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Sample text

Xi2 xi3 . a0 a1 ... f(xi ) = xi f(xi ) ... If the preceding solution is reduced to a linear approximation (n = 1), the matrix will be (n = 1) n xi xi xi2 a0 = a1 f(x)i xi f(xi ) and for a parabola (n = 2), the first three rows and columns. Another possible form is the exponential function F(t) = aebt and although partial differentiation will produce two equations in two unknowns, they will be nonlinear and cannot be written in matrix form. However, a change in variable form may produce a model that is linear, for example, for the preceding equation ln(F) = ln(a) + bt and if X is defined to be t, Y to be ln (F), a0 = ln (a), and a1 = b, the equation becomes Y(x) = a0 + a1 X and linear least squares analysis may be applied.

N! and f(x) is analytic near x = a. The preceding power series is called the Taylor series expansion of f(x) near x = a. If for some value of x as [x − a] is increased, the series is no longer convergent, then that value of x is outside the radius of convergence of the series. The error due to truncation of the series is partially due to [x − a] and partially due to the number of terms (n) to which the series is taken. The quantities [x − a] and n can be controlled, and the truncation error is said to be of the order of (x − a)n+1 or O(x − a)n+1 .

With the finite element approach, the continuum is divided into a number of “finite elements” that are assumed to be joined by a discrete number of points along their boundaries. A function is chosen to represent the variation of the quantity over each element in terms of the value of the quantity at the boundary points. A set of simultaneous equations can be obtained that will produce a large, banded matrix. 1-31 The three primary advantages of the finite element approach over finite difference methods are [1] 1.

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