Elements, finite

Several types of experiments have been carried out to investigate the stress state in the head of the bolt created by the body forces. The results of the finite element model experiment can be seen in Fig. 2, and those of the optical plane model experiment are presented in Fig. 3. 

This paper compares experimental data for aluminium and steel specimens with two methods of solving the forward problem in the thin-skin regime. The first approach is a 3D Finite Element / Boundary Integral Element method (TRIFOU) developed by EDF/RD Division (France). The second approach is specialised for the treatment of surface cracks in the thin-skin regime developed by the University of Surrey (England). In the thin-skin regime, the electromagnetic skin-depth is small compared with the depth of the crack. Such conditions are common in tests on steels and sometimes on aluminium. 

D Finite Element / Boundary Integral Element method (TRIFOU)... 

TRIFOU is a combined Finite Elements/Boundary Integral formulation code. The BIM formulation in vacuum is suitable for NDT simulation where the probe moves in the air around the test block. The FEM formulation needs more calculation time, but tetrahedral elements enable a large variety of specimens and defect geometries to be modelled. TRIFOU uses a formulation of Maxwell Equations using magnetic field vector h, where h is decomposed as h = hs + hr (hj source field, and hr reaction field). 

In order to describe inherited stress state of weldment the finite element modelling results are used. A series of finite element calculations were conducted to model step-by-step residual stresses as well as its redistribution due to heat treatment and operation [3]. The solutions for the reference weldment geometries are collected in the data base. If necessary (some variants of repair) the modelling is executed for this specific case. 

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. 

For the determination of the approximated solution of this equation the finite difference method and the finite element method (FEM) can be used. FEM has advantages because of lower requirements to the diseretization. If the material properties within one element are estimated to be constant the last term of the equation becomes zero. Figure 2 shows the principle discretization for the field computation. 

Finite Element Study of the Anomalous Behaviour of Remote Field Eddy Currents Proc. 7th Int. Conf on Offshore Mechanics, Houston, Texas 1988... 

In a first discretization step, we apply a suitable spatial discretization to Schrodinger s equation, e.g., based on pseudospectral collocation [15] or finite element schemes. Prom now on, we consider tjj, T, V and H as denoting the corresponding vector and matrix representations, respectively. The total... 

How can Equation (11.79) be solved Before computers were available only simple ihapes could be considered. For example, proteins were modelled as spheres or ellipses Tanford-Kirkwood theory) DNA as a uniformly charged cylinder and membranes as planes (Gouy-Chapman theory). With computers, numerical approaches can be used to solve the Poisson-Boltzmann equation. A variety of numerical methods can be employed, including finite element and boundary element methods, but we will restrict our discussion to the finite difference method first introduced for proteins by Warwicker and Watson [Warwicker and Watson 1982]. Several groups have implemented this method here we concentrate on the work of Honig s group, whose DelPhi program has been widely used. 

Pittman, J. F.T. and Nakazawa, S., 1984. Finite element analysis of polymer processing operations. In Pittman, J.F. T., Zienkiewicz, O.C., Wood, R.D. and Alexander, J. M. (eds), Num,erical Analysis of Forming Processes, Wiley, Chichester. 

Weighted Residual Finite Element Methods - an Outline... 

WEIGHTED RESIDUAL FINITE ELEMENT METHODS - AN OUTLINE... 

In most engineering problems the boundary of the problem domain includes curved sections. The discretization of domains with curved boundaries using meshes that consist of elements with straight sides inevitably involves some error. This type of discretization error can obviously be reduced by mesh refinements. However, in general, it cannot be entirely eliminated unless finite elements which themselves have curved sides are used. 

The discretization of a problem domain into a finite element mesh consisting of randomly sized triangular elements is shown in Figure 2,1. In the coarse mesh shown there are relatively large gaps between the actual domain boundary and the boundary of the mesh and hence the overall discretization error is expected to be large. 

The main consequence of the discretization of a problem domain into finite elements is that within each element, unknown functions can be approximated using interpolation procedures. 

Let be a well-defined finite element, i.e. its shape, size and the number and locations of its nodes are known. We seek to define the variations of a real valued continuous function, such as/, over this element in terms of appropriate geometrical functions. If it can be assumed that the values of /on the nodes of Oj, are known, then in any other point within this element we can find an approximate value for/using an interpolation method. For example, consider a one-dimensional two-node (linear) element of length I with its nodes located at points A(xa = 0) and B(a b = /) as is shown in Figure 2.2. 

Inherent in the development of approximations by the described interpolation models is to assign polynomial variations for function expansions over finite elements. Therefore the shape functions in a given finite element correspond to a... 

Shape functions of commonly used finite elements... 

Standard procedures for the derivation of the shape functions of common types of finite elements can be illustrated in the context of two-dimensional triangular and rectangular elements. Let us, first, consider a triangular element having three nodes located at its vertices as is shown in Figure 2.6. 

The number of terms of a complete polynomial of any given degree will hence correspond to the number of nodes in a triangular element belonging to this family. An analogous tetrahedral family of finite elements that corresponds to complete polynomials in terms of three spatial variables can also be constructed for three-dimensional analysis. 

The described direct derivation of shape functions by the formulation and solution of algebraic equations in terms of nodal coordinates and nodal degrees of freedom is tedious and becomes impractical for higher-order elements. Furthermore, the existence of a solution for these equations (i.e. existence of an inverse for the coefficients matrix in them) is only guaranteed if the elemental interpolations are based on complete polynomials. Important families of useful finite elements do not provide interpolation models that correspond to complete polynomial expansions. Therefore, in practice, indirect methods are employed to derive the shape functions associated with the elements that belong to these families. 

Finite elements that maintain inter-element compatibility of functions are called conforming elements . Finite elements that do not have this property are referred to as the non-conforming elements . Under certain conditions non-conforming elements can lead to accurate solutions and are more advantageous to use. 

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