Trefftz Finite Element Method and Its Applications

This paper presents an overview of the Trefftz finite element and its application in various engineering problems. Basic concepts of the Trefftz method are discussed, such as T-complete functions, special purpose elements, modified variational functionals, rank conditions, intraelement fields, and frame fields. The hybrid-Trefftz finite element formulation and numerical solutions of potential flow problems, plane elasticity, linear thin and thick plate bending, transient heat conduction, and geometrically nonlinear plate bending are described. Formulations for all cases are derived by means of a modified variational functional and T-complete solutions. In the case of geometrically nonlinear plate bending, exact solutions of the Lame-Navier equations are used for the in-plane intraelement displacement field, and an incremental form of the basic equations is adopted. Generation of elemental stiffness equations from the modified variational principle is also discussed. Some typical numerical results are presented to show the application of the finite element approach. Finally, a brief summary of the approach is provided and future trends in this field are identified. There are 151 references cited in this revised article. DOI: 10.1115/1.1995716

[1]  Qing Hua Qin,et al.  Dual variational formulation for Trefftz finite element method of elastic materials , 2004 .

[2]  Qing-Hua Qin,et al.  Variational formulations for TFEM of piezoelectricity , 2003 .

[3]  Q. Qin,et al.  Solving anti-plane problems of piezoelectric materials by the Trefftz finite element approach , 2003 .

[4]  Zi-Cai Li,et al.  Global superconvergence of Adini's elements coupled with the Trefftz method for singular problems , 2003 .

[5]  Y. Shlepnev,et al.  Trefftz finite elements for electromagnetics , 2002 .

[6]  Weichung Yeih,et al.  Applications of the direct Trefftz boundary element method to the free-vibration problem of a membrane. , 2002, The Journal of the Acoustical Society of America.

[7]  J-H. He.  Generalized variational principles for thermopiezoelectricity , 2002 .

[8]  J. A. Teixeira de Freitas,et al.  Mixed finite element formulation for the solution of parabolic problems , 2002 .

[9]  R. Yates,et al.  General theory of domain decomposition: Indirect methods , 2002 .

[10]  S. Reutskiy.  A boundary method of Trefftz type with approximate trial functions , 2002 .

[11]  V. Sladek,et al.  A Trefftz function approximation in local boundary integral equations , 2002 .

[12]  J. A. Teixeira de Freitas,et al.  Shape optimization with hybrid‐Trefftz displacement elements , 2002 .

[13]  R. Yates,et al.  General theory of domain decomposition: Beyond Schwarz methods , 2001 .

[14]  R. Van Keer,et al.  Application of Multi-Region Trefftz Method to Elasticity , 2000 .

[15]  Qing Hua Qin,et al.  The Trefftz Finite and Boundary Element Method , 2000 .

[16]  George C. Hsiao,et al.  Hybrid Coupled Finite-Boundary Element Methods for Elliptical Systems of Second Order , 2000 .

[17]  Ismael Herrera,et al.  Trefftz Method: A General Theory , 2000 .

[18]  Wim Desmet,et al.  A novel prediction technique for the steady-state dynamic analysis of coupled vibro-acoustic systems , 2000 .

[19]  O. Mahrenholtz,et al.  Diffraction loads on multiple vertical cylinders with rectangular cross section by Trefftz-type finite elements , 2000 .

[20]  D. Ostergaard,et al.  Hybrid p-element and Trefftz method for capacitance computation , 2000 .

[21]  F. Bussamra,et al.  Three-dimensional hybrid-Trefftz stress elements , 2000 .

[22]  P. Sas,et al.  An indirect Trefftz method for the steady-state dynamic analysis of coupled vibro-acoustic systems , 1999 .

[23]  Eckart Schnack,et al.  A hybrid coupled finite-boundary element method in elasticity , 1999 .

[24]  J. A. Teixeira de Freitas,et al.  Hybrid-Trefftz stress elements for elastoplasticity , 1998 .

[25]  Małgorzata Stojek,et al.  LEAST-SQUARES TREFFTZ-TYPE ELEMENTS FOR THE HELMHOLTZ EQUATION , 1998 .

[26]  Ivo Babuška,et al.  Solution of Helmholtz problems by knowledge-based FEM , 1997 .

[27]  Ivo Babuška,et al.  Approximation with harmonic and generalized harmonic polynomials in the partition of unity method , 1997 .

[28]  A. P. Zieliński,et al.  Special Trefftz Elements and Improvement of Their Conditioning , 1997 .

[29]  Q. Qin.  Postbuckling analysis of thin plates on an elastic foundation by HT FE approach , 1997 .

[30]  J. Jiroušek,et al.  Survey of Trefftz-type element formulations , 1997 .

[31]  S. Wakao,et al.  Hybrid Trefftz and finite element method applied to magnetic field analysis for rotating machines , 1997 .

[32]  J. Jirousek,et al.  T-elements: State of the art and future trends , 1996 .

[33]  S. Diao,et al.  Nonlinear analysis of thick plates on an elastic foundation by HT FE with p-extension capabilities , 1996 .

[34]  Qing Hua Qin,et al.  Nonlinear analysis of thick plates by HT FE approach , 1996 .

[35]  Qing Hua Qin,et al.  TRANSIENT PLATE BENDING ANALYSIS BY HYBRID TREFFTZ ELEMENT APPROACH , 1996 .

[36]  Joseph. Petrolito,et al.  Triangular thick plate elements based on a hybrid-Trefftz approach , 1996 .

[37]  E. Kita,et al.  A new boundary-type scheme for sensitivity analysis using Trefftz formulation , 1996 .

[38]  J. D. Freitas,et al.  HYBRID‐TREFFTZ EQUILIBRIUM MODEL FOR CRACK PROBLEMS , 1996 .

[39]  Qing Hua Qin,et al.  Application of hybrid-Trefftz element approach to transient heat conduction analysis , 1996 .

[40]  A. Venkatesh,et al.  Accurate representation of local effects due to concentrated and discontinuous loads in hybrid-Trefftz plate bending elements , 1995 .

[41]  Q. Qin.  Postbuckling analysis of thin plates by a hybrid Trefftz finite element method , 1995 .

[42]  J. Jiroušek,et al.  Numerical assessment of a new T-element approach , 1995 .

[43]  Qing Hua Qin,et al.  A family of quadrilateral hybrid-Trefftz p-elements for thick plate analysis , 1995 .

[44]  J. Jirousek,et al.  A new 12 DOF quadrilateral element for analysis of thick and thin plates , 1995 .

[45]  Qing Hua Qin,et al.  Hybrid-Trefftz finite element method for Reissner plates on an elastic foundation , 1995 .

[46]  Q. Qin,et al.  Hybrid Trefftz finite-element approach for plate bending on an elastic foundation , 1994 .

[47]  O. C. Zienkiewicz,et al.  Linked interpolation for Reissner‐Mindlin plate elements: Part III—An alternative quadrilateral , 1994 .

[48]  Werner Wagner,et al.  A new boundary-type finite element for 2-D- and 3-D-elastic structures , 1994 .

[49]  Adnan Ibrahimbegovic,et al.  Quadrilateral finite elements for analysis of thick and thin plates , 1993 .

[50]  J. Jiroušek.  Variational formulation of two complementary hybrid-Trefftz FE models , 1993 .

[51]  Robert L. Taylor,et al.  Linked interpolation for Reissner‐Mindlin plate elements: Part II—A simple triangle , 1993 .

[52]  Alf Samuelsson,et al.  Linked interpolation for Reissner-Mindlin plate elements: Part I—A simple quadrilateral , 1993 .

[53]  J. Jiroušek,et al.  Dual hybrid-Trefftz element formulation based on independent boundary traction frame , 1993 .

[54]  Richard E. Ewing,et al.  Eulerian‐Lagrangian localized adjoint method: The theoretical framework , 1993 .

[55]  O. C. Zienkiewicz,et al.  Trefftz method for Kirchhoff plate bending problems , 1993 .

[56]  Akshay Venkatesh,et al.  Comparative study of p-extensions based on conventional assumed displacement and hybrid-Trefftz FE models , 1993 .

[57]  A. Venkatesh,et al.  Hybrid trefftz plane elasticity elements with p ‐method capabilities , 1992 .

[58]  M. A. Aminpour,et al.  Direct formulation of a hybrid 4-node shell element with drilling degrees of freedom , 1992 .

[59]  Xu Zhongnian,et al.  A thick–thin triangular plate element , 1992 .

[60]  Reinhard E. Piltner,et al.  A quadrilateral hybrid‐Trefftz plate bending element for the inclusion of warping based on a three‐dimensional plate formulation , 1992 .

[61]  I. Herrera.  Localized adjoint methods: a new discretization methodology , 1992 .

[62]  O. C. Zienkiewicz,et al.  Solution of Helmholtz equation by Trefftz method , 1991 .

[63]  O. C. Zienkiewicz,et al.  Application of the Trefftz method in plane elasticity problems , 1990 .

[64]  Qing Hua Qin,et al.  BEM of postbuckling analysis of thin plates , 1990 .

[65]  张森文,et al.  THE NONLINEAR RESPONSES OF A FORCED TWO-DEGREE-OF-FREEDOM STRUCTURE , 1990 .

[66]  Joseph. Petrolito,et al.  Hybrid-Trefftz quadrilateral elements for thick plate analysis , 1990 .

[67]  R. Piltner.  On the representation of three-dimensional elasticity solutions with the aid of complex valued functions , 1989 .

[68]  O. C. Zienkiewicz,et al.  Direct solution procedure for solution of harmonic problems using complete, non-singular, Trefftz functions , 1989 .

[69]  Herbert Reismann,et al.  Elastic Plates: Theory and Application , 1988 .

[70]  A. Zieliński.  Trefftz method: elastic and elasto-plastic problems , 1988 .

[71]  Ismael Herrera,et al.  The algebraic theory approach for ordinary differential equations: Highly accurate finite differences , 1987 .

[72]  J. Jirousek,et al.  Hybrid‐Trefftz plate bending elements with p‐method capabilities , 1987 .

[73]  M. Celia,et al.  Solution of general ordinary differential equations using the algebraic theory approach , 1987 .

[74]  Ismael Herrera,et al.  Trefftz method: Fitting boundary conditions , 1987 .

[75]  A. Venkatesh,et al.  An improved 9 DOF Hybrid—Trefftz triangular element for plate bending , 1987 .

[76]  J. Jirouseka,et al.  The hybrid-Trefftz finite element model and its application to plate bending , 1986 .

[77]  G. Alduncin,et al.  Unified approach to numerical methods. III. Finite differences and ordinary differential equations , 1985 .

[78]  R. Piltner.  Special finite elements with holes and internal cracks , 1985 .

[79]  O. C. Zienkiewicz,et al.  Generalized finite element analysis with T-complete boundary solution functions , 1985 .

[80]  Barna A. Szabó,et al.  h- andp-version finite element analyses of a rhombic plate , 1984 .

[81]  J. Avilés,et al.  A boundary method for elastic wave diffraction: Application to scattering of SH waves by surface irregularities , 1982 .

[82]  P. Tong,et al.  Singular finite elements for the fracture analysis of V‐notched plate , 1980 .

[83]  I. Herrera.  Boundary methods: A criterion for completeness. , 1980, Proceedings of the National Academy of Sciences of the United States of America.

[84]  Ismael Herrera,et al.  Connectivity as an alternative to boundary integral equations: Construction of bases. , 1978, Proceedings of the National Academy of Sciences of the United States of America.

[85]  J. Jirousek,et al.  Basis for development of large finite elements locally satisfying all field equations , 1978 .

[86]  I. Herrera.  Theory of connectivity for formally symmetric operators. , 1977, Proceedings of the National Academy of Sciences of the United States of America.

[87]  J. Jirousek,et al.  A powerful finite element for plate bending , 1977 .

[88]  I. Herrera.  General variational principles applicable to the hybrid element method. , 1977, Proceedings of the National Academy of Sciences of the United States of America.

[89]  John Argyris,et al.  On the application of the SHEBA shell element , 1972 .

[90]  W. H. Wittrick,et al.  Skew Plates and Structures. L. S. D. Morley. Volume 5 of Division I in the International Series of monographs on aeronautics and astronautics. Pergamon Press, London. 1963. 129 pp. 45s. , 1963, The Journal of the Royal Aeronautical Society.

[91]  E. Sternberg.  ON SOME RECENT DEVELOPMENTS IN THE LINEAR THEORY OF ELASTICITY , 1958 .

[92]  M. Williams,et al.  Stress Singularities Resulting From Various Boundary Conditions in Angular Corners of Plates in Extension , 1952 .

[93]  Ismael Herrera.  Trefftz – Herrera Method , 2006 .

[94]  J. D. Freitas.  Hybrid-Trefftz displacement and stress elements for elastodynamic analysis in the frequency domain , 2006 .

[95]  O. Mahrenholtz,et al.  Combined time-stepping and Trefftz approach for nonlinear wave-structure interaction , 2006 .

[96]  B. Tirozzi,et al.  Trefftz spectral method for initial-boundary problems , 2006 .

[97]  Jan A. KoÃlodziej,et al.  Trefftz-type procedure for Laplace equation on domains with circular holes, circular inclusions, corners, slits, and symmetry , 2006 .

[98]  I. Herrera,et al.  Indirect Method of Collocation for the Biharmonic Equation , 2003 .

[99]  R. Yates,et al.  The Indirect Approach To Domain Decomposition , 2003 .

[100]  R. Yates,et al.  Parallel Implementation of Collocation Methods , 2003 .

[101]  H. Qin,et al.  A VARIATIONAL PRINCIPLE AND HYBRID TREFFTZ FINITE ELEMENT FOR THE ANALYSIS OF REISSNER PLATES , 2003 .

[102]  Marián Handrik,et al.  Finite displacements in reciprocity-based FE formulation , 2002 .

[103]  E. Maunder.  A Trefftz patch recovery method for smooth stress resultants and applications to Reissner--Mindlin equilibrium plate models , 2001 .

[104]  J. Jirousek,et al.  T-elements analysis of plated on unilateral elastic Winkler-type foundation , 2001 .

[105]  S. A. Diab.  The natural boundary conditions as a variational basis for finite element methods , 2001 .

[106]  P. Pimenta,et al.  Hybrid-Trefftz stress elements for three-dimensional elastoplasticity , 2001 .

[107]  I. Herrera.  On Jirousek method and its generalizations , 2001 .

[108]  E. Stein,et al.  General aspects of Trefftz method and relations to error estimation of finite element approximations , 2001 .

[109]  V. Leitão.  Comparison of Galerkin and collocation Trefftz formulations for plane elasticity , 2001 .

[110]  M. Vaško,et al.  Trefftz-polynomial reciprocity based FE formulations , 2001 .

[111]  A. Wróblewski,et al.  General strategy of h-p adaptive solutions in structural Trefftz-type element analysis , 2001 .

[112]  A. C. Mendes,et al.  T-functions for 2-D creeping flow on domains with circular cylinders, corners, and possessing symmetry , 2001 .

[113]  M. Ciałkowski.  Trefftz functions as basic functions of FEM in application to solution of inverse heat conduction problem , 2001 .

[114]  H. Sanecki,et al.  Effectiveness of the Trefftz method in different engineering optimization procedures , 2001 .

[115]  A. Zieliński,et al.  Parametric structural shape optimization using the global Trefftz approach , 2000 .

[116]  M. Aliabadi.  Plate bending analysis with boundary elements , 1998 .

[117]  Reinhard E. Piltner,et al.  On the Systematic Construction of Trial Functions for Hybrid Trefftz Shell Elements , 1997 .

[118]  E. Maunder,et al.  Hybrid-equilibrium elements with control of spurious kinematic modes , 1997 .

[119]  J. Jiroušek,et al.  Study of a family of hybrid-Trefftz folded plate p-version elements , 1997 .

[120]  E. Stein,et al.  Aspects of Trefitz' Method in BEM and FEM and their coupling , 1997 .

[121]  O. Zienkiewicz.  Trefitz type approximation and the generalized flnite element method | history and development , 1997 .

[122]  E. Kita,et al.  First- and second-order sensitivity analysis schemes by collocation-type Trefitz method , 1997 .

[123]  E. Kita,et al.  Trefftz method: an overview , 1995 .

[124]  J. Jiroušek,et al.  T-elements: a finite element approach with advantages of boundary solution methods , 1995 .

[125]  Yao Zhen-han,et al.  Some applications of the Trefftz method in linear elliptic boundary-value problems , 1995 .

[126]  J. Jiroušek,et al.  Least‐squares T‐elments: Equivalent FE and BE forms of a substructure‐oriented boundary solution approach , 1994 .

[127]  Q. Qin,et al.  Nonlinear analysis of reissner plates on an elastic foundation by the BEM , 1993 .

[128]  G. Vörös.  APPLICATION OF THE HYBRID-TREFFTZ FINITE ELEMENT MODEL TO THIN SHELL ANALYSIS , 1991 .

[129]  J. Smaill.  Large deflection response of annular plates on Pasternak foundations , 1991 .

[130]  E. Stein,et al.  A New Boundary-Type Finite Element for 2D- and 3D-Elastic Solids , 1991 .

[131]  J. Jirousek,et al.  New Trends in Hybrid-Treffz p-Element Approach , 1991 .

[132]  Akshay Venkatesh,et al.  A new fe approach for adaptive reliability assurance , 1990 .

[133]  J. Jirousek,et al.  Solution of orthotropic plates based on p-extension of the hybrid-Trefftz finite element model , 1990 .

[134]  A. Venkatesh,et al.  A simple stress error estimator for Hybrid‐Trefftz p‐version elements , 1989 .

[135]  J. Jirousek,et al.  Large finite elements method for the solution of problems in the theory of elasticity , 1982 .

[136]  H. Gourgeon,et al.  Boundary Methods. C-Complete Systems for the Biharmonic Equations , 1981 .

[137]  D. Colton.  Partial differential equations in the complex domain , 1976 .

[138]  L. E. Malvern.  Introduction to the mechanics of a continuous medium , 1969 .

[139]  Y. Fung.  Foundations of solid mechanics , 1965 .

[140]  H. Hu.  ON SOME PROBLEMS OF THE ANTISYMETRICAL SMALL DEFLECTION OF ISOTROPIC SANDWICH PLATES , 1963 .

[141]  L. Morley.  Skew plates and structures , 1963 .

[142]  M. Naimark,et al.  Some basic problems of the mathematical theory of elasticity , 1953 .

[143]  R. D. Mindlin,et al.  Influence of rotary inertia and shear on flexural motions of isotropic, elastic plates , 1951 .

[144]  E. Reissner.  The effect of transverse shear deformation on the bending of elastic plates , 1945 .

[145]  A. Love.  A treatise on the mathematical theory of elasticity , 1892 .