摘要
随着国民经济与国防技术需求的不断提高,工程机械系统的大型化、多构型化和高速化已成为发展趋势,多体系统的复杂性、强耦合性和非线性特征日益明显,系统动力学特性愈来愈复杂。传统的多体系统动力学分析方法已难于处理复杂工程对象的动力学问题。计算多体系统动力学作为经典力学与计算技术的结合应运而产生。
在计算多体系统动力学数值分析方面,由于非线性与时变耦合的多体系统动力学方程中,存在慢变大幅变量和快变微幅变量,导致了严重的数值病态问题。而且,拉格朗日乘子技术虽然较好地解决了复杂约束多体系统中的约束处理难题,但约束多体系统动力学方程的数学性质被改变,从微分方程组(ODEs)转变为微分代数方程组(DAEs)。由于DAEs的数值求解技术远不如ODEs的求解技术成熟,涉及一些在计算数学领域正在探索和研究的课题,为了获得准确、稳定和高效的多体系统数值解,迫切需要具有良好数值性态的计算方法。
鉴于此,本文基于多体系统动力学方程的两种等效变换途径,对约束多体系统数值算法展开研究,主要研究工作和结论如下。
在基于消去拉格朗日乘子项的ODEs等效变换类算法研究方面,本文基于离散零空间算法的改进,提出了约束多体系统的前离散零空间算法。首先,考虑到在离散零空间算法的等效变换公式中,零空间矩阵的确定通过离散导数实现,对数学离散积分方法有一定的依赖性。本文对零空间矩阵的计算方法进行了改进,提出了前离散零空间等效变换公式。该公式可不依赖于特定的积分方法,且能简洁、方便的与多种数值积分算法相结合,更有利于前离散零空间算法的推广。然后,以隐式中点法为数学离散方法,提出了约束多体系统的前离散零空间算法框架。推导了基于前离散零空间算法框架的单刚体动力学和多刚体动力学变量和算法参数。最后,考虑到已有离散零空间算法仅给出了多体动力学方程等效变换公式与中点法、特定变分积分法结合的数学离散积分公式,而在计算力学中,除中点法外,在数值积分算法中广泛使用和发展较快的龙格库塔法、变分积分法和Newmark法也均被广泛应用,成功地把隐式龙格库塔法、变分积分法和Newmark积分法与本文提出的约束多体系统前离散零空间算法框架相结合,降低了三种积分方法的计算复杂度,提高了计算效率。通过上述三种算法的构造、数值实验与分析,验证了前离散零空间等效变换公式的正确性,示例了前离散零空间等效变换公式与数值积分算法的良好结合性,说明了前离散零空间算法的有效性和可行性。
本文提出的前离散零空间算法,与约束多体系统的其他数值算法相比,它继承了离散零空间算法的优越性。采用前离散数学框架,降低了动力学方程的复杂度。利用零空间等效变换,实现了多体系统动力学方程的降维。通过每一单元的零空间矩阵的独立求解,较好的保持了系统动力学方程系数矩阵的带宽和相应的稀疏性,有利于稀疏矩阵求解技术的应用,从而可获得较高的计算效率。与已有离散零空间算法相比,前离散零空间算法的等效变换公式中零空间矩阵的确定方法更具一般性,不依赖于特定的积分方法,能简洁、方便的与多种数值积分算法相结合。基于前离散零空间等效变换公式构造的三种数值算法,拓宽了离散零空间算法在多体系统动力学计算中的应用,推广了零空间理论的应用范围。
在基于降指标技术的DAEs等效变换类算法研究方面,考虑到已有的Baumgarte约束违约稳定法,存在位移约束违约问题,计算结果不够准确。本文首先保持约束方程的位移约束形式,然后对DAEs形式的动力学方程进行降指标处理,将求解高阶微分代数方程的降阶GGL理论、ε嵌入处理方式与隐式龙格库塔法相结合,提出了约束多体系统的无违约算法。该算法始终直接满足位移约束方程,可避免位移约束违约问题,且能适应较大时间步长,在数值解的准确性、稳定性和计算效率方面均优于约束违约稳定法。
With the increasing needs of the national economy and the defense, to the mechanical systems in engineering, large-scale, multi-configuration and high speed has become a trend. The complexity, strong coupling and nonlinear char-acteristics for multibody systems have become increasingly evident, the system dynamics become more complex. The traditional multibody system dynamics method is difficult to deal with the dynamics of complex engineering objects. As a combination of classic mechanics and computing technology, the computa-tional of multibody system dynamics emerged.
In the numerical analysis of the computational dynamics of multibody sys-tems, since, in the motion equations of multibody systems with time-varying and nonlinear characteristics, the variables are changed both fast and greatly and gently and faintly, severe numerical ill-condition problems are caused. Moreover, although the Lagrange multiplier technique can solve the complex constrained multi-body systems, but the motion equations of constrained multibody system is changed, in the term of mathematics, from the ordinary differential equations (ODEs) into the differential algebraic equations (DAEs). The techniques ob-taining the numerical solution of DAEs, which involves a number of advanced areas in computational mathematics, is far less mature than that of ODEs. In order to obtain accurate, stable and efficient numerical solution of multibody system, the computational methods with good behavior is urgently needed.
In view of this, under the numerical algorithm framework for the con-strained multibody systems, based on the two ways of equivalent transformations for the motion equations of multibody systems, the numerical algorithm research is carried out in this paper.
In the respect of the equivalent transformation algorithm of ODEs based on the elimination of the Lagrange multiplier. based on the discretc null space method, the pre-discrete null space method (PDNS) for constrained multibody systems is proposed. First of all. taking into account the null-space matrix was obtained by the discrete derivative in the equivalent transformation formula in the discrete null space method, which is relied on the certain integral method to some extent, the formula is modified. This formula does not depend on the particular integration method, and can be concisely and conveniently combined with a variety of numerical integration algorithm, and is more conducive to promotion. Then, the implicit midpoint method as a mathematical discrete method, the framework of PDNS for constrained multibody systems is put for-ward. Finally, taking into account that only the transformation formula of the motion equations of multibody systems combining with the midpoint method was given in the discrete null space method and that, in addition to the mid-point method, the numerical integration algorithms such as the R.K method, the variational integration method and the Newmark method have been widely used in computational mechanics, to the promote of null space method, combining the framework of PDNS with the there methods, respectively, the PDNS is success-fully combined with the RK method, the variational integration method and the Newmark method. Through the construction of the three algorithms, numerical experiments and analyses, the equivalent transformation of PDNS is verified, the excellent unite of the PDNS with integration algorithm is illustrated, and the efficiency and feasibility of the PDNS is shown as well.
The proposed PDNS, comparing to other numerical algorithms used for con-strained multibody systems, inherits the advantages of the discrete null space method. By using the pre-discrete framework, the complexity of the motion equations is reduced. Through the null space equivalent transformation, the dimension of the dimensionality of multibody systems is reduced. Because the null space matrix of each unit is calculated independently, the bandwidth and sparsity of the coefficient matrix are kept better, which is conducive to the appli-cation of sparse matrix solution techniques and to higher efficiency. Comparing to the discrete null space method, this formula does not depend on the partic- ular integration method, and can be concisely and conveniently combined with a variety of numerical integration algorithm, and is more conducive to promo-tion. Through the three numerical algorithms based on PDNS, the discrete null space algorithm in multibody system dynamics is broadened, and the scope of application of null space method is extended.
In the respect of the equivalent transformation algorithm of ODEs based on the index-reduction, an algorithm without constraint violation for the mo-tion equations of multibody systems is proposed, combining the index-reduction theory of GGL and theεembedded approach with the implicit Runge-Kutta methods. The algorithm always directly meets the displacement constraint equa-tions, thus the displacement constraint violation problems can be avoided, and a larger time step can be adapted. As a result, the accuracy, stability and efficiency of the numerical solution are all better than constraint violation sta-bilization method.
引文
[1]洪嘉振.计算多体系统动力学.高等教育出版社,1999.
[2]齐朝晖.多体系统动力学.科学出版社,2008.
[3]A BO. Flexible Multibody Dynamics. Berlin, Heidelberg:Springer-Verlag, 2011.
[4]P F, J A, Pimenta Claro J, M LH. Kinematics and Dynamics of Multibody Systems with Imperfect Joints:Models and Case Studies. Springer,2008.
[5]Udwadia F, Kalaba R. What is the general form of the explicit equations of motion for constrained mechanical systems? Journal of Applied Mechanics-Transactions of the ASME,2002.69(3):335-339.
[6]Eun H, Park S, Lee E, Chung H. Analytical Method for Constrained Mechanical and Structural Systems. KSME International Journal,2004. 18(10):1691-1699.
[7]Falco D, Pennestri E, Vita L. Investigation of the Influence of Pseudoinverse Matrix Calculations on Multibody Dynamics Simulations by Means of the Udwadia-Kalaba Formulation. Journal of Aerospace Engineering,2009. 22(4):365-372.
[8]Braun DJ, Goldfarb M. Eliminating Constraint Drift in the Numerical Sim-ulation of Constrained Dynamical Systems. Computer Methods in Applied Mechanics and Engineering,2009.198(37-40):3151-3160.
[9]Gear C. Simultaneous Numerical Solution of Differential-Algebraic Equa-tions. IEEE Transactions on Circuit Theory,1971. CT-18(1):89-95.
[10]Gear C. Differential-Algebraic Equations. Computer Aided Analysis and Optimization of Mechanical Systems Dynamics,1984:323-334.
[11]Gear C, Petzold L. ODE Methods for the Solution of Differential/Algebraic Systems. SIAM Journal on Numerical Analysis,1984.21(4):716-728.
[12]Ascher U. On Symmetric Schemes and Differential-Algebraic Equations SIAM Journal on Scientific and Statistical Computing,1989.10(5):937-949.
[13]KE B, SL C, LR P. The Numerical Solution of Initial Value Problems in Differential-Algebraic Equations. North-Holland, New York,1989.
[14]Ercan C. On the Numerical Solution of Differential-Algebraic Equations with Index-2. Applied Mathematics and Computation,2004.156(2):541-550.
[15]Tseng F, Ma Z, Hulbert G. Efficient Numerical Solution of Constrained Multibody Dynamics Systems. Computer Methods in Applied Mechanics and Engineering,2003.192(3-4):439-472.
[16]Borri M, Trainelli L, Croce A. The Embedded Projection Method:A Gen-eral Index Reduction Procedure for Constrained System Dynamics. Com-puter Methods in Applied Mechanics and Engineering,2006.195(50-51): 6974-6992.
[17]Hairer E, Norsett SP, Wanner G. Solving Ordinary Differential Equations Ⅰ:Nonstiff Problems (Second Edition). Springer-Verlag, Berlin, Heidelberg, 2006.
[18]Hairer E, Wanner G. Solving Ordinary Differential Equations Ⅱ:Stiff and Differential-Algebraic Problems (Second Edition). Springer-Verlag, Berlin, Heidelberg,2006.
[19]Baumgarte E. Stabilization of Constraints and Integrals of Motion in Dy-namic Systems. Computer Methods in Applied Mechanics and Engineering, 1972.1:1-16.
[20]C CJ, D WS. Constraint Violation Stabilization Using Input-Output Feed-back Linearization in Multi-body Dynamic Analysis. Journal of Guidance, Control and Dynamics,1998.21(2):222-228.
[21]S L, M H. Stabilization method for numerical integration mechanical sys-tems. Journal of Mechanical Design,1998.120:565-572.
[22]Lin ST, Huang JN. Stabilization of Baumgarte's Method Using the Runge-Kutta Approach. Journal of Mechanical Design,2002.124:633-641.
[23]Lin ST, Huang JN. Numerical Integration of Multibody Mechanical Sys-tems Using Baumgarte's Constraint Stabilization Method. Journal of the Chinese Institute of Engineers,2002.25(2):243-252.
[24]付士慧,王琪.多体系统动力学方程违约修正的数值计算方法.计算力学学报,2007.24(1)：44-49.
[25]JK K, IS C, BH L. Determination of the feedback coefficients for the constraint violation stabilization method. Proceedings of Institution Me-chanical Engineers,1990.204:233-242.
[26]Weijia Z, Zhenkuan P, Yibing W. An Automatic Constraint Violation Sta-bilization Method for Differential/Algebraic Equations of Motion in Multi-body System Dynamics. Applied Mathematics and Mechanics (English Edition),2000.21(1):103-108.
[27]Betsch P. The Discrete Null Space Method for the Energy Consistent Integration of Constrained Mechanical Systems-Part Ⅰ:Holonomic Con-straints. Computer Methods in Applied Mechanics and Engineering,2005. 194(50-52):5159-5190.
[28]Betsch P, Leyendecker S. The Discrete Null Space Method for the En-ergy Consistent Integration of Constrained Mechanical Systems. Part Ⅱ: Multibody Synamics. International Journal for Numerical Methods in En-gineering,2006.67(4):499-552.
[29]Munoz JJ.Modelling Unilateral Frictionless Contact Using the Null-space Method and Cubic B-Spline Interpolation. Computer Methods in Applied Mechanics and Engineering,2008.197(9-12):979-993.
[30]Betsch P. Energy-Consistent Numerical Integration of Mechanical Systems with Mixed Holonomic and Nonholonomic Constraints. Computer Methods in Applied Mechanics and Engineering,2006.195(50-51):7020-7035.
[31]Betsch P, Uhlar S. Energy-Momentum Conserving Integration of Multi-body Dynamics. Multibody System Dynamics,2007.17(4):243-289.
[32]Leyendecker S, Marsden JE, Ortiz M. Variational Integrators for Con-strained Dynamical Systems. ZAMM,2008.88(9):677-708.
[33]Hesch C, Betsch P. A Mortar Method for Energy-Momentum Conserving Schemes in Frictionless Dynamic Contact Problems. International Journal for Numerical Methods in Engineering,2009.77(10):1468-1500.
[34]Leyendecker S, Ober-Bloebaum S, Marsden JE, Ortiz M. Discrete Me-chanics and Optimal Control for Constrained Systems. Optimal Control Applications & Methods,2010.31(6):505-528.
[35]Schmidt B, Leyendecker S, Ortiz M. Gamma-Convergence of Variational Integrators for Constrained Systems. Journal of Nonlinear Science,2009. 19(2):153-177.
[36]Betsch P, Hesch C, Saenger N, Uhlar S. Variational Integrators and Energy-Momentum Schemes for Flexible Multibody Dynamics. Journal of Com-putational and Nonlinear Dynamics,2010.5(3).
[37]Leyendecker S, Ober-Bloebaum S, Marsden JE, Ortiz M. Discrete mechan-ics and optimal control for constrained multibody dynamics. Proceedings of The ASME International Design Engineering Technical Conferences and Computers and Information in Engineering Conference.623-632.
Betsch P, Quasem M, Uhlar S. Numerical Integration of Discrete Mechan-ical Systems with Mixed Holonomic and Control Constraints. Journal of Mechanical Science and Technology,2009.23(4):1012-1018.
[39]Jung P, Leyendecker S, Linn J, Ortiz M. A Discrete Mechanics Approach to the Cosserat Rod Theory-Part 1:Static Equilibria. International Journal for Numerical Methods in Engineering,2011.85(1):31-60.
[40]Bauchau OA, Laulusa A. Enforcing Constraints in Multibody Systems:A Review (Draft). Proceedings of The ASME International Design Engineer-ing Technical Conferences and Computers and Information in Engineering Conference.73-96.
[41]Uhlar S, Betsch P. A Rotationless Formulation of Multibody Dynamics: Modeling of Screw Joints and Incorporation of Control Constraints. Multi-body System Dynamics,2009.22(1):69-95.
[42]Betsch P, Hesch C. Energy-Momentum Conserving Schemes for Friction-less Dynamic Contact Problems. Part I:NTS Method. Wriggers, P and Nackenhorst, U, editor,1UTAM Symposium on Computational Methods in Contact Mechanics.77-96.
[43]Uhlar S, Betsch P. Energy-Consistent Integration of Multibody Systems with Friction. Journal of Mechanical Science and Technology,2009.23(4): 901-909.
[44]Uhlar S, Betsch P. On the Derivation of Energy Consistent Time Stepping Schemes for Friction Afflicted Multibody Systems. Computers & Struc-tures,2010.88(11-12):737-754.
[45]Munoz JJ, Jelenic G. Sliding Joints in 3D Beams:Conserving Algorithms Using the Master-Slave Approach. Multibody System Dynamics,2006. 16(3):237-261.
[46]Munoz JJ. On the Modelling of Incompressibility in Linear and Non-linear Elasticity with the Master-Slave Approach. International Journal for Nu-merical Methods in Engineering.2008.74(2):269-293.
[47]Betsch P. Saenger N. On The Use of Geometrically Exact Shells in a Con-serving Framework for Flexible Multibody Dynamics. Computer Methods in Applied Mechanics and Engineering,2009.198(17-20):1609-1630.
[48]Betsch P, Siebert R. Rigid Body Dynamics in Terms of Quaternions:Hamil-tonian Formulation and Conserving Numerical Integration. International Journal for Numerical Methods in Engineering,2009.79(4):444-473.
[49]Gonzalez O. Time Integration and Discrete Hamiltonian Systems. Journal of Nonlinear Science,1996.6(5):449-467.
[50]Wang J, Rodriguez J, Keribar R. Integration of Flexible Multibody Systems Using Radau IIA Algorithms. Journal of Computational and Nonlinear Dynamics,2010.5(4).
[51]Bruls O, Cardona A. On the Use of Lie Group Time Integrators in Multi-body Dynamics. Journal of Computational and Nonlinear Dynamics,2010. 5(3).
[52]Nguyen VK, Nguyen PD, Hoang MC. Linearization and Parametric Vibra-tion Analysis of Some Applied Problems in Multibody Systems. Multibody System Dynamics,2009.22(2):163-180.
[53]Kanarachos S. Analysis of 2D Flexible Mechanisms Using Linear Finite Elements and Incremental Techniques. Computational Mechanics,2008. 42(1):107-117.
[54]Ferreirade Sousa PJSA, Pereira JCF, Allen JJ. Two-Dimensional Compact Finite Difference Immersed Boundary Method. International Journal for Numerical Methods in Fluids,2011.65(6):609-624.
[55]Xiao-feng P. The Collision Properties of Soliton Solutions of a Generally Nonlinear Scherodinger Equation. International Journal of Nonlinear Sci-ences and Numerical Simulation,2010.11(12):1069-1075.
[56]Trejo-Guerra R. Tlelo-Cuautle E, Munoz-Pacheco JM, Sanchez-Lopez C Cruz-Hernandez C. On the Relation between the Number of Scrolls and the Lyapunov Exponents in PWL-functions-based n-Scroll Chaotic Oscilla-tors. International Journal of Nonlinear Sciences and Numerical Simulation, 2010.11(11):903-910.
[57]Olagunju DO. Extensional Flow of a Viscoelastic Filament Governed by the FENE-CR Model. International Journal of Non-linear Mechanics,2011. 46(1):73-78.
[58]Goh SM, Al-Sawalha MM, Noorani MSM, Hashim I. Enhanced Varia-tional Iteration Method using Adomian Polynomials for Solving the Chaotic Lorenz System. International Journal of Nonlinear Sciences and Numerical Simulation,2010.11(9):689-700.
[59]Chang-Jian CW. Non-linear Dynamic Analysis of a HSFD Mounted Gear-bearing System. Nonlinear Dynamics,2010.62(1-2):333-347.
[60]Cao J, Ma C, Xie H, Jiang Z. Nonlinear Dynamics of Duffing System With Fractional Order Damping. Journal of Computational and Nonlinear Dynamics,2010.5(4).
[61]Rashidi R, Mohammadi AK, Nejad FB. Bifurcation and Nonlinear Dy-namic Analysis of a Rigid Rotor Supported by Two-lobe Noncircular Gas-Lubricated Journal Bearing System. Nonlinear Dynamics,2010.61(4): 783-802.
[62]Li Y, Song Z, Li Y, Lang Y, Li D. Theoretical Analysis and Numerical Simulation for the Spill Procedure of Liquid Fuel of Fuel Air Explosive with Shell. International Journal of Non-linear Mechanics,2010.45(7): 699-703.
[63]Schafer N, Negrut D. A Quantitative Assessment of the Potential of Im-plicit Integration Methods for Molecular Dynamics Simulation. Journal of Computational and Nonlinear Dynamics,2010.5(3).
[64]Rashidi R, Mohammadi AK. Xejad FB. Preload Effect on Nonlinear Dy-namic Behavior of a Rigid Rotor Supported by Noncircular Gas-Lubricated Journal Bearing Systems. Nonlinear Dynamics,2010.60(3):231-253.
[65]Zhang QH. Lin JZ. Wang CX. Orientation Distribution of Fibers Immersed in a Curved Expansion Duct. International Journal of Nonlinear Sciences and Numerical Simulation,2009.10(11-12):1585-1594.
[66]Jo H. Yabuno H. Stabilization of a 1/3-order Subharmonic Resonance Using Nonlinear Dynamic Vibration Absorber. Nonlinear Dynamics,2010. 59(4):747-758.
[67]Hong SW, Kim C. A New Finite Volume Method on Junction Coupling and Boundary Treatment for Flow Network System Analyses. International Journal for Numerical Methods in Fluids,2011.65(6):707-742.
[68]Halama J, Benkhaldoun F, Fort J. Flux Schemes Based Finite Volume Method for Internal Transonic Flow with Condensation. International Jour-nal for Numerical Methods in Fluids,2011.65(8):953-968.
[69]Chamkha A, Mansour MA, Aly A. Unsteady MHD Free Convective Heat and Mass Transfer from a Vertical Porous Plate with Hall Current, Ther-mal Radiation and Chemical Reaction Effects. International Journal for Numerical Methods in Fluids,2011.65(4):432-447.
[70]Zhang Y, Zou Q, Greaves D. Numerical Simulation of Free-surface Flow Using the Level-set Method with Global Mass Correction. International Journal for Numerical Methods in Fluids,2010.63(6):651-680.
[71]Kane C, Marsden JE, Ortiz M, West M. Variational Integrators and the Newmark Algorithm for Conservative and Dissipative Mechanical Systems. International Journal for Numerical Methods in Engineering,2000.49: 1295-1325.
[72]Lew A, Marsden JE, Ortiz M, West M. Variational Time Integrators. International Journal for Numerical Methods in Engineering.2004.60: 153-212.
[73]Ma Z, Rowley CW. Lie-Poisson integrators:A Hamiltonian, Variational Approach. International Journal for Numerical Methods in Engineering, 2010.82(13):1609-1644.
[74]Lee T, Leok M. McClamroch NH. Lagrangian Mechanics and Variational Integrators on Two-spheres. International Journal for Numerical Methods in Engineering,2009.79(9):1147-1174.
[75]Saccon A. Midpoint Rule for Variational Integrators on Lie Groups. In-ternational Journal for Numerical Methods in Engineering,2009.78(11): 1345-1364.
[76]Romero I. Formulation and Performance of Variational Integrators for Rotating Bodies. Computational Mechanics,2008.42(6):825-836.
[77]Lee T, Leok M, McClamroch NH. Computational Dynamics of a 3D Elastic String Pendulum Attached to a Rigid Body and an Inertially Fixed Reel Mechanism. Nonlinear Dynamics,2011.64(1-2):97-115.
[78]Scovazzi G, Shadid JN, Love E, Rider WJ. A Conservative Nodal Varia-tional Multiscale Method for Lagrangian Shock Hydrodynamics. Computer Methods in Applied Mechanics and Engineering,2010.199(49-52):3059-3100.
[79]Cho SS, Huh H, Park KC. A Time-discontinuous Implicit Variational Inte-grator for Stress Wave Propagation Analysis in Solids. Computer Methods in Applied Mechanics and Engineering,2011.200(5-8):649-664.
[80]Gates M, Matous K, Heath MT. Asynchronous Multi-domain Variational Integrators for Non-linear Problems. International Journal for Numerical Methods in Engineering,2008.76(9):1353-1378.
[81]Kale KG. Lew AJ. Parallel Asynchronous Variational Integrators. In-ternational Journal for Numerical Methods in Engineering,2007.70(3): 291-321.
[82]Romero I. Algorithms for Coupled Problems that Preserve Symmetries and the Laws of Thermodynamics Part I:Monolithic Integrators and their Ap-plication to Finite Strain Thermoelasticity. Computer Methods in Applied Mechanics and Engineering,2010.199(25-28):1841-1858.
[83]de Leon M, Martin de Diego D, Santamaria-Merino A. Discrete Varia-tional Integrators and Optimal Control Theory. Advances in Computa-tional Mathematics,2007.26(1-3):251-268.
[84]Betsch P. The Discrete Null Space Method for The Energy Consistent Inte-gration of Constrained Mechanical Systems Part I:Holonomic constraints. Computer Methods in Applied Mechanics and Engineering,2005.194: 5159-5190.
[85]Thai HT, Kim SE. Nonlinear Static and Dynamic Analysis of Cable Struc-tures. Finite Elements in Analysis and Design,2011.47(3):237-246.
[86]Boulaajine L, Farhloul M, Paquet L. A Priori Error Estimation for the Dual Mixed Finite Element Method of the Elastodynamic Problem in a Polygonal Domain. Journal of Computational and Applied Mathematics, 2011.235(5):1288-1310.
[87]Zhu C, Qin G, Zhang J. Implicit Chebyshev Spectral Element Method for Acoustics Wave Equations. Finite Elements in Analysis and Design,2011. 47(2):184-194.
[88]Laier JE. Spectral Analysis of a High-order Hermitian Algorithm for Struc-tural Dynamics. Applied Mathematical Modelling,2011.35(2):965-971.
[89]Biswal KC, Bhattacharyya SK. Dynamic Response of Structure Coupled with Liquid Sloshing in a Laminated Composite Cylindrical Tank with Baffle. Finite Elements in Analysis and Design,2010.46(11):966-981.
[90]Leontyev VA. Direct Time Integration Algorithm with Controllable Nu-merical Dissipation for Structural Dynamics:Two-step Lambda Method. Applied Numerical Mathematics,2010.60(3):277-292.
[91]Oyarzun P, Loureiro FS. Carrer JAM. Mansur WJ. A Time-stepping Scheme Based on Numerical Green's Functions for the Domain Boundary Element Method:The ExGA-DBEM Newmark Approach. Engineering Analysis with Boundary Elements,2011.35(3):533-542.
[92]Abdollahifar A, Shafiei AR, Mohyeddin A, Bozorg M. Using State-space Models for Solving Elastodynamic Problems Discretized by the MLPG Method. Engineering Analysis with Boundary Elements,2010.34(8):721-726.
[93]Lee K. A Stabilized Numerical Solution for the Dynamic Contact of the Bodies Having Very Stiff Constraint on the Contact Point. Computational Mechanics,2010.46(4):533-543.
[94]Assie AE, Eltaher MA, Mahmoud FF. Modeling of Viscoelastic Contact-impact Problems. Applied Mathematical Modelling,2010.34(9):2336-2352.
[95]Alonso FJ, Cuadrado J, Lugris U, Pintado P. A Compact Smoothing-differentiation and Projection Approach for the Kinematic Data Consis-tency of Biomechanical Systems. Multibody System Dynamics,2010.24(1): 67-80.
[96]Ibrahim HH, Yoo HH, Tawfik M, Lee KS. Thermo-acoustic Random Re-sponse of Temperature-dependent Functionally Graded Material Panels. Computational Mechanics,2010.46(3):377-386.
[97]Soares D Jr, Sladek J, Sladek V. Non-linear Dynamic Analyses by Meshless Local Petrov-Galerkin Formulations. International Journal for Numerical Methods in Engineering,2010.81(13):1687-1699.
[98]Ibrahim SM, Patel BP, Nath Y. Nonlinear Periodic Response of Composite Curved Beam Subjected to Symmetric and Antisymmetric Mode Excita-tion. Journal of Computational and Nonlinear Dynamics,2010.5(2).