95 Conference Record, New York, NY, USA, 1995,pp. 320-324
[4] S. Crary, Y.F. Zhang. CAEMEMS: An intergrated computer-aided engineering workbench for micro-electro mechanical systems. Proceedings-IEEE Micro Electro Mechanical Systems.Napa Valley, CA, USA. 1990: 113-114
[5] S.D.Senturia, CAD Challenges for Microsensor, Microactuators, and Microsystem, Pro.IEEE, 1998,86(8): 1611-1626
[6] http://www.coventor.com/coventorware/
[7] http://www.intellisuite.net
[8] http://www.memscap.com/products-cad.html
[9] 常洪龙,苑伟政,马炳和.MEMS的集成没计平台的关键技术研究.中国机械工程.2004,15(10):916-920
[10] 张海霞,郭辉.MEMS CAD系统IMEE的研究和开发.微纳电子技术.2003.40(7/8):92-95
[11] CoventorWare Help Document/Master Help.pdf
[12] S.D.Senturia, CAD Challenges for Microsensor, Microactuators, and Microsystem, Pro. IEEE, 1998, 86(8): 1611-1626
[13] S.D.Senturia. The future of microsensor and microactuator design [J]. Sensors and Actuators A. 1996, A (56): 125-127
[14] G.K.Fedder. Simulation of Microelectromechanical Systems. Ph.D Thesis. U.C.Berkeley, EECS, 1994
[15] V. I. Zhulin, S. J. Owen, D. F. Ostergaard. Finite Element Based Electrostatic-Structural Coupled Analysis with Automated Mesh Morphing. Proceedings of the International Conference on Modeling and Simulation of Microsystems MSM 2000. San Diego, CA, USA. 2000: 27-29
[16] Tilmans H A C. Equivalent circuit representation of electromechanical transducers: Lumped-parameter systems. J. Micromech. Microeng, 1996, 6(1): 157～176
[17] S. Orieux, C. Rossi, D. Esteve. Compact model based on a lumped parameter approach for the prediction of solid propellant micro-rocket performance [J]. Sensor and Actuators 2002 A (101): 383-391
[18] 李伟剑.微机电系统的多域耦合分析和多学科设计优化.[学位论文].西安:西北工业大学,2004
[19] Q.Jing and G.K.Fedder. NODAS 1.3-nodal design of actuators and sensors. IEEE/VIUF International Workshop on Bahavioral Modeling and Simulation, Orlando, Fla., October 27-28, 1998
[20] 霍鹏飞,微机电系统的多端口组件网络方法系统级建模研究.[学位论文]两安:西北工业大学,2004.
[21] L.D.Gabbay. Computer Aided Macromdoeling for MEMS. Ph.D Thesis, MIT, MA, 1998
[22] Zhaojun Bai. Krylov subspace techniques for reduced-order modeling of large-scale dynamical systems, Applied Numerical Mathematics [J]. 2002(43): 9-44
[23] W Z Lin, K H Lee, S P Lira, Y C Liang, Proper orthogonal decomposition and component mode synthesis in macromodel generation for the dynamic simulation of a complex MEMS device. Journal of micromechanics and microengineering. 2003 (13):646～654
[24] 陈兢、刘理天,李志坚.纹膜结构微麦克风的动态特性:使用EDA/CAD工具进行Top-down设计[J]半导体学报.2001,22(7):951-956
[25] 孟为民,李伟华,黄庆安.MEMS的设计方法与系统模拟.传感器技术2001,20(10):57-60
[26] 孙道恒.MEMS耦合场分析与系统级仿真[J].中国机械工程.2002,13(9):765-768
[27] 康建初,尹主林,高鹏.支持MEMS的CAD/CAE系统结构研究[J].北京航空航天大学学报.1998,24(4):475-478
[28] 卢桂章、赵新.基于部件库的MEMS设计方法[J].微纳电子技术.2003.40(7/8):36-39
[29] Mehner, J., Bennini, F., and Dotzel, W., "A Modal Decomposition Technique for Fast Harmonic and Transient Simulations of MEMS", International MEMS Workshop 2001 (IMEMS 2001), Singapore, pp. 477-484 (2001)
[30] Gabbay, L., Mehner, J., and Senturia, S.D., "Computer-Aided Generation of Nonlinear Reduced-Order Dynamic Macromodels-Ⅰ: Non-Stress-Stiffened Case", Journal of Microelectromechanical Systems, S. 262-269, (June 2000)
[31] H.S.Cheng, C.H.T.Pan, J. Basic Engineering, March 1965, pp. 185～192
[32] E.S.Hung, Y.-J.Yang, and S.D.Senturia, "Low-Order Models For Fast Dynamical Simulation of MEMS Microstructures," Transducers'97, pp. 1101-1104
[33] ANSYS, Inc. Theory Reference/Reduced Order Modeling of Coupled Domains
[34] 傅志方,华宏星,模态分析理论与应用.上海:上海交通大学出版社,2000.7
[35] 胡盘新.大学物理手册.上海:上海交通大学出版社,1999
[36] 乔大勇,基于MEMS技术的自适应光学微变形镜的设计与分析[硕十学位论文]西安:西北工业大学.2002
[37] ANSYS Tutorial on MEMS. The 2002 International Conference on Modeling and Simulation of Microsystems (MSM 2002). San Juan, PR. 2002
[38] 刘国庆,杨庆东.ANSYS工程应用教程——机械篇.北京:中国铁道出版社,2003
[39] 王永,冒小建等.面向微系统芯片的建模方法
[40] 胡寿松.自动控制原理(第3版).北京:国防工业出版社,1998
[41] R. W. Freund, "Krylov-subspace methods for reduced order modeling in circuit simulation", Journal of Computational and Applied Mathematics vol. 123, pp. 395-421, 2000
[42] E. B. Rudnyi, J. G. Korvink. "Model Order Reduction for Large Scale Engineering Models Developed in ANSYS". PARA'04, Workshop On State-of-the-art In Scientific Computing, Technical University of Denmark, June 20-23, 2004
[43] 同济大学数学教研室.高等数学(第四版).北京:高等教育出版社,1997
[44] 黄兑智,薛明德,陆明万.张量分析(第2版).北京:清华大学出版社,2003
[45] R.K.Gupta and S.D.Senturia, "Pull-in time dynamics as a measure of absolute pressure," in The Tenth Annual International Workshop on Micro Elctro Mechanical Systems, New York, NY, pp.290-294, 1997
[46] Y.J.Yang and S.D.Senturia, "Effect of air damping on the dynamics of nonuniform deformations of microstructures," in IEEE International Conference on Solid-State Sensors and Actuators, New York, NY, pp. 1993-1096,1997
[47] J.B.Starr, "Squeeze-fihn damping in solid-state accelerometers," in Tech. Dig. IEEE Solid-State Sensors and Actuators Workshop, Hilton Head Island, SC, pp.44-47, 1990
[48] 赵荣椿等.数字图像处理导论.西安:西北工业大学出版社,2000
[49] S.Banerjee, J.V. Cole, and K.F, Jensen, "Nonlinear model reduction strategies for rapid thermal processing systems", IEEE Transactions on Semiconductor Manufacturing, Vol. 11, no.2, pp.266-275, 1998
[50] 杨福生.随机信号分析.北京:清华大学出版社.1990.7
[51] 蒋孝煜.有限元素法基础.北京:清华大学出版社.1984.7
[52] M. Elwenspoek, R. Wiegernk著,陶家渠,李应选,万达,刘佑宝,沈文正,白丁译.硅微机械传感器.北京:中国宇航出版社.2002
[53] P. Pons, G. Blasquez, and N. Ratier, "Harmonic response of silicon capacitive pressure sensor". Sensors and Actuators. 1991, 26(1-3). pp.301-305
[54] 陈俊澳,顾昭朵.复变函数.北京:高等教育出版社.1987.7
[2] S.D. Senturia, R.M. Harris, et al. A Computer-Aided Design System for Microelectromechanical Systems. Journal of Microelectromechanical Systems. 1992, 1(1): 3-13.
[3] F. Maseeh, "IntelliCAD: the CAD for MEMS", in WESCON/95 Conference Record, New York, NY, USA, 1995,pp. 320-324
[4] S. Crary, Y.F. Zhang. CAEMEMS: An intergrated computer-aided engineering workbench for micro-electro mechanical systems. Proceedings-IEEE Micro Electro Mechanical Systems.Napa Valley, CA, USA. 1990: 113-114
[5] S.D.Senturia, CAD Challenges for Microsensor, Microactuators, and Microsystem, Pro.IEEE, 1998,86(8): 1611-1626
[6] http://www.coventor.com/coventorware/
[7] http://www.intellisuite.net
[8] http://www.memscap.com/products-cad.html
[9] 常洪龙,苑伟政,马炳和.MEMS的集成没计平台的关键技术研究.中国机械工程.2004,15(10):916-920
[10] 张海霞,郭辉.MEMS CAD系统IMEE的研究和开发.微纳电子技术.2003.40(7/8):92-95
[11] CoventorWare Help Document/Master Help.pdf
[12] S.D.Senturia, CAD Challenges for Microsensor, Microactuators, and Microsystem, Pro. IEEE, 1998, 86(8): 1611-1626
[13] S.D.Senturia. The future of microsensor and microactuator design [J]. Sensors and Actuators A. 1996, A (56): 125-127
[14] G.K.Fedder. Simulation of Microelectromechanical Systems. Ph.D Thesis. U.C.Berkeley, EECS, 1994
[15] V. I. Zhulin, S. J. Owen, D. F. Ostergaard. Finite Element Based Electrostatic-Structural Coupled Analysis with Automated Mesh Morphing. Proceedings of the International Conference on Modeling and Simulation of Microsystems MSM 2000. San Diego, CA, USA. 2000: 27-29
[16] Tilmans H A C. Equivalent circuit representation of electromechanical transducers: Lumped-parameter systems. J. Micromech. Microeng, 1996, 6(1): 157～176
[17] S. Orieux, C. Rossi, D. Esteve. Compact model based on a lumped parameter approach for the prediction of solid propellant micro-rocket performance [J]. Sensor and Actuators 2002 A (101): 383-391
[18] 李伟剑.微机电系统的多域耦合分析和多学科设计优化.[学位论文].西安:西北工业大学,2004
[19] Q.Jing and G.K.Fedder. NODAS 1.3-nodal design of actuators and sensors. IEEE/VIUF International Workshop on Bahavioral Modeling and Simulation, Orlando, Fla., October 27-28, 1998
[20] 霍鹏飞,微机电系统的多端口组件网络方法系统级建模研究.[学位论文]两安:西北工业大学,2004.
[21] L.D.Gabbay. Computer Aided Macromdoeling for MEMS. Ph.D Thesis, MIT, MA, 1998
[22] Zhaojun Bai. Krylov subspace techniques for reduced-order modeling of large-scale dynamical systems, Applied Numerical Mathematics [J]. 2002(43): 9-44
[23] W Z Lin, K H Lee, S P Lira, Y C Liang, Proper orthogonal decomposition and component mode synthesis in macromodel generation for the dynamic simulation of a complex MEMS device. Journal of micromechanics and microengineering. 2003 (13):646～654
[24] 陈兢、刘理天,李志坚.纹膜结构微麦克风的动态特性:使用EDA/CAD工具进行Top-down设计[J]半导体学报.2001,22(7):951-956
[25] 孟为民,李伟华,黄庆安.MEMS的设计方法与系统模拟.传感器技术2001,20(10):57-60
[26] 孙道恒.MEMS耦合场分析与系统级仿真[J].中国机械工程.2002,13(9):765-768
[27] 康建初,尹主林,高鹏.支持MEMS的CAD/CAE系统结构研究[J].北京航空航天大学学报.1998,24(4):475-478
[28] 卢桂章、赵新.基于部件库的MEMS设计方法[J].微纳电子技术.2003.40(7/8):36-39
[29] Mehner, J., Bennini, F., and Dotzel, W., "A Modal Decomposition Technique for Fast Harmonic and Transient Simulations of MEMS", International MEMS Workshop 2001 (IMEMS 2001), Singapore, pp. 477-484 (2001)
[30] Gabbay, L., Mehner, J., and Senturia, S.D., "Computer-Aided Generation of Nonlinear Reduced-Order Dynamic Macromodels-Ⅰ: Non-Stress-Stiffened Case", Journal of Microelectromechanical Systems, S. 262-269, (June 2000)
[31] H.S.Cheng, C.H.T.Pan, J. Basic Engineering, March 1965, pp. 185～192
[32] E.S.Hung, Y.-J.Yang, and S.D.Senturia, "Low-Order Models For Fast Dynamical Simulation of MEMS Microstructures," Transducers'97, pp. 1101-1104
[33] ANSYS, Inc. Theory Reference/Reduced Order Modeling of Coupled Domains
[34] 傅志方,华宏星,模态分析理论与应用.上海:上海交通大学出版社,2000.7
[35] 胡盘新.大学物理手册.上海:上海交通大学出版社,1999
[36] 乔大勇,基于MEMS技术的自适应光学微变形镜的设计与分析[硕十学位论文]西安:西北工业大学.2002
[37] ANSYS Tutorial on MEMS. The 2002 International Conference on Modeling and Simulation of Microsystems (MSM 2002). San Juan, PR. 2002
[38] 刘国庆,杨庆东.ANSYS工程应用教程——机械篇.北京:中国铁道出版社,2003
[39] 王永,冒小建等.面向微系统芯片的建模方法
[40] 胡寿松.自动控制原理(第3版).北京:国防工业出版社,1998
[41] R. W. Freund, "Krylov-subspace methods for reduced order modeling in circuit simulation", Journal of Computational and Applied Mathematics vol. 123, pp. 395-421, 2000
[42] E. B. Rudnyi, J. G. Korvink. "Model Order Reduction for Large Scale Engineering Models Developed in ANSYS". PARA'04, Workshop On State-of-the-art In Scientific Computing, Technical University of Denmark, June 20-23, 2004
[43] 同济大学数学教研室.高等数学(第四版).北京:高等教育出版社,1997
[44] 黄兑智,薛明德,陆明万.张量分析(第2版).北京:清华大学出版社,2003
[45] R.K.Gupta and S.D.Senturia, "Pull-in time dynamics as a measure of absolute pressure," in The Tenth Annual International Workshop on Micro Elctro Mechanical Systems, New York, NY, pp.290-294, 1997
[46] Y.J.Yang and S.D.Senturia, "Effect of air damping on the dynamics of nonuniform deformations of microstructures," in IEEE International Conference on Solid-State Sensors and Actuators, New York, NY, pp. 1993-1096,1997
[47] J.B.Starr, "Squeeze-fihn damping in solid-state accelerometers," in Tech. Dig. IEEE Solid-State Sensors and Actuators Workshop, Hilton Head Island, SC, pp.44-47, 1990
[48] 赵荣椿等.数字图像处理导论.西安:西北工业大学出版社,2000
[49] S.Banerjee, J.V. Cole, and K.F, Jensen, "Nonlinear model reduction strategies for rapid thermal processing systems", IEEE Transactions on Semiconductor Manufacturing, Vol. 11, no.2, pp.266-275, 1998
[50] 杨福生.随机信号分析.北京:清华大学出版社.1990.7
[51] 蒋孝煜.有限元素法基础.北京:清华大学出版社.1984.7
[52] M. Elwenspoek, R. Wiegernk著,陶家渠,李应选,万达,刘佑宝,沈文正,白丁译.硅微机械传感器.北京:中国宇航出版社.2002
[53] P. Pons, G. Blasquez, and N. Ratier, "Harmonic response of silicon capacitive pressure sensor". Sensors and Actuators. 1991, 26(1-3). pp.301-305
[54] 陈俊澳,顾昭朵.复变函数.北京:高等教育出版社.1987.7