Quadrilateral elements

Figure 2.17 Isoparametric mapping of an irregular quadrilateral element wilJi straight sides...

Consider the integration of a function/(xj, X2) over a quadrilateral element in a finite element mesh expressed as... 

The unknowns in this equation are the local coordinates of the foot (i.e. and 7]). After insertion of the global coordinates of the foot found at step 6 in the left-hand side, and the global coordinates of the nodal points in a given element in the right-hand side of this equation, it is solved using the Newton-Raphson method. If the foot is actually inside the selected element then for a quadrilateral element its local coordinates must be between -1 and +1 (a suitable criteria should be used in other types of elements). If the search is not successful then another element is selected and the procedure is repeated. 

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

Simple pressure/drag flow. Here we treat an idealization of the down-channel flow in a melt extruder, in which an incompressible viscous fluid constrained between two boundaries of infinite lateral extent (2). A positive pressure gradient is applied in the X-direction, and the upper boundary surface at y - H is displaced to the right at a velocity of u(H) - U this velocity is that of the barrel relative to the screw. This simple problem was solved by a 10x3 mesh of 4-node quadrilateral elements, as shown in Figure 1. 

To carry out a numerical solution, a single strip of quadrilateral elements is placed along the x-axis, and all nodal temperatures are set Initially to zero. The right-hand boundary is then subjected to a step Increase in temperature (T(H,t) - 1.0), and we seek to compute the transient temperature variation T(x,t). The flow code accomplishes this by means of an unconditionally stable time-stepping algorithm derived from "theta" finite differences a solution of ten time steps required 22 seconds on a PC/AT-compatible microcomputer operating at 6 MHz. 

Solution of Field Problems Using Isoparametric Quadrilateral Elements. 

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. 

Figure 9.15 Finite element mesh using isoparametric quadrilateral elements for the domain...

Figure 9.16 Isoparametric quadrilateral element in the xy-coordinate system.

Figure 10.15 3D single screw extruder barrel and mixing head discretization. White element delineations define the mixing head surface, and the black lines define the barrel surface representation. The mixing head surface is represented with triangular as well as quadrilateral elements. 

The solution domain was divided into quadrilateral elements that were finer near the tube walls and in regions where the two-phase interface was expected to pass, and coarser in regions where only... 

The MOTIF code is a three-dimensional finite-element code capable of simulating steady state or transient coupled/uncoupled variable-density, variable- saturation fluid flow, heat transport, and conservative or nonspecies radionuclide) transport in deformable fractured/ porous media. In the code, the porous medium component is represented by hexahedral elements, triangular prism elements, tetrahedral elements, quadrilateral planar elements, and lineal elements. Discrete fractures are represented by biplanar quadrilateral elements (for the equilibrium equation), and monoplanar quadrilateral elements (for flow and transport equations). 

In the numerical solution of the flow model, Eq. [9.7] and the solution domain are discretized. The part geometry, which is equivalent to the mold cavity, is discretized as a shell mesh in 3D using usually triangular and/or quadrilateral elements in 2D. The most common numerical method is to use a finite element/control volume (FE/CV) approach although boundary element and finite difference methods have been used. " ... 

Two test cases are used to validate the linear viscoelastic analysis capability implemented in the present finite-element program named NOVA. In the first case, the tensile creep strain in a single eight-noded quadrilateral element was computed for both the plane-stress and plane-strain cases using the program NOVA. The results were then compared to the analytical solution for the plane-strain case presented in Reference 49. A uniform uniaxial tensile load of 13.79 MPa was applied on the test specimen. A three-parameter solid model was used to represent the tensile compliance of the adhesive. The Poisson s ratio was assumed to remain constant with time. The following time-dependent functions were used in Reference 49 to represent the tensile compliance for FM-73M at 72 °C ... 

For each flow-field iteration, a Galerkin finite algorithm [19] is used to solve Eqs. (1) and (2). The domain, O, is tessellated into 20-noded quadrilateral elements. Over each of these elements, an approximate solution of the form... 

In laminar flows, the grid near boundaries should be refined to allow the solution to capture the boundary layer flow detail. A boundary layer grid should contain quadrilateral elements in 2D and hexahedral or prism elements in 3D, and should have at least five layers of cells. For turbulent flows, it is customary to use a wall function in the near-waU regions. This is due to the fact that the transport equation for the eddy dissipation has a singularity at the wall, where k [in the denominator in the source terms in eq. (5-14)] is zero. Thus, the equation for e must be treated in an alternative manner. Wall functions rely on the fact... 

When we consider non-linear material properties by a closed-form analysis such as Hart-Smith s, the limitation is how tractable is a realistic mathematical model of the stress-strain curve within an algebraic solution. With the finite-element techniques developed for adhesive joints by Adams and his co-workers, the limit becomes that of computing power. The high elastic stress and strain gradients at the ends of the adhesive layer need to be accommodated by three or four 8-node quadrilateral elements across the thickness. However, consideration of non-linear material behaviour requires a much larger computing effort on any given element. Thus, it becomes necessary to... 

Zhao C, Hobbs BE, MUhlhaus HB, Ord A (1999) A consistent point-searching algorithm for solution interpolation in unstructured meshes consisting of 4-node bilinear quadrilateral elements. Int J Numer... 

Fig. 2 Illustration of our developing procedure of automatic mesh subdivision on the basis of Voronoi Polygon and Delaunay Tessellation a) Initial geometric model, b) Automatic node generation, c) Intermediate search for new DT s in sequence by edge control, d) Automatic triangulation and e) Transformation to quadrilateral elements.

The first and perhaps the most basic step in the FE approach is flie selection of finite elements to uniformly cover the spatial dimensions of a physical problem. While many possible shapes could be considered for the fundamental spatial elements, in practice, fairly simple shapes are typically used. For two spatial dimensions, the elements typically considered are triangular elements or general quadrilateral elements as illustrated in Figure 13.2. (The three dimensional case will be briefly considered in a subsequent section). Three triangular elements are shown in (a) - (c) with specified nodes of 3, 6 and 10 where nodes are identified by the solid points in the figure and are the points at which the solution variable is... 

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