Errata : the list of errors in Finite element analysis and shape optimization of singular points in boundary value problems for partial differential equations
published in the journal Sugaku Expositions from American Mathematical Society


In the website, we introduce the concept of shape optimization of singular points of a weak solution in boundary value problem of partial differential equations (BVP) defined on a domain $\Omega $, including fracture mechanics, shape optimization of boundary and interface etc. Roughly speaking, singularity is a gap between weak and strong solution (see e.g., [Gr85, Gr92, Kne05, Na94, Sc91]).

We also introduce the integral formula $J_{\omega }(u,\mu )$ named the ``Generalized J-integral'' (GJ-integral) which is the sum of a surface integral $P_{\omega }(u,\mu )$ and a volume integral $R_{\omega }(u,\mu )$ defined on the domain $\omega \cap \Omega $ for an arbitrary domain $\omega $, in which $\omega $ is used to separate the singular points of the solution $u$, and $\mu $ is a vector field representing the perturbation of the singular points.

Generalized J-integral

The GJ-integral is proposed by the author when studying 3-dimensional fracture phenomena [Oh81] as an extension of the J-integral [Ch67, Ri68, Ri68-2] in fracture mechanics (in 2-dimensional).

There are two important properties of the J-integral:
The J-integral is equal to energy release rate (ERR) which is an important parameter in fracture mechanics.
The J-integral depends only on the coefficient of the singular term at the crack tip called the stress intensity factor (SIF) (see e.g., [Bu04, Ri68-2]).

The GJ-integral is defined in various BVPs and applicable to such shape sensitivity of boundary [Oh85] not only fracture mechanics [Oh81].

There are three important properties of the GJ-integral:
Independence from boundary conditions. The GJ-integral is made from partial differential equations only (see the proof of Theorem 2.2 ).
The GJ-integral takes zero if the solution is regular, that is, $J_{\omega }(u,\mu )=0$ for all vector field $\mu $ if $u$ is regular inside $\omega \cap \Omega $, which corresponds to (J-2).
The shape sensitivity of energy is almost equal to $-R_{\Omega }(u,\mu )$ which is well defined for the weak solution $u$ (see Corollary 2.13 ), corresponding to (J-1).

A simple example of the use of these properties is the following: If $u$ is regular inside $\Omega $, then from (GJ-3) and (GJ-2) the shape sensitivity of energy is almost equal to $P_{\Omega }(u,\mu )$, that is, the boundary expression is well obtained by shape optimization (see e.g., [Al07, N-S13, Sok92]).

The idea of separating singularity came from [Es56] and originated from the conservation laws (derived from Noether's theorem [Noe18]). A generalization of the Hadamard variational formula [F-O78, G-S52, Had68] is given in Theorem 3.1 and here is called the GJ-Hadamard formula. (GJ-3) is led using the two methods:

The first method is applicable if the material derivative become admissible function (Theorem 2.8 ). The second method is that sensitivity analysis at a minimizer of functional is expressed only by the shape derivative (Theorem 2.10 ), and is available in the nonlinear case. In the Stokes problem, due to the limitation of $\textrm{div}u=0$, it is difficult to use the mapping technique .

This problem can be solved partly by using the Lagrangian multiplier method (Section 2.4 ).


The second stage of shape optimization is how to search for the optimal shape using shape sensitivity analysis. Searching methods are classified as parametric or nonparametric. In the parametric method, the searching problem is reduced to a finite dimensional optimization problem [Sa99] which is defined by so-called control points on the boudnary or diameter, length, or thickness, etc.

On the other hand, in nonparametric shape optimization problems, a displacement vector denoting the domain variation from an initial domain to a new domain is selected as the design variable. Because the displacement vector is a function, the nonparametric shape optimization problem becomes a functional optimization problem. In this problem, the number of design parameters is infinite. Thus, we can expect a finer result than that for parametric optimization problems.

Mohammadi and Pironneau proposed a method of remaking a smooth boundary by using the Laplace operator on the boundary [M-P01], which was recently used in [Al07]. Azegami has developed the gradient method in the Hilbert space $H^1(\Omega ;\mathbb{R}^d)$ and called this method the traction method [Az94, A-W96], and later changed the name to the H1-gradient method [Az20]. In this method, domain variation is obtained as a solution to a boundary value problem of an elliptic partial differential equation, such as a linear elastic problem defined in the current domain (see [Az17, Az20] for applications). Since these methods are mathematically equivalent, they are collectively referred to as the ``H1-gradient method'' in this monograph, which is suitable for an optimal shape search using the GJ-integral.

Numerical analysis

The last stage is numerical analysis. Here, the finite element method (FEM) is used for numerical analysis. The stability of the H1-gradient method using the GJ-integral [Oh12, Oh14, Oh17, Oh18] by FEM is shown in Section 3.4 , and the calculation procedure for energy, mean compliance, least mean-square error, eigenvalue as a cost function is shown in Section 3.5 .

Unfortunately, the H1-gradient method is not suitable for crack growth as shown in Section 3.3 .

In actual calculation, it is necessary to create more optimal shapes (moving mesh) from the domain where calculation starts, mesh adaptation to prevent collapse of the mesh by moving mesh, and so on. Here we use FreeFem++ [A-P06, ffempp, Oh14] which can be realized by a mathematical description developed by O.Pironneau and F.Hecht of Laboratoire Jacques-Louis Lions in Paris 6, for the calculation of the test problems in Section 4 .

Information about the page: The current position is painted circle in the diagram below. Blue is the main MaKR and orange is a duplicate for MaKR's public use, where dashed line means the connection to the private area The dashed lines are only connections to main MaKR.



[Adams] Robert A.Adams, Sobolev spaces, Pure and Applied Mathematics, Vol. 65, Academic Press [A subsidiary of Harcourt Brace Jovanovich, Publishers], New York-London, 1975. MR0450957
[A-P06] G. Allaire and O. Pantz,Structural optimization with FreeFem++, Struct. Multidiscip. Optim.32 (2006), no. 3, 173--181, DOI 10.1007/s00158-006-0017-y. MR2252743
[Al07] Grégoire Allaire, Conception optimale de structures (French), Mathématiques \& Applications(Berlin) [Mathematics \& Applications], vol. 58, Springer-Verlag, Berlin, 2007. With the collaboration of Marc Schoenauer (INRIA) in the writing of Chapter 8. MR2270119
[Az94] H. Azegami, Solution to domain optimization problems, Trans. Japan Soc. Mech. Engrs. Series A, 60, No.574 (1994), 1479--1486. (in Japanese)
[A-W96] 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.
[Az17] 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.
[Az20] Hideyuki Azegami. Shape optimization problems, Springer Optimization and Its Applications, vol. 164. Springer, 2020. MR4179430.
[B-S04] M. P. Bends\o e and O. Sigmund, Topology optimization: Theory, methods and applications, Springer-Verlag, Berlin, 2003. MR2008524
[Bu04] H.D. Bui, Fracture mechanics -- Inverse problems and solutions, Springer, 2006.
[Ch67] G.P. Cherepanov, On crack propagation in continuous media, Prikl. Math. Mekh., 31 (1967), 476--493.
[Cir88] P.G. Ciarlet, Mathematical elasticity: Three-dimensional elasticity, North-Holland, 1988.
[Co85] Rafael Correa and Alberto Seeger, Directional derivative of a minimax function, Nonlinear Anal. 9 (1985), no. 1, 13--22, DOI 10.1016/0362-546X(85)90049-5. MR776359
[D-Z88] M. C. Delfour and J.-Zolésio, Shape sensitivity analysis via min max differentiability, SIAM J. Control Optim. 26 (1988), no. 4, 834--862, DOI 10.1137/0326048. MR948649
[D-D81] P. Destuynder and M. Djaoua, Sur une interprétation mathématique de l'intégrale de Rice en théorie de la rupture fragile (French, with English summary), Math. Methods Appl. Sci. 3 (1981), no. 1, 70-87, DOI 10.1002/mma.1670030106. MR606849
[D-L76] G. Duvaut J. L. Lions, Inequalities in Mechanics and Physics, Springer 1976.
[E-G04] Alexandre Ern and Jean-Luc Guermond, Theory and practice of finite elements, Applied Mathematical Sciences, vol. 159, Springer-Verlag, New York, 2004, DOI 10.1007/978-1-4757-4355-5. MR2050138
[Es56] J.D. Eshelby, The Continuum theory of lattice defects, Solid State Physics, 3 (1956), 79--144.
[F-O78] 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. MR517324
[G-S52] P. R. Garabedian and M. Schiffer, Convexity of domain functionals, J. Analyse Math. 2 (1953), 281--368, DOI 10.1007/BF02825640. MR60117
[Gr21] A.A. Griffith, The phenomena of rupture and flow in solids, Phil. Trans. Roy. Soc. London, Series A 221 (1921), 163--198.
[Gr24] A.A. Griffith, The theory of rupture, Proc. 1st.Intern. Congr. Appl. Mech., Delft (1924) 55--63.
[Gr85] P. Grisvard, Elliptic problems in nonsmooth domains, Monographs and Studies in Mathematics, vol. 24, Pitman (Advanced Publishing Program), Boston, MA, 1985. MR775683
[Gr92] P. Grisvard, Singularities in boundary value problems, Recherches en Mathématiques Appliquées [Research in Applied Mathematics], vol. 22, Masson, Paris; Springer-Verlag, Berlin, 1992. MR1173209
[Had68] J. Hadamard, Mémoire sur un problème d'analyse relatif à l'équilibre des plaques élastiques encastrées, Mémoire des savants étragers, 33 (1907), 515--629.
[Hau86] 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. MR860040
[ffempp] F. Hecht, New development in freefem++, J. Numer. Math. 20 (2012), no. 3-4, 251--265, DOI 10.1515/jnum-2012-0013. MR3043640
[Kato] Tosio Kato, Perturbation theory for linear operators. Second edition. Springer-Verlag, Berlin-New York, 1976. MR0407617
[K-W06] 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) \bibitem {K-W11} Masato Kimura and Isao Wakano, Shape derivative of potential energy and energy release rate in fracture mechanics, J. Math-for-industry, 3A (2011), 21--31. MR2788704
[Kne05] D. Knees, Regularity results for quasilinear elliptic systems of power-law growth in nonsmooth domains: boundary, transmission and crack problems. PhD thesis, Universität Stuttgart, 2005.
[Ko06] 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, DOI 10.1093/imamat/hxl014. MR2268880
[K-O18] 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, DOI10.1137/17M1125327. MR3864676
[Ku74] Hitoshi Kumano-Go, Pseudo-Differential Operators, Iwanami (in Japanese), 1974.; The MIT Press
[M-P01] B. Mohammadi and O. Pironneau, Applied shape optimization for fluids, Numerical Mathematics and Scientific Computation, Oxford Science Publications, The Clarendon Press, Oxford University Press, New York, 2001. MR1835648
[Li72] J.L.Lions and E.Magenes, Non-Homogeneous Boundary Value Problems and Applications, Springer, 1972.
[Na94] 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, DOI 10.1515/9783110848915.525. MR1283387.
[Nec67] Jindrich Necas, 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 Sárka Necasová and a contribution by Christian G. Simader, DOI 10.1007/978-3-642-10455-8. MR3014461
[Ne71] Umberto Neri, Singular integrals, Lecture Notes in Mathematics 200, Springer 1971.
[Noe18] E. Noether, Invariante variationsprobleme, göttinger nachrichten, Mathematisch-Physikalische Klasse (1918), 235--257.
[N-S13] Antonio André Novotny and Jan SokoM3:7owski, Topological derivatives in shape optimization, Interaction of Mechanics and Mathematics, Springer, Heidelberg, 2013, DOI 10.1007/978-3-642-35245-4. MR3013681
[Oh81] Kohji Ohtsuka, Generalized J-integral and three-dimensional fracture mechanics. I, Hiroshima Math. J. 11 (1981), no. 1, 21--52. MR606833
[Oh85] Kohji Ohtsuka, Generalized J-integral and its applications. I. Basic theory, Japan J. Appl. Math. 2 (1985), no. 2, 329--350, DOI 10.1007/BF03167081. MR839334
[O-K00] Kohji Ohtsuka and Alexander Khludnev, Generalized J-integral method for sensitivity analysis of static shape design, Control Cybernet. 29 (2000), no. 2, 513--533. MR1777698
[Oh02] Kohji Ohtsuka, Comparison of criteria on the direction of crack extension: Scientific and engineering computations for the 21st century-methodologies and applications (Shizuoka, 2001), J. Comput. Appl. Math. 149 (2002), no. 1, 335--339, DOI 10.1016/S0377-0427(02)00539-3. MR1952978
[Oh02-2] 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, DOI 10.1007/978-3-642-56288-4 17. MR1901670
[Oh09] K. Ohtsuka, Criterion for stable/unstable quasi-static crack extension by extended griffith energy balance theory, Theor. Appl. Mech. Japan, 57 (2009), 25--32.
[Oh12] K. Ohtsuka, Shape optimization for partial differential equations/system with mixed boundary conditions, RIMS K\^oky\^uroku 1791 (2012), 172--181.
[OT-K12] 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, DOI 10.1007/s13160-011-0049-6. MR2890353
[Oh14] K. Ohtsuka and T. Takaishi, Finite element anaysis using mathematical programming language FreeFem++, Kyoritsu Shuppan, 2014. (in Japanese)
[Oh17] K. Ohtsuka, Shape optimization by GJ-integral: Localization method for composite material, Mathematical Analysis of Continuum Mechanics and Industrial Applications, Springer, 2017, 73--109.
[Oh18] 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.
[Pr10] Andrew Pressley, Elementary differential geometry, 2nd ed., Springer Undergraduate Mathematics Series, Springer-Verlag London, Ltd., London, 2010, DOI 10.1007/978-1-84882-891-9. MR2598317
[Ri68] 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.
[Ri68-2] J.R. Rice, Mathematical analysis in the mechanics of fracture, Fracture Volume II, Academic Press, 1968, 191--311.
[Pi84] O. Pironneau, Optimal shape design for elliptic systems, Springer-Verlag, 1984.
[Sa99] J.A. Samareh, A survey of shape parameterization techniques, NASA Report CP-1999-209136 (1999), 333--343.
[Sc91] B.-W. Schulze, Pseudo-differential operators on manifolds with singularities, Studies in Mathematics and its Applications, vol. 24, North-Holland Publishing Co., Amsterdam, 1991. MR1142574
[Sok92] Jan Sokolowski and Jean-Paul Zolésio, Introduction to shape optimization: Shape sensitivity analysis, Springer Series in Computational Mathematics, vol. 16, Springer-Verlag, Berlin, 1992, DOI 10.1007/978-3-642-58106-9. MR1215733
[St14] K. Sturm, On shape optimization with non-linear partial differential equations, Doctoral thesis, Technische Universiltät of Berlin, 2014.
[Sumi] Yoichi Sumi, Mathematical and computational analyses of cracking formation: Fracture morphology and its evolution in engineering materials and structures, Mathematics for Industry (Tokyo), vol. 2, Springer, Tokyo, 2014, DOI 10.1007/978-4-431-54935-2. MR3234571
[Zei/2B] 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, DOI 10.1007/978-1-4612-0985-0. MR1033498