层合压电材料冲击问题的时域间断Galerkin有限元方法求解
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摘要
构建了压电耦合动力学问题的时域间断Galerkin有限元方法,在此基础上针对冲击作用下压电材料耦合问题进行了数值模拟。强间断、高梯度的冲击荷载作用下,传统的时域连续Galerkin有限元方法在压电耦合动力学问题模拟时,间断的波阵面及层合界面处会出现强烈的虚假数值振荡,这类振荡使得问题求解的精度大大降低。为了消除这类高频数值振荡,依据所发展的时域Galerkin有限元方法原理,在最终的有限元方法求解公式中引入了比例刚度阻尼。冲击作用下一维、二维压电动力学问题算例结果表明,所发展的时域间断Galerkin有限元方法,较好地消除了这类虚假的数值振荡,并捕捉了应力波及电势波的波前波后间断特性。
A time discontinuous Galerkin finite element method( DGFEM) is used to simulate the problem of laminated piezoelectric material subjected to the high-frequency impulsive load. At present,the traditional numerical method for solving the dynamic problem of piezoelectric solid subjected to high frequency impulsive load generally fails to properly capture discontinuities of waves in space and interface between two layers,and produces spurious numerical oscillations in the simulation of high modes and sharp gradients. In order to filter out the spurious oscillations,an artificial damping scheme,which based on the DGFEM,was implemented in the final finite element formula. Numerical results of one dimension and two dimension show that the developed DGFEM presents good abilities and provides much more accurate solutions for piezoelectric problem. It can capture the discontinuities effectively at the wave front and wave after and filter out the effects of spurious numerical oscillation induced by high-frequency impulsive load.
A time discontinuous Galerkin finite element method( DGFEM) is used to simulate the problem of laminated piezoelectric material subjected to the high-frequency impulsive load. At present,the traditional numerical method for solving the dynamic problem of piezoelectric solid subjected to high frequency impulsive load generally fails to properly capture discontinuities of waves in space and interface between two layers,and produces spurious numerical oscillations in the simulation of high modes and sharp gradients. In order to filter out the spurious oscillations,an artificial damping scheme,which based on the DGFEM,was implemented in the final finite element formula. Numerical results of one dimension and two dimension show that the developed DGFEM presents good abilities and provides much more accurate solutions for piezoelectric problem. It can capture the discontinuities effectively at the wave front and wave after and filter out the effects of spurious numerical oscillation induced by high-frequency impulsive load.
引文
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[25]LI D H,ZHENG X J,WU B,et al.Fracture analysis of a surface through-thickness crack in PZT thin film under a continuous laser irradiation[J].Engineering Fracture Mechanics,2009,76:525-532.
[2]AABDEL K,NAYFEH A H,HAJJ M R.Effects of nonlinear piezoelectric coupling on energy harvesters under direct excitation[J].Nonlinear Dynamics,2012,67(8):1221-1232.
[3]NANTHAKUMAR S S,LAHMER T,ZHUANG X Y,et al.Topology optimization of piezoelectric nanostructures[J].Journal of the Mechanics and Physics of Solids,2016,94(8):316-335.
[4]林鹏,王长利,李迅,等.压杆式压电应力传感器在爆炸冲量测试中的应用[J].应用数学和力学,2015,36:29-35.LIN Peng,WANG Changli,LI Xun,et al.Application of the piezoelectric pressure bar gauge to blast im pulse measurement[J].Applied Mathematics and Mechanics,2015,36:29-35.
[5]朱劲松,高志玉.基于PZT波传播法的混凝土内部钢筋锈蚀监测方法[J].建筑材料学报,2016,19(1):166-170.ZHU Jinsong,GAO Zhiyu.Monitoring method for corrosion of reinforcement in concrete based on wave propagation method with PZT[J].Journal of Building Materials,2016,19(1):166-170.
[6]SOROKIN B P,KVASHNIN G M,NOVOSELOV A S,et al.Excitation of hypersonic acoustic waves in diamond-based piezoelectric layered structure on the microwave frequencies up to 20 GHz[J].Ultrasonics,2017,78:162-165.
[7]ABDELKFI A.Aeroelastic energy harvesting:a review[J].International Journal of Engineering Science,2016,100:112-135.
[8]WILLIAMS R P,KIM D,GAWALT D P,et al.Surface micromachined differential piezoelectric shear-stress sensors[J].Journal of Micromechanics and Microengineering,2017,27(1):015011.
[9]HOSSEINI R,HAMEDI M,IM J,et al.Analytical and experimental investigation of partially covered piezoelectric cantilever energy harvester[J].International Journal of Precision Engineering and Manufacturing,2017,18(3):415-424.
[10]匡震邦.电弹性理论[M].上海:上海交通大学出版社,2010.
[11]MASIHOLLAH M,ABOLGHASSEM Z,MOHAMMAD V,et al.Design and analysis of an innovative light tracking device based on opto-thermo-electro-mechanical actuators[J].Microelectronic Engineering,2014,119:37-43.
[12]SAIRAJAN K K,AGLIETTI G S,MANI K M.A review of multifunctional structure technology for aerospace applications[J].Acta Astronautica,2016,120:30-42.
[13]SLADEK J,SLADEK V,STANAK P,et al.FEMformulation for size-dependent theory with application to micro coated piezoelectric and piezomagnetic fiber-composites[J].Comput Mech,2017,59(1):93-105.
[14]BUI T Q,ZHANG C Z.Extended finite element simulation of stationary dynamic cracks in piezoelectric solids under impact loading[J].Computational Materials Science,2012,62:243-257.
[15]GRZEGORZ D.Complex variable step method for sensitivity analysis of effective properties in multi-field micromechanics[J].Acta Mech,2016,227:11-28.
[16]MARQUEZ RENTERIA I A,BOLBORICI V.A dynamic model of the piezoelectric traveling wave rotary ultrasonic motor stator with the finite volume method[J].Ultrasonics,2017,77:69-78.
[17]BUI T Q.Extended isogeometric dynamic and static fracture analysis for cracks in piezoelectric materials using NURBS[J].Comput Methods Appl Mech Engrg,2015,295:470-509.
[18]HUGHES T J R,HULBERT G M.Space-time finite element methods for elastodynmaics:fomurlations and error estimates[J].Computer Methods in Applied Mechanics and Engineering,1988,66:339-363.
[19]WIBERG N E,LI X D.Adaptive finite element procedures for linear and non-linear dynamics[J].International Journal for Numerical Methods in Engineering,1999,46:1781-1802.
[20]MANCUSO M,UBERTINI F.An efficient time discontinuous Galerkin procedure for nonlinear structural dynamics[J].Computer Methods in Applied Mechanics and Engineering,2006,195:44-47.
[21]LI X K,YAO D M,LEWIS R W.A discontinuous Galerkin finite element method for dynamic and wave propagation problems in non-linear solids and saturated porous media[J].International Journal for Numerical Methods in Engineering,2003,57(3):1775-1800.
[22]WU Wenhua,LI Xikui.Application of the time discontinuous Galerkin finite element method to heat wave simulation[J].International Journal of Heat and Mass Transfer,2006,49:1679-1684.
[23]郭攀,武文华,吴志刚.改进时域间断Galerkin有限元方法在弹塑性波传播数值模拟中的应用[J].固体力学学报,2013,34(3):1-6.GUO Pan,WU Wenhua,WU Zhigang.Application of the modified time discontinuous Galerkin finite element method to elasto-plastic wave propagation simulation[J].Chinese Joural of Solid Mechanics,2013,34(3):1-6.
[24]GUO Pan,WU Wenhua,WU Zhigang.A time discontinuous Galerkin finite element method for generalized thermo-elastic wave analysis,considering non-Fourier effects[J].Acta Mechanica,2014,225:299-307.
[25]LI D H,ZHENG X J,WU B,et al.Fracture analysis of a surface through-thickness crack in PZT thin film under a continuous laser irradiation[J].Engineering Fracture Mechanics,2009,76:525-532.