Numerical Solutions Finite Element Method

Despite the existence of powerful analytical tools that allow for explicit solution of certain problems of interest, in general, the modeler cannot count on the existence of analytic solutions to most questions. To remedy this problem, one must resort to numerical approaches, or further simplify the problem so as to refine it to the point that analytic progress is possible. In this section, we discuss one of the key numerical engines used in the continuum analysis of boundary value problems, namely, the finite element method. The finite element method replaces the search for unknown fields (i.e. the solutions to the governing equations) with the search for a discrete representation of those fields at a set of points known as nodes, with the values of the field quantities between the nodes determined via interpolation. From the standpoint of the principle of minimum potential energy introduced earlier, the finite element method effects the replacement 

From the perspective of variational principles, the idea embodied in the finite element approach is similar to that in other schemes identify that particular linear combination of basis functions that is best in a sense to be defined below. The approximate solution of interest is built up as linear combinations of basis functions, and the crucial question becomes how to formulate the original boundary value problem in terms of the relevant expansion coefficients. 

Recall that within the continuum mechanics setting set forth in section 2.3.2, the equations of equilibrium may be written as 

Our present task is to rewrite these equations in integral (or weak) form which will simplify the formulation in terms of finite elements. We introduce a series of test functions Wi (x) that satisfy the relevant displacement boundary conditions which are hence denoted as admissible fields and now contract these test functions with the equation given above and integrate over the spatial region of interest. This strategy results in 

If we now use integration by parts on the first term in this equation, we are left with 

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

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. 

Although only approximate analytical solutions to this partial differential equation have been available for x(s,D,r,t), accurate numerical solutions are now possible using finite element methods first introduced by Claverie and coworkers [46] and recently generalized to permit greater efficiency and stabihty [42,43] the algorithm SEDFIT [47] employs this procedure for obtaining the sedimentation coefficient distribution. 

Finite element methods are one of several approximate numerical techniques available for the solution of engineering boundary value problems. Analysis of materials processing operations lead to equations of this type, and finite element methods have a number of advantages in modeling such processes. This document is intended as an overview of this technique, to include examples relevant to polymer processing technology. 

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. One simply combines the methods for ordinary differential equations (see Ordinary Differential Equations—Boundary Value Problems ) with the methods for initial-value problems (see Numerical Solution of Ordinary Differential Equations as Initial Value Problems ). Fast Fourier transforms can also be used on regular grids (see Fast Fourier Transform ). 

The modeling of the internal pore diffusion of a wax-filled cylindrical single catalyst pore was accomplished by the software Comsol Multiphysics (from Comsol AB, Stockholm, Sweden) as well as by Presto Kinetics (from CiT, Rastede, Germany). Both are numerical differential equation solvers and are based on a three-dimensional finite element method. Presto Kinetics displays the results in the form of diagrams. Comsol Multiphysics, instead, provides a three-dimensional solution of the problem. 

The nonlinearity of the system of partial differential equations (51) and (52) poses a serious obstacle to finding an analytical solution. A reported analytical solution for the nonlinear problem of diffusion coupled with complexation kinetics was erroneous [12]. Thus, techniques such as the finite element method [53-55] or appropriate change of variables (applicable in some cases of planar diffusion) [56] should be used to find the numerical solution. One particular case of the nonlinear problem where an analytical solution can be given is the steady-state for fully labile complexes (see Section 3.3). However, there is a reasonable assumption for many relevant cases (e.g. for trace elements such as... 

The equations are normally solved with a control volume finite element method. With these methods no boundary conditions are necessary at the outlets (i.e., it is implicitly assumed that there will always be an outlet at the point where the flow front merges and fills the last part of the mold). For a further discussion of boundary conditions and details about the numerical solution of the field equations (Eqs. 12.5 and 12.6) see [15,21,22-24]. 

The first group of methods then manipulate a very small subset of vector elements Vi at a time, and a direct method continually updates the affected elements rj. Such methods are collectively known as relaxation methods, and they are primarily used in situations where, for each elements >< to be changed, the set of affected rjt and the matrix elements Ay, are immediately known. This applies in particular to difference approximations and also to the so-called Finite Element Method for obtaining tabular descriptions of the wave function, i.e. a list of values of the wave function at a set of electron positions. (Far some reason, such a description is commonly referred to as a numerical solution to the Schrodinger equation). Relaxation methods have also been applied to the Cl problem in the past, but due to their slow convergence they have been replaced by analytical methods. Even for numerical problems, the relaxation methods are slowly yielding to analytical methods. 

All numerical techniques require application of sampling theory. Briefly stated, one chooses a representative sample of points within the region of interest and at each point attempts to calculate iteratively the most accurate solution possible, guided by self-consistency of local solutions with each other and with the specified boundary conditions. We describe two seemingly contrasting techniques finite-difference and finite-element methods (1,2). 

There are different approaches for carrying out the numerical solution of the differential equations involved in electrochemical problems, with the most popular being the Finite Difference Method (FDM) and the Finite Element Method (FEM) [1-3]. This appendix will be focused on the first one. 

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... 

Example 5.3 The Semi-infinite Solid with Variable Thermophysical Properties and a Step Change in Surface Temperature Approximate Analytical Solution We have stated before that the thermophysical properties (k, p, Cp) of polymers are generally temperature dependent. Hence, the governing differential equation (Eq. 5.3-1) is nonlinear. Unfortunately, few analytical solutions for nonlinear heat conduction exist (5) therefore, numerical solutions (finite difference and finite element) are frequently applied. There are, however, a number of useful approximate analytical methods available, including the integral method reported by Goodman (6). We present the results of Goodman s approximate treatment for the problem posed in Example 5.2, for comparison purposes. 

Numerical micromagnetics, which may be based either on the finite difference or finite element method, resolve the local arrangement of the magnetization which arises from the interaction between intrinsic magnetic properties such as the magnetocrystalline anisotropy and the physical and chemical microstructure of the material. The numerical solution of the equation of motion also provides information on how the magnetization evolves in time. The time and space resolution of numerical micromagnetic simulations is in the order of nanometers and nanoseconds, respectively. 

This chapter has mainly been devoted to the solution of the boundary layer form of the governing equations. While these boundary layer equations do adequately describe a number of problems of great practical importance, there are many other problems that can only be adequately modeled by using the full governing equations. In such cases, it is necessary to obtain the solution numerically and also almost always necessary to use a more advanced type of turbulence model [6],[12],[28],[29]. Such numerical solutions are most frequently obtained using the commercially available software based on the finite volume or the finite element method. 

To understand the behavior of the movement of the contaminant in ground-water, people solve Eq. (1) forward in time. In solving this equation forward in time, one assumes that the plume is originated from somewhere and will travel through the porous media due to advection and dispersion. The conventional procedure to solve Eq. (1) is to use finite difference or finite element methods. For simple cases, closed-form solutions exist. Quantitative descriptions of the processes forward in time are well understood. Multidimensional models of these processes have been used successfully in practice [50]. Numerical solute transport models were first developed about 25 years ago. When properly applied, these models can provide useful information about transport processes and can assist in the design of remedial programs. 

Finite element methods — The finite element method is a powerful and flexible numerical technique for the approximate solution of (both ordinary and partial) differential equations involving replacing the continuous problem with unknown solution by a system of algebraic equations. The method was first introduced by Richard Courant in 1943 [i], and over the next three decades, and particularly in the 1960s, a comprehensive mathematical framework was developed to underpin the method. 

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