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Numerical implementation of meshless methods for beam problems

Rosca V. E., Leitao V. M. A.

Archives of Civil Engineering

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2012

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Vol. 58, nr 2

175-184

EN

For solving a partial different equation by a numerical method, a possible alternative may be either to use a mesh method or a meshless method. A flexible computational procedure for solving 1D linear elastic beam problems is presented that currently uses two forms of approximation function (moving least squares and kernel approximation functions) and two types of formulations, namely the weak form and collocation technique, respectively, to reproduce Element Free Galerkin (EFG) and Smooth Particle Hydrodynamics (SPH) meshless methods. The numerical implementation for beam problems of these two formulations is discussed and numerical tests are presented to illustrate the difference between the formulations.

2

Thin plate bending analysis using an indirect Trefftz collocation method

Tiago C. M., Leitao V. M. A., Vasarhelyi A.

Computer Assisted Mechanics and Engineering Sciences

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2001

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Vol. 8, No. 1

1--16

EN

In this work, the application of an indirect Trefftz collocation method to the analysis of bending of thin plates (Kirchhoff's theory) is described. The deflection field approximation is obtained with the use of a set of functions satisfying a priori the homogeneous part of the differential equation of the problem. Each of the approximating functions is derived from a known thick plate solution. The boundary conditions are imposed by means of continuous (integral) and discrete (collocation) least squares methods. Numerical examples are presented and the accuracy of the proposed technique is assessed.

3

Comparison of Galerkin and collocation Trefftz formulations for plane elasticity

Leitao V. M. A.

Computer Assisted Mechanics and Engineering Sciences

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2001

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Vol. 8, No. 2-3

397-407

EN

The purpose of this work is to compare and assess, more in terms of computational efficiency than in terms of accuracy, three alternative implementations of a boundary formulation based on the Trefftz method for linear elastostatics, namely a collocation-based and two Galerkin-based approaches. A finite element approach is used in the derivation of the formulation for the Galerkin-based alternative implementations. The coefficients of the structural matrices and vectors are defined either by regular boundary integral expressions or determined by direct collocation of the trial functions. Numerical tests are performed to assess the relative performance of the different alternative implementations.