Abstract
This paper is concerned by numerical sensitivity analysis in problems governed by boundary-value problems described by partial differential equations (P.D.E.).
The P.D.E. solution y is approximated by the solution of finite element method (F.E.M) yh.
In shape optimization, we are concerned by a cost functional J(Ωh)=h(Ωh, Mh, yh), where Ωh is the discretized geometrical domain and Mh the nodes of the mesh.
This work is devoted to the numerical calculation of the derivative with respect to the coordinates of the nodes of the cost J associated to Mh we have a Ql Lagrange F.E.M and y is solution of a variational problem. For simplicity, we shall restrict the presentation to a second order problem.
The domain Ωh is only described by the boundary nodes, we also discuss the use of the derivatives with respect to the internal nodes. The case P1 Lagrange finite element is treated by Zolesio [6]. In this work, we treat the general case of Ql Lagrange finite element.
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Souli, M., Zolesio, J.P. (1991). Shape derivative of discretized problems. In: Hoffmann, KH., Krabs, W. (eds) Optimal Control of Partial Differential Equations. Lecture Notes in Control and Information Sciences, vol 149. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0043226
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DOI: https://doi.org/10.1007/BFb0043226
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