The Finite-Element Method

The FEM has been an attractive method for analyzing heat and mass transport, in solidification processing of materials. This is because of its ability to handle problems with complex geometries and inherent nonlinear properties arising out of dependence of these properties on the field variables as well as the presence of nonlinear boundary conditions. The FEM is popular for two main reasons. First, it is able to reproduce using fewer nodes than does FDM, the shape of complex domains. Second, large commercial or public domain codes have already been developed. The FEM has been successfully applied to solidification problems [50-59]. 

Early attempts at applying finite element analysis to solidification problems focused only on heat conduction. The most important phenomena taken into account are the release of latent heat due to phase change. If this is incorporated in the governing equations as a variation in the specific heat of material, it is evident that there occurs a jump at the phase-change temperature in the specific heat curve. This is analogous to the peak of a Dirac delta function. In order that this peak is not missed in the analysis, an alternate averaging procedure on the smoother enthalpy-temperature curve was suggested [60]. 

Applying the splitting techniques to discretize the diffusion and convection terms, the momentum and energy equations are split according to physical processes, and contribution from each of these processes is calculated separately in the time integration procedure. Ramaswamy and Jue [62] reported successful use 

In the finite element solution procedure, the nodal values of quantities are interpolated to approximate the velocity, temperature and pressure in the domain thus  

To discretize the equations, the Galerkin s method is usually employed. In the Galerkin s procedure, the second-order diffusion terms in the momentum and energy equations and the pressure term are reduced to first-order terms and a surface integral, by the application of temperature dependent properties (enthalpy or specific heat). It is readily applied to coarse grids with large time step. The temperature is held constant until all the latent heat associated with the nodal volume is completely released. This ensures that no latent heat is lost, but this approach has been found to be unstable in some problems [65]. 

To illustrate the finite element method, the basic steps in the formulation of the standard Galerkin finite element method for solving a one-dimensional Poisson equation is outlined in the following. 

The governing equations with boundary conditions are called the strong form of the problem, and can be defined by  

Multiplying the residual function with a weighting function W(z) and integrate, the weak form is established  

The ai,i= 1. IV are constants and y , are linearly independent basis functions of z, which are chosen to satisfy the boundary conditions and N is the total number of basis functions used. 

In the Galerkin method, the weights Wi (z) are taken to be the basis functions (z) used in the approximate solution of (12.41). In this case, the continuous residual function (12.39) can be approximated by  

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 finite element method is similar to the finite volume method in that it describes the behavior of a continuum using a subdivision of the continuum domain into smaller subregions. In the FEM nomenclature each subregion of the domain is referred to as an element and the process of subdividing a domain into a finite number of simple elements is referred to as discretization. In ID line element can be used. In 2D the elements are usually triangles or quadrilaterals, while in 3D tetrahedrals or hexahedrals are most often used. 

Computational costs, on the other hand, may increase significantly as the number of elements in a model increases. Thus, it may be more practical to use fewer elements, and ones that have a higher order of interpolation. 

The finite-element technique is based on dividing the cell domain into polygonal sections. The potential within each of the elements is assumed to be a linear combination of the value at the vertices. However, unlike the finite-dilference method, which solves the finite-difference approximation of the Laplace equation, the finite-elements method seeks a solution for the potential distribution within the cell, which best fits the Laplace equation and the boundary conditions. The degree of accuracy is similar to that of the finite-difference method however, curved boundaries and narrow corners can be described with more precision and ease. On the other hand, the presence of electrochemical nonlinear boundary conditions leads to ill-conditioned matrix equations which are more difficult to solve than the finite-difference system. 

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For the determination of the approximated solution of this equation the finite difference method and the finite element method (FEM) can be used. FEM has advantages because of lower requirements to the diseretization. If the material properties within one element are estimated to be constant the last term of the equation becomes zero. Figure 2 shows the principle discretization for the field computation. 

Mesh refinement - h- and p-versions of the finite element method... 

The family of hierarchical elements are specifically designed to minimize the computational cost of repeated computations in the p-version of the finite element method (Zienkiewicz and Taylor, 1994). Successive approximations based on hierarchical elements utilize the derivations of a lower step to generate the solution for a higher-order approximation. This can significantly reduce the... 

Note that in equation system (2.64) the coefficients matrix is symmetric, sparse (i.e. a significant number of its members are zero) and banded. The symmetry of the coefficients matrix in the global finite element equations is not guaranteed for all applications (in particular, in most fluid flow problems this matrix will not be symmetric). However, the finite element method always yields sparse and banded sets of equations. This property should be utilized to minimize computing costs in complex problems. 

Ciarlet, P.G., 1978, The Finite Element Method for Elliptic Problems, North-Holland, Amsterdam. 

Johnson, C., 1987. Numerical Solution of Partial Deferential Equations by the Finite Element Method, Cambridge University Press, Cambridge. 

Mitchell, A.R. and Wait, R., 1977. The Finite Element Method in Partial Differential Ecjualions, Wiley, London. 

Reddy, J. N., 1993. An Introduction to the Finite Element Method, 2nd edn, McGraw-HHl, New York. 

Strang, G. and Fix, G. J., 1973. An Analysis of the Finite Element Method, Prentice Hall, Englewood Cliffs, NJ. 

Zienkiewicz, O. C. and Taylor, R.L., 1994. The Finite Element Method, 4th edn, Vols 1 and 2, McGraw-Hill, London. 

Further details of the BB, sometimes referred to as Ladyzhenskaya-Babuska-Brezi (LBB) condition and its importance in the numerical solution of incompressible flow equations can be found in textbooks dealing with the theoretical aspects of the finite element method (e.g. see Reddy, 1986), In practice, the instability (or checker-boarding) of pressure in the U-V-P method can be avoided using a variety of strategies. 

Christie, I. et al., 1981. Product approximation for non-linear problems in the finite element method. IMA J. Numer. Anal 1, 253-266. 

Ghassemieli, E. and Nassehi, V., 2001a. Stiffness analysis of polymeric composites using the finite element method. Adv. Poly. Tech. 20, 42-57. 

Hughes, T.. 1. R, 1987. The Finite Element Method, Prenticc-Hall, Englewood Cliffs N.I,... 

The most successful models are based on the finite element method. The flow is discretized into small subregions (elements) and mass and force balances are appHed in each. The result is a large system of equations, the solution of which usually gives the speed of the coating Hquid in each element, pressure, and the location of the unknown free surfaces. The smaller the elements, the more the equations which are often in the range of 10,000 to upward of 100,000. 

Mitchell, A. R., and R. Wait. The Finite Element Method in Fartial Differential Equations, Wiley, New York (1977). 

Reddy, J. N., and D. K. Gartling. The Finite Element Method in Heat Transfer and Fluid Dynamics. CRC Press (1994). 

Galerldn Finite Element Method In the finite element method, the domain is divided into elements and an expansion is made for the solution on each finite element. In the Galerldn finite element method an additional idea is introduced the Galerldn method is used to solve the equation. The Galerldn method is explained before the finite element basis set is introduced, using the equations for reaction and diffusion in a porous catalyst pellet. 

Other methods can be used in space, such as the finite element method, the orthogonal collocation method, or the method of orthogonal collocation on finite elements (see Ref. 106). Spectral methods employ Chebyshev polynomials and the Fast Fourier Transform and are quite useful for nyperbohc or parabohc problems on rec tangular domains (Ref. 125). 

Elliptic Equations Elhptic equations can be solved with both finite difference and finite element methods. One-dimensional elhptic problems are two-point boundary value problems. Two- and three-dimensional elliptic problems are often solved with iterative methods when the finite difference method is used and direct methods when the finite element method is used. So there are two aspects to consider howthe equations are discretized to form sets of algebraic equations and howthe algebraic equations are then solved. 

If the boundaiy is parallel to a coordinate axis any derivative is evaluated as in the section on boundary value problems, using either a onesided, centered difference or a false boundary. If the boundary is more irregular and not parallel to a coordinate line then more complicated expressions are needed and the finite element method may be the better method. 

One nice feature of the finite element method is the use of natural boundaiy conditions. It may be possible to solve the problem on a domain that is shorter than needed to reach some limiting condition (such as at an outflow boundaiy). The externally applied flux is still apphed at the shorter domain, and the solution inside the truncated domain is still vahd. Examples are given in Refs. 67 and 107. The effect of this is to allow solutions in domains that are smaller, thus saving computation time and permitting the solution in semi-infinite domains. 

A square matrix is one in which the number of columns is equal to the number of rows. An important type of square matrix which arises quite often in the finite element method is a symmetric matrix. Such matrices possess the property that aij = aji- An example of such a matrix is given below ... 

This information is supported by stress-strain behavior data collected in actual materials evaluations. With computers the finite element method (FEA) has greatly enhanced the capability of the structural analyst to calculate displacement, strain, and stress values in complicated plastic structures subjected to arbitrary loading conditions (Chapter 2). FEA techniques have made analyses much more precise, resulting in better and more optimum designs. 

The mechanical layout of molded parts and molds—ways of achieving optimum results with the Finite Element Method (FEM) ... 

Although many interface models have been given so far, they are too qualitative and we can hardly connect them to the mechanics and mechanism of carbon black reinforcement of rubbers. On the other hand, many kinds of theories have also been proposed to explain the phenomena, but most of them deal only with a part of the phenomena and they could not totally answer the above four questions. The author has proposed a new interface model and theory to understand the mechanics and mechanism of carbon black reinforcement of rubbers based on the finite element method (FEM) stress analysis of the filled system, in journals and a book. In the new model and theory, the importance of carbon gel (bound rubber) in carbon black reinforcement of rubbers is emphasized repeatedly. Actually, it is not too much to say that the existence of bound rubber and its changeable and deformable characters depending on the magnitude of extension are the essence of carbon black reinforcement of rubbers. 

Zienkiewlcz, O.C. The Finite Element Method, McGraw-Hill Co., London, 1977. 

When the coordinate functions y>iix) = y x — x )/h) are chosen by an approved rule as suggested before, the Ritz and the Bubnov-Galerkin methods coincide with the finite element method. 

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