Supernatural QUAD4: A template formulation

Computer Methods in Applied Mechanics and Engineering

Volumn 195, Issue 41-43, 2006, Pages 5316-5342

  Felippa, Carlos A ^a  

^a University of Colorado   (United States)

Author keywords

Finite elements; Free formulation; Individual element test; Natural equations; Natural fields; Patch test; Quadrilateral panel; Templates

Indexed keywords

ALGEBRA; COMPUTATIONAL GEOMETRY; COMPUTATIONAL METHODS; FINITE ELEMENT METHOD; MATHEMATICAL MODELS; STRUCTURAL PANELS;

INDIVIDUAL ELEMENT TEST; NATURAL EQUATIONS; PATCH TEST; TEMPLATES;

STRUCTURAL ANALYSIS;

ALGEBRA; COMPUTATIONAL GEOMETRY; COMPUTATIONAL METHODS; FINITE ELEMENT METHOD; MATHEMATICAL MODELS; STRUCTURAL ANALYSIS; STRUCTURAL PANELS;

EID: 33745389582     PISSN: 00457825     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.cma.2005.12.007     Document Type: Article

References (64)

1. Allman D.J. Evaluation of the constant strain triangle with drilling rotations. Int. J. Numer. Methods Engrg. 26 (1988) 2645-2655
2. Alvin K., de la Fuente H.M., Haugen B., and Felippa C.A. Membrane triangles with corner drilling freedoms: I. The EFF element. Finite Elem. Anal. Des. 12 (1992) 163-187
3. Argyris J.H., and Kelsey S. Energy Theorems and Structural Analysis (1960), Butterworths, London Part I reprinted from Aircr. Engrg., 26, October-November 1954 and 27, April-May 1955
4. Argyris J.H., Kelsey S., and Kamel H. Matrix methods of structural analysis-a precise of recent developments. In: Fraeijs de Veubeke B.M. (Ed). AGARDograph 72: Matrix Methods of Structural Analysis (1964), Pergamon Press, Oxford 1-164
5. Argyris J.H. Continua and discontinua. In: Przemieniecki J., et al. (Ed). Proc. 1st Conf. Matrix Meth. Struct. Mech. (1965), Air Force Institute of Technology, Dayton, OH 10-170 AFFDL-TR-66-80
6. Belytschko T., Liu W.K., and Engelmann B.E. In: Hughes T.J.R., and Hinton E. (Eds). Finite Element Methods for Plate and Shell Structures. Element Technology vol. I (1986), Pineridge Press, Swansea, UK 316-347
7. Bergan P.G., and Hanssen L. A new approach for deriving 'Good' finite elements. In: Whiteman J.R. (Ed). The Mathematics of Finite Elements and Applications vol. II (1976), Academic Press, London 483-497
8. Bergan P.G. Finite elements based on energy orthogonal functions. Int. J. Numer. Methods Engrg. 15 (1980) 1141-1555
9. Bergan P.G., and Nygård M.K. Finite elements with increased freedom in choosing shape functions. Int. J. Numer. Methods Engrg. 20 (1984) 643-664
10. Bergan P.G., and Felippa C.A. A triangular membrane element with rotational degrees of freedom. Comput. Methods Appl. Mech. Engrg. 50 (1985) 25-69
11. Bergan P.G., and Felippa C.A. Efficient implementation of a triangular membrane element with drilling freedoms. In: Hughes T.J.R., and Hinton E. (Eds). Finite Element Methods for Plate and Shell Structures. Element Technology vol. I (1986), Pineridge Press, Swansea, UK 128-152
12. Clough R.W. The finite element method-a personal view of its original formulation. In: Bell K. (Ed). From Finite Elements to the Troll Platform-the Ivar Holand 70th Anniversary Volume (1994), Tapir, Norway 89-100
13. W.P. Doherty, E.L. Wilson, R.L. Taylor, Stress analysis of axisymmetric solids utilizing higher order quadrilateral finite elements, SESM Report 69-3, Department of Civil Engineering, University of California, Berkeley, 1969.
14. Felippa C.A., and Bergan P.G. A triangular plate bending element based on an energy-orthogonal free formulation. Comput. Methods Appl. Mech. Engrg. 61 (1987) 129-160
15. Felippa C.A. The extended free formulation of finite elements in linear elasticity. J. Appl. Mech. 56 (1989) 609-616
16. Felippa C.A., and Militello C. Membrane triangles with corner drilling freedoms: II. The ANDES element. Finite Elem. Anal. Des. 12 (1992) 189-201
17. Felippa C.A., and Alexander S. Membrane triangles with corner drilling freedoms: III. Implementation and performance evaluation. Finite Elem. Anal. Des. 12 (1992) 203-239
18. Felippa C.A. A survey of parametrized variational principles and applications to computational mechanics. Comput. Methods Appl. Mech. Engrg. 113 (1994) 109-139
19. Felippa C.A., Haugen B., and Militello C. From the individual element test to finite element templates: evolution of the patch test. Int. J. Numer. Methods Engrg. 38 (1995) 199-222
20. Felippa C.A. Parametrized unification of matrix structural analysis: classical formulation and d-connected mixed elements. Finite Elem. Anal. Des. 21 (1995) 45-74
21. Felippa C.A. Recent developments in parametrized variational principles for mechanics. Comput. Mech. 18 (1996) 159-174
22. Felippa C.A., and Militello C. Construction of optimal 3-node plate bending elements by templates. Comput. Mech. 24 (1999) 1-13
23. Felippa C.A. Recent developments in basic finite element technologies. In: Cheng F.Y., and Gu Y. (Eds). Computational Mechanics in Structural Engineering-Recent Developments (1999), Elsevier, Amsterdam 141-156
24. Felippa C.A. Recent advances in finite element templates. In: Topping B.J.V. (Ed). Computational Mechanics for the Twenty-First Century (2000), Saxe-Coburn Pub., Edinburgh 71-98
25. C.A. Felippa, K.C. Park, Fitting strains and displacements by minimizing dislocation energy, in: Proceedings of the Sixth International Conference on Computational Structures Technology, Prague, Czech Republic, 2002, pp. 49-51. Complete text in CDROM. Available from: http://caswww.colorado.edu/Felippa.d/FelippaHome.d/Publications.d/Report.CU-CAS-02-25.pdf.
26. Felippa C.A. A study of optimal membrane triangles with drilling freedoms. Comput. Methods Appl. Mech. Engrg. 192 (2003) 2125-2168
27. Felippa C.A. A template tutorial. Chapter 3. In: Mathisen K.M., Kvamsdal T., and Okstad K.M. (Eds). Computational Mechanics: Theory and Practice (2004), CIMNE, Barcelona 29-68. http://caswww.colorado.edu/Felippa.d/FelippaHome.d/Publications.d/Report.CU-CAS-05-04.pdf Available from:
28. Gallaguer R.H. A Correlation Study of Methods of Matrix Structural Analysis (1964), Pergamon Press, Oxford
29. Hanssen L., Bergan P.G., and Syversten T.J. Stiffness derivation based on element convergence requirements. In: Whiteman J.R. (Ed). The Mathematics of Finite Elements and Applications vol. III (1979), Academic Press, London 83-96
30. Hughes T.J.R., and Malkus D.S. Mixed finite element methods-reduced and selective integration techniques: a unification of concepts. Comput. Methods Appl. Mech. Engrg. 15 (1978) 63-81
31. Hughes T.J.R. Generalization of selective integration procedures to anisotropic and nonlinear media. Int. J. Numer. Methods Engrg. 15 (1980) 1413-1418
32. Hughes T.J.R., and Malkus D.S. A general penalty mixed equivalence theorem for anisotropic, incompressible finite elements. In: Atluri S.N., Gallagher R.H., and Zienkiewicz O.C. (Eds). Hybrids and Mixed Finite Element Methods (1983), Wiley, London
33. Hughes T.J.R. The Finite Element Method: Linear Static and Dynamic Finite Element Analysis (1987), Prentice-Hall, Englewood Cliffs, N.J. Reprinted by Dover, NY, 2000
34. Irons B.M. Engineering application of numerical integration in stiffness methods. AIAA J. 4 (1966) 2035-2037
35. Irons B.M., and Ahmad S. Techniques of Finite Elements (1980), Ellis Horwood Ltd., Chichester, UK
36. Kavanagh K., and Key S.W. A note on selective and reduced integration techniques in the finite element method. Int. J. Numer. Methods Engrg. 4 (1972) 148-150
37. Lautersztajn-S N., and Samuelsson A. Further discussion on four-node isoparametric elements in plane bending. Int. J. Numer. Methods Engrg. 47 (2000) 129-140
38. MacNeal R.H. A simple quadrilateral shell element. Comput. Struct. 8 (1978) 175-183
39. MacNeal R.H., and Harder R.L. A proposed standard set of problems to test finite element accuracy. Finite Elem. Anal. Des. 1 (1985) 3-20
40. MacNeal R.H. The evolution of lower order plate and shell elements in MSC/NASTRAN. In: Hughes T.J.R., and Hinton E. (Eds). Finite Element Methods for Plate and Shell Structures. Element Technology vol. I (1986), Pineridge Press, Swansea, UK 85-127
41. MacNeal R.H. A theorem regarding the locking of tapered four noded membrane elements. Int. J. Numer. Methods Engrg. 24 (1987) 1793-1799
42. R.F. Melosh, Development of the stiffness method to define bounds on the elastic behavior of structures, Ph.D. Dissertation, University of Washington, Seattle, 1962.
43. Melosh R.F. Bases for the derivation of matrices for the direct stiffness method. AIAA J. 1 (1963) 1631-1637
44. Militello C., and Felippa C.A. The first ANDES elements: 9-DOF plate bending triangles. Comput. Methods Appl. Mech. Engrg. 93 (1991) 217-246
45. Militello C., and Felippa C.A. The individual element patch revisited. In: Oñate E., Periaux J., and Samuelsson A. (Eds). The Finite Element Method in the 1990s-a book dedicated to O.C. Zienkiewicz (1991), CIMNE, Springer-Verlag, Barcelona, Berlin 554-564
46. M.K. Nygård, The Free Formulation for nonlinear finite elements with applications to shells, Ph.D. Dissertation, Division of Structural Mechanics, NTH, Trondheim, Norway, 1986.
47. 0 shell element based on assumed natural-coordinate strains. J. Appl. Mech. 53 (1986) 278-290
48. Pawsey S.F., and Clough R.W. Improved numerical integration of thick shell finite elements. Int. J. Numer. Methods Engrg. 3 (1971) 545-586
49. Pian T.H.H. Derivation of element stiffness matrices by assumed stress distributions. AIAA J. 2 (1964) 1333-1336
50. Pian T.H.H. Element stiffness matrices for boundary compatibility and for prescribed boundary stresses. Proc. 1st Conf. on Matrix Methods in Structural Mechanics, AFFDL-TR-66-80 (1966), Air Force Institute of Technology, Dayton, OH 457-478
51. Pian T.H.H., and Tong P. Basis of finite element methods for solid continua. Int. J. Numer. Methods Engrg. 1 (1969) 3-29
52. Pian T.H.H., and Sumihara K. Rational approach for assumed stress finite elements. Int. J. Numer. Methods Engrg. 20 (1984) 1685-1695
53. Pian T.H.H. Some notes on the early history of hybrid stress finite element method. Int. J. Numer. Methods Engrg. 47 (2000) 419-425
54. Simo J.C., and Rifai M.S. A class of mixed assumed strain methods and the method of incompatible modes. Int. J. Numer. Methods Engrg. 29 (1990) 1595-1638
55. G. Skeie, The Free Formulation: linear theory and extensions with applications to tetrahedral elements with rotational freedoms, Ph.D. Dissertation, Division of Structural Mechanics, NTH, Trondheim, Norway, 1991.
56. Strang G. Variational crimes in the finite element method. In: Aziz A.K. (Ed). The Mathematical Foundations of the Finite Element Method with Applications to Partial Differential Equations (1972), Academic Press, New York 689-710
57. Strang G., and Fix G. An Analysis of the Finite Element Method (1973), Prentice-Hall
58. Stanley G.M., Park K.C., and Hughes T.J.R. Continuum based resultant shell elements. In: Hughes T.J.R., and Hinton E. (Eds). Finite Element Methods for Plate and Shell Structures. Element Technology vol. I (1986), Pineridge Press, Swansea, UK 1-45
59. Taig I.C., and Kerr R.I. Some problems in the discrete element representation of aircraft structures. In: Fraeijs de Veubeke B.M. (Ed). Matrix Methods of Structural Analysis (1964), Pergamon Press, London
60. Taylor R.L., Wilson E.L., and Beresford P.J. A nonconforming element for stress analysis. Int. J. Numer. Methods Engrg. 10 (1976) 1211-1219
61. Turner M.J., Clough R.W., Martin H.C., and Topp L.J. Stiffness and deflection analysis of complex structures. J. Aero. Sci. 23 (1956) 805-824
62. Wilson E.L., Taylor R.L., Doherty W.P., and Ghaboussi J. Incompatible displacement models. In: Fenves S.J., Perrone N., Robinson A.R., and Schnobrich W.C. (Eds). Numerical and Computer Models in Structural Mechanics (1973), Academic Press, New York 43-57
63. Wu C.C., and Cheung Y.K. On optimization approaches of hybrid stress elements. Finite Elem. Anal. Des. 21 (1995) 111-128
64. Zienkiewicz O.C., Taylor R.L., and Too J.M. Reduced integration technique in general analysis of plates and shells. Int. J. Numer. Methods Engrg. 3 (1971) 275-290