Finite element analysis and shape optimization of singular points in boundary value problems for partial differential equations

Ohtsuka, Kohji

doi:10.1090/suga/471

Finite element analysis and shape optimization of singular points in boundary value problems for partial differential equations
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by Kohji Ohtsuka
Translated by: the author

Sugaku Expositions 35 (2022), 167-196

DOI: https://doi.org/10.1090/suga/471

Published electronically: August 9, 2022

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Abstract:

We introduce the concept of shape optimization of singular points of a weak solution in boundary value problems of partial differential equations (BVP) including fracture mechanics, shape optimization of boundary and interface, etc. Roughly speaking, singularity is a gap between weak and strong solutions. We also introduce the integral formula named the GJ-integral defined with an arbitrary subdomain as an extension of the J-integral in fracture mechanics. The GJ-integral is defined locally, is made from partial differential equations in BVP only, takes zero if the solution is regular, and expresses energy shape sensitivity. A generalization of the Hadamard variational formula is proposed and various cost functions are rewritten using the GJ-integral. In the last section there are numerical calculations using the GJ-integral with the finite element method.

References

Robert A. Adams, Sobolev spaces, Pure and Applied Mathematics, Vol. 65, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1975. MR 0450957
G. Allaire and O. Pantz, Structural optimization with FreeFem++, Struct. Multidiscip. Optim. 32 (2006), no. 3, 173–181. MR 2252743, DOI 10.1007/s00158-006-0017-y
Grégoire Allaire, Conception optimale de structures, Mathématiques & Applications (Berlin) [Mathematics & Applications], vol. 58, Springer-Verlag, Berlin, 2007 (French). With the collaboration of Marc Schoenauer (INRIA) in the writing of Chapter 8. MR 2270119
H. Azegami, Solution to domain optimization problems, Trans. Japan Soc. Mech. Engrs. Series A, 60, No.574 (1994), 1479–1486. (in Japanese)
H. Azegami and Z. Wu, Domain optimization analysis in linear elastic problems: Approach using traction method, JSME Inter. J. Series A, 39 (1996), 272–278.
H. Azegami. Solution of shape optimization problem and its application to product design, Mathematical Analysis of Continuum Mechanics and Industrial Applications, Springer, 2017, 83–98.
Hideyuki Azegami, Shape optimization problems, Springer Optimization and Its Applications, vol. 164. Springer, 2020. MR4179430.
M. P. Bendsøe and O. Sigmund, Topology optimization, Springer-Verlag, Berlin, 2003. Theory, methods and applications. MR 2008524
H. D. Bui, Fracture mechanics – Inverse problems and solutions, Springer, 2006.
G. P. Cherepanov, On crack propagation in continuous media, Prikl. Math. Mekh., 31 (1967), 476–493.
P.G. Ciarlet, Mathematical elasticity: Three-dimensional elasticity, North-Holland, 1988.
Rafael Correa and Alberto Seeger, Directional derivative of a minimax function, Nonlinear Anal. 9 (1985), no. 1, 13–22. MR 776359, DOI 10.1016/0362-546X(85)90049-5
M. C. Delfour and J.-P. Zolésio, Shape sensitivity analysis via min max differentiability, SIAM J. Control Optim. 26 (1988), no. 4, 834–862. MR 948649, DOI 10.1137/0326048
P. Destuynder and M. Djaoua, Sur une interprétation mathématique de l’intégrale de Rice en théorie de la rupture fragile, Math. Methods Appl. Sci. 3 (1981), no. 1, 70–87 (French, with English summary). MR 606849, DOI 10.1002/mma.1670030106
Alexandre Ern and Jean-Luc Guermond, Theory and practice of finite elements, Applied Mathematical Sciences, vol. 159, Springer-Verlag, New York, 2004. MR 2050138, DOI 10.1007/978-1-4757-4355-5
J. D. Eshelby, The Continuum theory of lattice defects, Solid State Physics, 3 (1956), 79–144.
Daisuke Fujiwara and Shin Ozawa, The Hadamard variational formula for the Green functions of some normal elliptic boundary value problems, Proc. Japan Acad. Ser. A Math. Sci. 54 (1978), no. 8, 215–220. MR 517324
P. R. Garabedian and M. Schiffer, Convexity of domain functionals, J. Analyse Math. 2 (1953), 281–368. MR 60117, DOI 10.1007/BF02825640
A. A. Griffith, The phenomena of rupture and flow in solids, Phil. Trans. Roy. Soc. London, Series A 221 (1921), 163–198.
A. A. Griffith, The theory of rupture, Proc. 1st.Intern. Congr. Appl. Mech., Delft (1924) 55–63.
P. Grisvard, Elliptic problems in nonsmooth domains, Monographs and Studies in Mathematics, vol. 24, Pitman (Advanced Publishing Program), Boston, MA, 1985. MR 775683
P. Grisvard, Singularities in boundary value problems, Recherches en Mathématiques Appliquées [Research in Applied Mathematics], vol. 22, Masson, Paris; Springer-Verlag, Berlin, 1992. MR 1173209
J. Hadamard, Mémoire sur un problème d’analyse relatif à l’équilibre des plaques élastiques encastrées, Mémoire des savants étragers, 33 (1907), 515–629.
Edward J. Haug, Kyung K. Choi, and Vadim Komkov, Design sensitivity analysis of structural systems, Mathematics in Science and Engineering, vol. 177, Academic Press, Inc., Orlando, FL, 1986. MR 860040
F. Hecht, New development in freefem++, J. Numer. Math. 20 (2012), no. 3-4, 251–265. MR 3043640, DOI 10.1515/jnum-2012-0013
Tosio Kato, Perturbation theory for linear operators, 2nd ed., Grundlehren der Mathematischen Wissenschaften, Band 132, Springer-Verlag, Berlin-New York, 1976. MR 0407617
M. Kimura. and I. Wakano, New mathematical approach to the energy release rate in crack extension, Trans. Japan Soc. Indust. Appl. Math., 16(2006) 345–358. (in Japanese)
Masato Kimura and Isao Wakano, Shape derivative of potential energy and energy release rate in fracture mechanics, J. Math-for-Ind. 3A (2011), 21–31. MR 2788704
D. Knees, Regularity results for quasilinear elliptic systems of power-law growth in nonsmooth domains: boundary, transmission and crack problems, Ph.D. thesis, Universität Stuttgart, 2005. http://elib.uni-stuttgart.de/opus/volltexte/2005/2191/.
V. A. Kovtunenko, Primal-dual methods of shape sensitivity analysis for curvilinear cracks with nonpenetration, IMA J. Appl. Math. 71 (2006), no. 5, 635–657. MR 2268880, DOI 10.1093/imamat/hxl014
V. A. Kovtunenko and K. Ohtsuka, Shape differentiability of Lagrangians and application to Stokes problem, SIAM J. Control Optim. 56 (2018), no. 5, 3668–3684. MR 3864676, DOI 10.1137/17M1125327
B. Mohammadi and O. Pironneau, Applied shape optimization for fluids, Numerical Mathematics and Scientific Computation, The Clarendon Press, Oxford University Press, New York, 2001. Oxford Science Publications. MR 1835648
Sergey A. Nazarov and Boris A. Plamenevsky, Elliptic problems in domains with piecewise smooth boundaries, De Gruyter Expositions in Mathematics, vol. 13, Walter de Gruyter & Co., Berlin, 1994. MR 1283387, DOI 10.1515/9783110848915.525
Jindřich Nečas, Direct methods in the theory of elliptic equations, Springer Monographs in Mathematics, Springer, Heidelberg, 2012. Translated from the 1967 French original by Gerard Tronel and Alois Kufner; Editorial coordination and preface by Šárka Nečasová and a contribution by Christian G. Simader. MR 3014461, DOI 10.1007/978-3-642-10455-8
Emmy Noether, Invariante variationsprobleme, göttinger nachrichten, Mathematisch-Physikalische Klasse (1918), 235–257.
Antonio André Novotny and Jan Sokołowski, Topological derivatives in shape optimization, Interaction of Mechanics and Mathematics, Springer, Heidelberg, 2013. MR 3013681, DOI 10.1007/978-3-642-35245-4
Kohji Ohtsuka, Generalized $J$-integral and three-dimensional fracture mechanics. I, Hiroshima Math. J. 11 (1981), no. 1, 21–52. MR 606833
Kohji Ohtsuka, Generalized $J$-integral and its applications. I. Basic theory, Japan J. Appl. Math. 2 (1985), no. 2, 329–350. MR 839334, DOI 10.1007/BF03167081
Kohji Ohtsuka and Alexander Khludnev, Generalized $J$-integral method for sensitivity analysis of static shape design, Control Cybernet. 29 (2000), no. 2, 513–533. MR 1777698
Kohji Ohtsuka, Comparison of criteria on the direction of crack extension, J. Comput. Appl. Math. 149 (2002), no. 1, 335–339. Scientific and engineering computations for the 21st century—methodologies and applications (Shizuoka, 2001). MR 1952978, DOI 10.1016/S0377-0427(02)00539-3
Kohji Ohtsuka, Theoretical and numerical analysis on 3-dimensional brittle fracture, Mathematical modeling and numerical simulation in continuum mechanics (Yamaguchi, 2000) Lect. Notes Comput. Sci. Eng., vol. 19, Springer, Berlin, 2002, pp. 233–251. MR 1901670, DOI 10.1007/978-3-642-56288-4_{1}7
K. Ohtsuka, Criterion for stable/unstable quasi-static crack extension by extended griffith energy balance theory, Theor. Appl. Mech. Japan, 57 (2009), 25–32.
K. Ohtsuka, Shape optimization for partial differential equations/system with mixed boundary conditions, RIMS Kôkyûroku 1791 (2012), 172–181.
K. Ohtsuka and M. Kimura, Differentiability of potential energies with a parameter and shape sensitivity analysis for nonlinear case: the $p$-Poisson problem, Jpn. J. Ind. Appl. Math. 29 (2012), no. 1, 23–35. MR 2890353, DOI 10.1007/s13160-011-0049-6
K. Ohtsuka and T. Takaishi, Finite element anaysis using mathematical programming language FreeFem++, Kyoritsu Shuppan, 2014. (in Japanese)
K. Ohtsuka, Shape optimization by GJ-integral: Localization method for composite material, Mathematical Analysis of Continuum Mechanics and Industrial Applications, Springer, 2017, 73–109.
K. Ohtsuka, Shape optimization by Generalized J-integral in Poisson’s equation with a mixed boundary condition, Mathematical Analysis of Continuum Mechanics and Industrial Applications II, Springer, 2018, 73–83.
Andrew Pressley, Elementary differential geometry, 2nd ed., Springer Undergraduate Mathematics Series, Springer-Verlag London, Ltd., London, 2010. MR 2598317, DOI 10.1007/978-1-84882-891-9
J. R. Rice, A path-independent integral and the approximate analysis of strain concentration by notches and cracks, J. Appl. Mech., 35(1968), 379–386.
J. R. Rice, Mathematical analysis in the mechanics of fracture, Fracture Volume II, Academic Press, 1968, 191–311.
J. A. Samareh, A survey of shape parameterization techniques, NASA Report CP-1999-209136 (1999), 333–343.
B.-W. Schulze, Pseudo-differential operators on manifolds with singularities, Studies in Mathematics and its Applications, vol. 24, North-Holland Publishing Co., Amsterdam, 1991. MR 1142574
Jan Sokołowski and Jean-Paul Zolésio, Introduction to shape optimization, Springer Series in Computational Mathematics, vol. 16, Springer-Verlag, Berlin, 1992. Shape sensitivity analysis. MR 1215733, DOI 10.1007/978-3-642-58106-9
K. Sturm, On shape optimization with non-linear partial differential equations, Doctoral thesis, Technische Universiltät of Berlin, 2014. https://d-nb.info/106856959X/34
Yoichi Sumi, Mathematical and computational analyses of cracking formation, Mathematics for Industry (Tokyo), vol. 2, Springer, Tokyo, 2014. Fracture morphology and its evolution in engineering materials and structures. MR 3234571, DOI 10.1007/978-4-431-54935-2
Eberhard Zeidler, Nonlinear functional analysis and its applications. II/B, Springer-Verlag, New York, 1990. Nonlinear monotone operators; Translated from the German by the author and Leo F. Boron. MR 1033498, DOI 10.1007/978-1-4612-0985-0