Abstract
How do modeling errors influence the output values? How do displacements or stresses change if the coefficients of the differential equation change? Any change in the underlying equation translates into a change of the stiffness matrix and its inverse. We apply a 'direct' method to calculate the solution vector of the modified system. In this approach it is not the stiffness matrix which gets modified but the right-hand side. This allows to operate on the modified system with the Green's functions of the original system. Each entry of the inverse of the stiffness matrix changes if one coefficient of the stiffness matrix changes and so and so such a change requires a complete reanalysis. In an alternative formulation integration needs only to be done over the defective element. This integral is the strain energy product between the Green's function and the modified solution. It is a weak influence function. Various techniques are discussed how best to calculate this form in the context of the FE-method. By solving small auxiliary problems the effects caused by changes in an element stiffness can be calculated exactly.
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Notes
- 1.
The negative sign in \((-f^+,v)\) allows to write \(-d(u_c,v) = (f^+,v)\).
- 2.
The same holds true when you drive the first half of a distance with 50Â mph and the second half with 70Â mph. Better to keep a constant speed of 60Â mph: you arrive earlier.
- 3.
\(\varvec{S} \cdot \varvec{E} = \sigma _{11}\,\varepsilon _{11} + \sigma _{12}\,\varepsilon _{12} + \sigma _{21}\,\varepsilon _{21} + \sigma _{22}\,\varepsilon _{22}\) (scalar product).
- 4.
But \(u_c\) is not zero on \(\varOmega _e\).
- 5.
A stress \(\sigma = E\) implies because of \(\sigma = E\, \varepsilon \) that \(\varepsilon = 1\) and so \(\varDelta l = \varepsilon \cdot l = l\).
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Hartmann, F. (2013). Modeling Error. In: Green's Functions and Finite Elements. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-29523-2_5
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DOI: https://doi.org/10.1007/978-3-642-29523-2_5
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