Isoparametric element

In conjunction with the use of isoparametric elements it is necessary to express the derivatives of nodal functions in terms of local coordinates. This is a straightforward procedure for elements with C continuity and can be described as follows Using the chain rule for differentiation of functions of multiple variables, the derivative of a function in terms of local variables ij) can be expressed as... 

Differentiation of locally defined shape functions appearing in Equation (2.34) is a trivial matter, in addition, in isoparametric elements members of the Jacobian matrix are given in terms of locally defined derivatives and known global coordinates of the nodes (Equation 2.27). Consequently, computation of the inverse of the Jacobian matrix shown in Equation (2.34) is usually straightforward. 

The previous section used the constant strain three-noded element to solve Poisson s equation with steady-state as well as transient terms. The same problems, as well as any field problems such as stress-strain and the flow momentum balance, can be formulated using isoparametric elements. With this type of element, the same (as the name suggests) shape functions used to represent the field variables are used to interpolate between the nodal coordinates and to transform from the xy coordinate system to a local element coordinate system. The first step is to discretize the domain presented in Fig. 9.12 using the isoparametric quadrilateral elements as shown in Fig. 9.15. 

If we pick an arbitrary element we can see that it is represented by the xy-coordinates of the four nodal points, as depicted in Fig. 9.16. The figure also shows a -coordinate system embedded within the element. In the r/, or local, coordinate system, we have a perfectly square element of area 2x2, where the element spreads between —1 > < 1 and — 1 > rj < 1. This attribute allows us to easily allows us to use Gauss quadrature as a numerical integration scheme, where the limits vary between -1 and 1. The isoparametric element described in the //-coordinate system is presented in Fig. 9.17. 

Solution of the two-dimensional Poisson s equation compression molding. To illustrate the use of the four-noded isoparametric element, we can solve for the pressure distribution and velocity field during compression molding of an L-shaped polymer charge, shown in Fig. 9.18, with the physical and numerical data presented in Table 9.3. 

A finite element program with four-noded isoparametric elements was used to solve the above governing equations and boundary conditions. Algorithm 10 presents the scheme used to evaluate the element stiffness matrices and force vectors using numerical integration. 

The isoparametric element works quite well to formulate the finite element equations for flow problems, such as flows with non-Newtonian shear thinning viscosity. Due to the flexibility that exists to integrate variables throughout the elements, the method lends itself... 

Algorithm 10 Computing the element stiffness matrix and force vector for a four-noded isoparametric element by numerical integration... 

At this point, we can proceed to the finite element formulation of the above governing equations. For this, we will use the isoparametric element presented in the last sections. 

Similar to the two-dimensional isoparametric element, for three dimensional elements we use a mapping of the normalized coordinates, , ty, C> (Li volume coordinates for a tetrahedral element), in such a way that the cartesian coordinates will appear as a curvilinear set. 

In 3D we also need the two transformations used with the 2D isoparametric element. In the first place, the global derivatives of the formulation, dNi/dx, must be expresses in terms of local derivatives, dNi/d . Second, the integration of volume (or surface) needs to be performed in the appropriate coordinate system with the correct limits of integration. The global and local derivatives are related through a Jacobian transformation matrix as follows... 

For three-dimensional problems the integral formulations previously obtained are also valid and are implemented into two-dimensional elements that cover the domain surface as shown in Fig. 10.15. Here, we use triangular and rectangular elements as used with FEM. Again, depending on the number of nodes per element, we can have constant, linear and quadratic elements. To be able to represent any geometry it is best to use curvilinear isoparametric elements as schematically illustrated in Fig. 10.16. 

Solution of Equation 14 is obtained by the finite element model RMA-4 for any stipulated flow field u x,t) and v y,t) over a continuum of elements, usually of triangular or rectangular shape (although isoparametric elements with quadratic functions defining the sides are allowed). Depths may be fixed or variable, both in time and space. RMA-4 has been applied successfully to a wide variety of practical problems (13). The version used here is a modification, RMA-4A, that is designed specifically to deal with the kinetics of copper as described. 

The powder compact is modelled using three dimensional 20-noded isoparametric elements and only isothermal heating condition is considered. The problem is firstly solved using the full... 

While this algorithm can conceivably fail in the case of an isoparametric element for which one or more of its sides has extreme curvature, in practice, the probability of such an occurrence is vanishing. 

This chapter presents the linear and non-linear dynamic finite element analysis intended to be used for nuclear facilities. Plasticity and cracking models are included. Solid isoparametric elements, panel and line elements are included which represent various materials. Solution procedures are recommended. Programs ISOPAR, F-BANG and other computer packages are recommend for the dynamic non-linear analysis of structures for nuclear facilities with and without cracking. 

A 3D finite element analysis is developed in which a provision has been made for time-dependent plasticity and rupturing in steel and cracking in materials such as concrete. The influence of studs, tugs and connectors is included. Concrete steel liners and studs are represented by solid isoparametric elements, shell elements and line elements with or without bond hnkages. To begin with, a displacement finite element is adopted. 

Fig. 3.1 (a) Isoparametric elements (i) parent element, 3D isoparametric derived element (ii) solid element (20-noded) (iii) 32-noded solid element, (b) Line elements within the body of the solid isoparametric elements (ISS - isoperametric solid element)... 

Finite Element Data 400 elements in the cylindrical part 20 noded isoparametric Elements 351 dome part mixiture of 8-noded and 82 noded elements... 

A 3D finite element analysis has been carried out using, respectively, isoparametric element for solid concrete and bar (noded) element for reinforcement. Figure AIA.4 shows a finite element meshscheme for adjacent anchorages. Dimension loading and other parameters are kept the same of example under Section A.2.2.1. the 3D displacements are shown in Fig. AIA.5 from the output opf ratio of stress fyjfc are plotted in Fig. AIA.2 against the depths of the immediate blocks. The results are also compared experimental and other theories prominently shown in Fig. AIA.2. 

Felicelli et al. [54] used a bilinear Lagrangian isoparametric element to discretize the transport equations. The convective terms are dealt with using a Petrov-Galerkin formulation in which the weighting function is perturbed in the convective term. The perturbed weighting function is expressed as ... 

FeUceUi et al. [57] extended the model to three-dimensional problems and applied it to solidification of a Pb-Sn alloy in a parallelepiped and in a cylinder. The transport equations were discretized and integrated in time using a FEM based on the bilinear Lagrangian isoparametric element. A Petrov-Galerkin technique was used to... 

Adams et al (1978b) studied solid and annular butt joints loaded in torsion and tension, and examined the effects of adherend flexibility in the spew fillet. They used an eight-node parabolic isoparametric element which gives a good estimation of the stresses in regions of high stress gradient. Their results were confined to the linearly elastic behaviour of aluminium adherends and an epoxy based adhesive which... 

In isoparametric elements, the same interpolation functions are used to define the variation of the primary variable across the element and the shape of the element. In other words, [hJ] is used to determine the element shape from the nodal coordinates [x ] (hence the term shape function, which takes on added significance for isoparametric elements) and to determine element displacement from nodal displacements. 

The finite element model used here was a nine-node isoparametric element which is much more efficient in handling high stress gradients than the constant strain elements used... 

Relating to these components, the 3-dimensional isoparametric element must exist in finite element analysis software package. 

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