Finite element computing methods

R. A. Sauer, X.T. Duong, and C.J. Corbett, A computational formulation for constrained sohd and liquid membranes considering isogeometric finite elements, Comput. Method Appl. Mech. Engrg. 271,48-68 (2014). 

Algorithms based on the last approach usually provide more flexible schemes than the other two methods and hence are briefly discussed in here. Hughes et al. (1986) and de Sampaio (1991) developed Petrov-Galerkin schemes based on equal order interpolations of field variables that used specially modified weight functions to generate stable finite element computations in incompressible flow. These schemes are shown to be the special cases of the method described in the following section developed by Zienkiewicz and Wu (1991). 

The most important direct solution algorithms used in finite element computations are based on the Gaussian elimination method. 

As mentioned earlier, overall accuracy of finite element computations is directly detennined by the accuracy of the method employed to obtain the numerical solution of the global system of algebraic equations. In practical simulations, therefore, computational errors which are liable to affect the solution of global stiffness equations should be carefully analysed. 

One example of non-IRC trajectory was reported for the photoisomerization of cA-stilbene.36,37 In this study trajectory calculations were started at stilbene in its first excited state. The initial stilbene structure was obtained at CIS/6-31G, and 2744 argon atoms were used as a model solvent with periodic boundary conditions. In order to save computational time, finite element interpolation method was used, in which all degrees of freedom were frozen except the central ethylenic torsional angle and the two adjacent phenyl torsional angles. The solvent was equilibrated around a fully rigid m-stilbene for 20 ps, and initial configurations were taken every 1 ps intervals from subsequent equilibration. The results of 800 trajectories revealed that, because of the excessive internal potential energy, the trajectories did not cross the barrier at the saddle point. Thus, the prerequisites for common concepts of reaction dynamics such TST or RRKM theory were not satisfied. 

Several numerical methods, such as finite volume, finite difference, finite element, spectral methods, etc., are widely used for solving the complex set of partial differential equations. The latest computer technology allows us to obtain solutions with a mesh resolution on the order of millions of nodes. More-detailed discussion on numerical methodology is provided later. 

Wasfy, A. C. West, and V. Modi, Parallel Finite Element Computation of Unsteady Incompressible Flows, Int. J. Numerical Methods in Fluids, Vol. 26 (1998) 17. 

Traditional methods of simulation in hydrodynamics are based on the description of a fluid field obeying to partial differential equations. Finite difference, finite elements, spectral methods are generally used to approximate the equations and they are represented in the computer by floating point numbers. The implementation of the boundary conditions is the main difficulty of these methods. 

Szabo, B. A. (1994), Geometric Idealizations in Finite Element Computations, Communications in Applied Numerical Methods, Vol. 4, No. 3, pp. 393-400. 

Sharan S K. 2007. A finite element perturbation method for the prediction of rockburst. Computers Structures S5(n-IS) 1 304-1 309. 

Finally, computational aspects and numerical issues are considered. This includes a short description of the finite difference, the finite volume, and the finite element discretization methods. The chapter ends with some general comments. 

Ihrahimhegovic, A. 1995. On finite element implementation of geometrically nonlinear Reiss-ners beam theory three-dimensional curved beam elements. Computer Methods in Applied Mechanics and Engineering 122 11-26. 

The present discussion has a twofold objective First, to review the literature in the stress analysis of adhesive joints using the finite-element method. Second, to present a finite-element computational procedure for adhesive joints experiencing two-dimensional deformation and stress fields. The adherends are linear elastic and can undergo large deformations, and the adhesive experiences linear strains but nonlinear viscoelastic behavior. Following these general comments, a review of the literature is presented. The technical discussion given in the subsequent sections comes essentially from the authors research(i 2> conducted for the Oifice of Naval Research. 

This equation can be solved using finite element analytical method. In numerical calculations, computer reads in the digitized documented data of p(0), k(9), and Cp(0) for the b(0)- The skin s( ) is obtained by multiplying b(0) with a coefficient that is 4/3 that is the inverse of skin mass density. Figures 41.2 and 41.3 compare the examination results of the following parameters. 

Abstract The aim of this chapter is to introduce special numerical techniques. The first part covers special finite element techniques which reduce the size of the computational models. In the case of the substructuring technique, internal nodes of parts of a finite element mesh can be condensed out so that they do not contribute to the size of the global stiffiiess matrix. A post computational step allows to determine the unknowns of the condensed nodes. In the case of the submodel technique, the results of a finite element computation based on a coarse mesh are used as input, i.e., boundary conditions, for a refined submodel. The second part of this chapters introduces alternative approximation methods to solve the partial differential equations which describe the problem. The boundary element method is characterized by the fact that the problem is shifted to the boundary of the domain and as a result, the dimensionality of the problem is reduced by one. In the case of the finite difference method, the differential equation and the boundary conditions are represented by finite difference equations. Both methods are introduced based on a simple one-dimensional problem in order to demonstrate the major idea of each method. Furthermore, advantages and disadvantages of each alternative approximation methods are given in the light of the classical finite element simulation. Whenever possible, examples of application of the techniques in the context of adhesive joints are given. 

The development of methods of measuring the fundamental properties of polymer mortar surfacing materials from initial stages of cure and of a finite element computer model which can take into account time dependent effects, will enable stresses and strains in a surfacing material, concrete substrate and their interface to be determined. The sensitivity of these parameters will be assessed and engineering judgement can then be used to select PC systems which are unlikely to induce delamination failures. 

The most current state-of-the-art in analysis methods is the finite element analysis (FEA) method. The FEA method of analysis is conducted using finite element computer programs. Because of the nonlinear thermomechanical behavior of... 

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