线 和 面 的积分。。。

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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:

  1. 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 $t_0$ and$t_1$ are values of the parameter$t$ at the endpoints of the arc.

  2. 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 $(s_0,s_1)$ and$(t_0,t_1)$ are the ranges of integration for$s$ and$t$.

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 $(x_i,y_i)$ 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 $E$ in theglobal system onto node 3 in the parent element, and node $F$ mapsonto node 4. The line $EF$ 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 $x$ and$y$ 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.
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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


  • Basic Concepts of the Finite Element Method
    • Introduction
    • One Dimensional Problems
      • A Simple Example
      • The Elastic Rod
      • Bending of a Beam
      • Heat Conduction
    • Two Dimensional Problems
      • A Simple Example
      • Plane Elasticity
      • Inviscid Potential Flow of a Fluid
      • Transient Fluid Flow
    • Minimum Energy Principles
      • Beam Element
      • Plane Elasticity
      • Potential Flow of a Fluid
    • Conclusions
  • Programming Techniques
    • Introduction
    • Contents of the Finite Element Library
      • The Level 0 Library - Basic Routines
      • The Level 1 Library - Example Programs
    • Overall Structure of a Level 1 Program
    • Initialisation and Data Input
      • Programming Conventions
      • Nodal Coordinate Array -COORD
      • The Element Topology Array -ELTOP
      • Nodal Freedom Array -NF
    • Element Matrix Contruction
      • Shape Functions and Local Coordinate Systems
      • Isoparametric Elements
      • Numerical Integration
      • Plane Strain of Elastic Solid
      • Axisymmetric Problems
      • Steady Potential Fluid Flow
      • Mass Matrix Formation
      • Higher Order Elements
      • Three Dimensional Elements
    • Assembly of System Matrices
      • An Example of the Assembly Process
      • Programming the Element Matrix Assembly
      • Efficient Mesh and Freedom Numbering
    • Solution Techniques
  • Time-Dependent Problems
    • Introduction
    • First Order Equations
      • Eigenvalues for First Order Problems
      • Direct Integration of First Order Problems
    • Second Order Equations
      • Eigenvalues for Second Order Problems
      • Modal Superposition
      • Newmark's Method of Direct Integration
    • The Generalised Eigenvaule Problem
      • Transformation to Standard Form
      • Consistent and Lumped Mass Matrix Approximations
      • Programming the Lumped Mass Approximation
      • Programming the Consistent Mass Approximation
      • The Standard Eigenvalue Problem
  • The Numerical Inclusion of Boundary Conditions
    • Introduction
    • Dirichlet Conditions
      • Direct Elimination
      • Payne-Irons Method
    • Calculation of Direction Cosines
    • Neuman and Cauchy Conditions
      • Performing Line and Surface Integrals
      • Programming Boundary Integrals
    • Lagrange Multipliers
  • Bibliography
  • About this document ...

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