UPWIND

Development of weighted residual finite element schemes that can yield stable solutions for hyperbolic partial differential equations has been the subject of a considerable amount of research. The most successful outcome of these attempts is the development of the streamline upwinding technique by Brooks and Hughes (1982). The basic concept in the streamline upwinding is to modify the weighting function in the Galerkin scheme as... 

In the earlier versions of the streamline upwinding scheme the modified weight function was only applied to the convection tenns (i.e. first-order derivatives in the hyperbolic equations) while all other terms were weighted in the usual manner. This is called selective or inconsistent upwinding. Selective upwinding can be interpreted as the introduction of an artificial diffusion in addition to the physical diffusion to the weighted residual statement of the differential equation. This improves the stability of the scheme but the accuracy of the solution declines. 

The first order derivative in Equation (2.80) corresponds to the convection in a field problem and the examples shown in Figure 2.26 illustraTes the ina bility of the standard Galerkin method to produce meaningful results for convection-dominated equations. As described in the previous section to resolve this difficulty, in the solution of hyperbolic (convection-dominated) equations, upwind-ing or Petrov-Galerkin methods are employed. To demonstrate the application of upwinding we consider the case where only the weight function applied to the first-order derivative in the weak variational statement of the problem, represented by Equation (2.82), is modified. 

In the simple one-dimensional example considered here the upwinded weight function found using Equation (2.89) is reduced to W = N + j3 dNldx). Therefore, the modified weight functions applied to the first order derivative term in Equation (2.91) can be written as... 

After the aussembly of elemental equations into a global set and imposition of the boundary conditions the final solution of the original differential equation with respect to various values of upwinding parameter jS can be found. The analytical solution of Equation (2.80) with a = 50 is found as... 

The finite element results obtained for various values of (3 are compared with the analytical solution in Figure 2.27. As can be seen using a value of /3 = 0.5 a stable numerical solution is obtained. However, this solution is over-damped and inaccurate. Therefore the main problem is to find a value of upwinding parameter that eliminates oscillations without generating over-damped results. To illustrate this concept let us consider the following convection-diffusion equation... 

Figure 2.28 Comparison of the analytical solution with the finite element result obtained using optimal upwinding...

Brooks, A. N, and Hughes, T. J.R., 1982. Streamline-upwind/Petrov Galerldn formulations for convection dominated hows with particular emphasis on the incompressible Navier -Stokes equations. Cornput. Methods Appl Meek Eng. 32, 199-259. 

Hughes, T. J.R. and Brooks, A.N., 1979, A multidimensional upwind scheme with no cross-wind diffusion. In Hughes, I . J. R. (ed.), Finite Element Methods for Convection Dominated Flows, AMD Vol. 34, ASME, New York. 

The first finite element schemes for differential viscoelastic models that yielded numerically stable results for non-zero Weissenberg numbers appeared less than two decades ago. These schemes were later improved and shown that for some benchmark viscoelastic problems, such as flow through a two-dimensional section with an abrupt contraction (usually a width reduction of four to one), they can generate simulations that were qualitatively comparable with the experimental evidence. A notable example was the coupled scheme developed by Marchal and Crochet (1987) for the solution of Maxwell and Oldroyd constitutive equations. To achieve stability they used element subdivision for the stress approximations and applied inconsistent streamline upwinding to the stress terms in the discretized equations. In another attempt, Luo and Tanner (1989) developed a typical decoupled scheme that started with the solution of the constitutive equation for a fixed-flow field (e.g. obtained by initially assuming non-elastic fluid behaviour). The extra stress found at this step was subsequently inserted into the equation of motion as a pseudo-body force and the flow field was updated. These authors also used inconsistent streamline upwinding to maintain the stability of the scheme. 

The integrals in Equation (3.32) are found using a quadrature over the element domain The viscoelastic constitutive equations used in the described model are hyperbolic equations and to obtain numerically stable solutions the convection terms in Equation (3.32) are weighted using streamline upwinding as (inconsistent upwinding)... 

Derivation of the working equations of upwinded schemes for heat transport in a polymeric flow is similar to the previously described weighted residual Petrov-Galerkm finite element method. In this section a basic outline of this derivation is given using a steady-state heat balance equation as an example. 

In the consistent streamline upwind Petrov-Galerkin (SUPG) scheme all of the terms in Equation (3.52) are weighted using the function defined by Equation (3.53) and hence Wjj = Wj. 

Extension of the streamline Petrov -Galerkin method to transient heat transport problems by a space-time least-squares procedure is reported by Nguen and Reynen (1984). The close relationship between SUPG and the least-squares finite element discretizations is discussed in Chapter 4. An analogous transient upwinding scheme, based on the previously described 0 time-stepping technique, can also be developed (Zienkiewicz and Taylor, 1994). 

Luo, X. L, and Tanner, R. L, 1989. A decoupled finite element streamline-upwind scheme for viscoelastic flow problems. J. Non-Newtonian Fluid Mech. 31, 143-162. 

Working equations of the streamline upwind (SU) scheme for the steady-state energy equation in Cartesian, polar and axisymmetric coordinate systems... 

Following the procedure described in in Chapter 3, Section 3 the streamlined-upwind weighted residual statement of the energy equation is formulated as... 

Least-square.s and streamline upwind Petrov-Galerkin (SUPG) schemes... 

The inconsistent streamline upwind scheme described in the last section is fonuulated in an ad hoc manner and does not correspond to a weighted residual statement in a strict sense. In tins seetion we consider the development of weighted residual schemes for the finite element solution of the energy equation. Using vector notation for simplicity the energy equation is written as... 

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