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Published in:
Basic Equations of Continuum Mechanics Read chapter Read first chapter

2016 | OriginalPaper | Chapter

4. Automation of Discretization Techniques

Authors : Jože Korelc, Peter Wriggers

Published in: Automation of Finite Element Methods

Publisher: Springer International Publishing

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Abstract

Finite elements will be applied to discretize weak forms of the nonlinear continuum or structural problems. Within this approach the primary variables, like displacements or rotations etc., have to be approximated within a finite element by interpolation functions. The emphasis of this chapter is to describe how the automation technique introduced in Chap. 2 can be employed to generate finite element vectors and matrices automatically as well as providing efficient source code for different finite element programs. Hence this chapter is focused on the essential details needed to apply the systems AceGen to a standard nonlinear analysis of continua using finite elements. To achieve this goal, first the necessary interpolation functions and related mappings are discussed then two-dimensional continuum elements are derived using AceGen. The derivation starts from different configurations and also as well from the functional and weak forms presented in the previous chapter.

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Footnotes
1
In an Computer algebra system like Mathematica the scalar product of two one-dimensional matrices is denoted by a \( `` \cdot \)” as it is used also in this book for the scalar product of vectors. However since we have here one-dimensional matrices of length \(n_{en}\) the scalar product is defined by the multiplication of the transposed vector \({{\mathbf N}}\) with the vector containing the coordinates.
 
2
In an analogous way the transformations between the gradients of different configurations, see (1.​23), are derived e.g. for the gradient of a single scalar shape function \(N_I\)
 
3
For classical beam and shell theories, due to the fact that the mathematical models are of higher order, different interpolation functions are needed which are e.g. \(C^1\)-continuous. The associated formulations and specifications of the interpolation functions will be provided in the sections where the beam and shell theories are described.
 
4
A pure displacement triangular element derived on the basis of the linear shape functions has two degrees of freedom per node. Hence the element has in total \(2 \times 3 = 6\) unknowns. Of these, three are needed to describe the rigid body motions (two translations and one rotation) and three needed to model the constant strain states.
Besides that the element is very simple and also very robust in nonlinear applications it does not perform very well since its approximation properties are not too good. Due to that the element is very “stiff”, meaning its approximation will yield smaller displacements than an analytical solution. This is especially negative when a structure undergoes bending deformations or when incompressible material has to be considered. In such cases higher order elements or special elements have to be applied, see also Chap. 6.
 
5
The formulation can easily be extended to higher order or triangular elements. In that case only the ansatz functions have to be exchanged, see e.g. Sect. 6.​2.​1.
 
6
For a stress calculation within an actual design the 2nd Piola–Kirchhoff stresses have to be transformed to real stresses, e.g. the Cauchy stresses using (3.​72).
 
7
The first part of Eq. (4.62), partial derivative of W, is easier to use within AceGen when the strain energy function is known.
 
8
The scalar product can also be written as \( {{\varvec{S}}} \cdot \frac{\partial {{\varvec{C}}}}{\partial p_m} =\text{ tr }\,[{{\varvec{S}}}^T \frac{\partial {{\varvec{C}}}}{\partial p_m}]\) by using the trace operator, see Appendix B.1.3 and B.1.5.
 
9
It is of course also possible to formulate the pseudo potential \(W^P\) using the strain energy function W, see (4.50). Then the 2nd Piola–Kirchhoff stress is computed from
$${\varvec{S}} = 2\,\frac{\partial W}{\partial {\varvec{C}}}$$
and can be introduced as stress in the pseudo potential.
 
10
Note that a dependency of W on the left Cauchy–Green tensor cannot be used for anisotropic materials.
 
11
The formulation can easily be extended to higher order or triangular elements. In that case only the ansatz functions have to be exchanged.
 
Literature
Bathe, K.J. 1996. Finite element procedures. Englewood Cliffs: Prentice-Hall.MATH
Braess D. 2007. Finite elements: theory, fast solvers, and applications in solid mechanics. Cambridge: Cambridge University Press.
Dhatt, G., and G. Touzot. 1985. The finite element method displayed. Chichester: Wiley.MATH
Hughes, T.R.J. 1987. The finite element method. Englewood Cliffs, New Jersey: Prentice Hall.MATH
Irons, B. 1971. Quadrature rules for brick based finite elements. International Journal for Numerical Methods in Engineering 3: 293–294.CrossRefMATH
Oñate, E. 2009. Structural analysis with the finite element method. linear statics: volume 1: basis and solids. Springer: Heidelberg.
Simo, J.C., and F. Armero. 1992. Geometrically non-linear enhanced strain mixed methods and the method of incompatible modes. International Journal for Numerical Methods in Engineering 33: 1413–1449.MathSciNetCrossRefMATH
Wriggers, P. 2008. Nonlinear finite elements. Berlin: Springer.MATH
Zienkiewicz, O.C., and R.L. Taylor. 1989. The finite element method, vol. 1, 4th ed. London: McGraw Hill.
Zienkiewicz, O.C., and R.L. Taylor. 1991. The finite element method, vol. 2, 4th ed. London: McGraw Hill.
Zienkiewicz, O.C., and R.L. Taylor. 2000a. The finite element method, vol. 1, 5th ed. Oxford: Butterworth-Heinemann.
Zienkiewicz, O.C., and R.L. Taylor. 2000b. The finite element method, vol. 2, 5th ed. Oxford: Butterworth-Heinemann.
Metadata
Title
Automation of Discretization Techniques
Authors
Jože Korelc
Peter Wriggers
Copyright Year
2016
Book
Automation of Finite Element Methods

Print ISBN: 978-3-319-39003-1

Electronic ISBN: 978-3-319-39005-5

Copyright Year: 2016

DOI
https://doi.org/10.1007/978-3-319-39005-5_4

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