内域波动数值模拟的显式方法
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摘要
如何更精确高效地模拟大型、复杂系统内域的波动是发展和完善近场波动数值模拟技术的一个重要研究课题。内域波动的数值模拟通常采用计算量较小的显式方法,但现有的时空解耦显式有限元方法的精度只有二阶;低精度不仅影响数值模拟的精度,而且制约着计算效率的提高。鉴于此,本文旨在探索内域波动数值模拟具有更高精度且稳定的显式方法。作者发展了内域波动数值模拟的现有显式有限元解耦技术,提出了一种高精度且稳定的显式数值模拟方法。
第一,依据波速有限的概念,从波动方程的精确解出发提出了一种新的显式数值模拟方法。此方法与现有有限元技术相似之处在于:适于处理非规则网格节点,且节点递推公式是具有显式和时空解耦特征。其不同之处在于:新方法可给出时空离散精度皆为2M阶的稳定格式,M为正整数。本文通过一维模型详细论述了这一方法的可行性:建立了非规则网格节点递推公式,详细分析了均匀网格标量波动数值模拟的精度和稳定性,提出了构建时空精度皆为2M阶( M为正整数)的稳定递推公式的技术途径,并以构建二阶( M = 1)和四阶( M = 2)公式为例予以说明。
第二,将一维情形研究结果推广到高维情形,分别建立了二维、三维非规则网格节点的递推公式。针对二维正方形均匀网格详细论述了时空离散精度皆为2M阶的稳定递推公式的构建方法,给出了二阶和四阶递推公式的具体结果;并对三维立方体均匀网格模型作了简要的讨论。
第三,本文通过一维、二维模型算例检验了波动方程数值模拟显式方法的精度和稳定性等理论结果,指出了高阶公式对提高计算效率的价值。
第四,从空间解耦有限元常微分方程组(结构动力学方程)出发,通过被积函数的拉格朗日多项式内插和分部积分导出了一组具有高阶精度的时域显式积分格式,并将其推广为适用于一般数学物理方程的显式积分格式。作者以一个简单的线性时不变系统为例,初步考察了此积分格式的稳定性。
How to simulate the wave motion in the interior of a complicated system of large scale accurately and effectively is a very important research subject for developing and improving techniques for numerical simulation of near-field wave motion. Traditionally, the explicit method with small computational cost is employed for the numerical simulation of wave motion within the computational region. However, the existing time-space decoupling explicit finite element method is only second order accurate. The low order of accuracy not only affects precision of the numerical simulation, but also restricts improving efficiency of the computation. Accordingly, this paper is devoted to explore stable explicit methods with high order of accuracy for the numerical simulation of wave motion within the computational region. The paper further develops the existing time-space decoupling explicit numerical simulation technique, and proposes a new highly accurate stable explicit method.
First, a new explicit method of numerical simulation is proposed based on the exact solution of wave equations. The method shares similarity with the lumped mass FEM, which is suitable for dealing with irregular grids and the recursion formulas derived are explicit and decoupling both in time and space. But, the recursion formulas developed by the method are 2M-order (M is the positive integer) accurate both in time and space. In this paper, the feasibility of the method is demonstrated via 1-D model: the recursion formulas at nodal points of an irregular grid are constructed, meanwhile, the stability and accuracy of the recursion formulas for a uniform grid are discussed in detail; according, an approach is proposed to construct the stable formulas which are of 2M-order of accuracy both in time and space with M being a positive integer and illustrated by constructing formulas of the second order (M=1) and the fourth order (M=2).
Second, the method is generalized to the multi-dimensional cases and the corresponding recursion formulas for an irregular grid in 2-D and 3-D are constructed respectively. For a uniform quadrate grid, the approach used to construct the stable formulas of 2M-order of accuracy is deliberated in detail, and the stable formulas of the second order (M=1) and the fourth order (M=2) are presented; for a 3-D cubic grid, the recursion formulas are discussed briefly.
Third, the stability and accuracy of the method are demonstrated by a series of 1-D and 2-D numerical tests, and the value of high order formulas for the improvement of calculation efficiency is pointed out.
Last, starting from the space decoupling FEM ordinary differential equation system (the equations of motion of the structure), a group of explicit time integration schemes are derived via the Lagrange polynomial interpolation and integration by parts, which are of high order of accuracy; furthermore, the expanding form of the schemes are given for any mathematical- physical model; the stability of the schemes are investigated preliminarily by a simple linear time-invariant system.
第一,依据波速有限的概念,从波动方程的精确解出发提出了一种新的显式数值模拟方法。此方法与现有有限元技术相似之处在于:适于处理非规则网格节点,且节点递推公式是具有显式和时空解耦特征。其不同之处在于:新方法可给出时空离散精度皆为2M阶的稳定格式,M为正整数。本文通过一维模型详细论述了这一方法的可行性:建立了非规则网格节点递推公式,详细分析了均匀网格标量波动数值模拟的精度和稳定性,提出了构建时空精度皆为2M阶( M为正整数)的稳定递推公式的技术途径,并以构建二阶( M = 1)和四阶( M = 2)公式为例予以说明。
第二,将一维情形研究结果推广到高维情形,分别建立了二维、三维非规则网格节点的递推公式。针对二维正方形均匀网格详细论述了时空离散精度皆为2M阶的稳定递推公式的构建方法,给出了二阶和四阶递推公式的具体结果;并对三维立方体均匀网格模型作了简要的讨论。
第三,本文通过一维、二维模型算例检验了波动方程数值模拟显式方法的精度和稳定性等理论结果,指出了高阶公式对提高计算效率的价值。
第四,从空间解耦有限元常微分方程组(结构动力学方程)出发,通过被积函数的拉格朗日多项式内插和分部积分导出了一组具有高阶精度的时域显式积分格式,并将其推广为适用于一般数学物理方程的显式积分格式。作者以一个简单的线性时不变系统为例,初步考察了此积分格式的稳定性。
How to simulate the wave motion in the interior of a complicated system of large scale accurately and effectively is a very important research subject for developing and improving techniques for numerical simulation of near-field wave motion. Traditionally, the explicit method with small computational cost is employed for the numerical simulation of wave motion within the computational region. However, the existing time-space decoupling explicit finite element method is only second order accurate. The low order of accuracy not only affects precision of the numerical simulation, but also restricts improving efficiency of the computation. Accordingly, this paper is devoted to explore stable explicit methods with high order of accuracy for the numerical simulation of wave motion within the computational region. The paper further develops the existing time-space decoupling explicit numerical simulation technique, and proposes a new highly accurate stable explicit method.
First, a new explicit method of numerical simulation is proposed based on the exact solution of wave equations. The method shares similarity with the lumped mass FEM, which is suitable for dealing with irregular grids and the recursion formulas derived are explicit and decoupling both in time and space. But, the recursion formulas developed by the method are 2M-order (M is the positive integer) accurate both in time and space. In this paper, the feasibility of the method is demonstrated via 1-D model: the recursion formulas at nodal points of an irregular grid are constructed, meanwhile, the stability and accuracy of the recursion formulas for a uniform grid are discussed in detail; according, an approach is proposed to construct the stable formulas which are of 2M-order of accuracy both in time and space with M being a positive integer and illustrated by constructing formulas of the second order (M=1) and the fourth order (M=2).
Second, the method is generalized to the multi-dimensional cases and the corresponding recursion formulas for an irregular grid in 2-D and 3-D are constructed respectively. For a uniform quadrate grid, the approach used to construct the stable formulas of 2M-order of accuracy is deliberated in detail, and the stable formulas of the second order (M=1) and the fourth order (M=2) are presented; for a 3-D cubic grid, the recursion formulas are discussed briefly.
Third, the stability and accuracy of the method are demonstrated by a series of 1-D and 2-D numerical tests, and the value of high order formulas for the improvement of calculation efficiency is pointed out.
Last, starting from the space decoupling FEM ordinary differential equation system (the equations of motion of the structure), a group of explicit time integration schemes are derived via the Lagrange polynomial interpolation and integration by parts, which are of high order of accuracy; furthermore, the expanding form of the schemes are given for any mathematical- physical model; the stability of the schemes are investigated preliminarily by a simple linear time-invariant system.
引文
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[2]王勖成.有限单元法[M].北京:清华大学出版社,2003
[3]刘晶波.波动的有限元模拟及复杂场地对地震动的影响[D].国家地震局博士学位论文,1989
[4]刘晶波,廖振鹏.离散网格中的弹性波动(Ⅰ)[J],地震工程与工程振动,1986:6(2),1-16
[5]刘晶波,廖振鹏,离散网格中的弹性波动(Ⅱ)[J],地震工程与工程振动,1989:9(2),1-11.
[6]向新民.谱方法的数值分析[M].北京:科学出版社,2000
[7]吴斌,韩强等.结构中的应力波[M].北京:科学出版社,2001
[8]张雄,刘岩.无网格法[M].北京:清华大学出版社,Springer出版社,2004.
[9]李小军,廖振鹏,杜修力.有阻尼动力问题的一种显式差分解法[J].工程力学(下册),北京:科学技术出版社,1992:471~477
[10]李小军,廖振鹏.非线性结构动力方程求解的显式差分格式的特性分析[J].世界地震工程,2000:16(2),8~12
[11]杨柏坡,陈庆彬.显式有限元法在地震工程中的应用[J].世界地震工程,1992:8(4),31~40
[12]易大义,陈道琦.数值分析引论[M].杭州:浙江大学出版社,1998
[13]胡建伟,汤怀民.微分方程数值方法[M].北京:科学出版社,2001
[14]钟万勰,暂态历程的精细计算方法[J].计算结构力学及其应用,1995:12(1),1~6
[15]梁昆淼.数学物理方法[M].北京:高等教育出版社,1995
[16]程建春.数学物理方程及其渐进方法[M].北京:科学出版社,2004
[17]廖振鹏.工程波动理论导论[M].第二版,科学出版社,2002
[18]廖振鹏,杨柏坡.频域透射边界[J].工程振动与振动工程,1986:6 (4),1～9.
[19]廖振鹏,杨柏坡等.暂态弹性波分析人工边界的研究[J].地震工程与工程振动,1982:2 (1),1～11.
[20]廖振鹏.近场波动问题的有限元解法[J].地震工程与工程振动1984:4(2), 1~14
[21]廖振鹏,周正华和张艳红.波动数值模拟中透射边界的稳定实现[J].地球物理学报,2002:45(4),533~545
[22]廖振鹏.法向透射边界条件[J].中国科学(E辑), 1996:26(2),186~192
[23]廖振鹏.结构动力反应的数值模拟[J].中国科学技术前沿,2001:4,247~258
[24]廖振鹏.透射边界和无穷远辐射条件[J].中国科学(E辑),2001:31(2), 454-261
[25]廖振鹏,黄孔亮,杨柏坡和袁一凡.暂态波透射边界[J].中国科学(A辑):1984:26(2),556~564
[26]廖振鹏和刘晶波.波动有限元模拟的基本问题[J].中国科学(B辑),1992:8, 874~882
[27]黎在良,刘殿魁.固体中波动[M].北京:科学出版社,1995
[28] A. Bayliss, K.E. Jordan, B. Lemesurier, and E. Turkel.A Fourth Accurate Finite Difference Scheme for the Computation of Elastic Waves[J].Bull. Seim. Soc. Amer., 1988:76,1115~1132
[29] Allen Taflove.A survey of the finite-difference time-domain literature[J], Computational Electrodynamics:The Finite-Difference Time-Domain Method[M] .Boston: Artech House, 1998:Chapter 1
[30] B. Gustafsson and E, Mossberg. Time Compact High Order Difference Methods for Wave Propagation[J].Report 2003-016, Dept of Information Technology, Uppsala University, SIAM J. Scient. Comp., 2004:26, 259~271
[31] B. Gustafsson and P, Wahlund. Time Compact High Order Difference Methods for Wave Propagation in Discontinuous Media[J].Report 2003-023, Dept of Information Technology, Uppsala University, SIAM J. Scient. Comp., 2004:26, 272~293
[32] B. Gustafsson and P. Wahlund. Time Compact High Order Difference Methods for Wave Propagation,2D[J], J. Scient. Comp.,2005:25(1),195~211
[33] Chen Houqun,Du Xiuli and Hou Shunzi. Appliciation of transmitting boundaries to nonlinear dynamic analysis of an arch dam-foundation-reservoir system[J],Dynamic Soil-Structure Interaction (edited by Zhang Chuham and J. P. Wolf),Beijing :International Academic Publishers,1997:115-124
[34] Cohen G C. Higher order numerical methods for transient wave equations[M]. Berlin, Heidelberg, New York: Springer, 2002:169-179
[35] Courant R, Friedrichs K, Lewy H.On the partial difference equations of mathematical Physics[J].Mathematics Annalen 100,1928:32~74
[36] Courant R. Variational Method for Solutions of Problems of Equilibrium and Vibrations[J].Bull.Am.Math.Soc.,1943:49,1~23
[37] G. Cohen and P. Joly.Construction and Analysis of Fourth-Order Finite Difference Schemes for the Acoustic Wave Equation in Nonhomogeneous Media[J].SIAM J. Numer. Anal., 1996:33,1266~1302
[38] G. R. Shubin and J. B. Bell.A Modified Equation Approach to ConstructingFourth Order Methods for Acoustic Wave Propagation[J].SIAM J. Sci. Statist. Comp., 1987:2, 135~151
[39] H.-O. Kriess and J. Oliger. Comparison of accurate methods for the integration of hyperbolic equations[J].Tellus,1972: 199~215
[40] Hilber,H,M,etc.Collocation,Dissipation and‘Overshoot for Time Integration Dynamic[J].Earthquake Engineering and Structural Dynamics,1978:6, 99~117
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