Model finite-element method

Mori-Tanaka model Kalpin-Tsai model Lattice-spring model Finite element method Equivalent continuum approach Seif-similar approach... 

The AUGUR information on defect configuration is used to develop the three-dimensional solid model of damaged pipeline weldment by the use of geometry editor. The editor options provide by easy way creation and changing of the solid model. This model is used for fracture analysis by finite element method with appropriate cross-section stress distribution and external loads. 

WEIGHTED RESIDUAL FINITE ELEMENT METHODS - AN OUTLINE 2.1.1 Interpolation models... 

Weighted residual finite element methods described in Chapter 2 provide effective solution schemes for incompressible flow problems. The main characteristics of these schemes and their application to polymer flow models are described in the present chapter. 

The most successful models are based on the finite element method. The flow is discretized into small subregions (elements) and mass and force balances are appHed in each. The result is a large system of equations, the solution of which usually gives the speed of the coating Hquid in each element, pressure, and the location of the unknown free surfaces. The smaller the elements, the more the equations which are often in the range of 10,000 to upward of 100,000. 

In reality, heat is conducted in all three spatial dimensions. While specific building simulation codes can model the transient and steady-state two-dimensional temperature distribution in building structures using finite-difference or finite-elements methods, conduction is normally modeled one-... 

These apparent restrictions in size and length of simulation time of the fully quantum-mechanical methods or molecular-dynamics methods with continuous degrees of freedom in real space are the basic reason why the direct simulation of lattice models of the Ising type or of solid-on-solid type is still the most popular technique to simulate crystal growth processes. Consequently, a substantial part of this article will deal with scientific problems on those time and length scales which are simultaneously accessible by the experimental STM methods on one hand and by Monte Carlo lattice simulations on the other hand. Even these methods, however, are too microscopic to incorporate the boundary conditions from the laboratory set-up into the models in a reahstic way. Therefore one uses phenomenological models of the phase-field or sharp-interface type, and finally even finite-element methods, to treat the diffusion transport and hydrodynamic convections which control a reahstic crystal growth process from the melt on an industrial scale. 

Although many interface models have been given so far, they are too qualitative and we can hardly connect them to the mechanics and mechanism of carbon black reinforcement of rubbers. On the other hand, many kinds of theories have also been proposed to explain the phenomena, but most of them deal only with a part of the phenomena and they could not totally answer the above four questions. The author has proposed a new interface model and theory to understand the mechanics and mechanism of carbon black reinforcement of rubbers based on the finite element method (FEM) stress analysis of the filled system, in journals and a book. In the new model and theory, the importance of carbon gel (bound rubber) in carbon black reinforcement of rubbers is emphasized repeatedly. Actually, it is not too much to say that the existence of bound rubber and its changeable and deformable characters depending on the magnitude of extension are the essence of carbon black reinforcement of rubbers. 

Finite element methods are one of several approximate numerical techniques available for the solution of engineering boundary value problems. Analysis of materials processing operations lead to equations of this type, and finite element methods have a number of advantages in modeling such processes. This document is intended as an overview of this technique, to include examples relevant to polymer processing technology. 

The beauty of finite-element modelling is that it is very flexible. The system of interest may be continuous, as in a fluid, or it may comprise separate, discrete components, such as the pieces of metal in this example. The basic principle of finite-element modelling, to simulate the operation of a system by deriving equations only on a local scale, mimics the physical reality by which interactions within most systems are the result of a large number of localised interactions between adjacent elements. These interactions are often bi-directional, in that the behaviour of each element is also affected by the system of which it forms a part. The finite-element method is particularly powerful because with the appropriate choice of elements it is easy to accurately model complex interactions in very large systems because the physical behaviour of each element has a simple mathematical description. 

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... 

While most authors have used the finite-difference method, the finite element method has also been used—e.g., a two-dimensional finite element model incorporating shrinkable subdomains was used to de.scribe interroot competition to simulate the uptake of N from the rhizosphere (36). It included a nitrification submodel and found good agreement between ob.served and predicted uptake by onion on a range of soil types. However, while a different method of solution was used, the assumptions and the equations solved were still based on the Barber-Cushman model. 

C. Abbes, J. L. Robert, and L. E. Parent, Mechanistic modeling of coupled ammonium and nitrate uptake by onions using the finite element method. Soil Sci. Soc. Am. J. 60 1160 (1996). 

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... 

An alternative approach to the finite element approach is one, introduced as a concept by Courant as early as 1943 [197], in which the total energy functional, implicit in the finite element method, is directly minimized with respect to all nodal positions. The approach is conjugate to the finite element method and merely differs in its procedural approach. It parallels, however, methods often used in atomistic modeling schemes where the potential energy functional of a system (e. g., given by the force field ) is minimized with respect to the position of all (or at least many) atoms of the system. A simple example of this emerging technique is given below. 

Taking these effects into account, internal pore diffusion was modeled on the basis of a wax-filled cylindrical single catalyst pore by using experimental data. The modeling was accomplished by a three-dimensional finite element method as well as by a respective differential-algebraic system. Since the Fischer-Tropsch synthesis is a rather complex reaction, an evaluation of pore diffusion limitations... 

The modeling of the internal pore diffusion of a wax-filled cylindrical single catalyst pore was accomplished by the software Comsol Multiphysics (from Comsol AB, Stockholm, Sweden) as well as by Presto Kinetics (from CiT, Rastede, Germany). Both are numerical differential equation solvers and are based on a three-dimensional finite element method. Presto Kinetics displays the results in the form of diagrams. Comsol Multiphysics, instead, provides a three-dimensional solution of the problem. 

The focus of the remainder of this chapter is on interstitial flow simulation by finite volume or finite element methods. These allow simulations at higher flow rates through turbulence models, and the inclusion of chemical reactions and heat transfer. In particular, the conjugate heat transfer problem of conduction inside the catalyst particles can be addressed with this method. 

With the above-described heat transfer model and rapid solidification kinetic model, along with the related process parameters and thermophysical properties of atomization gases (Tables 2.6 and 2.7) and metals/alloys (Tables 2.8,2.9,2.10 and 2.11), the 2-D distributions of transient droplet temperatures, cooling rates, achievable undercoolings, and solid fractions in the spray can be calculated, once the initial droplet sizes, temperatures, and velocities are established by the modeling of the atomization stage, as discussed in the previous subsection. For the implementation of the heat transfer model and the rapid solidification kinetic model, finite difference methods or finite element methods may be used. To characterize the entire size distribution of droplets, some specific droplet sizes (forexample,.D0 16,Z>05, andZ)0 84) are to be considered in the calculations of the 2-D motion, cooling and solidification histories. 

Finite element methods (FEM) are capable of incorporating complex variations in materia stresses in the time varying response. While these methods are widely available, they are quite complex and, in many cases, their use is not warranted due to uncertainties in blast load prediction. The dynamic material properties presented in this section can be used in FEM calculations however, the simplified response limits in the next section may not be suitable. Most FEM codes contain complex failure models which are better indicators of acceptable response. See Chapter 6, Dynamic Analysis Methods, for additional information. 

Although the finite element method can provide the most accurate means for analyzing structures for blast loads, the uncertainty associated with determination of loads generally does not justify its use. Also, the effort associated with finite element model development and interpretation of results is often greater that what is required by the simplified methods outlined above. The simpler SDOF based analytical methods are recommended for use except in those cases, as described above, where the inaccuracies associated with SDOF approximations may be unacceptable. 

Kaluarachchi, J. J. and Parker, J. C., 1989, An Efficient Finite Element Method for Modeling Multiphase How Water Resources Research, Vol. 25, pp. 43-54. 

As indicated above, there are a large number of modeling packages on the market. Some of those are mentioned below. In the vast majority, differential equations that describe the electrochemical setup are solved using numeric methods. Two of the most common methods are the finite-difference method and the finite-elements method. These are discussed in some detail in this chapter, including example calculations in Section 15.3. We begin with a few general remarks. 

Mechanical properties of PNCs can also be estimated by using computer modeling and simulation methods at a wide range of length and time scales. Seamless movement from one scale to another, for example, from the molecular scale (e.g., MD) and microscale (e.g., Halphin-Tsai) to macroscale (e.g., finite element method, FEM), and the combination of scales (or the so-called multiscale methods) is the most important prerequisite for the efficient transfer and extrapolation of calculated parameters, properties, and numerical information across length scales. 

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