For the development of a new family of higher-order time integration algorithms for structural dynamics problems, the displacement vector is approximated over a typical time interval using the pth-degree Hermite interpolation functions in time. The residual vector is defined by substituting the approximated displacement vector into the equation of structural dynamics. The modified weighted-residual method is applied to the residual vector. The weight parameters are used to restate the integral forms of the weighted-residual statements in algebraic forms, and then, these parameters are optimized by using the single-degree-of-freedom problem and its exact solution to achieve improved accuracy and unconditional stability. As a result of the pth-degree Hermite approximation of the displacement vector, pth-order (for dissipative cases) and (p + 1)st-order (for the nondissipative case) accurate algorithms with dissipation control capabilities are obtained. Numerical examples are used to illustrate performances of the newly developed algorithms.

References

References
1.
Hulbert
,
G. M.
,
1994
, “
A Unified Set of Single-Step Asymptotic Annihilation Algorithms for Structural Dynamics
,”
Comput. Methods Appl. Mech. Eng.
,
113
(
1
), pp.
1
9
.
2.
Fung
,
T. C.
,
1999
, “
Weighting Parameters for Unconditionally Stable Higher-Order Accurate Time Step Integration Algorithms—Part 2: Second-Order Equations
,”
Int. J. Numer. Methods Eng.
,
45
(
8
), pp.
971
1006
.
3.
Idesman
,
A. V.
,
2007
, “
A New High-Order Accurate Continuous Galerkin Method for Linear Elastodynamics Problems
,”
Comput. Mech.
,
40
(
2
), pp.
261
279
.
4.
Fung
,
T. C.
,
2003
, “
Numerical Dissipation in Time-Step Integration Algorithms for Structural Dynamic Analysis
,”
Prog. Struct. Eng. Mater.
,
5
(
3
), pp.
167
180
.
5.
Ma
,
J.
,
2015
, “
A New Space-Time Finite Element Method for the Dynamic Analysis of Truss-Type Structures
,”
Ph.D. thesis
, Edinburgh Napier University, Edinburgh, Scotland.
6.
Kim
,
W.
, and
Reddy
,
J. N.
,
2017
, “
A New Family of Higher-Order Time Integration Algorithms for the Analysis of Structural Dynamics
,”
ASME J. Appl. Mech.
, epub.
7.
Zienkiewicz
,
O. C.
, and
Taylor
,
R. L.
,
2005
,
The Finite Element Method for Solid and Structural Mechanics
,
Butterworth-Heinemann
, Oxford, UK.
8.
Fung
,
T. C.
,
1996
, “
Unconditionally Stable Higher-Order Accurate Hermitian Time Finite Elements
,”
Int. J. Numer. Methods Eng.
,
39
(
20
), pp.
3475
3495
.
9.
Fung
,
T. C.
,
2001
, “
Solving Initial Value Problems by Differential Quadrature Method—Part 2: Second-and Higher-Order Equations
,”
Int. J. Numer. Methods Eng.
,
50
(
6
), pp.
1429
1454
.
10.
Putcha
,
N. S.
, and
Reddy
,
J. N.
,
1986
, “
A Refined Mixed Shear Flexible Finite Element for the Nonlinear Analysis of Laminated Plates
,”
Comput. Struct.
,
22
(
4
), pp.
529
538
.
11.
Kim
,
W.
,
2008
, “
Unconventional Finite Element Models for Nonlinear Analysis of Beams and Plates
,”
Master's thesis
, Texas A&M University, College Station, TX.
12.
Kim
,
W.
, and
Reddy
,
J. N.
,
2010
, “
Novel Mixed Finite Element Models for Nonlinear Analysis of Plates
,”
Latin Am. J. Solids Struct.
,
7
(
2
), pp.
201
226
.
13.
Idesman
,
A. V.
,
Schmidt
,
M.
, and
Sierakowski
,
R. L.
,
2008
, “
A New Explicit Predictor–Multicorrector High-Order Accurate Method for Linear Elastodynamics
,”
J. Sound Vib.
,
310
(
1
), pp.
217
229
.
14.
Zienkiewicz
,
O. C.
,
Taylor
,
R. L.
, and
Zhu
,
J. Z.
,
2005
,
The Finite Element Method: Its Basis and Fundamentals
,
Butterworth-Heinemann
,
Burlington, VT
.
15.
Argyris
,
J.
, and
Mlejnek
,
H. P.
,
1991
, “
Dynamics of Structures
,”
Texts on Computational Mechanics
, Vol.
5
,
North-Holland
,
Amsterdam
, The Netherlands.
16.
Howard
,
G. F.
, and
Penny
,
J.
,
1978
, “
The Accuracy and Stability of Time Domain Finite Element Solutions
,”
J. Sound Vib.
,
61
(
4
), pp.
585
595
.
17.
Reddy
,
J. N.
,
2017
,
Energy Principles and Variational Methods in Applied Mechanics
,
3rd ed.
,
Wiley
, New York.
18.
Gellert
,
M.
,
1978
, “
A New Algorithm for Integration of Dynamic Systems
,”
Comput. Struct.
,
9
(
4
), pp.
401
408
.
19.
Bathe
,
K. J.
, and
Noh
,
G.
,
2012
, “
Insight Into an Implicit Time Integration Scheme for Structural Dynamics
,”
Comput. Struct.
,
98–99
, pp.
1
6
.
20.
Hilber
,
H. M.
,
Hughes
,
T. J. R.
, and
Taylor
,
R. L.
,
1977
, “
Improved Numerical Dissipation for Time Integration Algorithms in Structural Dynamics
,”
Earthquake Eng. Struct. Dyn.
,
5
(
3
), pp.
283
292
.
21.
Hughes
,
T. J. R.
,
1983
, “
Analysis of Transient Algorithms With Particular Reference to Stability Behavior
,”
Computational Methods for Transient Analysis
, Vol.
1
,
North Holland Publishing
,
Amsterdam
, The Netherlands, pp.
67
155
.
22.
Leon
,
S. J.
,
1980
,
Linear Algebra With Applications
,
Macmillan
,
New York
.
23.
Fung
,
T. C.
, and
Chow
,
S. K.
,
2002
, “
Solving Non-Linear Problems by Complex Time Step Methods
,”
Commun. Numer. Methods Eng.
,
18
(
4
), pp.
287
303
.
24.
Hilber
,
H. M.
,
1976
, “
Analysis and Design of Numerical Integration Methods in Structural Dynamics
,” Ph.D. thesis, University of California, Berkeley, CA.
25.
Chung
,
J.
,
1992
, “
Numerically Dissipative Time Integration Algorithms for Structural Dynamics
,”
Ph.D. thesis
, University of Michigan, Ann Arbor, MI.
26.
Newmark
,
N. M.
,
1959
, “
A Method of Computation for Structural Dynamics
,”
J. Eng. Mech. Div.
,
85
(
3
), pp.
67
94
.
27.
Baig
,
M. M. I.
, and
Bathe
,
K. J.
,
2005
, “
On Direct Time Integration in Large Deformation Dynamic Analysis
,”
Third MIT Conference on Computational Fluid and Solid Mechanics
, Boston, MA, June 14–17, pp.
1044
1047
.
28.
Kim
,
W.
,
Park
,
S.
, and
Reddy
,
J. N.
,
2014
, “
A Cross Weighted-Residual Time Integration Scheme for Structural Dynamics
,”
Int. J. Struct. Stab. Dyn.
,
14
(
6
), p.
1450023
.
29.
Reddy
,
J. N.
,
2006
,
An Introduction to the Finite Element Method
,
3rd ed.
,
McGraw-Hill
,
New York
.
30.
Fried
,
I.
, and
Malkus
,
D. S.
,
1975
, “
Finite Element Mass Matrix Lumping by Numerical Integration With no Convergence Rate Loss
,”
Int. J. Solids Struct.
,
11
(
4
), pp.
461
466
.
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