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Trefftz radial basis functions (TRBF)

Kompiš V., Štiavnicky M., Žmindák M., Murčinková Z.

Computer Assisted Mechanics and Engineering Sciences

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2008

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Vol. 15, No. 3/4

239-249

EN

The TRBF's are radial functions satisfying governing equation in the domain. They can be used as interpolation functions of the field variables especially in boundary methods. In present paper discrete dipoles are used to simulate composite material reinforced by stiff particles using with boundary point collocation method which does not require any meshing and any integration. The better the interpolation (unction satisfies also the boundary conditions, the more efficient it is. In examples it is shown that a triple dipole (which is a TRBK) located into the center of the particle can approximate the inter-domain boundary conditions very good, if the particles are not very close to each other and their size is not very different. In general problem the model can be used as very good start point for international improvements in refined model. (Composite reinforced by short fibres with very large aspect ratio continuous TRBF were developed. They enable to reduce problem considerably and to simulate complicated interactions for investigation such composites.

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Trefftz functions for 3D stress concentration problems

Murčinková Z., Kompiš V., Štiavnicky M.

Computer Assisted Mechanics and Engineering Sciences

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2008

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Vol. 15, No. 3/4

305-318

EN

The paper deals with solution of 3D problems with stress concentration using the Trefftz functions. The modelled stress concentrators are holes and cavities of spherical and ellipsoidal shapes. Moreover, the random spherical cavity microstructure is modelled. The Method of External Finite Element Approximation (MEFEA) is applied to simulate detailed stress state of mentioned stress concentrators. This boundary-type method was developed to build special approximation fund ions that are associated with surface which causes the stress concentration. The method does not need discretization by classical finite elements, however, instead of elements the domain is divided into Trefftz type subdomains. The displacement and force boundary conditions are met only approximately whereas the governing equations are fulfilled exactly in the volume for linear elasticity, making it possible to assess accuracy in terms of error in boundary conditions.

3

Trefftz functions in FEM, BEM and meshless methods

Kompiš V., Štiavnicky M.

Computer Assisted Mechanics and Engineering Sciences

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2006

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Vol. 13, No. 3

417-426

EN

The paper contains three different multi-domain formulations using Trefftz (T-) displacement approximation/interpolation, namely the hybrid-displacement FEM, reciprocity based FEM (multi-domain BEM) and the Boundary Meshless Method (BMM) for a single and multi-domain (MD) formulation. All three methods can lead to compatible formulation with the isoparametric FEM, when the displacements along the common boundaries are defined by same interpolation function. All three T-formulations enable to define more complicated elements/subdomains (the T-element can be also a multiply connected region) with integration along the element boundaries, only.

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Trefftz polynomials reciprocity based boundary element formulations for elastodynamics

Konkol F., Kompiš V.

Computer Assisted Mechanics and Engineering Sciences

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2004

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Vol. 11, No. 2-3

239-245

EN

In this paper Trefftz polynomials are used for the BEM (Boundary Element Method) based on the reciprocity relations. BEM provides a powerful tool for the calculation of dynamic structural response in the frequency and time domains. Field equations of motion and boundary conditions are cast into boundary integral equations (BIE), which are discretized only on the boundary [1]. Trefftz polynomials or other non-singular (e.g. harmonic), Trefftz functions [2] (i.e. functions satisfying all governing differential equations but not the boundary conditions) used in the Betti's reciprocity relations lead to corresponding BIE that do not contain any (weak, strong, hyper) singularities. Fundamental solutions are not needed and evaluation of the field variables inside the domain is simpler.

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Trefftz functions using the fundamental solution with the singularity outside the domain

Jakubovičová L., Vaško M., Kompiš V.

Computer Assisted Mechanics and Engineering Sciences

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2003

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Vol. 10, No. 4

515-521

EN

Two types of Trefftz (T-) functions are often used - fundamental solutions with their singularities outside the given region and general solutions of homogenous differential equations. For elasticity problems the general solution of the homogeneous differential equation (equilibrium equation in displacements known as Lame-Navier equations) can be found in the polynomial form. In this paper we present the first type of T-functions. The paper deals with the investigation of accuracy and stability of the resulting system of discretized equations in relation to the position of the source (singularity) point. In this way non-singular reciprocity based boundary integral equations relate the boundary tractions and the boundary displacements of the searched solution to corresponding quantities of the known solutions. It was found that there exist an optimal relation of the distance of the singularity to the distance of the collocation points where both the integration accuracy and numerical stability are good.

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Finite displacements in reciprocity-based FE formulation

Kompiš V., Novák P., Handrik M.

Computer Assisted Mechanics and Engineering Sciences

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2002

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Vol. 9, No. 4

469-480

EN

In this paper, Trefftz polynomials are used for the development of FEM based on the reciprocity relations. Such reciprocity principles are known from the Boundary Element formulations, however, using the Trefftz polynomials in the reciprocity relations instead of the fundamental solutions yields the non-singular integral equations for the evaluation of corresponding sub-domain (element) relations. A weak form satisfaction of the equilibrium is used for the inter-domain connectivity relations. For linear problems, the element stiffness matrices are defined in the boundary integral equation form. In non-linear problems the total Lagrangian formulation leads to the evaluation of the boundary integrals over the original (related) domain evaluated only once during the solution and to the volume integrals containing the non-linear terms. Also, Trefftz polynomials can be used in the post-processing phase of the FEM computations for small strain problems. By using the Trefftz polynomials as local interpolators, smooth fields of the secondary variables (strains, stresses, etc.) can be found in the whole domain (if it is homogeneous). This approach considerably increases the accuracy of the evaluated fields while maintaining the same rate of convergence as that of the primary fields. Stress smoothing for large displacements will be the aim of further research. Considering the examples of simple tension, pure bending and tension of fully clamped rectangular plate (2D stress/strain problems) for large strain-large rotation problems, the use of the initial stiffness, the Newton-Raphson procedure, and the incremental Newton-Raphson procedure will be discussed.

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Trefftz-polynomial reciprocity based FE formulations

Kompiš V., Konkol F., Vaško M.

Computer Assisted Mechanics and Engineering Sciences

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2001

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Vol. 8, No. 2-3

385-395

EN

The paper contains a general procedure for obtaining of Trefftz polynomials of arbitrary order for 2D or 3D problems by numerical or analytical way. Using Trefftz polynomials for displacement and tractions the unknown displacements and tractions are related by non-singular boundary integral equations. For a multi-domain (element) formulation we suppose the displacements to be continuous between the sub-domains and the tractions are connected in a weak (integral) sense by a variational formulation of inter-element equilibrium. The stiffness matrix defined in this way is nonsymmetric and positive semi-definite. The finite elements can be combined with other well known elements. The form of the elements can be, however, more general (the multiply connected form of the element is possible, transition elements which can be connected to more elements along one side are available). It is also very easy and simply possible to assess the local errors of the solution from the traction incompatibilities (the inter-element equilibrium, which is satisfied in a weak sense only, is the only incompatibility in the solution of the linear problem). The stress smoothing is a very useful tool in the post-processing stage. It can improve the accuracy of the stress field by even one order or more comparing to the simple averaging, if the stress gradients in the element are large. Also the convergence of the so obtained stress field increases. The examples with high order gradient field and crack modelling document the efficiency of this FEM formulation. The extension to the solution of other field problems is very simple.