Abstract
Our goal is to introduce a Newton method in computing the stationary points of a total energy with respect to the shape. We formulated a precise description of the second order shape derivative. It is given by a symmetrical boundary integral operator, useful for numerical calculations. This method is applied to a particular shape optimisation problem, the electromagnetic casting problem.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
REFERENCES
R. Dautray and J.-L. Lions, Analyse Mathématique et Calcul Numérique, Masson, Paris, 1988.
C. Farhat and F.-X. Roux, Implicit parallel processing in structural mechanics, Comput. Mech. Adv., 2:1 (1994), pp.1–124.
R. Fletcher, Practical Methods of Optimization, John Wiley, Chichester, 1987.
E. Giusti, Minimal Surfaces and Functions of Bounded Variation, Birkhäuser, Boston, MA, 1984.
G. H. Golub and C. F. Van Loan, Matrix Computations, 2nd ed., Johns Hopkins University Press, Baltimore, MD., 1989.
Y. Goto and N. Fujii, Second order numerical method for domain optimization problems, J. Optim. Theory Appl., 67 (1990), pp. 533–550.
W. Gropp, E. Lusk and A. Skjellum, Using MPI, Portable Parallel Programming with the Message-Passing Interface, MIT Press, Cambridge, MA, 1994.
A. Henrot and M. Pierre, Un probléme inverse en formage de métaux liquides, M2AN, 23 (1989), pp. 155–177.
R. Kress, Linear Integral Equations, Springer-Verlag, Berlin, 1989.
S. Murat and J. Simon, Sur le contrôle par un domaine géométrique, Rapport du Laboratoire d'Analyse Numérique, Université de Paris, 1976.
S. G. Nash and A. Sofer, Block truncated-Newton methods for parallel optimization, Math. Programming, Ser. B, 45:1 (1989), pp. 529–546.
J.-C. Nedelec, Approximation des équations intégrales en mécanique et en physique, Rapport Interne, Ecole Polytechnique, 1977.
A. Novruzi, Contribution en Optimisation de Formes et Applications, Thése de l'Université Henri Poincaré, Nancy 1, 1997.
A. Novruzi and J. R. Roche, Second order derivatives, Newton method, application to shape optimization, Rapport INRIA, 1995.
A. Novruzi and J. R. Roche, Newton method in 3-dimensional shape optimization problems. Application to electromagnetic casting, Rapport INRIA, 1998.
M. Pierre and J. R. Roche, Numerical simulation of tridimensional electromagnetic shaping of liquid metals, Numer. Math., 65 (1993), pp. 203–217.
J. Simon, Differentiation with respect to the domain in boundary value problems, Numer. Funct. Anal. Optim., 2 (1980), pp. 649–687.
J. Sokolowski and J. P. Zolesio, Introduction to Shape Optimization,Shape Sensitivity Analysis, Springer Verlag, Berlin, 1992.
E. F. Van de Velde, Concurrent Scientific Computing, Springer-Verlag, New York, 1994.
T. J. Ypma, The effect of rounding errors on Newton-like methods, IMA J. Numer. Anal., 3 (1983) pp. 109–118.
A. S. Zenios and J. M. Mulvey, Vectorization and multitasking of nonlinear network programming algorithms, Math. Programming, Ser. B, 42:2 (1988), pp. 449–470.
A. S. Zenios and M. C. Pinar, Parallel block-partitioning of truncated Newton for nonlinear network optimization, SIAM J. Sci. Stat. Comput., 12:5 (1992), pp. 1173–1193.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Novruzi, A., Roche, J.R. Newton's Method in Shape Optimisation: A Three-Dimensional Case. BIT Numerical Mathematics 40, 102–120 (2000). https://doi.org/10.1023/A:1022370419231
Issue Date:
DOI: https://doi.org/10.1023/A:1022370419231