THE K-VERSION OF FINITE ELEMENT METHOD FOR NONLINEAR OPERATORS IN BVP

Abstract

International Journal of Computational Engineering ScienceVol. 05, No. 01, pp. 133-207 (2004) No AccessTHE K-VERSION OF FINITE ELEMENT METHOD FOR NONLINEAR OPERATORS IN BVPK. S. SURANA, A. R. AHMADI, and J. N. REDDYK. S. SURANADepartment of Mechanical Engineering, University of Kansas, Lawrence, KS 66044, USA Search for more papers by this author , A. R. AHMADIDepartment of Mechanical Engineering, University of Kansas, Lawrence, KS 66044, USA Search for more papers by this author , and J. N. REDDYDepartment of Mechanical Engineering, Texas A&M University, College Station, TX 77843-3123, USA Search for more papers by this author https://doi.org/10.1142/S1465876304002307Cited by:29 PreviousNext AboutSectionsPDF/EPUB ToolsAdd to favoritesDownload CitationsTrack CitationsRecommend to Library ShareShare onFacebookTwitterLinked InRedditEmail AbstractIn the companion papers [1,2], authors introduced the concepts of k-version of finite element method and k, hk, pk, hkp-processes of the finite element method for boundary value problems described by self-adjoint and non-self adjoint operators using Ĥk,p(Ω) spaces with specific details including numerical studies for weak forms and least square processes. It was demonstrated that a variationally consistent (VC) weak form is possible when the differential operator is self-adjoint, however, in case of non-self-adjoint operators the weak forms are variationally inconsistent (VIC) which lead to degenerate computational processes that can produce spurious oscillations in the computed solutions. In this paper we demonstrate that when the boundary value problems are described by non-linear differential operators, Galerkin processes and weak forms can never be variationally consistent and hence result in degenerate computational processes and suffer from same problems as in the case of non-self-adjoint operators plus more due to the presence of non-linearity. In the proposed mathematical and computational frame-work, upwinding methods are neither required nor used. The k-version of FEM over Ĥk,p spaces for the Galerkin method with weak forms, though meritorious in comparison to Ĥ1,p(Ω) spaces, but it is plagued due to problems arising from variational inconsistency.We demonstrate that the order of the space k in Ĥk,p(Ω) Hilbert spaces is an independent parameter in all computational processes in addition to the characteristic length h of the discretizations and the degree p of the local approximations. This gives rise to k-version of finite element method and thus, associated k, hk, pk, and hpk processes. The global differentiability of a finite element solution is only dependent on k. The h, p and hp-adaptive processes can not yield global differentiability of order higher than the order of the space containing the local approximations. It is shown that variational consistency of the integral forms and higher order global differentiability of a computed solution by increasing k in Ĥk,p(Ω) spaces are two most important features of mathematical and computational frame work if one wishes the computational process to (1) be non-degenerate and (2) yield solution with the same characteristics in terms of global differentiability as the theoretical solution.In this paper, we illustrate the important variational aspects of the least squares finite element processes for non-linear partial differential equations of stationary processes. Variationally consistent least squares finite element method (LSFEM) using p-version basis functions in Ĥk,p(Ω) spaces provides a remarkably general framework for numerical simulation of any BVP described by non-linear differential operators in which any desired order of global smoothness or global differentiability is achievable. The proposed mathematical framework indeed allows one to numerically simulate characteristics of the theoretical solutions of any non-linear BVP regardless of the nature of the differential operator. Stationary one-dimensional Burgers equation is used as a model problem to present supporting numerical studies. Many issues and concepts related to upwinding methods and the behaviors of commonly encountered problems in Galerkin method for non-linear differential operators in BVP are considered, discussed and explained. References K. S. Surana, A. R. Ahmadi and J. N. Reddy, Int. J. Comp. Engng. Sci. 3(2), 155 (2002). Link, Google Scholar Surana, K. S., Ahmadi, A. R., and Reddy, J. N., "k-version of finite element method in Ĥk,p" spaces for non-self-adjoint operators in boundary value problems, Int. J. Comp. Engng. Sci., to appear . Google Scholar G. F. Carey and J. T. Oden , Finite Elements: A Second Course II ( Prentice Hall , Englewood Cliffs, N.J. , 1983 ) . Google Scholar J. T. Oden and G. F. Carey , Finite Elements: Mathematical Aspects ( Prentice Hall , Englewood Cliffs, N.J. , 1983 ) . Google Scholar C. Johnson , Numerical Solution of Partial Differential Equations by Finite Element Method ( Cambridge University Press , 1987 ) . Google Scholar B. A. Finlayson , Numerical Methods for Problems with Moving Fronts ( Ravema Park Publishing Company, Seattle , Washington , 1992 ) . Google ScholarT. J. R. Hughes, Int. J. Num. Meth. Eng. 12, 1359 (1978). Crossref, Google ScholarA. N. Brooks and T. J. R. Hughes, Comp. Meth. Appl. Mech. Engrg. 32, 199 (1982). Crossref, Google ScholarT. J. R. Hughes and M. A. Mallet, Comp. Meth. Appl. Mech. Engrg. 58, 305 (1986). Crossref, Google ScholarT. J. R. Hughes and M. Mallet, Computer Methods in Applied Mechanics and Engineering 58, 329 (1986). Crossref, Google ScholarT. J. R. Hughes, M. Mallet and A. Mizukami, Comp. Meth. Appl. Mech. Engrg. 54, 341 (1986). Crossref, Google ScholarG. L. LeBeau and T. E. Tezduyar, Advances in Finite Element Analysis in Fluid Dynamics 123 (1991) pp. 21–27. Google ScholarF. Shakib, T. J. R. Hughes and Z. Johan, Comp. Meth. Appl. Mech. Engrg. 89, 141 (1991). Crossref, Google ScholarL. P. Franca, S. L. Frey and T. J. R. Hughes, Computer Methods in Applied Mechanics and Engineering 95, 253 (1992). Crossref, Google ScholarI. Christieet al., Int. J. Num. Meth. Engrg. 10, 1389 (1976). Crossref, Google ScholarJ. C. Hienrich, P. S. Huyakorn and O. C. Zienkiewicz, Int. J. Num. Meth. Engrg. 11, 131 (1977). Crossref, Google ScholarJ. C. Hienrich and O. C. Zienkiewicz, Int. J. Num. Meth. Engrg. 11, 1831 (1977). Crossref, Google ScholarJ. C. Hienrich, Int. J. Num. Meth. Engrg. 15, 1041 (1980). Crossref, Google ScholarD. W. Kellyet al., Int. J. Num. Meth. Engrg. 15, 1705 (1980). Crossref, Google Scholar Winterscheidt, D., p-version least squares finite element methods for fluid dynamics, Ph.D. Thesis, University of Kansas, 1992 . Google Scholar Van Dyne, D. G., Non-weak/strong Solutions in Gas Dynamics, Ph.D. Thesis, University of Kansas, 1999 . Google ScholarP. P. Lynn and K. Alani, Int. J. Num. Meth. Engrg. 10, 809 (1976). Crossref, Google ScholarJ. F. Polk and P. P. Lynn, Int. J. Num. Meth. Engrg. 12, 3 (1978). Google Scholar Jiang, B. N., Least-square finite element methods with element-by-element solution including adaptive refinement, Ph.D. Dissertation, University of Texas at Austin, 1986 . Google ScholarB. N. Jiang and C. F. Carey, Int. J. Num. Meth. Fluids 8, 933 (1988). Crossref, Google ScholarI. Kececioglu and B. Rubinsky, Int. J. Num. Meth. Engrg. 28, 2583 (1989). Crossref, Google ScholarI. Kececioglu and B. Rubinsky, Int. J. Num. Meth. Engrg. 28, 2609 (1989). Crossref, Google ScholarI. Kececioglu and B. Rubinsky, Int. J. Num. Meth. Engrg. 28, 2715 (1989). Crossref, Google ScholarC. L. Chang and B. N. Jiang, Comp. Meth. Appl. Mech. Engrg. 84, 247 (1980). Crossref, Google ScholarB. N. Jiang and L. A. Povinelli, Comp. Meth. Appl. Mech. Engrg. 81, 13 (1980). Crossref, Google ScholarD. Wintershiedt and K. S. Surana, Int. J. Num. Meth. Engrg. 36, 3629 (1993). Crossref, Google ScholarD. Wintershiedt and K. S. Surana, Int. J. Num. Meth. Engrg. 18, 43 (1994). Crossref, Google ScholarD. Wintershiedt and K. S. Surana, Int. J. Num. Meth. Engrg. 36, 111 (1993). Crossref, Google ScholarB. C. Bell and K. S. Surana, Int. J. Num. Meth. Engrg. 18, 127 (1993). Crossref, Google Scholar Bell, B. C., and Surana, K. S., "p-version space-time coupled least squares finite element formulation for two dimensional unsteady incompressible Newtonian fluid flow," Presented at 1993 ASME Winter Annual Meeting, 1993 . Google ScholarB. C. Bell and K. S. Surana, Int. J. Num. Meth. Engrg. 37, 3545 (1994). Crossref, Google ScholarB. C. Bell and K. S. Surana, Int. J. Num. Meth. Engrg. 39, 2593 (1996). Crossref, Google ScholarN. B. Edgar and K. S. Surana, Comp. Meth. Appl. Mech. Engrg. 113, 271 (1994). Crossref, Google ScholarK. S. Surana and J. S. Sandhu, Comp. Mech. 16, 151 (1995). Crossref, Google ScholarK. S. Surana and M. Bona, Int. J. Computational Engg. Sci. 1(2), 299 (2000). Link, Google ScholarK. S. Surana and David G. Van Dyne, Int. J. Num. Meth. Engrg. 53, 1051 (2002). Crossref, Google ScholarK. S. Surana and David G. Van Dyne, Int. J. Comp. Engg. Sci. (2002). Google ScholarK. S. Surana and David G. Van Dyne, Int. J. Num. Meth. Engrg. 53, 1025 (2002). Crossref, Google ScholarK. S. Surana and David G. , Int. J. Num. Meth. Engrg. (2001). Google ScholarK. S. Suranaet al., Int. J. Comp. Eng. Sci. (2002). Google Scholar I. M. Gelfand and S. V. Fomin , Calculus of Variations ( Dover Publications , New York , 2000 ) . Google Scholar FiguresReferencesRelatedDetailsCited By 29A Thermodynamically Consistent Formulation for Dynamic Response of Thermoviscoelastic Plate/Shell Based on Classical Continuum Mechanics (CCM)K. S. Surana and S. S. C. 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