An integrated numerical technique for static and dynamic nonlinear structural problems adopting the equilibrium iteration is proposed. The differential quadrature finite element method (DQFEM), which uses the differential quadrature (DQ) techniques to the finite element discretization, is used to analyze the static and dynamic nonlinear structural mechanics problems. Numerical time integration in conjunction with the use of equilibrium iteration is used to update the response history. The equilibrium iteration can be carried out by the accelerated iteration schemes. The global secant relaxation-based accelerated constant stiffness and diagonal stiffness-based predictor-corrector equilibrium iterations which are efficient and reliable are used for the numerical computations. Sample problems are analyzed. Numerical results demonstrate the algorithm.

Issue Section:
Technical Papers
Keywords:
failure (mechanical), elastoplasticity, nonlinear dynamical systems
Topics:
Equilibrium (Physics), Newton's method, Stiffness, Finite element analysis, Finite element methods, Displacement, Algorithms
1.
Chen, C. N., 1990, “Improved Constant Stiffness Algorithms for the Finite Element Analysis,” Proc., NUMETA 90, Swansea, UK, pp. 623–628.
2.
Chen
,
C. N.
,
1992
, “
Efficient and Reliable Accelerated Constant Stiffness Algorithms for the Solution of Non-linear Problems
,”
Int. J. Numer. Methods Eng.
,
35
, pp.
481
–
490
.
3.
Ponthot, J. P. and Hogge, M., 1994, “On Relative Merits of Implicit/Explicit Algorithms for Transient Problems in Metal Forming Simulation,” Proc., International Conference on Numerical Methods for Metal Forming in Industry, Baden-Baden, GERMANY, 2, pp. 128–148.
4.
Chen, C. N., 1998, “The Differential Quadrature Finite Element Method,” Applied Mechanics in the Americas, D. Pamplona et al. eds., American Academy of Mechanics, 6, pp. 309–312.
5.
Bellman
,
R. E.
, and
Casti
,
J.
,
1971
, “
Differential Quadrature and Long-term Integration
,”
J. Math. Anal. Appl.
,
34
, pp.
235
–
238
.
6.
Chen
,
C. N.
,
1999
, “
Generalization of Differential Quadrature Discrtization
,”
Numer. Algorithms
,
22
, pp.
167
–
182
.
7.
Chen, C. N., 1998, “Extended Differential Quadrature,” Applied Mechanics in the Americas, D. Pamplona et al., eds., American Academy of Mechanics, 6, pp. 309–312.
8.
Chen, C. N., 1998, “A Differential Quadrature Finite Difference Method,” Proc., International Conference on Advanced Computational Methods in Engineering, Ghent, Belgium, pp. 713–720.
9.
Chen, C. N., 1995, “A Differential Quadrature Element Method,” Proc., 1st International Conference on Engineering Computation and Computer Simulation, Changsha, China, 1, pp. 25–34.
10.
Chen, C. N., 1998, “A Generalized Differential Quadrature Element Method,” Proc., International Conference on Advanced Computational Methods in Engineering, Ghent, Belgium, pp. 721–728.
11.
Chung
,
J.
, and
Hulbert
,
G. M.
,
1993
, “
A Time Integration Algorithm for Structural Dynamics with Improved Numerical Dissipations: the Generalized-α Method
,”
ASME J. Appl. Mech.
,
60
, pp.
371
–
375
.
12.
Hilber
,
H. M.
et al.,
1977
, “
Improved Numerical Dissipation for Time Integration Algorithms in Structural Dynamics
,”
Earthquake Eng. Struct. Dyn.
,
5
, pp.
283
–
292
.
13.
Wood
,
W. L.
et al.,
1980
, “
An Alpha Modification of Newmark’s Method
,”
Int. J. Numer. Methods Eng.
,
15
, pp.
1562
–
1566
.
14.
Newmark, N. M., 1959, “A Method of Computation for Structural Dynamics,” Proc., ASCE, 85, EM3, pp. 67–94.
15.
Hogge, M. and Ponthot, J. P., 1996, “Efficient Implicit Schemes for Transient Problems in Metal Forming Simulation,” Numerical and Physical Study of Material Forming Processes, Paris, France.
16.
Cahill, E., 1992, “Acceleration Techniques for Functional Iteration of Non-linear Equations,” IMACS Conference on Mathematical Modelling and Scientific Computing, Bangalove, India.
17.
Crisfield
,
M. A.
,
1984
, “
Accelerated and Damping the Modified Newton-Raphson Method
,”
Comput. Struct.
,
18
, pp.
267
–
278
.
18.
Chen, C. N., 1998, “Newton-Raphson Techniques in Finite Element Methods for Nonlinear Structural Problems,” International Series in Engineering, Technology and Applied Science, Structural Dynamic Systems Computational Techniques and Optimization, C. T. Leondes, eds., Gordon and Breach, pp. 149–242.
19.
Saint-Georges
,
P.
, et al.,
1996
, “
High-Performance PCG Solvers for FEM Structural Analysis
,”
Int. J. Numer. Methods Eng.
,
39
, pp.
1313
–
1340
.
20.
Chen
,
C. N.
,
1995
, “
A Global Secant Relaxation (GSR) Method-Based Predictor-Corrector Procedure for the Iterative Solution of Finite Element Systems
,”
Comput. Struct.
,
54
, pp.
199
–
205
.
21.
Stenger
,
F.
,
1981
, “
Numerical Methods Based on Whittaker Cardinal, or Sinc Functions
,”
SIAM Review
,
23
, pp.
165
–
224
.
Copyright © 2001
 by ASME
You do not currently have access to this content.