Computational Physics

446 The finite element method for partial differential equations

y

a x

c

a'

c'

b'

b

1

1

Figure 13.9. Archetypical triangle with two angles of 45◦and sides 1, oriented

along thex- andy-axes. Another triangle is shown, which can be obtained from

the archetypical one through a linear transformation.

wheref(x,y)=x, say, and wherex′,y′are the images ofx,yunder the

transformationU. It now is straightforward to verify that the expressions for the

natural coordinates (13.11) are correct.

References

[1] O. C. Zienkiewicz and R. L. Taylor,The Finite Element Method: Its Basis and Fundamentals,
6th edn. Oxford/Burlington (MA), Elsevier Butterworth-Heinemann, 2005.
[2] O. C. Zienkiewicz and R. L. Taylor,The Finite Element Method for Solid and Structural
Mechanics, 6th edn. Oxford/Burlington (MA), Elsevier Butterworth-Heinemann, 2005.
[3] O. C. Zienkiewicz and R. L. Taylor,The Finite Element Method for Fluid Dynamics, 6th edn.
Oxford/Burlington (MA), Elsevier Butterworth-Heinemann, 2005.
[4] K. J. Bathe,Finite Element Procedures. Upper Saddle River, NJ, Prentice Hall, 1996.
[5] J. Mackerle,A Primer for Finite Elements in Elastic Structures. New York, Wiley, 1999.
[6] W. Kaplan,Advanced Calculus, 4th edn. Reading (MA), Addison–Wesley, 1991.
[7] J. T. Oden and H. J. Brauchli, ‘On the calculation of consistent stress distributions in finite
element calculations,’Int. J. Numer. Meth. Eng., 3 (1971), 317–25.
[8] E. Hinton and J. S. Campbell, ‘Local and global smoothing of discontinuous finite element
functions using a least squares method,’Int. J. Numer. Meth. Eng., 8 (1974), 461–80.
[9] O. C. Zienkiewicz and J. Z. Zhu, ‘A simple error estimator and adaptive procedures for practical
engineering analysis,’Int. J. Numer. Meth. Eng., 24 (1987), 337–57.
[10] O. C. Zienkiewicz and R. L. Taylor,The Finite Element Method, vol. I. London, McGraw-Hill,
1988.
[11] J. Barlow, ‘Optimal stress locations in finite element models,’Int. J. Numer. Meth. Eng., 10
(1976), 243–51.
[12] J. Barlow, ‘More on optimal stress points, reduced integration, element distortions and error
estimation,’Int. J. Numer. Meth. Eng., 28 (1989), 1487–504.