Finite Element Analysis. Barna Szabó
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Название: Finite Element Analysis

Автор: Barna Szabó

Издательство: John Wiley & Sons Limited

Жанр: Физика

Серия:

isbn: 9781119426462

isbn:

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      We seek an approximation to u in the form:

      On computing the elements of

,
and
we get

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

      The choice of basis functions

      By definition, a set of functions

,
are linearly independent if

      implies that

for
. It is left to the reader to show that if the basis functions are linearly independent then matrix
is invertible.

      Given a set of linearly independent functions

,
, the set of functions that can be written as

      is called the span and

are basis functions of S.

      We could have defined other polynomial basis functions, for example;

      (1.15)

      When one set of basis functions

can be written in terms of another set
in the form:

      (1.16)

      where

is an invertible matrix of constant coefficients then both sets of basis functions are said to have the same span. The following exercise demonstrates that the approximate solution depends on the span, not on the choice of basis functions.

,
and show that the resulting approximate solution is identical to the approximate solution obtained in Example 1.1. The span of the basis functions in this exercise and in Example 1.1 is the same: It is the set of polynomials of degree less than or equal to 3 that vanish in the points
and
.

      Summary of the main points

      1 The definition of the integral by eq. (1.8) made it possible to find an approximation to the exact solution u of eq. (1.5) without knowing u.

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