Читать книгу Finite Element Analysis - Barna Szabó - Страница 15

We introduce the finite element method through approximating the exact solution of the following second order ordinary differential equation

(1.5)

with the boundary conditions

(1.6)

where the prime indicates differentiation with respect to x. It is assumed that where α and β are real numbers, on , and are defined such that the indicated operations are meaningful on I. For example, the indicated operations would not be meaningful if , c or f would not be finite in one or more points on the interval . The function f is called a forcing function.

We seek an approximation to u in the form:

(1.7)

where are fixed functions, called basis functions, and aj are the coefficients of the basis functions to be determined. Note that the basis functions satisfy the zero boundary conditions.

Let us find aj such that the integral defined by

(1.8)

is minimum. While there are other plausible criteria for selecting aj, we will see that this criterion is fundamentally important in the finite element method. Differentiating with respect to ai and letting the derivative equal to zero, we have:

(1.9)

Using the product rule: we write

(1.10)

The underbraced terms vanish on account of the boundary conditions, see eq. (1.7). On substituting this expression into eq. (1.9), we get

which will be written as

(1.11)

We define

(1.12)

and write eq. (1.11) in the following form

(1.13)

which represents n simultaneous equations in n unknowns. It is usually written in matrix form:

(1.14)

On solving these equations we find an approximation un to the exact solution u in the sense that un minimizes the integral .

Example 1.1 Let , , and

With these data the exact solution of eq. (1.5) is

Figure 1.1 Exact and approximate solutions for the problem in Example 1.1.

We seek an approximation to u in the form:

On computing the elements of , and we get

The solution of this problem is , . These coefficients, together with the basis functions, define the approximate solution un. The exact and approximate solutions are shown in Fig. 1.1.