A two-step unconditionally stable explicit method with controllable numerical dissipations
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摘要
A family of unconditionally stable direct integration algorithm with controllable numerical dissipations is proposed. The numerical properties of the new algorithms are controlled by three parameters α, β and γ. By the consistent and stability analysis, the proposed algorithms achieve the second-order accuracy and are unconditionally stable under the condition that α≥-0.5, β≤ 0.5 and γ≥-(1+α)/2. Compared with other unconditionally stable algorithms, such as Chang's algorithms and CR algorithm, the proposed algorithms are found to be superior in terms of the controllable numerical damping ratios. The unconditional stability and numerical damping ratios of the proposed algorithms are examined by three numerical examples. The results demonstrate that the proposed algorithms have a superior performance and can be used expediently in solving linear elastic dynamics problems.
A family of unconditionally stable direct integration algorithm with controllable numerical dissipations is proposed. The numerical properties of the new algorithms are controlled by three parameters α, β and γ. By the consistent and stability analysis, the proposed algorithms achieve the second-order accuracy and are unconditionally stable under the condition that α ≥-0.5, β ≤ 0.5 and γ ≥-(1+α)/2. Compared with other unconditionally stable algorithms, such as Chang's algorithms and CR algorithm, the proposed algorithms are found to be superior in terms of the controllable numerical damping ratios. The unconditional stability and numerical damping ratios of the proposed algorithms are examined by three numerical examples. The results demonstrate that the proposed algorithms have a superior performance and can be used expediently in solving linear elastic dynamics problems.
A family of unconditionally stable direct integration algorithm with controllable numerical dissipations is proposed. The numerical properties of the new algorithms are controlled by three parameters α, β and γ. By the consistent and stability analysis, the proposed algorithms achieve the second-order accuracy and are unconditionally stable under the condition that α ≥-0.5, β ≤ 0.5 and γ ≥-(1+α)/2. Compared with other unconditionally stable algorithms, such as Chang's algorithms and CR algorithm, the proposed algorithms are found to be superior in terms of the controllable numerical damping ratios. The unconditional stability and numerical damping ratios of the proposed algorithms are examined by three numerical examples. The results demonstrate that the proposed algorithms have a superior performance and can be used expediently in solving linear elastic dynamics problems.
引文
Belytschko T,Hughes TJR and Burgers P(1983),Computational Methods for Transient Analysis,NorthHolland.
Chang SY(1997),“Improved Numerical Dissipation for Explicit Methods in Pseudodynamic Tests,”Earthquake Engineering and Structural Dynamics,26(9):917?929.
Chang SY(2002),“Explicit Pseudodynamic Algorithm with Unconditional Stability,”Journal of Engineering Mechanics,128(9):935?947.
Chang SY(2004),“Unconditional Stability for Explicit Pseudodynamic Testing,”Structural Engineering and Mechanics,18(4):411?428.
Chang SY(2007),“Improved Explicit Method for Structural Dynamics,”Journal of Engineering Mechanics,133(7):748?760.
Chang SY(2010),“A New Family of Explicit Methods for Linear Structural Dynamics,”Computers and Structures,88(11-12):755?772.
Chang SY and Huang C-L(2014),“Non-Iterative Integration Methods with Desired Numerical Dissipation,”International Journal for Numerical Methods in Engineering,100(1):62?86.
Chang SY(2015),“Dissipative,Noniterative Integration Algorithms with Unconditional Stability for Mildly Nonlinear Structural Dynamic Problems,”Nonlinear Dynamics,79(2):1625?1649.
Chen C and Ricles JM(2008),“Development of Direct Integration Algorithms for Structural Dynamics Using Discrete Control Theory,”Journal of Engineering Mechanics,134(8):676?683.
Chopra AK(2011),Dynamics of Structures:Theory and Applications to Earthquake Engineering,Prentice Hall.Chung J and Hulbert GM(1993),“A Time Integration Algorithm for Structural Dynamics with Improved Numerical Dissipation:The Generalized-αMethod,”Journal of Applied Mechanics,60:371.
Goudreau GL and Taylor RL(1973),“Evaluation of Numerical Methods in Elastodynamics,”Computer Methods in Applied Mechanics and Engineering,2(1):69?97.
Hilber HM,Hughes TJR and Taylor RL(1977),“Improved Numerical Dissipation for Time Integration Algorithms in Structural Dynamics,”Earthquake Engineering and Structural Dynamics,5(3):283?292.
Hilber HM and Hughes TJR(1978),“Collocation,Dissipation and‘Overshoot’for Time Integration Schemes in Structural Dynamics,”Earthquake Engineering and Structural Dynamics,6(1):99?117.
Hughes TJR(1987),The Finite Element Method:Linear Static and Dynamic Finite Element Analysis,PrenticeHall,New York.
Kolay C and Ricles JM(2014),“Development of a Family of Unconditionally Stable Explicit Direct Integration Algorithms with Controllable Numerical Energy Dissipation,”Earthquake Engineering and Structural Dynamics,43(9):1361?1380.
Kolay C and Ricles JM(2016),“Assessment of Explicit and Semi-Explicit Classes of Model-Based Algorithms for Direct Integration in Structural Dynamics,”International Journal for Numerical Methods in Engineering,107(1):49?73.
Li C,Jie J,Jiang L and Yang TY(2016),“Theory and Implementation of a Two-Step Unconditionally Stable Explicit Integration Algorithm for Vibration Analysis of Structures,”Shock and Vibration,2016(3):1?8.
Newmark NM(1959),“A Method of Computation for Structural Dynamics,”Proc Asce,85(1):67?94.
Yu K(2008),“A New Family of Generalized-αTime Integration Algorithms without Overshoot for Structural Dynamics,”Earthquake Engineering and Structural Dynamics,37(12):1389?1409.
Zhou X,Tamma KK,Kanapady R and Sha D(2000),“The Time Dimension:New and Recent Advances and a Unified Framework towards Design of Time Discretized Operators for Structural Dynamics,”in European Congress on Computational Methods in Applied Sciences and Engineering,ECCOMAS,Barcelona,Spain.
Zhou X,Tamma KK and Sha D(2001),“An Optimal Family of Controllable Numerically Dissipative Time Integration Algorithms for Structural Dynamics,”Aiaa Applied Aerodynamics Conference,5:3050?3060.
Zhou X and Tamma KK(2004a),“Design,Analysis,and Synthesis of Generalized Single Step Single Solve and Optimal Algorithms for Structural Dynamics,”International Journal for Numerical Methods in Engineering,59(5):597-668.
Zhou X and Tamma KK(2004b),“A New Unified Theory Underlying Time Dependent Linear FirstOrder Systems:A Prelude to Algorithms by Design,”International Journal for Numerical Methods in Engineering,60(10):1699?1740.
Zhou X,Tamma KK and Sha D(2005),“Design Spaces,Measures and Metrics for Evaluating Quality of Time Operators and Consequences Leading to Improved Algorithms by Design-Illustration to Structural Dynamics,”Nternational Journal for Numerical Methods in Engineering,64(14):1841?1870.
Chang SY(1997),“Improved Numerical Dissipation for Explicit Methods in Pseudodynamic Tests,”Earthquake Engineering and Structural Dynamics,26(9):917?929.
Chang SY(2002),“Explicit Pseudodynamic Algorithm with Unconditional Stability,”Journal of Engineering Mechanics,128(9):935?947.
Chang SY(2004),“Unconditional Stability for Explicit Pseudodynamic Testing,”Structural Engineering and Mechanics,18(4):411?428.
Chang SY(2007),“Improved Explicit Method for Structural Dynamics,”Journal of Engineering Mechanics,133(7):748?760.
Chang SY(2010),“A New Family of Explicit Methods for Linear Structural Dynamics,”Computers and Structures,88(11-12):755?772.
Chang SY and Huang C-L(2014),“Non-Iterative Integration Methods with Desired Numerical Dissipation,”International Journal for Numerical Methods in Engineering,100(1):62?86.
Chang SY(2015),“Dissipative,Noniterative Integration Algorithms with Unconditional Stability for Mildly Nonlinear Structural Dynamic Problems,”Nonlinear Dynamics,79(2):1625?1649.
Chen C and Ricles JM(2008),“Development of Direct Integration Algorithms for Structural Dynamics Using Discrete Control Theory,”Journal of Engineering Mechanics,134(8):676?683.
Chopra AK(2011),Dynamics of Structures:Theory and Applications to Earthquake Engineering,Prentice Hall.Chung J and Hulbert GM(1993),“A Time Integration Algorithm for Structural Dynamics with Improved Numerical Dissipation:The Generalized-αMethod,”Journal of Applied Mechanics,60:371.
Goudreau GL and Taylor RL(1973),“Evaluation of Numerical Methods in Elastodynamics,”Computer Methods in Applied Mechanics and Engineering,2(1):69?97.
Hilber HM,Hughes TJR and Taylor RL(1977),“Improved Numerical Dissipation for Time Integration Algorithms in Structural Dynamics,”Earthquake Engineering and Structural Dynamics,5(3):283?292.
Hilber HM and Hughes TJR(1978),“Collocation,Dissipation and‘Overshoot’for Time Integration Schemes in Structural Dynamics,”Earthquake Engineering and Structural Dynamics,6(1):99?117.
Hughes TJR(1987),The Finite Element Method:Linear Static and Dynamic Finite Element Analysis,PrenticeHall,New York.
Kolay C and Ricles JM(2014),“Development of a Family of Unconditionally Stable Explicit Direct Integration Algorithms with Controllable Numerical Energy Dissipation,”Earthquake Engineering and Structural Dynamics,43(9):1361?1380.
Kolay C and Ricles JM(2016),“Assessment of Explicit and Semi-Explicit Classes of Model-Based Algorithms for Direct Integration in Structural Dynamics,”International Journal for Numerical Methods in Engineering,107(1):49?73.
Li C,Jie J,Jiang L and Yang TY(2016),“Theory and Implementation of a Two-Step Unconditionally Stable Explicit Integration Algorithm for Vibration Analysis of Structures,”Shock and Vibration,2016(3):1?8.
Newmark NM(1959),“A Method of Computation for Structural Dynamics,”Proc Asce,85(1):67?94.
Yu K(2008),“A New Family of Generalized-αTime Integration Algorithms without Overshoot for Structural Dynamics,”Earthquake Engineering and Structural Dynamics,37(12):1389?1409.
Zhou X,Tamma KK,Kanapady R and Sha D(2000),“The Time Dimension:New and Recent Advances and a Unified Framework towards Design of Time Discretized Operators for Structural Dynamics,”in European Congress on Computational Methods in Applied Sciences and Engineering,ECCOMAS,Barcelona,Spain.
Zhou X,Tamma KK and Sha D(2001),“An Optimal Family of Controllable Numerically Dissipative Time Integration Algorithms for Structural Dynamics,”Aiaa Applied Aerodynamics Conference,5:3050?3060.
Zhou X and Tamma KK(2004a),“Design,Analysis,and Synthesis of Generalized Single Step Single Solve and Optimal Algorithms for Structural Dynamics,”International Journal for Numerical Methods in Engineering,59(5):597-668.
Zhou X and Tamma KK(2004b),“A New Unified Theory Underlying Time Dependent Linear FirstOrder Systems:A Prelude to Algorithms by Design,”International Journal for Numerical Methods in Engineering,60(10):1699?1740.
Zhou X,Tamma KK and Sha D(2005),“Design Spaces,Measures and Metrics for Evaluating Quality of Time Operators and Consequences Leading to Improved Algorithms by Design-Illustration to Structural Dynamics,”Nternational Journal for Numerical Methods in Engineering,64(14):1841?1870.