Finite element modelling assumptions

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. 

Figure 5.27 shows a finite element model of the joint. The members are modelled by 8-node Serendipity elements under the assumption of plane stress conditions. The two steel plates of thickness h are modelled by one steel plate of thickness 2h for an accurate overall stiffness representation. The fasteners and holes are represented by Unite spring elements which are connected to node points on the members. The stiffness constants of the spring elements (Kffe) are determined according to the procedure described in section 5.2.2.4 of the EUROCOMP Handbook. 

Of particular importance is the assumption of thin-walled geometry. From Eq. (7.3) we see that the pressure is independent of the z coordinate. Consequently, the finite element utilized for pressure calculation need have no thickness. That is, the element is a plane shell—generally a triangle or quadrilateral. This has great implications for users of plastics CAE. It means that a finite element model of the component is required that has no thickness. In the past this was not a problem. Almost all common CAD systems were using surface or wireframe modeling and thickness was never shown explicitly. The path from the CAD model to the FEA model was clear and direct. 

With the use of this MDFEA framework, nonlinearities are accounted for in the individual computational modules, such as in ZEUS-NL and VecTorl. The different features of these modules, including finite element model resolutions, theoretical algorithms and numerical techniques, will lead to different accuracy levels and different deviations of strain and stress resultants. Hence, the actual movements and reaction feedbacks at control points will contain errors combined from multiple modules that are difficult to eliminate. Another error source originates from the interface modeling, such as in this case study example in which either rigid or flexible slab assumptions were used. 

SKORTON In Janz s finite element model that you showed there were 4, 5, or 6 elements distributed across the wall. Was the assumption made throughout the cardiac cycle, or the portion that was studied, that those elements maintained the same relative sizes throughout the cycle ... 

Although the mild slope equation was obtained with the assumption of mild slope, the work by Booji showed that the regular mild slope equation is applicable for bottom slope as large as 1/3. Since the mild slope equation can be conveniently implemented in a finite element model, we will apply it to real harbors in this presentation. The basics of the model will be shown in the following section. 

Hbaieb et al. [24] compared the utility of modeling polymer-montmorillonite nanocomposites by finite element analysis (TEA) in relation to the Mori-Tanaka model. The three-dimensional finite element model (FEM) was found to be superior to the two-dimensional one. For the calculations, the aspect ratio (A) was chosen to be 50, YfjYp = 100, the Poisson ratio for the polymer (Pp) was assumed to be 0.35, and the Poisson ratio for the montmorillonite was assumed to be Pf = 0.2. The FEA was performed using the commercial package, ABAQUS. The morphology of the montmorillonite was assumed to be disk-shaped. The limitations of this assumption are exposed in the discussion above by Lee and Paul. 

Nowadays, commercial finite element programs are quite powerful and user friendly codes which are available for all platforms of operating systems. However, reliable results can only be obtained when the user of such codes understands the problem, how to model it, the behavior of finite elements, the assumptions and limitations of the code, and when the analyst is able to check for errors at all stages. It is not necessary that the end-user understands in detail all procedures and routines, but incorrect results, which may cause serious damage such as production delay, redesign or even collapse, can be avoided only by those who understand the basic ideas and single steps of the FEM (cf. O Chap. 25). 

In the three layers finite element model discussed above, a linear relationship relating the SIF to the longitudinal displacement near the crack tip was assumed in the prediction of the SIF of the unpatched and the patched side of the cracked plate. In order to examine this assumption, a modified three layers finite element model is proposed. The modified model uses 3-D brick elements to model the cracked plate and shell elements to model both the CFRP patching and the adhesive layer. Since... 

Given a model, the analysis can be performed mathematically. A finite element computer code (Ref 38) for the analysis of finite or infinitesimal strains is now available, modified (Ref 68) to account for shock induced stresses, temp rise (by the assumption of a constant Grueneisen parameter), heat generated by the decompn of the expl and transient heat transfer. Later in this article we report an empirical treatment of propint initiation data (Fig 4)-Analy tic ally obtained data are in fair agreement with exptl results so that further effort along these lines appears justified (Ref 68)... 

At this point, a distinction should be made between cellular and finite difference/element models. The latter are finite approximations of continuous equations [e.g., Eq. (11)], with the implicit assumption that the width of the reaction zone is larger than other pertinent length scales (diffusion, heterogeneity of the medium, etc.). However, no such assumptions need to be made for cellular... 

The solution to (P12) gives us the optimal separation profile as a function of age within the reactor. However, except in the case of reactive phase equilibrium, the assumption of a continuous separation profile is not really required. Furthermore, a continuous separation profile may not be implementable in practice. To address this, we take advantage of the structure of a discretization procedure for the differential equation system. In this case, we choose orthogonal collocation on finite elements to discretize the above model. This results... 

The periodic unit cell results are directly comparable to the IMT predictions, because both approaches represent the same matrix/inclusion type microstructure. However, such comparisons have to be done carefully since some assumptions regarding the finite element calculations are not equivalent for the extended unit cell approaches and the present mean-field method. The plane stress analysis of the unit cell models does not take into account the constraints in the out-of-plane direction. In contrast, within the present IMT formulation the inclusions are enclosed three-dimensionally by the matrix material. In contrast to the plane stress unit cell models, the constraint in the out-of plane direction is accounted for. Accordingly, these predictions are denoted as full internal constraint. To overcome this internal constraint in order to simulate the plane stress model assump-... 

In geochemical models, these quantities represent the smallest time period for incremental steps in a simulated titration, or the smallest distance between grid points in a finite element or finite difference grid, if LEQ is to be a valid assumption. Or, as Knapp puts it, reactive transport calculations assuming LEQ are good approximations only if teq is less than the size of the time step, and /eq is less than the distance between adjacent grid points. 

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