The embedded discontinuous Galerkin method: Application to linear shell problems

International Journal for Numerical Methods in Engineering

Volumn 70, Issue 7, 2007, Pages 757-790

  Guzey S ^a   Cockburn, B ^a   Stolarski, H K ^a  

^a University of Minnesota   (United States)

Author keywords

Discontinuous Galerkin method; Finite elements; Hybrid methods; Locking free formulation; Shells

Indexed keywords

FINITE ELEMENT METHOD; LINEAR EQUATIONS; MATRIX ALGEBRA; NUMERICAL METHODS; PARTIAL DIFFERENTIAL EQUATIONS; PROBLEM SOLVING;

DISCONTINUOUS GALERKIN METHOD; ELEMENT MATRICES; HYBRID METHODS; LINEAR SHELL PROBLEMS; LOCKING-FREE FORMULATION;

GALERKIN METHODS;

FINITE ELEMENT METHOD; GALERKIN METHODS; LINEAR EQUATIONS; MATRIX ALGEBRA; NUMERICAL METHODS; PARTIAL DIFFERENTIAL EQUATIONS; PROBLEM SOLVING;

EID: 34249338763     PISSN: 00295981     EISSN: 10970207     Source Type: Journal    
DOI: 10.1002/nme.1893     Document Type: Article

References (43)

1. Reed WH, Hill TR. Triangular mesh methods for the neutron transport equation. Technical Report LA-UR-73-479, Los Alamos Scientific Laboratory, 1973.
2. Cockburn B, Shu C-W. Runge-Kutta discontinuous Galerkin methods for convection-dominated problems. SIAM Journal on Scientific Computing 2001; 16(3):173-261.
3. Arnold DN, Brezzi F, Cockburn B, Marini LD. Unified analysis of discontinuous Galerkin methods for elliptic problems. SIAM Journal on Numerical Analysis 2001/2002; 39(5):1749-1779 (electronic).
4. Engel G, Garikipati K, Hughes TJR, Larson MG, Mazzei L, Taylor RL. Continuous/discontinuous finite element approximations of fourth-order elliptic problems in structural and continuum mechanics with applications to thin beams and plates, and strain gradient elasticity. Computer Methods in Applied Mechanics and Engineering 2002; 191(34):3669-3750.
5. Hansbo P, Larson MG. A discontinuous Galerkin method for the plate equation. Calcolo 2002; 39(1):41-59.
6. Brezzi F, Marini LD. A nonconforming element for the Reissner-Mindlin plate. Computers and Structures 2003; 81(8-11):515-522. In honour of Klaus-Jürgen Bathe.
7. Cockburn B, Schötzau D, Wang J. Discontinuous Galerkin methods for incompressible elastic materials. Computer Methods in Applied Mechanics and Engineering 2006; 195:3184-3204 (Special issue on Discontinuous Galerkin Methods for Structural Mechanics, Dawson C (ed.)).
8. Arnold DN, Brezzi F, Marini LD. A family of discontinuous Galerkin finite elements for the Reissner-Mindlin plate. Journal of Scientific Computing 2005; 22:25-45.
9. Celiker F, Cockburn B, Stolarski H. Locking-free DG methods for the Timoshenko beams. SIAM Journal on Numerical Analysis, in press.
10. Stolarski H, Belytschko T, Lee SH. A review of shell finite elements and corotational theories. Computational Mechanics Advances 1995; 2:125-212.
11. Hughes TJR, Tezduyar T. Finite elements based upon Mindlin plate theory with particular reference to the four-node bilinear isoparametric element. Journal of Applied Mechanics 1981; 48:587-596.
12. Stolarski H, Belytschko T, Carpenter N, Kennedy JM. A simple triangular curved shell element. Engineering Computations 1984; 1:210-218.
13. Batoz JL, Bathe KJ, Ho LW. A study of three-node triangular plate bending element. International Journal for Numerical Methods in Engineering 1980; 15:1771-1812.
14. Ramm E, Wall WA. Shell structures a sensitive interrelation between physics and numerics. International Journal for Numerical Methods in Engineering 2004; 60:381-427.
15. Belytschko T, Tsay CS. A stabilization procedure for the quadrilateral plate bending element with one point quadrature. International Journal for Numerical Methods in Engineering 1983; 19:405-420.
16. Belytschko T, Leviathan I. Physical stabilization of the 4-node shell element with one point quadrature. Computer Methods in Applied Mechanics and Engineering 1994; 113(3-4):321-350.
17. Simo JC, Fox DD, Rifai MS. On a stress resultant geometrically exact shell model. Part II: the linear theory; computational aspects. Computer Methods in Applied Mechanics and Engineering 1989; 73:53-92
18. Hughes TJR, Cohen M, Haroun M. Reduced and selective integration techniques in the finite element analysis of plates. Nuclear Engineering and Design 1978; 46:203-222.
19. Dvorkin EN, Bathe KJ. A continuum mechanics based four-node shell element for general nonlinear analysis. Engineering Computations 1984; 1:77-88.
20. o triangular plate element with one-point quadrature. International Journal for Numerical Methods in Engineering 1984; 20:787-802.
21. o bilinear and serendipity plate elements. Computer Methods in Applied Mechanics and Engineering 1985; 50:121-145.
22. Wempner G, Talaslidis D, Hwang CM. Simple and efficient approximation of shells via finite quadrilateral elements. Journal of Applied Mechanics-Transactions of the ASME 1982; 49(1):115-120.
23. Pian THH, Sumihara K. Rational approach for assumed stress finite elements. International Journal for Numerical Methods in Engineering 1984; 20:1685-1695.
24. Argyris JH, Papadrakakis M, Apostolopoulou C, Koutsourelakis S. The TRIC shell element: theoretical and numerical investigation. Computer Methods in Applied Mechanics and Engineering 2000; 182:217-245.
25. Simo JC, Rifai MS. A class of mixed assumed strain methods and the method of incompatible modes. International Journal for Numerical Methods in Engineering 1990; 29(8):1595-1638.
26. Güzey S, Stolarski HK, Cockburn B, Tamma KK. Design and development of a discontinuous Galerkin method for shells. Computer Methods in Applied Mechanics and Engineering 2006; 195:3528-3548 (Special issue on Discontinuous Galerkin Methods for Structural Mechanics, Dawson C (ed.)).
27. Cockburn B, Gopalakrishnan J, Lazarov R. Unified hybridization of discontinuous Galerkin, mixed and continuous Galerkin methods for second-order elliptic problems, submitted.
28. Fraeijs de Veubeke BM. Displacement and equilibrium models. In Stress Analysis, Zienkiewicz OC, Hollister G (eds). Wiley: London, 1965; 145-197.
29. Reprinted in International Journal for Numerical Methods in Engineering 2001; 52:287-342.
30. Arnold DN, Brezzi F. Mixed and nonconforming finite element methods: implementation, postprocessing and error estimates. Mathematical Modelling and Numerical Analysis 1985; 7-32.
31. Cockburn B, Gopalakrishnan J. A characterization of hybridized mixed methods for second order elliptic problems. SIAM Journal on Numerical Analysis 2004; 42(1):283-301.
32. Cockburn B, Gopalakrishnan J. Error analysis of variable degree mixed methods for elliptic problems via hybridization. Mathematics of Computation 2005; 74:1653-1677.
33. Cockburn B, Gopalakrishnan J. Incompressible finite elements via hybridization. Part I: the Stokes system in two space dimensions. SIAM Journal on Numerical Analysis 2005; 43:1627-1650.
34. Cockburn B, Gopalakrishnan J. Incompressible finite elements via hybridization. Part II: the Stokes system in there space dimensions. SIAM Journal on Numerical Analysis 2005; 43:1651-1672.
35. Carrero J, Cockburn B, Schötzau D. Hybridized globally divergence-free LDG methods. Part I: the Stokes problem. Mathematics of Computation 2006; 75:533-563.
36. Cockburn B, Gopalakrishnan J. New hybridization techniques. GAMM-Mitteilungen 2005; 154-183.
37. Cockburn B, Gopalakrishnan J, Wang H. Can the continuous Galerkin method yield conservative fluxes? Submitted.
38. Hughes TJR, Scovazzi G, Bochev PB, Buffa A. A multiscale discontinuous Galerkin method with the computational structure of a continuous Galerkin method. Computer Methods in Applied Mechanics and Engineering 2006; 195:2761-2787.
39. Farhat C, Harari I, Franca LP. The discontinuous enrichment method. Computer Methods in Applied Mechanics and Engineering 2001; 190(48):6455-6479.
40. Ahmad S, Irons BM, Zienkiewicz OC. Analysis of thick and thin shell structures by curved finite elements. International Journal for Numerical Methods in Engineering 1970; 2(3):419-451.
41. Green AE, Zerna W. Theoretical Elasticity. Dover: New York, 1992.
42. MacNeal RH, Harder RL. Proposed standard set of problems to test finite element accuracy. Finite Elements in Analysis and Design 1985; 1(1):3-20.
43. Scordelis AC, Lo KS. Computer analysis of cylindrical shells. American Concrete Institute - Journal 1964; 61(5):539-561.