Finite Difference Methods and Interpolation

X he most commonly encountered mathematical models in engineering and science are in the form of differential equations. The dynamics of physical systems that have one independent variable can be modeled by ordinary differential equations, whereas systems with two, or more, independent variables require the use of partial differential equations. Several types of ordinary differential equations, and a few partial differential equations, render themselves to analytical (closed-form) solutions. These methods have been developed thoroughly in differential calculus. However, the great majority of differential equations, especially the nonlinear ones and those that involve large sets of 

Several numerical methods for differentiation, integration, and the solution of ordinary and partial differential equations are discussed in Chaps. 4-6 of this book. These methods are based on the concept of finite differences. Therefore, the purpose of this chapter is to develop the systematic terminology used in the calculus of finite differences and to derive the relationships between finite differences and differential operators, which are needed in the numerical solution of ordinary and partial differential equations. 

The calculus of finite differences may be characterized as a two-way street that enables the user to take a differential equation and integrate it numerically by calculating the values of the function at a discrete (finite) number of points. Or, conversely, if a set of finite values is available, such as experimental data, these may be differentiated, or integrated, using the calculus of finite differences. It should be pointed out, however, thatnumerical differentiation is inherently less accurate than numerical integration. 

Another very useful application of the calculus of finite differences is in the derivation of interpolation/extrapolation formulas, the so-called interpolating polynomials, which can be used to represent experimental data when the actual functionality of these data is not known. A very common example of the application of interpolation is in the extraction of physical properties of water from the steam tables. Interpolating polynomials are also used to estimate numerical derivative and integral of the tabulated data (see Chap. 4). The discussion of several interpolating polynomials is given in Secs. 3.7-3.10. 

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Finite difference methods, and interpolation of equally and unequally spaced points... 

Two different formulas for quadratic interpolation can be compared Equation (5.8), the finite difference method, and Equation (5.12). 

The Galerkin finite element method results when the Galerkin method is combined with a finite element trial function. The domain is divided into elements separated by nodes, as in the finite difference method. The solution is approximated by a linear (or sometimes quadratic) function of position within the element. These approximations are substituted into Eq. (3-80) to provide the Galerkin finite element equations. For example, with the grid shown in Fig. 3-48, a linear interpolation would be used between points x, and, vI+1. 

A finite element method based on these functions would have an error proportional to Ax2. The finite element representations for the first derivative and second derivative are the same as in the finite difference method, but this is not true for other functions or derivatives. With quadratic finite elements, take the region from x,.i and x,tl as one element. Then the interpolation would be... 

Other analysis methods that use discretization include the finite difference method, the boundary element method, and the finite volume method. However, FEA is by far the most commonly used method in structural mechanics. The finite difference method approximates differential equations using difference equations. The method works well for two-dimensional problems but becomes cumbersome for regions with irregular boundaries (Segerlind 1984). Another difference between the finite element and finite difference methods is that in the finite difference method, the field variable is only computed at specific points while in the finite element method, the variation of a field variable within a finite element is available from the assumed interpolation function (Hutton 2003). Thus, the derivatives of a field variable can be directly determined in finite element method as opposed to the finite difference method where only data concerning the field variable is available. The boundary element method is also not general in terms of structural shapes (MacNeal 1994). 

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