Numerical finite element method

Numerical Method Finite volume Finite volume Finite volume Finite volume Finite Element... 

To be successful in solving applied and mostly differential problems numerically, we must know how to implement our physico-chemical based differential equations models inside standard numerical ODE solvers. The numerical ODE solvers that we use in this book are integrators that work only for first-order differential equations and first-order systems of differential equations. [Other DE solvers, for which we have no need in this book, are discretization methods, finite element methods, multigrid methods etc.]... 

It is also worth mentioning that numerical solutions of the Schrodinger equation frequently enclose the atom in a spherical box of finite radius for example, discrete variable methods, finite elements methods and variational methods which employ expansions in terms of functions of finite support, such as -splines, all assume that the wave function vanishes for r > R, which is exactly the situation we deal with here. For such solutions to give an accurate description of the unconfined system it is, of course, necessary to choose R sufficiently large that there is negligible difference between the confined and unconfined atoms. 

We note that the area of modehng the self-assembly of quantum dots is very active. Various numerical methods have been developed for the solution of the evolution equation for the film surface shape that include non-local elastic effects and anisotropy, such as phase-field methods, finite element methods and others [31]. Also, numerous investigations are devoted to atomistic modeling of self-assembly of quantum dots, as well as to the combination of modeling at small and large scales. These investigations are reviewed in [32]. 

Once we have stated or derived the mathematical equations which define the physics of the system, we must figure out how to solve these equations for the particular domain we are interested in. Most numerical methods for solving boundary value problems require that the continuous domain be broken up into discrete elements, the so-called mesh or grid, which one can use to approximate the governing equation (s) using the particular numerical technique (finite element, boundary element, finite difference, or multigrid) best suited to the problem. 

There are numerous coupled atomistic-continuum modeling strategies, for example, peridynamics, a family of quasi-continuum (QC) methods, finite element adaptive techniques (FEAt), coupling-of-length scales (CLS) method, and coupled atomistic-and-discrete-dislocalion (CADD) method. The grand challenge of multiscale modeling is to connect atomic-scale processes to mesoscale and bulk continuum models. 

There are four kinds of numerical methods which are generally used finite difference method, finite element method, charge simulation method, and surface charge method. We apply the third one, charge simulation method, to our problems because it has following three advantages ... 

Experimental and numerical analysis methods have both been used to analyze the effects of post-core system. Since experimental methods may be too time consuming, expensive and require sophisticated procedure with coarse results, numerical analysis provides an effective tool. For numerical analysis, finite element method (FEM) was normally employed. Several studies investigated stress distribution in... 

Keywords Thermoelectric module. Optimization, Numerical calculation. Finite element method. 

Numerical Methods for Fluid Flow Simulation The main classes of traditional numerical methods are finite difference methods, finite element methods, finite volume methods, spectral methods and pseudo-spectral methods. When applied to the saturation equation in the pressure-saturation formulation, the important device of a moving coordinate system has sometimes been tried. 

However, integrabihty imposes a criterion for obtaining DBs analytically. DBs are obtained analytically for integrable systems, while for non-integrable systems it is obtained by various numerical methods viz. spectral collocation method, finite-difference method, finite element method, Floquet analysis, etc. As evident from many numerical experiments, DBs mobility is achieved by an appropriate perturbation [42]. From the practical application perspective, dissipative DBs are more relevant than their Hamiltonian counterparts. The latter with the character of an attractor for different initial conditions in the corresponding basin of attraction may appear whenever power balance, instead of energy conservation, governs the nonlinear lattice dynamics. The attractor character for dissipative DBs allows for the existence of quasi-periodic and even chaotic DBs [54, 55]. 

How can Equation (11.79) be solved Before computers were available only simple ihapes could be considered. For example, proteins were modelled as spheres or ellipses Tanford-Kirkwood theory) DNA as a uniformly charged cylinder and membranes as planes (Gouy-Chapman theory). With computers, numerical approaches can be used to solve the Poisson-Boltzmann equation. A variety of numerical methods can be employed, including finite element and boundary element methods, but we will restrict our discussion to the finite difference method first introduced for proteins by Warwicker and Watson [Warwicker and Watson 1982]. Several groups have implemented this method here we concentrate on the work of Honig s group, whose DelPhi program has been widely used. 

The weighted residual method provides a flexible mathematical framework for the construction of a variety of numerical solution schemes for the differential equations arising in engineering problems. In particular, as is shown in the followmg section, its application in conjunction with the finite element discretizations yields powerful solution algorithms for field problems. To outline this technique we consider a steady-state boundary value problem represented by the following mathematical model... 

The comparison between the finite element and analytical solutions for a relatively small value of a - 1 is shown in Figure 2.25. As can be seen the standard Galerkin method has yielded an accurate and stable solution for the differential Equation (2.80). The accuracy of this solution is expected to improve even further with mesh refinement. As Figmre 2.26 shows using a = 10 a stable result can still be obtained, however using the present mesh of 10 elements, for larger values of this coefficient the numerical solution produced by the standard... 

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. 

Bell, B.C. and Surana, K. S, 1994. p-version least squares finite element formulations for two-dimensional, incompressible, non-Newtonian isothermal and non-isothcmial fluid flow. hit. J. Numer. Methods Fluids 18, 127-162. 

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

Kheshgi, H. S. and Scriven, L. E., 1985. Variable penalty method for finite element analysis of incompressible flow. Int. J. Numer. Methods Fluids 5, 785-803. 

Petera, J. and Nassehi, V., 1996. Finite element modelling of free surface viscoelastic flows with particular application to rubber mixing. Int. J. Numer. Methods Fluids 23, 1117-1132. 

As mentioned in Chapter 2, the numerical solution of the systems of algebraic equations is based on the general categories of direct or iterative procedures. In the finite element modelling of polymer processing problems the most frequently used methods are the direet methods. 

As mentioned earlier, overall accuracy of finite element computations is directly detennined by the accuracy of the method employed to obtain the numerical solution of the global system of algebraic equations. In practical simulations, therefore, computational errors which are liable to affect the solution of global stiffness equations should be carefully analysed. 

Irons, B, M., 1970. A frontal solution for finite element analysis, hit. J. Numer. Methods Eng. 2, 5-32,... 

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