Weighted residuals method

The weighted residual method provides a flexible mathematical framework for the construction of a variety of numerical solution schemes for the differential equations arising in engineering problems. In particular, as is shown in the followmg section, its application in conjunction with the finite element discretizations yields powerful solution algorithms for field problems. To outline this technique we consider a steady-state boundary value problem represented by the following mathematical model... 

The residual, Rq, is a function of position in Q. The weighted residual method is based on the elimination of this residual, in some overall manner, over the entire domain. To achieve this the residual is weighted by an appropriate number of position dependent functions and a summation is carried out. This is written as... 

Despite the simplicity of the outlined weighted residual method, its application to the solution of practical problems is not straightforward. The main difficulty arises from the lack of any systematic procedure that can be used to select appropriate basis and weight functions in a problem. The combination of finite element approximation procedures with weighted residual methods resolves this problem. This is explained briefly in the forthcoming section. 

Application of the weighted residual method to the solution of incompressible non-Newtonian equations of continuity and motion can be based on a variety of different schemes. Tn what follows general outlines and the formulation of the working equations of these schemes are explained. In these formulations Cauchy s equation of motion, which includes the extra stress derivatives (Equation (1.4)), is used to preseiwe the generality of the derivations. However, velocity and pressure are the only field unknowns which are obtainable from the solution of the equations of continuity and motion. The extra stress in Cauchy s equation of motion is either substituted in terms of velocity gradients or calculated via a viscoelastic constitutive equation in a separate step. 

For both the finite difference and weighted residual methods a set of coupled ordinary differential equations results which are integrated forward in time using the method of lines. Various software packages implementing Gear s method are popular. 

Fourier-Galerkin method. To illustrate the weighted residuals method we chose a Fourier-Galerkin method to solve a PDE of the form,... 

For a collocation solution, we need first to chose a set of points within the domain where we want to find the solution. At these locations, we will place delta functions for the weighted residual method. Here, we select N terms for the approximation and we need N points located at (A = 1,..., iV). The collocation weighted residual... 

Next, we apply Galerkin s weighted residual method and reduce the order of integration of the various terms in the above equations using the Green-Gauss Theorem (9.1.2) for each element. For a simpler presentation we will deal with each term in the above equations separately. The terms of the x-component (eqn. (9.95)) of the penalty formulation momentum balance become... 

The selected mathematical model is represented by a discretization method for approximating the differential equations by a system of algebraic equations for the variables at some set of discrete locations in space and time. Many different approaches are used in reactor engineering , but the most important of them are the simple finite difference methods (FDMs), the flrrx conservative finite volume methods (FVMs), and the accurate high order weighted residual methods (MWRs). 

The finite approximations to be used in the discretization process have to be selected. In a finite difference method, approximations for the derivatives at the grid points have to be selected. In a finite volume method, one has to select the methods of approximating surface and volume integrals. In a weighted residual method, one has to select appropriate trail - and weighting functions. A compromise between simplicity, ease of implementation, accuracy and computational efficiency has to be made. For the low order finite difference- and finite volume methods, at least second order discretization schemes (both in time and space) are recommended. For the WRMs, high order approximations are normally employed. 

This group of methods consists of a number of weighted residual methods that are based on spectral solution strategies, but only the least squares spectral method is discussed. 

Lage, P. L. C. 2011 On the representation of QMOM as a weighted-residual method -the dual-quadrature method of generalized moments. Computers Chemical Engineering 35(11), 2186-2203. 

The orthogonal collocation technique is a simple numerical method which is easy to program for a computer and which converges rapidly. Therefore it is useful for the solution of many types of second order boundary value problems. This method in its simplest form as presented in this section was developed by Villadsen and Stewart (1967) as a modification of the collocation methods. In collocation methods, trial function expansion coefficients are typically determined by variational principles or by weighted residual methods (Finlayson, 1972). The orthogonal collocation method has the advantage of ease of computation. This method is based on the choice of suitable trial series to represent the solution. The coefficients of trial series are determined by making the residual equation vanish at a set of points called collocation points , in the solution domain. 

Rubinow defined a normalized cell maturation variable, p. such that a cell will divide when / = 1. Now /x is related to the biochemical events occurring during the cell cycle in some undetermined fashion. It is, therefore, a semiempirical variable and is operationally defined in terms of measured cell cycle times. This implies that all cells divide after the cell cycle time of t hours. It is observed, however, that cell division times are scattered about a mean value. This randomness must be accounted for suitably. Most models account for this with an explicit operation in the mathematical solution by averaging cell division times over the entire population. This scheme leads to the solution of a rather difficult integral equation (see, e.g., Trucco (4)). Recently Subramarian et al. (8) have considered weighted-residual methods for more easily solving these problems. 

In connection with the solutions of the equation of change, there will from time to time be found some new and useful analytical solutions. The emphasis surely will be put, however, on the development of more and more efficient numerical techniques, such as collocation and weighted residual methods, implemented with high-speed computers. 

The Galerkin weighted residual method is employed to formulate the finite element discretisation. An implicit mid-interval backward difference algorithm is implemented to achieve temporal discretisation. With appropriate initial and boundary conditions the set of non-linear coupled governing differential equations can be solved. 

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