Fragile points method for Euler–Bernoulli beams

The finite volume method in the context of the finite element method

In the previous chapters, with an exception of GPG, the finite element formulations are based on the Galerkin methods in which test functions are chosen to be the same as the trial functions. This is not required in the weighted residual methods. Weighted residual methods other than the Galerkin methods include spectral element methods (SEM), least square methods (LSM), moment methods, or collocation methods, in which the test functions or weighting functions are not necessarily the same as the trial functions. In spectral element methods (SEM), polynomials in terms of nodal values of the variables are combined with special functions such as Chebyshev or Legendre polynomials. For least square methods, the test functions are constructed by the derivative of the residual with respect to the nodal values of the variables. Some arbitrary functions are chosen as test functions for the moment and collocation methods. Recently, the weighted residual concept has been used in meshless configurations, known as the finite point method (FPM), partition of unity method, meshless cloud method, or element-free method.

An unsymmetric 8-node hexahedral solid-shell element based on ANS and incompatible concepts for thin shell analysis

The classical 8-node isoparametric serendipity element uses parametric shape functions for both test and trial functions. Although this element performs well in general, it yields poor results under severe mesh distortions. The distortion sensitivity is caused by the lack of continuity and/or completeness of shape functions used for test and trial functions. A recent element using parametric and metric shape functions for constructing the test and trial functions exhibits distortion immunity. This paper discusses the choice of parametric or metric shape functions as the basis for test and/or trial functions, satisfaction of continuity and completeness requirements, and their connection to distortion sensitivity. Also, the performances of four types of elements, viz., parametric, metric, parametric-metric, and metric-parametric, are compared for distorted meshes, and their merits and demerits are discussed.

A meshless fragile points method for rule-based definition of myocardial fiber orientation

In this report we present a new approach to simulations on complex shaped domains. The method uses a Discontinuous Galerkin discretization and a structured grid to construct the test and trial functions. Boundary and transmission conditions along the complex shape of the domains are imposed weakly via the Discontinuous Galerkin formulation. This method o ers a discretization where the minimal number of unknowns is independent of the possibly very complicated shape of the domain.

A meshless local natural neighbour interpolation method to modeling of functionally graded viscoelastic materials

Application of localized meshless methods to 2D shallow water equation problems

Time periodic finite element solutions for sinusoidally excited electromagnetic field problems in moving media are presented. Solutions by the Galerkin method contain spurious oscillations when the grid Peclet number is more than one. To suppress these osillations an upwind finite element method using two different time periodic test functions is introduced. One is multiplied to second and first order space derivative terms and the other to the time derivative term. Test functions are obtained from trial functions by adding or subtracting quadratic bias functions with appropriate scaling factors. Phase differences are considered between trial functions and bias functions. For simple interpretations of the phase differences complex scaling factors are used. The proposed method is developed to give nodally exact solutions for uniform grid spacing in one dimensional problems. Based on the one dimensional results a two dimensional upwinding scheme is also derived.

A new approach for modelling discrete cracks in meshfree methods is described. In this method, the crack can be arbitrarily oriented, but its growth is represented discretely by activation of crack surfaces at individual particles, so no representation of the crack's topology is needed. The crack is modelled by a local enrichment of the test and trial functions with a sign function (a variant of the Heaviside step function), so that the discontinuities are along the direction of the crack. The discontinuity consists of cylindrical planes centred at the particles in three dimensions, lines centred at the particles in two dimensions. The model is applied to several 2D problems and compared to experimental data. Copyright © 2004 John Wiley & Sons, Ltd.

The present work is a comparative study of different data transfer techniques in the context of the finite cell method (FCM) in combination with remeshing for hyperelastic problems undergoing large deformations. The FCM is an immersed-boundary method that uses Cartesian grids for the discretization so as to avoid the generation of boundary conforming meshes. To overcome problems with heavily distorted meshes at large deformation states, we apply a remeshing procedure. During the remeshing, the data containing the deformation history has to be transferred between the meshes. In the present study, different methods are considered and compared: radial basis functions without and with polynomial extension, inverse distance weighting, and $$\textit{L}_\text {2}$$ L 2 -projection applying the shape functions used in the FCM for the trial and test functions.

Compact RBF meshless methods for photonic crystal modelling

We consider a Galerkin method for functional equations in one space variable which uses periodic cardinal splines as trial functions and trigonometric polynomials as test functions. We analyze the method applied to the integral equation of the first kind arising from a single layer potential formulation of the Dirichlet problem in the interior or exterior of an analytic plane curve. In constrast to ordinary spline Galerkin methods, we show that the method is stable, and so provides quasioptimal approximation, in a large family of Hilbert spaces including all the Sobolev spaces of negative order. As a consequence we prove that the approximate solution to the Dirichlet problem and all its derivatives converge pointwise with exponential rate.

Fast formation of isogeometric Galerkin matrices by weighted quadrature

Pomraning2 introduced the variational principle of Roussopoulos3 into transport theory. Nelson4 presented a Generalized Roussopoulos Principle (GRP) suitable for discontinuous trial functions, and initiated a program of systematically using this principle to derive various types of functionals. In Ref. 4 the GRP was used to obtain functionals admitting trial functions con- taining an interface value which is independent of the limiting values from either side. Our purpose in this work is to continue the program initiated in Ref. 4 by applying the GFP to functionals of classical type, by which we mean functionals in which there are no approximations at an interface other than the limiting values from either side.