×

Computational error estimates and adaptive processes for some nonlinear structural problems. (English) Zbl 0528.65047


MSC:

65L60 Finite element, Rayleigh-Ritz, Galerkin and collocation methods for ordinary differential equations
65H10 Numerical computation of solutions to systems of equations
65L10 Numerical solution of boundary value problems involving ordinary differential equations
74S05 Finite element methods applied to problems in solid mechanics
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
65N15 Error bounds for boundary value problems involving PDEs
Full Text: DOI

References:

[1] SIAM Rev., 22, 186-187 (1980), correction: · Zbl 0465.73050
[2] Babuška, I.; Dorr, M. R., Error estimates for the combined \(H\) and \(P\) versions of the finite element method, (Inst. Phys. Sci. and Techn., Technical Note BN-951 (1980), Univ. of Maryland), to appear in Numer. Math · Zbl 0487.65058
[3] Babuška, I.; Luskin, M., An adaptive time discretization procedure for parabolic problems, (Vichnevetsky, R.; Stepleman, R. S., Proc. of the Fourth IMACS Internat. Symp.. Proc. of the Fourth IMACS Internat. Symp., IMACS (1981)), 5-9
[4] Babuška, I.; Miller, A., A posteriori error estimates and adaptive techniques for the finite element method, (Inst. Phys. Sci. and Techn., Technical Note BN-968 (1981), Univ. of Maryland) · Zbl 0571.73074
[5] Babuška, I.; Rheinboldt, W., Error estimates for adaptive finite element computations, SIAM J. Numer. Anal., 15, 736-754 (1978) · Zbl 0398.65069
[6] Babuška, I.; Rheinboldt, W., A posteriori error estimates for the finite element method, Internat. J. Numer. Meths. Engrg., 12, 1597-1615 (1978) · Zbl 0396.65068
[7] Babuška, I.; Rheinboldt, W., A posteriori bounds and adaptive procedures for the finite element method, (Sierakowski, R. L., Recent Advances in Engineering Science (1978), Univ. of Florida Press: Univ. of Florida Press Gainesville, FL), 413-418
[8] Babuška, I.; Rheinboldt, W., On a system for adaptive, parallel finite element computations, (Proc. 1978 Annual Conference (1978), Association for Computing Mach: Association for Computing Mach New York), 480-489
[9] Babuška, I.; Rheinboldt, W., Analysis of optimal finite element meshes in \(R^1\), Math. Comput., 33, 435-463 (1979) · Zbl 0431.65055
[10] Babuška, I.; Rheinboldt, W., On the reliability and optimality of the finite element method, Comput. and Structures, 10, 87-94 (1979) · Zbl 0395.65058
[11] Babuška, I.; Rheinboldt, W., Adaptive approaches and reliability estimations in finite element analysis, Comput. Meths. Appl. Mech. Engrg., 17/18, 519-540 (1979) · Zbl 0396.73077
[12] Babuška, I.; Rheinboldt, W., Reliable error estimation and mesh adaptation for the finite element method, (Oden, J. T., Computational Methods in Nonlinear Mechanics (1980), North-Holland: North-Holland Amsterdam), 67-108 · Zbl 0451.65078
[13] Babuška, I.; Rheinboldt, W., A posteriori error analysis of finite element solutions for one-dimensional problems, SIAM J. Numer. Anal., 18, 565-589 (1981) · Zbl 0487.65060
[14] Babuška, I.; Szabo, B. A., On the rates of convergence of the finite element method, (Rept. WU/CCM-80/2 (1980), Washington Univ., Center for Comp. Mechanics), to appear in Internat. J. Numer. Meth. Engrg.
[15] Babuška, I.; Szabo, B. A.; Katz, I. N., The \(P\)-version of the finite element method, SIAM J. Numer. Anal., 18, 515-545 (1981) · Zbl 0487.65059
[16] Brezzi, F.; Rappaz, J.; Raviart, P. A., Finite dimensional approximation of nonlinear problems, Part I: Branches of nonsingular solutions, Ecole Polytechn. Centre de Math. Appl., Tech. Rept., 52 (1979) · Zbl 0488.65021
[17] Brezzi, F.; Rappaz, J.; Raviart, P. A., Finite dimensional approximation of nonlinear problems, Part II: Limit points, Ecole Polytechn. Rept., 64 (1980) · Zbl 0488.65021
[18] Brezzi, F.; Rappaz, J.; Raviart, P. A., Finite dimensional approximation of nonlinear problems, Part III: Simple bifurcation points, Ecole Polytechn., Centre de Math. Appl., Technical Rept., 65 (1980) · Zbl 0488.65021
[19] den Heijer, C.; Rheinboldt, W., On steplength algorithms for a class of continuation methods, SIAM J. Numer. Anal., 18, 925-947 (1981) · Zbl 0472.65042
[20] Gaines, R., Difference equations associated with boundary value problems for second order nonlinear ordinary differential equations, SIAM J. Numer. Anal., 11, 411-434 (1974) · Zbl 0279.65068
[21] Gonnet, G. H., On the structure of zero finders, BIT, 17, 170-183 (1977) · Zbl 0365.65029
[22] Holland, J. H., Adaptation in Natural and Artificial Systems (1975), Univ. of Michigan Press: Univ. of Michigan Press Ann Arbor · Zbl 0317.68006
[23] Keller, H. B., Numerical solution of bifurcation and nonlinear eigenvalue problems, (Rabinowitz, P., Applications of Bifurcation Theory (1977), Academic Press: Academic Press New York), 359-384 · Zbl 0581.65043
[24] Keller, H. B., Global Homotopies and Newton methods, (de Boor, C.; Golub, G. H., Recent Advances in Numerical Analysis (1978), Academic Press: Academic Press New York), 73-94 · Zbl 0116.33301
[25] Kerr, A. D.; Soifer, M. T., The linearization of the prebuckling state and its effect on the determined instability load, J. Appl. Mech., 36, 775-783 (1969), Trans. ASME
[26] Mau, S. T.; Gallagher, R. H., A finite element procedure for nonlinear prebuckling and initial postbuckling analysis, NASA Contractor Rept. NASA-CR-1936 (1972)
[27] Moore, G.; Spence, A., The calculation of turning points of nonlinear equations, SIAM J. Numer. Anal., 17, 567-576 (1980) · Zbl 0454.65042
[28] Peitgen, H. O.; Saupe, D.; Schmitt, K., Nonlinear elliptic boundary value problems versus their finite difference approximations: Numerically irrelevant solutions, J. Reine Angew. Math., 322, 74-117 (1981) · Zbl 0449.65071
[29] Pönisch, G.; Schwetlick, H., Computing turning points of curves implicity defined by nonlinear equations depending on a parameter, Comput., 26, 107-121 (1981) · Zbl 0463.65036
[30] Poston, T.; Stewart, I., Catastrophe Theory and its Applications (1978), Pitman: Pitman London · Zbl 0382.58006
[31] Rheinboldt, W., Numerical methods for a class of finite dimensional bifurcation problems, SIAM J. Numer. Anal., 15, 1-11 (1978) · Zbl 0389.65024
[32] Rheinboldt, W., Solution fields of nonlinear equations and continuation methods, SIAM J. Numer. Anal., 17, 221-237 (1980) · Zbl 0431.65035
[33] Rheinboldt, W., Adaptive mesh refinement processes for finite element solutions, Internat. J. Numer. Meths. Engrg., 17, 649-662 (1981) · Zbl 0485.65057
[34] Rheinboldt, W., Numerical analysis of continuation methods for nonlinear structural problems, Comput. and Structures, 13, 103-114 (1981) · Zbl 0465.65030
[35] Rheinboldt, W., Computation of critical boundaries on equilibrium manifolds, (Inst. Comp. Math. and Appl., Technical Rept. ICMA-80-20 (1980), Univ. of Pittsburgh), to appear in SIAM J. Numer. Anal. · Zbl 0489.65033
[36] Rheinboldt, W.; Mesztenyi, C. K., On a data-structure for adaptive finite element mesh refinements, ACM Trans. Math. Software, 6, 166-187 (1980) · Zbl 0437.65081
[37] Sewell, M. J., On the connection between stability and the shape of the equilibrium surface, J. Mech. Phys. Solids, 14, 203-230 (1966)
[38] Sewell, M. J., Some global equilibrium surfaces, Internat. J. Mech. Engrg. Educ., 6, 163-174 (1978)
[39] Seydel, R., Numerical computation of branch points in nonlinear equations, Numer. Math., 33, 339-352 (1979) · Zbl 0396.65023
[40] Simpson, R. B., A method for the numerical determination of bifurcation states of nonlinear systems of equations, SIAM J. Numer. Anal., 12, 439-451 (1975) · Zbl 0318.65022
[41] Szabo, B. A., Some recent developments in finite element analysis, Comput. Math. Appl., 5, 99-115 (1979) · Zbl 0408.65066
[42] Tsypkin, Ya. Z., Adaptation and learning in automatic systems (1971), Academic Press: Academic Press NY · Zbl 0221.68049
[43] Walker, A. C., A nonlinear finite element analysis of shallow circular arches, Internat. J. Solids and Structures, 5 (1969) · Zbl 0164.26504
[44] Zadeh, L. A., On the definition of adaptivity, (Proc. I.R.E., 51 (1963)), 3-4 · Zbl 0377.04002
[45] P. Zave and G.E. Cole, Jr., A quantitative evaluation of the feasibility of, and suitable hardware architectures for, an adaptive, parallel finite element system, Univ. of Maryland, Dept. of Comp. Science, Technical Rept., in preparation.; P. Zave and G.E. Cole, Jr., A quantitative evaluation of the feasibility of, and suitable hardware architectures for, an adaptive, parallel finite element system, Univ. of Maryland, Dept. of Comp. Science, Technical Rept., in preparation.
[46] Zave, P.; Rheinboldt, W., Design of an adaptive parallel finite element system, ACM Trans. Math. Software, 5, 1-17 (1979) · Zbl 0401.65066
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.