Supernatural QUAD4: A template formulation

Abstract A universal template for a 4-node quadrilateral in plane stress is constructed using a combination of old and new techniques. The use of natural quantities, strongly advocated by J.H. Argyris since his Continua and Discontinua 1965 exposition, is further expanded. The qualifier ‘supernatural’ means that all governing equations: kinematic, constitutive and equilibrium, are expressed in both Cartesian and natural forms. These two sets are used to build different components of the template. With timely help from a computer algebra system, a template that include all possible quadrilateral elements that pass the Individual Element Test of Bergan and Hanssen emerges. It yields an infinite number of hitherto undiscovered instances that may be customized to fit special needs. A striking example is the construction of a two-trapezoid macroelement that is bending exact about one direction, for any amount of distortion. This concludes a five decade search that begins with the formulation of the wing-cover rectangular panel in Argyris’ 1954 seminal serial on Energy Methods and Structural Analysis .

[1]  B. Irons,et al.  Techniques of Finite Elements , 1979 .

[2]  A. Aziz.  The Mathematical Foundations of the Finite Element Method with Applications to Partial Differential Equations , 1972 .

[3]  G. Strang.  VARIATIONAL CRIMES IN THE FINITE ELEMENT METHOD , 1972 .

[4]  R. K. Livesley,et al.  Matrix Methods of Structural Analysis , 1964 .

[5]  P. Bergan,et al.  Finite elements with increased freedom in choosing shape functions , 1984 .

[6]  Ray W. Clough,et al.  Improved numerical integration of thick shell finite elements , 1971 .

[7]  D. J. Allman,et al.  Evaluation of the constant strain triangle with drilling rotations , 1988 .

[8]  R. Melosh.  BASIS FOR DERIVATION OF MATRICES FOR THE DIRECT STIFFNESS METHOD , 1963 .

[9]  B. Irons,et al.  Engineering applications of numerical integration in stiffness methods. , 1966 .

[10]  C. Felippa.  Recent advances in finite element templates , 2000 .

[11]  Carlos A. Felippa,et al.  Membrane triangles with corner drilling freedoms III: implementation and performance evaluation , 1992 .

[12]  Chang-Chun Wu,et al.  On optimization approaches of hybrid stress elements , 1995 .

[13]  T. Pian.  Derivation of element stiffness matrices by assumed stress distributions , 1964 .

[14]  Carlos A. Felippa,et al.  A triangular bending element based on an energy-orthogonal free formulation , 1987 .

[15]  Carlos A. Felippa,et al.  The individual element test revisited , 1991 .

[16]  M. Turner.  Stiffness and Deflection Analysis of Complex Structures , 1956 .

[17]  Wing Kam Liu.  The Gamma-Elements and Related Developments: Element Technology , 1986 .

[18]  Richard H. Macneal,et al.  A theorem regarding the locking of tapered four‐noded membrane elements , 1987 .

[19]  Carlos A. Felippa,et al.  Construction of optimal 3-node plate bending triangles by templates , 1999 .

[20]  Carlos A. Felippa,et al.  From the individual element test to finite element templates: Evolution of the patch test , 1995 .

[21]  Carlos A. Felippa,et al.  A triangular membrane element with rotational degrees of freedom , 1985 .

[22]  Theodore H. H. Pian,et al.  Basis of finite element methods for solid continua , 1969 .

[23]  P. Hajela,et al.  COMPUTATIONAL MECHANICS FOR THE TWENTY- FIRST CENTURY , 2007 .

[24]  Carlos A. Felippa,et al.  The first ANDES elements: 9-dof plate bending triangles , 1991 .

[25]  C. Felippa.  Parametric unification of matrix structural analysis: classical formulation and d-connected mixed elements , 1995 .

[26]  T. Hughes.  Generalization of selective integration procedures to anisotropic and nonlinear media , 1980 .

[27]  Carlos A. Felippa,et al.  A survey of parametrized variational principles and applications to computational mechanics , 1994 .

[28]  C. L. Morgan,et al.  Continua and Discontinua , 1916 .

[29]  T. Pian,et al.  Rational approach for assumed stress finite elements , 1984 .

[30]  Edward L. Wilson,et al.  Incompatible Displacement Models , 1973 .

[31]  A. Samuelsson,et al.  Further discussion on four-node isoparametric quadrilateral elements in plane bending , 2000 .

[32]  J. H. Argyris,et al.  Energy theorems and structural analysis , 1960 .

[33]  C. Felippa.  A study of optimal membrane triangles with drilling freedoms , 2003 .

[34]  K. Park,et al.  A Curved C0 Shell Element Based on Assumed Natural-Coordinate Strains , 1986 .

[35]  C. Felippa.  Recent Developments In Basic Finite Element Technologies , 1999 .

[36]  C. Felippa,et al.  Fitting strains and displacements by minimizing dislocation energy , 2002 .

[37]  Theodore H. H. Pian.  Some notes on the early history of hybrid stress finite element method , 2000 .

[38]  J. C. Simo,et al.  A CLASS OF MIXED ASSUMED STRAIN METHODS AND THE METHOD OF INCOMPATIBLE MODES , 1990 .

[39]  Carlos A. Felippa,et al.  Membrane triangles with corner drilling freedoms II: the ANDES element , 1992 .

[40]  O. C. Zienkiewicz,et al.  Reduced integration technique in general analysis of plates and shells , 1971 .

[41]  Carlos A. Felippa.  The extended free formulation of finite elements in linear elasticity , 1989 .

[42]  C. Felippa.  Recent developments in parametrized variational principles for mechanics , 1996 .

[43]  P. G. Bergan,et al.  Finite elements based on energy orthogonal functions , 1980 .

[44]  Richard H. Macneal,et al.  A simple quadrilateral shell element , 1978 .

[45]  J. Whiteman.  The Mathematics of Finite Elements and Applications. , 1983 .

[46]  E. Wilson,et al.  A non-conforming element for stress analysis , 1976 .

[47]  Richard H. MacNeal,et al.  The evolution of lower order plate and shell elements in MSC/NASTRAN , 1989 .

[48]  D. Malkus,et al.  Mixed finite element methods—reduced and selective integration techniques: a unification of concepts , 1990 .

[49]  R. L. Harder,et al.  A proposed standard set of problems to test finite element accuracy , 1985 .

[50]  Franklin Y. Cheng,et al.  Computational Mechanics in Structural Engineering: Recent Developments , 1999 .

[51]  Carlos A. Felippa,et al.  A TEMPLATE TUTORIAL , 2003 .