线和面的积分。。。_摆线求积分上限的a为什么直接去掉-优快云博客

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Performing Line and Surface Integrals

The basic problem is performing the line or surfaceintegrals over the sides or faces of elements. In manycases these can be done analytically but in the mostgeneral situation they can only be performed numerically.

Evaluating these integrals both analytically andnumerically relies on two results from differentialgeometry:

If the equation of a curve in two dimensional spaceis given in the parametric form

$\begin{displaymath}x = x(t); ~y = y(t)\end{displaymath}$ (4.28)

then the length of any arc on the curve is given by

$\begin{displaymath}L = \int_{t_o}^{t_1} \sqrt {\left[\frac{\partial x}{\parti......} \right]^2\left[\frac{\partial y}{\partial t} \right]^2}~dt\end{displaymath}$ (4.29)

where and are values of the parameter at the endpoints of the arc.
If the equation of a surface in three dimensional spaceis given in the parametric form

$\begin{displaymath}x = x(s,t); ~y = y(s,t); ~z = z(s,t)\end{displaymath}$ (4.30)

then the area of any facet on the surface is given by

$\begin{displaymath}A = \int_{s_0}^{s_1}\int_{t_0}^{t_1} \sqrt{ J^2(y,z) + J^2(z,x) + J^2(x,y)} ~dt\,ds\end{displaymath}$ (4.31)

where

$\begin{displaymath}J(x,y) = \frac{\partial x}{\partial s} \frac{\partial y}{\pa......- \frac{\partial x}{\partial t} \frac{\partial y}{\partial s}\end{displaymath}$ (4.32)

and and are the ranges of integration for and.

These results can be found in most texts on integrationor differential geometry (see, for example, Gillespie, 1959).

Performing these finite element integrals rests on parameterisingthe boundary with some convenient parameters. It should be apparentthat in the context of the finite element method theparameterisation of the element boundary can be done most simply in termsof the local coordinates of the parent elements. Thus theparameterisation is the isoparametric transformationused to transform the parent element into elements of generalorientation and shape.

To illustrate these ideas consider the mesh in Figure 4.2.Suppose that the Neumann condition (4.3) is to be imposed on theboundary line EF.

The boundary integral (4.27) becomes

$\begin{displaymath}I = \int_{F}^{E} N_i p(x,y) ~d\Gamma\end{displaymath}$

(4.33)

The isoparametric transformation from the parent element is:

$\begin{displaymath}x = \sum_{i=1} N_i(\xi,\eta) x_i; y = \sum_{i=1} N_i(\xi,\eta) y_i\end{displaymath}$

(4.34)

where

are the global coordinates of the nodes on theelement. The parent element is shown in Figure 4.3:

**Figure 4.3:**Four-noded quadrilateral element
$\begin{figure}\vspace*{60mm}\special{psfile=fig4.3.eps hoffset=150 voffset=10 vscale=60 hscale=60}\end{figure}$

Assume that this transformation maps node

in theglobal system onto node 3 in the parent element, and node

mapsonto node 4. The line

will map onto the linejoining node 3 to node 4 ( $\eta=+1$ )in the parent element. ( 4.34) now degenerates into:

$\displaystyle x$	$\textstyle =$	$\displaystyle \sum_{i=1} N_i^{*}(\nu) x_i = x(\nu)$
$\displaystyle ~$	$\textstyle ~$	$\displaystyle ~$	(4.35)
$\displaystyle y$	$\textstyle =$	$\displaystyle \sum_{i=1} N_i^{*}(\nu) y_i = y(\nu)$

where $N_i^{*}(\nu) = N_i(\xi=+1,\eta)$ . This is now in the form of ( 4.28) and the boundary integral ( 4.33) becomes

$\begin{displaymath}I = \int_{-1}^{+1} N_i^{*}(\nu) p(x,y)\sqrt{\left( \frac{dx}{d\nu} \right)^2 \left( \frac{dy}{d\nu}\right)^2}~d\nu\end{displaymath}$

(4.36)

where

and

are given by ( 4.35).

This integral is now in a formto which quadrature can easily be applied.Hence

$\begin{displaymath}I = \sum_{j} w_j N_j^{*}(\nu_j) p(x_j,y_j) h_j\end{displaymath}$

(4.37)

where

$\begin{displaymath}h_j = \sqrt{\left( \frac{dx(\nu_j)}{d\nu} \right)^2 +\left( \frac{dy(\nu_j)}{d\nu} \right)^2}\end{displaymath}$

(4.38)

Clearly the same process can be extended into three dimensionswith little difficulty.

Next: Programming Boundary Integrals Up: Neuman and Cauchy Conditions Previous: Neuman and Cauchy Conditions

Chris Greenough (c.greenough@rl.ac.uk): September 2001

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The Finite Element Library

Theory and Programming Techniques

Dr C Greenough and Dr K Robinson
Rutherford Appleton Laboratory
Computational Science & Engineering

December 2000

PREFACE - RELEASE 4

The Finite Element Library has been around for some time and hasformed an important part of many research projects using numerical methods. The usage of the Libraryhas grown greatly with the help of the Numerical Algorithms Group Ltdand Release 4 is well due.

It is pleasing to report that over the past years only one or two serious bugs have been found and there have been many useful comments passed on about the functionality of the Library routines and the subjects covered by the Level 1 programs.

Release 4 introduces six new programs to the Level 1 Library and many more Level 0 routines. An important addition is the treatment of two simple non-linear problems. It is hoped that all these additions will be found useful to users of the Library and they will provide to starting point to new areas of application.

The authors of the Library are very interested in suggestions fromusers about new areas to be covered and comments about the existing material.The authors can be contacted directly at the Rutherford Appleton Laboratory.

C Greenough
Rutherford Appleton Laboratory - October 2000

Tel: +44 (1235) 445307
Fax: +44 (1235) 446626
Email: c.greenough@rutherford.ac.uk