Abstract
We treat Zolésio’s velocity method of shape calculus using the formalism of differential forms, in particular, the notion of Lie derivative. This provides a unified and elegant approach to computing even higher-order shape derivatives of domain and boundary integrals and avoids the tedious manipulations entailed by classical vector calculus. Hitherto unknown expressions for shape Hessians can be derived with little effort. The perspective of differential forms perfectly fits second-order boundary value problems (BVPs). We illustrate its power by deriving the shape derivatives of solutions to second-order elliptic BVPs with Dirichlet, Neumann and Robin boundary conditions. A new dual mixed variational approach is employed in the case of Dirichlet boundary conditions.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Adams R.A.: Sobolev Spaces. Academic Press, New York, NY (1975)
Arnold D.N., Falk R.S., Winther R.: Finite element exterior calculus, homological techniques, and applications. Acta Numerica 15, 1–155 (2006)
Bucur D., Zolésio J.-P.: Anatomy of the shape hessian via lie brackets. Annali di Matematica Pura ed Applicata 173, 127–143 (1997)
Cartan H.: Differential Forms. Hermann, Paris (1970)
Delfour M.C., Zolésio J.P.: Anatomy of the shape hessian. Annali di Matematica Pura ed Applicata 159, 315–339 (1991)
Delfour M.C., Zolésio J.-P.: Shapes and Geometries: Analysis, Differential Calculus, and Optimization. SIAM, Philadelphia (2001)
Flanders H.: Differential Forms with Applications to the Physical Sciences. Academic Press, New York, NY (1963)
Frankel T.: The Geometry of Physics: An Introduction. Cambridge University Press, Cambridge (1997)
Hadamard J.: Lessons on the Calculus of Variation (in French). Gauthier-Villards, Paris (1910)
Hettlich F.: Frechet derivatives in inverse obstacle scattering. Inverse Probl. 11, 371–382 (1995)
Hettlich F.: Frechet derivatives in inverse obstacle scattering. Inverse Probl. 14, 209–210 (1998)
Hiptmair R.: Discrete hodge operators. Numer. Math. 90, 265–289 (2001)
Hiptmair R.: Finite elements in computational electromagnetism. Acta Numerica 11, 237–339 (2002)
Hiptmair, R., Li, J.: Shape Derivatives in Differential Forms II: Applications to Acoustic and Electromagnetic Scattering Problems. Technical report, SAM, ETH, Zürich, Switzerland (2012, in preparation)
Kolář I., Michor P., Slovák J.: Natural Operations in Differential Geometry. Springer, Berlin (1993)
Nédélec J.-C.: Acoustic and Electromagnetic Equations: Integral Representations for Harmonic Problems, vol. 144 of Applied Mathematical Sciences. Springer, New York, NY (2001)
O’Neill B.: Elementary Differential Geometry, 2nd edn. Academic Press, New York, NY (1997)
Sokolowski J., Zolésio J.-P.: Introduction to Shape Optimization: Shape Sensitivity Analysis. Springer, Berlin (1992)
Stein E.M.: Singular Integrals and Differentiability Properties of Functions. Princeton University Press, Princeton (1970)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Hiptmair, R., Li, J. Shape derivatives in differential forms I: an intrinsic perspective. Annali di Matematica 192, 1077–1098 (2013). https://doi.org/10.1007/s10231-012-0259-9
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10231-012-0259-9