Finite elements method

The finite element has proved to be ideal when simulating the mold filling, fiber orientation, shrinkage, and warpage of thin plastic parts. With more difficulty, it has 

The finite element method proceeds in a slightly different way. The unknown solution y(x) is expanded in a series of functions, called trial functions  

The functions in the finite element method are constant, linear, or quadratic functions of position in a small region (called a finite element). The idea is explained for interpolation of a known function, Eq. (F.31)  

Indeed, with 16 elements, the curve - while composed of straight-line segments -looks like a smooth curve. Whatever the trial function, the approximation is better when more elements are used, and mesh rehnement is the easiest way to ensure that the approximation is accurate enough. 

It is beyond the scope of this book to describe the method used to obtain the coefficients in Eq. (F.30), and how the boundary conditions are included, but complete details are available (Finlayson, 1972, 1980). There is a variety of books available about the finite element method. A book focusing on flow and convection/diffusion is by Gresho and Sani (1998). The representation of the second derivative is the same as given by the finite difference method, but the representation of the function is different. The finite 

If quadratic functions are used, there are more differences in fact, the only time the representations are the same is for first and second derivatives that are second-order and when using linear trial functions in the finite element method. 

Various formulations of hnite element methods have been proposed. For an exhaustive account on hnite element methods, the reader is referred to Chen (2005), Donea (2003), Reddy (2005), etc. We present here one of the popular formulations known as the weak formulation of the governing differential equation that, instead of requiring the solution to be twice continuously differentiable, requires that the derivative of the solution be square integrable. We illustrate the weak formulation of the boundary-value problem. Equation (2.157). 

To derive the weak formulation, multiplying Equation (2.157) by a test function, (()(x) (any member of a family of suitably smooth functions), and integrating by parts over an arbitrary element (Xi,X2) we get 

Selecting ( ) = / and substituting the approximate solution into the weak formulation, one obtains the local Galerkin finite element equation, 

The above equation may be written in matrix form with the following matrix elements  

The main advantage of using a Finite Element Method to verify slope stabihty failure is that a potential failure mechanism does not have to be predefined. The critical failure surface automatically results from the FEM calculations, provided that soil stratification and soil characteristics are correctly modelled. A disadvantage of the software is that sometimes non-relevant local failure mechanisms are found which break off the calculation process. The designer has to avoid these local effects by adjusting the soil characteristics in some of the elements of the FE model. 

The Finite Element Method is essentially a method in which the stress-strain behaviour of soil is respected and, in the first instance, deformations are calculated. The Factor of Safety is normally defined by performing a c-reduction the shear strength of the soil defined by tan (p) and c is gradually decreased until failure occurs in the model. Failure is reached when the deformations become very large under a very small further reduction of the shear strength. 

Contrary to Eimit Equilibrium models, the effect of load spreading is also taken into account. 

In general 2D models are used although 3D models exist as well. In almost all cases found in literature the stability factor for a 3D model exceeds that of the 2D geometry (although the difference also depends on specific conditions and the geometry). 

The general description of these methods is provided in the present section. More detailed information on the finite differences and boundary elements will be provided in later sections of the article. It will become clear that in electrochemistry applications (of interest to us), the BEM could be the most efficient numerical technique. 

The domain under consideration (the electrolyte) is covered with a grid. For the sake of simplicity, we use a two-dimensional Cartesian grid. The discrete version of the Laplace equation at each grid point is represented by (64)  

The subscripts (i,j) refer to the location xi,yj) where discretization takes place. Furthermore (pij is the approximation to 

In the FEM, the variational formulation (or weak formulation) of the boimdary value problem is employed. As a concrete example, suppose the strong form of the boundary value problem is described by (65)-(67)  

It can be shown (although it is neither trivial, nor intuitive) that the solution to the above boundary value problem, among all functions satisfying (66), also minimizes the functional J((p) described by (68)  

A round man cannot be expected to fit in a square hole right away. He must have time to modify his shape. 

The finite element method (FEM) was first developed in 1956 to numerically analyze stress problems [16] for the design of aircraft structures. Since then it has been modified to solve more general problems in solid mechanics, fluid flow, heat transfer, among others. In fact, due to its versatility, the method is being used to study coupled problems for applications with complex geometries where the solutions are highly non-linear. 

Instead of starting with a rigorous and mathematical development of the finite element technique, we proceed to present the finite element method through a solution of onedimensional applications. To illustrate the technique, we will first find a numerical solution to a heat conduction problem with a volumetric heat source 

Our first step in the development of the technique is to discretize the domain by creating a mesh. Here, we will divide the domain into elements as illustrated in Fig. 9.1. The figure presents nodes and elements (element numbers are shown inside circles). 

Since we chose a discretization with two-noded elements, we are assuming that within each element the unknown temperature function T (x) is a simple linear polynomial that can be written as 

For a Lorentz factor y = 1.5, the probability for pair creation turns out to be rather small. However, in this energy region we can study excitation, ionization and charge transfer into the ground state of the projectile. For collisions of 

In the case of the collision system Au79+(y = 2) -f- U9,+ calculations have been carried out with high precision. The time-step At is set to 25.4 fm/c. In order to achieve an error in the norm of the wave function less than 10 12, the time-evolution 

Some aspects of the so-called fermion doubling and how to avoid them in standard finite-element or finite-difference methods have been addressed recently by Busic et al. (1999b). 

The examples presented above demonstrate that the finite-element method can be applied successfully to the investigation of processes such as excitation, ioniza- 

Maxwell s equations are discussed in Section 9.2.2. Any problem in regard to the distribution of current or electric field in a homogeneous or composite material can be solved with these equations if the excitation and electrical properties of the materials are known. There are in general two different types of problems to be solved in differential equations initial value problems and boundary value problems. Initial value problems arises when the values of the unknowns are given at a particular point (e.g., at a given time), and die values at future times are to be computed. Boundary value problems arise when the values on the boundary of a material are known, and the values of the interior are to be computed. The latter is a common situation in bioimpedance research where, for example, the current distribution in tissue is to be computed from a given excitation from surface electrodes. 

Because the differential equations that describe tbe behavior of our system, in this case the Maxwell equations, basically describe an infinite-dimensional object, we must use a finitedimensional approximation to represent the solution. There are two main forms of such approximation, differences aoA finite elements. In the finite difference method, the 

Earlier in Chapter 6, we treated some simple electrode/tissue geometries with mathematical analjrtical solutions. Of course, tissue morphology and composition is so that analytical solutions most often cannot be found. It is therefore necessary to take a more engineering approach to make a realistic geometrical model of the tissue and electrode system, inamittance distribution included. Then let a computer calculate current density vectors and equipotential lines on the basis of a chosen mesh. 

In principle, the computer programs may calculate complex 3D models with frequency dependent tissue parameters. However, the computing time may be long, and accordingly time-consuming to experiment with the model. It is important to define the problem in the simplest possible way, and in a way adapted to the software used. If the 3D problem can be reduced by symmetry to two dimensions, and if a DC/purely resistive problem definition can be used, results are more easily obtained. 

Often uses less memory than UMFPACK 

The shape (or interpolation) fimctions mentioned in the previous algorithm are mostly considered to be polynomial type due to their ease of integration and differentiation. Another advantage of the polynomial shape functions is the possibility of obtaining a more accurate approximation of the exact solution simply by increasing the order of the 

Fundamentals and theory of the FEM are covered extensively in the literature and will not he covered in this chapter. However, the reader is directed to Refs. [8] and [13] for a concise explanation of FEJVl theory and to Ref. [ 14] for in-depth description and formulation of shape fimctions, elements, and applications of FEJVl to various heat transfer problems. 

COJVISOL JVlultiphysics is one of the commercial computing software that uses FEJVl technique to solve continuum equations. The solvers used by COJVISOL JVlultiphysics for the solution of the matrices assembled in the FEJVl break down the linear (or nonlinear) problems into one or several linear systems of equations by approximating the pertinent problem with a linearized one [15]. The so-called linear system solvers available in COJVISOL JVlulti-physics and their key features are presented in Table 11.1. 

The direct solvers listed in Table 11.1 solve the assembled set of equations by the Gaussian elimination method [1, 8] and deliver solutions often faster than the iterative solvers for ID and 2D problems. Direct solvers can be used for 3D models if the degrees of freedom is less than 10 . Iterative solvers, on the other hand, are used in models with degrees of freedom above 10 and in the solution of 3D problems, for which the memory requirements of the direcrt solvers are excessive. The readers are directed to Ref. [15] for further details of the solvers listed in Table 11.1. 

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The AUGUR information on defect configuration is used to develop the three-dimensional solid model of damaged pipeline weldment by the use of geometry editor. The editor options provide by easy way creation and changing of the solid model. This model is used for fracture analysis by finite element method with appropriate cross-section stress distribution and external loads. 

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. 

Weighted Residual Finite Element Methods - an Outline... 

WEIGHTED RESIDUAL FINITE ELEMENT METHODS - AN OUTLINE... 

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... 

WEIGHTED RESIDUAI. FINITE ELEMENT METHODS - AN OEITLINE Therefore for e,. L = 1... 

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. 

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... 

Brenner, S. C. and Scott, L. R., 1994. The Mathematical Theory of Finite Element Methods, Springer-Verlag, New York. 

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

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. 

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. 

Piroimeau, O., 1989. Finite element methods for fiuids. Wiley, Chichester. 

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. 

Weighted residual finite element methods described in Chapter 2 provide effective solution schemes for incompressible flow problems. The main characteristics of these schemes and their application to polymer flow models are described in the present chapter. 

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. 

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. 

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

Donea, J., 1992. Arbitrary Lagrangian-Eulerian finite element methods. In Belytschko, T. and Hughes, T. J. R. (eds), Computational Methods for Transient Analysis, Elsevier Science, Amsterdam. 

Papaiiastasiou, T. C., Scriven, L. E. and Macoski, C. W., 1987. A finite element method for liquid with memory. J. Non-Newtonian Fluid Mech 22, 271-288. 

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