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September  2021, 11(3): 653-679. doi: 10.3934/mcrf.2021017

Second order directional shape derivatives of integrals on submanifolds

Anton Schiela , and  Julian Ortiz

Universität Bayreuth, 95440 Bayreuth, Germany

* Corresponding author: Anton Schiela

Received  February 2019 Revised  October 2019 Published  September 2021 Early access  March 2021

Fund Project: This work was supported by DFG grant SCHI 1379/3-1

We compute first and second order shape sensitivities of integrals on smooth submanifolds using a variant of shape differentiation. The result is a quadratic form in terms of one perturbation vector field that yields a second order quadratic model of the perturbed functional. We discuss the structure of this derivative, derive domain expressions and Hadamard forms in a general geometric framework, and give a detailed geometric interpretation of the arising terms.

Keywords: Shape optimization, shape derivative, shape hessian, submanifolds.
Mathematics Subject Classification: Primary: 53A07, 49Q10, 49Q12.
Citation: Anton Schiela, Julian Ortiz. Second order directional shape derivatives of integrals on submanifolds. Mathematical Control & Related Fields, 2021, 11 (3) : 653-679. doi: 10.3934/mcrf.2021017
References:
[1]

D. Bucur and J.-P. Zolésio, Anatomy of the shape Hessian via Lie brackets, Ann. Mat. Pura Appl., 173 (1997), 127-143.  doi: 10.1007/BF01783465.  Google Scholar

[2]

M. C. Delfour and J.-P. Zolésio, Shapes and Geometries: Metrics, Analysis, Differential Calculus, and Optimization, 22 of Advances in Design and Control, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, second edition, 2011. doi: 10.1137/1.9780898719826.  Google Scholar

[3]

F. R. Desaint and J.-P. Zolésio, Manifold derivative in the Laplace-Beltrami equation, J. Funct. Anal., 151 (1997), 234-269.  doi: 10.1006/jfan.1997.3130.  Google Scholar

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A. Henrot and M. Pierre, Variation et Optimisation de Formes, Une Analyse Géométrique, 48 of Mathématiques & Applications (Berlin), Springer, Berlin, 2005. doi: 10.1007/3-540-37689-5.  Google Scholar

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M. Hintermüller and W. Ring, A second order shape optimization approach for image segmentation, SIAM J. Appl. Math., 64 (2003/04), 442-467.  doi: 10.1137/S0036139902403901.  Google Scholar

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R. Hiptmair and J. Li, Shape derivatives in differential forms I: An intrinsic perspective, Ann. Mat. Pura Appl., 192 (2013), 1077-1098.  doi: 10.1007/s10231-012-0259-9.  Google Scholar

[7]

S. Lang, Fundamentals of Differential Geometry, 191 of Graduate Texts in Mathematics., Springer-Verlag, New York, 1999. doi: 10.1007/978-1-4612-0541-8.  Google Scholar

[8]

F. Murat and J. Simon, Etudes de problems d'optimal design, Lectures Notes in Computer Science, 41 (1976), 54-62.   Google Scholar

[9]

A. Novruzi and M. Pierre, Structure of shape derivatives, J. Evol. Equ., 2 (2002), 365-382.  doi: 10.1007/s00028-002-8093-y.  Google Scholar

[10]

V. H. Schulz, A Riemannian view on shape optimization, Found. Comput. Math., 14 (2014), 483-501.  doi: 10.1007/s10208-014-9200-5.  Google Scholar

[11]

J. Simon, Second variation for domain optimization problems. control and estimation of distributed parameter systems, International Series of Numerical Mathematics, 91, Birkhäuser, Basel, 1989,361–378.  Google Scholar

[12]

J. Sokołowski and J.-P. Zolésio, Introduction to Shape Optimization, Shape Sensitivity Analysis, 16 of Springer Series in Computational Mathematics., Springer-Verlag, Berlin, 1992. doi: 10.1007/978-3-642-58106-9.  Google Scholar

[13]

M. Spivak, A comprehensive Introduction to Differential Geometry. Vol. III, Publish or Perish, Inc., Wilmington, Del., second edition, 1979.  Google Scholar

[14]

K. Sturm, Convergence of Newton's method in shape optimisation via approximate normal functions, Technical Report arXiv: 1608.02699, arXiv, 2016. Google Scholar

[15]

K. Sturm, A structure theorem for shape functions defined on submanifolds, Interfaces Free Bound., 18 (2016), 523-543.  doi: 10.4171/IFB/372.  Google Scholar

[16]

L. Younes, Shapes and Diffeomorphisms, 171 of Applied Mathematical Sciences., Springer-Verlag, Berlin, 2010. doi: 10.1007/978-3-642-12055-8.  Google Scholar

[17]

J.-P. Zolésio, Un résultat d'existence de vitesse convergente dans des problèmes d'identification de domaine, C. R. Acad. Sci. Paris Sér. A-B, 283 (1976), Aiii, A855–A858.  Google Scholar

[18]

J.-P. Zolésio, The material derivative (or speed) method for shape optimization, In Optimization of distributed parameter structures, Vol. II (Iowa City, Iowa, 1980), 50 of NATO Adv. Study Inst. Ser. E: Appl. Sci., Nijhoff, The Hague, 1981, 1089–1151.  Google Scholar

show all references

References:
[1]

D. Bucur and J.-P. Zolésio, Anatomy of the shape Hessian via Lie brackets, Ann. Mat. Pura Appl., 173 (1997), 127-143.  doi: 10.1007/BF01783465.  Google Scholar

[2]

M. C. Delfour and J.-P. Zolésio, Shapes and Geometries: Metrics, Analysis, Differential Calculus, and Optimization, 22 of Advances in Design and Control, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, second edition, 2011. doi: 10.1137/1.9780898719826.  Google Scholar

[3]

F. R. Desaint and J.-P. Zolésio, Manifold derivative in the Laplace-Beltrami equation, J. Funct. Anal., 151 (1997), 234-269.  doi: 10.1006/jfan.1997.3130.  Google Scholar

[4]

A. Henrot and M. Pierre, Variation et Optimisation de Formes, Une Analyse Géométrique, 48 of Mathématiques & Applications (Berlin), Springer, Berlin, 2005. doi: 10.1007/3-540-37689-5.  Google Scholar

[5]

M. Hintermüller and W. Ring, A second order shape optimization approach for image segmentation, SIAM J. Appl. Math., 64 (2003/04), 442-467.  doi: 10.1137/S0036139902403901.  Google Scholar

[6]

R. Hiptmair and J. Li, Shape derivatives in differential forms I: An intrinsic perspective, Ann. Mat. Pura Appl., 192 (2013), 1077-1098.  doi: 10.1007/s10231-012-0259-9.  Google Scholar

[7]

S. Lang, Fundamentals of Differential Geometry, 191 of Graduate Texts in Mathematics., Springer-Verlag, New York, 1999. doi: 10.1007/978-1-4612-0541-8.  Google Scholar

[8]

F. Murat and J. Simon, Etudes de problems d'optimal design, Lectures Notes in Computer Science, 41 (1976), 54-62.   Google Scholar

[9]

A. Novruzi and M. Pierre, Structure of shape derivatives, J. Evol. Equ., 2 (2002), 365-382.  doi: 10.1007/s00028-002-8093-y.  Google Scholar

[10]

V. H. Schulz, A Riemannian view on shape optimization, Found. Comput. Math., 14 (2014), 483-501.  doi: 10.1007/s10208-014-9200-5.  Google Scholar

[11]

J. Simon, Second variation for domain optimization problems. control and estimation of distributed parameter systems, International Series of Numerical Mathematics, 91, Birkhäuser, Basel, 1989,361–378.  Google Scholar

[12]

J. Sokołowski and J.-P. Zolésio, Introduction to Shape Optimization, Shape Sensitivity Analysis, 16 of Springer Series in Computational Mathematics., Springer-Verlag, Berlin, 1992. doi: 10.1007/978-3-642-58106-9.  Google Scholar

[13]

M. Spivak, A comprehensive Introduction to Differential Geometry. Vol. III, Publish or Perish, Inc., Wilmington, Del., second edition, 1979.  Google Scholar

[14]

K. Sturm, Convergence of Newton's method in shape optimisation via approximate normal functions, Technical Report arXiv: 1608.02699, arXiv, 2016. Google Scholar

[15]

K. Sturm, A structure theorem for shape functions defined on submanifolds, Interfaces Free Bound., 18 (2016), 523-543.  doi: 10.4171/IFB/372.  Google Scholar

[16]

L. Younes, Shapes and Diffeomorphisms, 171 of Applied Mathematical Sciences., Springer-Verlag, Berlin, 2010. doi: 10.1007/978-3-642-12055-8.  Google Scholar

[17]

J.-P. Zolésio, Un résultat d'existence de vitesse convergente dans des problèmes d'identification de domaine, C. R. Acad. Sci. Paris Sér. A-B, 283 (1976), Aiii, A855–A858.  Google Scholar

[18]

J.-P. Zolésio, The material derivative (or speed) method for shape optimization, In Optimization of distributed parameter structures, Vol. II (Iowa City, Iowa, 1980), 50 of NATO Adv. Study Inst. Ser. E: Appl. Sci., Nijhoff, The Hague, 1981, 1089–1151.  Google Scholar

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