Finite element solutions

All numerical computations inevitably involve round-off errors. This error increases as the number of calculations in the solution procedure is increased. Therefore, in practice, successive mesh refinements that increase the number of finite element calculations do not necessarily lead to more accurate solutions. However, one may assume a theoretical situation where the rounding error is eliminated. In this case successive reduction in size of elements in the mesh should improve the accuracy of the finite element solution. Therefore, using a P C" element with sufficient orders of interpolation and continuity, at the limit (i.e. when element dimensions tend to zero), an exact solution should be obtaiiied. This has been shown to be true for linear elliptic problems (Strang and Fix, 1973) where an optimal convergence is achieved if the following conditions are satisfied ... 

The finite element solution of differential equations requires function integration over element domains. Evaluation of integrals over elemental domains by analytical methods can be tedious and impractical and is not attempted in... 

The simplicity gained by choosing identical weight and shape functions has made the standard Galerkin method the most widely used technique in the finite element solution of differential equations. Because of the centrality of this technique in the development of practical schemes for polymer flow problems, the entire procedure of the Galerkin finite element solution of a field problem is further elucidated in the following worked example. 

As an illustrative example we consider the Galerkin finite element solution of the following differential equation in domain Q, as shown in Figure 2.20. 

Figure 2,22 Comparison of the analytical and finite element solutions...

In the finite element solution of engineering problems the global set of equations obtained after the assembly of elemental contributions will be very large (usually consisting of several thousand algebraic equations). They may also be... 

Figure 2.26 Comparison of the finite element solution of low and high Peclet mimber problems...

The standard least-squares approach provides an alternative to the Galerkin method in the development of finite element solution schemes for differential equations. However, it can also be shown to belong to the class of weighted residual techniques (Zienkiewicz and Morgan, 1983). In the least-squares finite element method the sum of the squares of the residuals, generated via the substitution of the unknown functions by finite element approximations, is formed and subsequently minimized to obtain the working equations of the scheme. The procedure can be illustrated by the following example, consider... 

In the finite element solution of the energy equation it is sometimes necessary to impose heat transfer across a section of the domain wall as a boundary condition in the process model. This type of convection (Robins) boundary condition is given as... 

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... 

Solution of the flow equations has been based on the application of the implicit 0 time-stepping/continuous penalty scheme (Chapter 4, Section 5) at a separate step from the constitutive equation. The constitutive model used in this example has been the Phan-Thien/Tanner equation for viscoelastic fluids given as Equation (1.27) in Chapter 1. Details of the finite element solution of this equation are published elsewhere and not repeated here (Hou and Nassehi, 2001). The predicted normal stress profiles along the line AB (see Figure 5.12) at five successive time steps are. shown in Figure 5.13. The predicted pattern is expected to be repeated throughout the entire domain. 

The values of the exact solution at the same finite element nodes are = 0.2658, c<) = 0.3271, C3 = 0.5392, and C4 = 1, indicating that the three-element finite element solution is accurate within 3 percent. When the exact solution is not known, the problem must he solved several times, each with a different niimher of elements, so that convergence is seen as the nuniher of elements increases. 

Strictly speaking, finite difference or finite element solutions to differential equations are simply multiplying the number of comparments many times, but the mathematical rules for linking cells in difference calculations are rigorously set by the form of the equations. 

Holst, M.J. Baker, N.A. Wang, F., Adaptive multilevel finite element solution of the Poisson-Boltzmann equation I algorithms and examples, J. Comp. Chem. 2000, 21, 1319-1342... 

M. Holst, N. Baker, and F. Wang, Adaptive multilevel finite element solution of the... 

The polycarbonate glazing is modeled as a simply supported plate subjected to nonlinear center deflections up to 15 times the pane thickness. Using the finite element solution of Moore (Reference 4), the resistance function is generated for each pane under consideration. Typically, the resistance is concave up, as illustrated for typical pane sizes in Figure 1. This occurs because membrane stresses induced by the stretching of the neutral axis of the pane become more pronounced as the ratio of the center pane deflection to the pane... 

B.M.A. Rahman and J.B. Davies, Finite-element solution of integrated optical waveguides, / Lightwave Technol. 2, 682-688 (1984). 

The Neumann (natural) boundary condition qx = 0 is automatically satisfied. The above system of algebraic equations can easily be solved to give T) = 200, T2 = 275 and I s = 300. A comparison between the analytical finite element solutions is shown in Fig. 9.7. As can be seen, the agreement is excellent. 

The first step when formulating the finite element solution to the above equations, is to discretize the domain of interest into triangular elements, as schematically depicted in Fig. 9.13. In the constant strain triangles, represented in Fig. 9.14, the field variable within the element is approximated by,... 

M. L. Hami and J. F. T. Pittman, Finite Element Solutions for Flow in a Single-Screw Extruder, Including Curvature Effects , Polym. Eng. Sci., 20, 339 (1980). 

E. Finite Element Solutions of Free Surface Flows 91... 

W.R. Bowen and A.O. Sharif, Adaptive finite element solution of the non-linear Poisson-Boltzmann equation—a charged spherical particle at various distances from a charged cylindrical pore in a charged planar surface, J. Colloid Interface Sci. 187 (1997)... 

Rodi, W. L., 1976, A technique for improving the accuracy of finite element solutions for magnetotelluric data Geophys. J. R. Astr. Soc., 44, 483-506. 

Finite element solution of boundary-value problems... 

N. Baker, D. Sept, M. Holst, and J. A. McCammon, The adaptive multilevel finite element solution of the Poisson-Boltzmann equation on massively parallel computers, IBM J. Res. Dev. in press (2001). 

The finite element solution is constructed by dividing the space coordinate into Ne successive intervals, with Ne-hi nodes having for coordinates S, such that... 

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