Explicit/implicit partitioning and a new explicit form of the generalized alpha method

Most finite element packages use the Newmark algorithm for time integration of structural dynamics. Various algorithms have been proposed to better optimize the high frequency dissipation of this algorithm. Hulbert and Chung proposed both implicit and explicit forms of the generalized alpha method....

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Veröffentlicht in:	Communications in numerical methods in engineering Jg. 19; H. 11; S. 909 - 920
1. Verfasser:	Daniel, W. J. T.
Format:	Journal Article
Sprache:	Englisch
Veröffentlicht:	Chichester, UK John Wiley & Sons, Ltd 01.11.2003 Wiley
Schlagworte:	Computational techniques direct integration Exact sciences and technology explicit/implicit partition finite elements Finite-element and galerkin methods Fundamental areas of phenomenology (including applications) generalized alpha method Mathematical methods in physics Physics Solid mechanics Structural and continuum mechanics Vibration, mechanical wave, dynamic stability (aeroelasticity, vibration control...) Vibrations and mechanical waves generalized alpha method explicit/implicit partition direct integration finite elements
ISSN:	1069-8299, 1099-0887
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Abstract	Most finite element packages use the Newmark algorithm for time integration of structural dynamics. Various algorithms have been proposed to better optimize the high frequency dissipation of this algorithm. Hulbert and Chung proposed both implicit and explicit forms of the generalized alpha method. The algorithms optimize high frequency dissipation effectively, and despite recent work on algorithms that possess momentum conserving/energy dissipative properties in a non‐linear context, the generalized alpha method remains an efficient way to solve many problems, especially with adaptive timestep control. However, the implicit and explicit algorithms use incompatible parameter sets and cannot be used together in a spatial partition, whereas this can be done for the Newmark algorithm, as Hughes and Liu demonstrated, and for the HHT‐αalgorithm developed from it. The present paper shows that the explicit generalized alpha method can be rewritten so that it becomes compatible with the implicit form. All four algorithmic parameters can be matched between the explicit and implicit forms. An element interface between implicit and explicit partitions can then be used, analogous to that devised by Hughes and Liu to extend the Newmark method. The stability of the explicit/implicit algorithm is examined in a linear context and found to exceed that of the explicit partition. The element partition is significantly less dissipative of intermediate frequencies than one using the HHT‐αmethod. The explicit algorithm can also be rewritten so that the discrete equation of motion evaluates forces from displacements and velocities found at the predicted mid‐point of a cycle. Copyright © 2003 John Wiley & Sons, Ltd.
AbstractList	Most finite element packages use the Newmark algorithm for time integration of structural dynamics. Various algorithms have been proposed to better optimize the high frequency dissipation of this algorithm. Hulbert and Chung proposed both implicit and explicit forms of the generalized alpha method. The algorithms optimize high frequency dissipation effectively, and despite recent work on algorithms that possess momentum conserving/energy dissipative properties in a non‐linear context, the generalized alpha method remains an efficient way to solve many problems, especially with adaptive timestep control. However, the implicit and explicit algorithms use incompatible parameter sets and cannot be used together in a spatial partition, whereas this can be done for the Newmark algorithm, as Hughes and Liu demonstrated, and for the HHT‐αalgorithm developed from it. The present paper shows that the explicit generalized alpha method can be rewritten so that it becomes compatible with the implicit form. All four algorithmic parameters can be matched between the explicit and implicit forms. An element interface between implicit and explicit partitions can then be used, analogous to that devised by Hughes and Liu to extend the Newmark method. The stability of the explicit/implicit algorithm is examined in a linear context and found to exceed that of the explicit partition. The element partition is significantly less dissipative of intermediate frequencies than one using the HHT‐αmethod. The explicit algorithm can also be rewritten so that the discrete equation of motion evaluates forces from displacements and velocities found at the predicted mid‐point of a cycle. Copyright © 2003 John Wiley & Sons, Ltd. abstract Most finite element packages use the Newmark algorithm for time integration of structural dynamics. Various algorithms have been proposed to better optimize the high frequency dissipation of this algorithm. Hulbert and Chung proposed both implicit and explicit forms of the generalized alpha method. The algorithms optimize high frequency dissipation effectively, and despite recent work on algorithms that possess momentum conserving/energy dissipative properties in a non‐linear context, the generalized alpha method remains an efficient way to solve many problems, especially with adaptive timestep control. However, the implicit and explicit algorithms use incompatible parameter sets and cannot be used together in a spatial partition, whereas this can be done for the Newmark algorithm, as Hughes and Liu demonstrated, and for the HHT‐αalgorithm developed from it. The present paper shows that the explicit generalized alpha method can be rewritten so that it becomes compatible with the implicit form. All four algorithmic parameters can be matched between the explicit and implicit forms. An element interface between implicit and explicit partitions can then be used, analogous to that devised by Hughes and Liu to extend the Newmark method. The stability of the explicit/implicit algorithm is examined in a linear context and found to exceed that of the explicit partition. The element partition is significantly less dissipative of intermediate frequencies than one using the HHT‐αmethod. The explicit algorithm can also be rewritten so that the discrete equation of motion evaluates forces from displacements and velocities found at the predicted mid‐point of a cycle. Copyright © 2003 John Wiley & Sons, Ltd. Most finite element packages use the Newmark algorithm for time integration of structural dynamics. Various algorithms have been proposed to better optimize the high frequency dissipation of this algorithm. Hulbert and Chung proposed both implicit and explicit forms of the generalized alpha method. The algorithms optimize high frequency dissipation effectively, and despite recent work on algorithms that possess momentum conserving/energy dissipative properties in a non-linear context, the generalized alpha method remains an efficient way to solve many problems, especially with adaptive timestep control. However, the implicit and explicit algorithms use incompatible parameter sets and cannot be used together in a spatial partition, whereas this can be done for the Newmark algorithm, as Hughes and Liu demonstrated, and for the HHT- alpha algorithm developed from it. The present paper shows that the explicit generalized alpha method can be rewritten so that it becomes compatible with the implicit form. All four algorithmic parameters can be matched between the explicit and implicit forms. An element interface between implicit and explicit partitions can then be used, analogous to that devised by Hughes and Liu to extend the Newmark method. The stability of the explicit/implicit algorithm is examined in a linear context and found to exceed that of the explicit partition. The element partition is significantly less dissipative of intermediate frequencies than one using the HHT- alpha method. The explicit algorithm can also be rewritten so that the discrete equation of motion evaluates forces from displacements and velocities found at the predicted mid-point of a cycle. (Author)
Author	Daniel, W. J. T.
Author_xml	– sequence: 1 givenname: W. J. T. surname: Daniel fullname: Daniel, W. J. T. email: bill@mech.uq.edu.au organization: Department of Mechanical Engineering, The University of Queensland, St Lucia Campus, 4072, Brisbane, Queensland, Australia
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Cites_doi	10.1002/(SICI)1097-0207(19990620)45:5<569::AID-NME595>3.0.CO;2-A 10.1115/1.3424305 10.1007/s00466-001-0273-z 10.1016/S0045-7825(96)01036-5 10.1016/S0045-7825(00)00256-5 10.1115/1.2900803 10.1016/S0045-7825(98)00176-5 10.1115/1.3424304 10.1002/eqe.4290050306
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References	Hughes TJR, Liu WK. Implicit-explicit finite elements in transient analysis: implementation and numerical examples. ASME Journal of Applied Mechanics 1978; 45:375-378. Erlicher S, Bonaventura L, Bursi OS. The analysis of the generalized-αmethod for non-linear dynamic problems. Computational Mechanics 2002; 28:83-104. Kuhl D, Crisfield MA. Energy-conserving and decaying algorithms in non-linear structural dynamics. International Journal for Numerical Methods in Engineering 1999; 45:569-599. Hulbert GM, Chung JP. Explicit time integration algorithms for structural dynamics with optimal numerical dissipation. Computer Methods in Applied Mechanics and Engineering 1996; 137:175-188. Chung J, Hulbert GM. A time integration method for structural dynamics with improved numerical dissipation: the generalized-α method. Journal of Applied Mechanics 1993; 60:371-375. Armero F, Romero I. On the formulation of high-frequency dissipative time-stepping algorithms for non-linear dynamics. Part 1: low-order methods for two model problems and non-linear elastodynamics. Computer Methods in Applied Mechanics and Engineering 2001; 190:2603-2649. Bauchau OA, Botasso CL. On the design of energy preserving and decaying schemes for flexible, non-linear multi-body systems. Computer Methods in Applied Mechanics and Engineering 1999; 169:61-79. Hughes TJR, Liu WK. Implicit-explicit finite elements in transient analysis: stability theory. ASME Journal of Applied Mechanics 1978; 45:371-374. Hilber HM, Hughes TJR, Taylor RL. Improved numerical dissipation for time integration algorithms in structural dynamics. Earthquake Engineering and Structural Dynamics 1977; 5:283-292. 1999; 45 2002; 28 1996 1999; 169 2001 1996; 137 1978; 45 2001; 190 1993; 60 1977; 5 Hulbert GM (e_1_2_1_7_2) 2001 e_1_2_1_6_2 e_1_2_1_4_2 e_1_2_1_5_2 e_1_2_1_2_2 e_1_2_1_11_2 e_1_2_1_3_2 e_1_2_1_12_2 e_1_2_1_10_2 e_1_2_1_13_2 e_1_2_1_14_2 e_1_2_1_8_2 e_1_2_1_9_2
References_xml	– reference: Armero F, Romero I. On the formulation of high-frequency dissipative time-stepping algorithms for non-linear dynamics. Part 1: low-order methods for two model problems and non-linear elastodynamics. Computer Methods in Applied Mechanics and Engineering 2001; 190:2603-2649. – reference: Hilber HM, Hughes TJR, Taylor RL. Improved numerical dissipation for time integration algorithms in structural dynamics. Earthquake Engineering and Structural Dynamics 1977; 5:283-292. – reference: Hulbert GM, Chung JP. Explicit time integration algorithms for structural dynamics with optimal numerical dissipation. Computer Methods in Applied Mechanics and Engineering 1996; 137:175-188. – reference: Erlicher S, Bonaventura L, Bursi OS. The analysis of the generalized-αmethod for non-linear dynamic problems. Computational Mechanics 2002; 28:83-104. – reference: Hughes TJR, Liu WK. Implicit-explicit finite elements in transient analysis: implementation and numerical examples. ASME Journal of Applied Mechanics 1978; 45:375-378. – reference: Kuhl D, Crisfield MA. Energy-conserving and decaying algorithms in non-linear structural dynamics. International Journal for Numerical Methods in Engineering 1999; 45:569-599. – reference: Chung J, Hulbert GM. A time integration method for structural dynamics with improved numerical dissipation: the generalized-α method. Journal of Applied Mechanics 1993; 60:371-375. – reference: Bauchau OA, Botasso CL. On the design of energy preserving and decaying schemes for flexible, non-linear multi-body systems. Computer Methods in Applied Mechanics and Engineering 1999; 169:61-79. – reference: Hughes TJR, Liu WK. Implicit-explicit finite elements in transient analysis: stability theory. ASME Journal of Applied Mechanics 1978; 45:371-374. – year: 1996 – volume: 28 start-page: 83 year: 2002 end-page: 104 article-title: The analysis of the generalized‐αmethod for non‐linear dynamic problems publication-title: Computational Mechanics – volume: 45 start-page: 375 year: 1978 end-page: 378 article-title: Implicit‐explicit finite elements in transient analysis: implementation and numerical examples publication-title: ASME Journal of Applied Mechanics – volume: 169 start-page: 61 year: 1999 end-page: 79 article-title: On the design of energy preserving and decaying schemes for flexible, non‐linear multi‐body systems publication-title: Computer Methods in Applied Mechanics and Engineering – start-page: 1515 year: 2001 end-page: 1520 – volume: 137 start-page: 175 year: 1996 end-page: 188 article-title: Explicit time integration algorithms for structural dynamics with optimal numerical dissipation publication-title: Computer Methods in Applied Mechanics and Engineering – volume: 45 start-page: 371 year: 1978 end-page: 374 article-title: Implicit‐explicit finite elements in transient analysis: stability theory publication-title: ASME Journal of Applied Mechanics – volume: 60 start-page: 371 year: 1993 end-page: 375 article-title: A time integration method for structural dynamics with improved numerical dissipation: the generalized‐α method publication-title: Journal of Applied Mechanics – volume: 5 start-page: 283 year: 1977 end-page: 292 article-title: Improved numerical dissipation for time integration algorithms in structural dynamics publication-title: Earthquake Engineering and Structural Dynamics – volume: 45 start-page: 569 year: 1999 end-page: 599 article-title: Energy‐conserving and decaying algorithms in non‐linear structural dynamics publication-title: International Journal for Numerical Methods in Engineering – year: 2001 – volume: 190 start-page: 2603 year: 2001 end-page: 2649 article-title: On the formulation of high‐frequency dissipative time‐stepping algorithms for non‐linear dynamics. Part 1: low‐order methods for two model problems and non‐linear elastodynamics publication-title: Computer Methods in Applied Mechanics and Engineering – ident: e_1_2_1_3_2 doi: 10.1002/(SICI)1097-0207(19990620)45:5<569::AID-NME595>3.0.CO;2-A – ident: e_1_2_1_14_2 – volume-title: Computational Aspects of Nonlinear Systems with Large Rigid Body Motion year: 2001 ident: e_1_2_1_7_2 – ident: e_1_2_1_13_2 doi: 10.1115/1.3424305 – ident: e_1_2_1_8_2 doi: 10.1007/s00466-001-0273-z – ident: e_1_2_1_10_2 doi: 10.1016/S0045-7825(96)01036-5 – ident: e_1_2_1_5_2 doi: 10.1016/S0045-7825(00)00256-5 – ident: e_1_2_1_2_2 doi: 10.1115/1.2900803 – ident: e_1_2_1_4_2 doi: 10.1016/S0045-7825(98)00176-5 – ident: e_1_2_1_6_2 – ident: e_1_2_1_9_2 – ident: e_1_2_1_11_2 doi: 10.1115/1.3424304 – ident: e_1_2_1_12_2 doi: 10.1002/eqe.4290050306
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SubjectTerms	Computational techniques direct integration Exact sciences and technology explicit/implicit partition finite elements Finite-element and galerkin methods Fundamental areas of phenomenology (including applications) generalized alpha method Mathematical methods in physics Physics Solid mechanics Structural and continuum mechanics Vibration, mechanical wave, dynamic stability (aeroelasticity, vibration control...) Vibrations and mechanical waves
Title	Explicit/implicit partitioning and a new explicit form of the generalized alpha method
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