Weighted-residual formulation

This form is referred to as the weighted-residual formulation of the problem. What is required to be determined is the potential inner surface of the box. (One may constrain w to zero on both surfaces, since this has no effect on the equivalence of the two expressions.) Equation (15.11) may be used as the launching... 

The most common methodology when solving transient problems using the finite element method, is to perform the usual Garlerkin weighted residual formulation on the spatial derivatives, body forces and time derivative terms, and then using a finite difference scheme to approximate the time derivative. The development, techniques and limitations that we introduced in Chapter 8 will apply here. The time discretization, explicit and implicit methods, stability, numerical diffusion etc., have all been discussed in detail in that chapter. For a general partial differential equation, we can write... 

The nodal unknows a are to be chosen so as to satisfy the governing equations in an integral sense this can be done by using a Galerkin weighted residual formulation of the conservation equations for momentum and energy transport ... 

A weighted residual formulation [89, 90] is used in order to solve Eq. (41). This approach seeks to approximate the variation of concentration over a region in space using a residual R x) such that... 

As already discussed, variations of a field unknown within a finite element is approximated by the shape functions. Therefore finite element discretization provides a nat ural method for the construction of piecewise approximations for the unknown functions in problems formulated in a global domain. This is readily ascertained considering the mathematical model represented by Equation (2.40). After the discretization of Q into a mesh of finite elements weighted residual statement of Equ tion (2.40), within the space of a finite element T3<, is written as... 

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. 

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

Solvent Effects in the Sn Spectra of Poly(TBTM/MMA). Samples of poly(MMA/TBTM) synthesized by the free-radical copolymerization of the appropriate monomers were solutions in benzene with approximately 33% solids (weight to volume). The particular formulation chosen as representative of the class contained a 1 1 ratio of pendant methyl to tri-n-butyltin groups. In preparing the dry polymer, the benzene was removed in vacuo with nominally 5% by weight residual solvent. 

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 next step will be to formulate the Galerkin-weighted residual for each of the governing equations... 

In the early 1970s, the standard finite element approximations were based upon the Galerkin formulation of the method of weighted residuals. This technique did emerge as a powerful numerical procedure for solving elliptic boundary value problems [102, 75, 53, 84, 50, 89, 17, 35]. The Galerkin finite element methods are preferable for solving Laplace-, Poisson- and and diffusion equations because they do not require that a variational principle exists for the problem to be analyzed. However, the power of the method is still best utilized in systems for which a variational principle exists, and it... 

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. 

Variational methods [5] are a class of high-order weighted residual techniques that combines the high spatial accuracy and rapid convergence of spectral methods with the generality and geometric flexibility of finite-element methods. Consider a variational method on Q for mie-dimensional Helmholtz Eq. 22. A variational formulation of this problem is that u(x) should be the solution to... 

The FEM is part of a larger group of techniques that exploit the method of weighted residuals (MWR) [76]. These use a set of weighting functions to allow the approximation of a variable over a domain. The choice of weighting function leads to a number of different alternative formulations, including the collocation method, subdomain method, method of moments, and the Galerkin method. 

The formulation procedure begins in an analogous maimer to that of the FEM. For the case of the diffusional operator Eq. (41), the weighted residual form is given by Eq. (51) as before, integration (by parts) yields... 

One-dimensional models for inviscid, incompressible, axisymmetric, armular liquid jets falling under gravity have been obtained by means of methods of regular perturbations for slender or long jets, integral formulations, Taylor s series expansions, weighted residuals, and variational principles [27, 47]. 

The Least-Squares Method as Derived Through the Weighted Residual Framework (Weak Formulation)... 

In the following, the general implementation issues of the generalized problem formulation (12.402) are presented. Based on variational or weighted residual principles (weak formulation), the algebraic equation system (12.401) where A andF are defined by (12.462) and (12.463) or alternatively (12.475) and (12.476) as obtained by the weak formulation of the least-squares technique can be presented as ... 

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