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T-elements: State of the art and future trends

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Summary

Trefftz-type elements, or T-elements, are finite elements the internal field of which fulfils the governing differential equations of the problema priori whereas the interelement continuity and the boundary conditions are enforced in an integral weighted residual sense or pointwise. Although the key ideas of such elements can be traced back to Jirousek and Leon in 1977, the T-element approach has received serious attention only for the past ten years. The T-element approach makes it possible to generate highly accurate h- or p-elements exhibiting many important advantages over their more conventional counterpart. The paper surveys existing T-element formulations (including some yet unpublished ones) and assesses critically their performance (accuracy, h- and p-convergence, sensitivity to mesh distortions, handling of singularities and geometry or load induced local effects, etc.). The available applications include plane elasticity, thin and thick plates, cylindrical shells, axisymmetric 3-D elasticity, Poisson's equation and transient heat conduction analysis. Existing approaches to adaptive reliability assurance based on p-extension are also discussed and future trends in the T-element research shortly outlined.

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Abbreviations

c :

vector of coefficients of a set of T-functions involved in the internal field of the element

d :

vector of degrees of freedom of the element

d u :

vector of generalized boundary displacementsv in collocation points

k :

element stiffness matrix

m :

number of T-functions in internal element fieldu

m RIG :

number of rigid body modes of element

m STR :

number of strain modes in fieldu (number of T-functions without rigid body modes)

r:

vector of equivalent nodal reactions

\(\dot r\) :

particular part ofr

t :

vector of the boundary tractions of the element, derived from its internal displacement fieldu

\(\dot t\) :

particular part of t

\(\bar t\) :

imposed boundary tractions

\(\hat t\) :

auxiliary boundary traction functions

\(\tilde t\) :

traction frame (independent oft), expressed in terms of nodal stress parametersr'

u :

intraelement vector of displacements, expressed in terms of parametersc

ů :

particular part ofu

v :

vector of generalized boundary displacements derived fromu

\(\dot v\) :

particular part ofv

\(\bar v\) :

imposed generalized boundary displacements

\(\tilde v\) :

displacement frame (independent ofv), expressed in terms of nodal DOFd

CIQ :

conventional isoparametric quadratic element

E :

Young modulus

M :

number of hierarchic degrees of freedom associated with mid-side nodes

N :

total number of degree of freedom of the element assembly (imposed degree of freedom included)

N a :

number of active degrees of freedom of the element assembly

N DOF :

number of degrees of freedom of element

N :

matrix of homogeneousT-functions in intraelement displacement field

V :

matrix of homogeneousT-function inv

\(\tilde v\) :

matrix of shape functions in displacement frame\(\tilde v\)

T :

Matrix of boundary tractions due to homogeneous T-functionsN

\(\tilde T\) :

matrix of shape functions in traction frame\(\tilde t\)

U, U EX,U FE :

generalized energy, its exact value and value calculated by FEM

ε:

relative error energy norm

\(\Gamma ^e \) :

element boundary

\(\Gamma _t^e \) :

part of element boundary on which tractions are imposed

\(\Gamma _v^e \) :

part of element boundary on which displacements are imposed

\(\Gamma _I^e \) :

interelement part of\(\Gamma ^e \)

\(\Omega ^e \) :

element domain

CT-T:

element formulation based on collocation

HT-D:

hybrid T-element with displacement frame

HT-T:

hybrid T-element with traction frame

HTLS1-D:

hybrid least squares displacement frame T-element

HTLS2-D:

modified hybrid least squares displacement frame T-element

HTLS1-T:

hybrid least squares T-element with traction frame

HT-DI:

hybrid T-element with interelement displacement frame

HT-TI:

hybrid T-element with interelement traction frame

HTLS-DI:

hybrid least squares T-element with interelement displacement frame

LST:

direct least squares T-element formulation

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  1. J. Jirousek (Professor)

    Present address: LSC-DGC, Swiss Federal Institute of Technology, Sweden

Authors and Affiliations

  1. Départment de Génic Civil, École Polytecnique Fédérale de Lausanne, CH-1015, Lausanne, Switzerland

    J. Jirousek (Professor)

  2. Technical University of Cracow, Cracow, Poland

    A. Wróblewski

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Research engineer, on leave from the Institute of Mechanics and Machine Design

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Jirousek, J., Wróblewski, A. T-elements: State of the art and future trends. ARCO 3, 323–434 (1996). https://doi.org/10.1007/BF02818934

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