Trefftz method

Abstract: Part 1: finite element technique shape functions and element stiffness matrix brief historical background basic relationships in engineering problems modified variational principles the concept of T-complete solution comparison of T-elements with conventional finite elements comparison of T-elements with boundary elements. Part 2: potential problems - introduction statement of the problem T-complete functions assumed fields generation of element matrix equation rank condition special purpose functions sensitivity to mesh distortion orthotropic case the Helmholtz equation HT-element with boundary "traction" frame frameless T-elements. Part 3: linear elastostatics - introduction linear theory of elasticity assumed fields in plane elasticity T-complete functions variational formulations element stiffness equation special-purpose elements p-extension approach three-dimensional elasticity numerical examples. Part 4: thin plates - introduction thin plate theory assumed field T-complete functions and particular solutions variational formulations for plate bending generation of element stiffness matrix p-method elements special purpose functions Extension to thin plates on elastic foundation Two alternative plate bending p-elements Numerical examples and assessment. Chapter 5 - Thick Plates - Introduction Basic equations for Reissner-Mindlin plate theory Assumed fields and particular solution Variational formulation for HT thick plate elements Implementation of the new family of HT elements A 12 DOF quadrilateral element free of shear locking Extension to thick plates on elastic foundation Sensitivity to mesh distortion Numerical assessment. Chapter 6 - Transient Heat Conduction - Introduction Elements of heat conduction Time step formula Element matrix formulations T-complete functions and particular solutions Numerical examples. Chapter 7 - Geometrically Nonlinear Analysis of Plate Bending Problems - Introduction Basic equations of nonlinear thin plate bending Assumed fields and Trefftz functions Particular solutions Modified variational principle Element matrix Iterative scheme Extension to post-buckling thin plates on elastic foundation Geometrically nonlinear analysis of thick plates Numerical examples. Chapter 8 - Elastoplasticity - Introduction Time discretization Basic relations Assumed fields Constraints on the approximation functions Finite element equilibrium and compatibility equations Finite element equations Finite element governing system. Chapter 9 - Dynamics of Plate Bending Problems - Introduction Basic equations Time-stepping formulation Numerical examples. Chapter 10 - Trefftz Boundary Element Method - Introduction Potential problems Plane elasticity Thin plate bending Moderately thick plates.

Abstract: Tutorial introduction Algorithms of CM, TM, and CTM Coupling techniques Boundary element methods Other kinds of boundary methods Comparisons Part I: Collocation Trefftz method 1 - Basic algorithms and theory Notations and preliminaries Approximation problems Error estimates Debye-Huckel equation Stability analysis 2 - Motz's problem and its variants Introduction Basic algorithms of CTM Error bounds and integration approximation Variants of Motz's problem Concluding remarks 3 - Coupling techniques Introduction Description of generalized TMs Penalty TMs Simplified hybrid TMs Penalty plus hybrid TMs Lagrange multiplier TM Effective condition number Numerical experiments 4 - Biharmonic equations with singularities Introduction The Green formulas of A2u The collocation Trefftz methods Error bounds Part II: Collocation methods 5 - Collocation methods Introduction Description of collocation methods Error analysis Robin boundary conditions Inverse inequalities Final remarks 6 - Combinations of collocation and finite element methods Introduction Combinations of FEMs Linear algebraic equations of combination of FEM and CM Uniformly Vh0-elliptic inequality Uniformly Vh0-elliptic inequality involving integration approximation Final remarks 7 - Radial basis function collocation methods Introduction Radial basis functions Description of radial basis function collocation methods Inverse estimates for radial basis functions Numerical experiments Comparisons and conclusions Part III: Advanced topics 8 - Combinations with high-order FEMs Introduction Combinations of TM and Lagrange FEMs Global superconvergence Adini's elements 9 - Eigenvalue problems Introduction New numerical algorithms for eigenvalue problems Error bounds of eigenvalues Error bounds of eigenfunctions Computational models and numerical experiments Eigenvalues for the singularity problem Summaries and discussions 10 - The Helmholtz equation Introduction The Trefftz method Error analysis Summaries and discussions 11 - Explicit harmonic solutions of Laplace's equation Introduction Harmonic functions Harmonic solutions involving Neumann conditions Extensions and analysis on singularity New models of singularities for Laplace's equation Concluding remarks Appendix - Historic review of boundary methods Potential theory Existence and uniqueness Reduction in dimensions and Green's formula Integral equations Extended Green's formula Pre-electronic computer era Electronic computer era Boundary integral equation and boundary element methods

Abstract: This paper presents a new hybrid element approach and applies it to plate bending. In contrast to more conventional models, the formulation is based on displacement fields which fulfil a priori the non-homogeneous Lagrange equation (Trefftz method). The interelement continuity is enforced by using a stationary principle together with an independent interelement displacement. The final unknowns are the nodal displacements and the elements may be implemented without any difficulty in finite element libraries of standard finite element programs. The formulation only calls for integration along the element boundaries which enables arbitrary polygonal or even curve-sided elements to be generated. Where relevant, known local solutions in the vicinity of a singularity or stress concentration may be used as an optional expansion basis to obtain, for example, particular singular corner elements, elements presenting circular holes, etc. Thus a high degree of accuracy may be achieved without a troublesome mesh refinement. Another important advantage of the formulation is the possibility of generating by a single element subroutine a large number of various elements (triangles, quadrilaterals, etc.), presenting an increasing degree of accuracy. The paper summarizes the results of numerical studies and shows the excellent accuracy and efficiency of the new elements. The conclusions present some ideas concerning the adaptive version of the new elements, extension to nonlinear problems and some other developments.

Abstract: A precise definition of Trefftz method is proposed and, starting with it, a general theory is briefly explained. This leads to formulating numerical methods from a domain decomposition perspective. An important feature of this approach is the systematic use of “fully discontinuous functions” and the treatment of a general boundary value problem with prescribed jumps. Usually finite element methods are developed using splines, but a more general point of view is obtained when they are formulated in spaces in which the functions together with their derivatives may have jump discontinuities and in the general context of boundary value problems with prescribed jumps. Two broad classes of Trefftz methods are obtained: direct (Trefftz—Jirousek) and indirect (Trefftz—Herrera) methods. In turn, each one of them can be divided into overlapping and nonoverlapping. The generality of the resulting theory is remarkable, because it is applicable to any partial (or ordinary) differential equation or system of such equations, which is linear. The article is dedicated to Professor Jiroslav Jirousek, who has been a very important driving force in the modern development of Trefftz method. © 2000 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 16: 561–580, 2000