Bi-quadratic element

WEIGHTED RESIDUAL FINIl E ELEMENT METHODS - AN OUTLINE Nine-node bi-quadratic element... 

The momentum and continuity equations give rise to a 22 x 22 elemental stiffness matrix as is shown by Equation (3.31). In Equation (3.31) the subscripts I and / represent the nodes in the bi-quadratic element for velocity and K and L the four corner nodes of the corresponding bi-linear interpolation for the pressure. The weight functions. Nr and Mf, are bi-qiiadratic and bi-linear, respectively. The y th component of velocity at node J is shown as iPj. Summation convention on repeated indices is assumed. The discretization of the continuity and momentum equations is hence based on the U--V- P scheme in conjunction with a Taylor-Hood element to satisfy the BB condition. 

Figure 5.2 shows the finite element mesh corresponding to the configuration shown in Figure 5.1. This mesh consists of 225 nine-node bi-quadratic elements and its utihzation in the present model is based on the application of isoparametric mapping, described in Chapter 2.

Figure 5.2 The mesh used in this example consisting of 225 nine-node bi-quadratic elements...

Depending on the type of elements used appropriate interpolation functions are used to obtain the elemental discretizations of the unknown variables. In the present derivation a mixed formulation consisting of nine-node bi-quadratic shape functions for velocity and the corresponding bi-linear interpolation for the pressure is adopted. To approximate stres.ses a 3 x 3 subdivision of the velocity-pressure element is considered and within these sub-elements the stresses are interpolated using bi-linear shape functions. This arrangement is shown in Edgure 3.1. 

SHAPE. Gives the shape functions in terms of local coordinates for bi-linear or bi-quadratic quadrilateral elements. 

As a complementary approach a mean-field method is used in combination with the finite element method to investigate the FGM. To compare the predictions with the periodic unit cell simulations, the FGM part is divided into nine sublayers. Each of them consists of two bi-quadratic 8-node plane elements over the thickness, and only one element is used in the horizontal direction. The parts of pure alumina and pure nickel are modeled by three and 12 elements, respectively. In each sublayer the volume fractions of the phases are constant. In addition, the center sublayer can be split into a metal-matrix and a ceramic-matrix half sublayer. The properties of the particular material on the meso-structural level within the finite element calculation are described via a con-... 

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