Computational methods finite element method

This information is supported by stress-strain behavior data collected in actual materials evaluations. With computers the finite element method (FEA) has greatly enhanced the capability of the structural analyst to calculate displacement, strain, and stress values in complicated plastic structures subjected to arbitrary loading conditions (Chapter 2). FEA techniques have made analyses much more precise, resulting in better and more optimum designs. 

The family of hierarchical elements are specifically designed to minimize the computational cost of repeated computations in the p-version of the finite element method (Zienkiewicz and Taylor, 1994). Successive approximations based on hierarchical elements utilize the derivations of a lower step to generate the solution for a higher-order approximation. This can significantly reduce the... 

Donea, J., 1992. Arbitrary Lagrangian-Eulerian finite element methods. In Belytschko, T. and Hughes, T. J. R. (eds), Computational Methods for Transient Analysis, Elsevier Science, Amsterdam. 

Donea, J. and Quartapelle, L., 1992. An introduction to finite element methods for transient advection problems. Comput. Methods Appl Meek Eng. 95, 169-203. 

One nice feature of the finite element method is the use of natural boundaiy conditions. It may be possible to solve the problem on a domain that is shorter than needed to reach some limiting condition (such as at an outflow boundaiy). The externally applied flux is still apphed at the shorter domain, and the solution inside the truncated domain is still vahd. Examples are given in Refs. 67 and 107. The effect of this is to allow solutions in domains that are smaller, thus saving computation time and permitting the solution in semi-infinite domains. 

J. Donea, Arbitrary Lagrangian-Eulerian Finite Element Methods, Computational Methods for Transient Analysis (edited by J.D. Achenbach), North-Holland, Amsterdam, 1983. 

Gelbard and Seinfeld (1978), Nicmanis and Hounslow (1998) and Wulkow etal. (2001) propose alternative finite element methods with improved precision and reduced computational time. 

When required, combined with the use of computers, the finite element analysis (FEA) method can greatly enhanced the capability of the structural analyst to calculate displacement and stress-strain values in complicated structures subjected to arbitrary loading conditions. In its fundamental form, the FEA technique is limited to static, linear elastic analysis. However, there are advanced FEA computer programs that can treat highly nonlinear dynamic problems efficiently. 

Correspondingly all calculations are finite element method (FEM)-based. Furthermore, the flow channel calculations are based on computer fluid dynamics (CFD) research test for the optimization of the mbber flow. 

Walter et al. studied the flow distribution in simple multichannel geometries by means of the finite-element method [112]. In order to reduce the computational effort, a 2-D model was set up to mimic the 3-D multichannel geometry. Even at a comparatively small Reynolds number of 30 they found recirculation zones in the flow distribution chamber and corresponding deviations from the mean flow rate inside the channels of about 20%. They also investigated the influence of contact time variation on a simple two-step reaction. 

TeGrotenhuis et al. studied a counter-current heat-exchanger reactor for the WGS reaction with integrated cooling gas channels for removal of the reaction heat. The computational domain of their 2-D model on the basis of the finite-element method... 

The modeling of complex solids has greatly advanced since the advent, around 1960, of the finite element method [196], Here the material is divided into a number of subdomains, termed elements, with associated nodes. The elements are considered to consist of materials, the constitutive equations of which are well known, and, upon change of the system, the nodes suffer nodal displacements and concomitant generalized nodal forces. The method involves construction of a global stiffness matrix that comprises the contributions from all elements, the relevant boundary conditions and body and thermal forces a typical problem is then to compute the nodal displacements (i. e., the local strains) by solving the system K u = F, where K is the stiffness matrix, u the... 

In the finite element method, Petrov-Galerkin methods are used to minimize the unphysical oscillations. The Petrov-Galerkin method essentially adds a small amount of diffusion in the flow direction to smooth the unphysical oscillations. The amount of diffusion is usually proportional to Ax so that it becomes negligible as the mesh size is reduced. The value of the Petrov-Galerkin method lies in being able to obtain a smooth solution when the mesh size is large, so that the computation is feasible. This is not so crucial in one-dimensional problems, but it is essential in two- and three-dimensional problems and purely hyperbolic problems. 

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. 

A. Madzvamuse, A.J. Wathen, P. Maini, A moving grid finite element method applied to a model biological pattern generator, J. Comput. Phys. 190 (2003) 478-500. 

M. Kawahara and N. Takeuchi. Mixed finite element method for analysis of viscoelastic fluid flow. Comput. Fluids., 5 33, 1977. 

Masud, A. and Hughes, T.J.R. (2002) A stabilized finite element method for darcy flow. Comput. Methods Appl. Mech. Eng. 191, 4341-4370... 

C. Braudo, A. Fortin, T. Coupez, Y. Demay, B. Vergnes, and J. F. Agassant, A Finite Element Method for Computing the Flow of Multi-mode Viscoelastic Fluids Comparison with Experiments, J. Non-Newt. Fluid Meek, 75, 1 (1998)... 

With the great strides in computational fluid mechanics made over the past decades, the current trend is toward applying sophisticated finite element methods. These include both two- and three-dimensional (10-15) methods, which in principle allow the computation of two- or three-dimensional velocity and temperature fields with a variety of boundary... 

Over the past few years, however, techniques have been developed to enable continuous reinforcement of thermoplastics. The simplest way is to put a cloth and a plastic sheet on top of each other in a heated press and to carry out impregnation under pressure. More difficult is the forming of an end-product from the sheet produced with conventional sheet-forming techniques the position of the fibres will be distorted in an unacceptable way. As in nearly all processing techniques, the modern finite-element methods with advanced computers are able to present solutions to this problem in principle they can predict the position of the fibres in the sheet-forming operation, so that optimum reinforcement is realised in the end product. 

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