Contents

# Shape sensitivity of energy

We describe the definition of the GJ-integral and the relation with fracture mechanics in Section 2.2 , and prove properties (GJ-1) and (GJ-2) in Theorem 2.2 and (GJ-3) is given in Theorem 2.8 in linear case using the first method and is given in Corollary 2.13 after showing the differentiability of functional (Theorem 2.10 ) with respect to a parameter in an abstract framework. Let $\widehat{W}(\xi ,z,\zeta )$ be a function defined for almost all $\xi \in \mathbb{R}^d$ and for all $z\in \mathbb{R}^m (m\ge 1), \zeta \in \mathbb{R}^{m\times d}$. For a measurable functions $v_i=v_i(x) (x\in \Omega , i=1,…m)$, the operator $[v\mapsto \widehat{W}(x,v(x),\nabla v(x))]$ is defined by the formula $$\widehat{W}(x,v(x),\nabla v(x))=\widehat{W}(\xi ,z,\zeta ), \xi =x, z=v(x), \zeta =\nabla v(x) \textrm{for }x\in \Omega$$ In the followsing we shall use $\widehat{W}(v)$ for $\widehat{W}(x,v,\nabla v)$ when there is no ambiguity. For a given functions $f\in L^2(\Omega ;\mathbb{R}^m), g\in H^1(\Omega ;\mathbb{R}^m)$, the weak solution $u$ is the minimization of the energy functional \begin{eqnarray} \mathcal{E}(u;f,g,\Omega )&=&\min _{v\in V(\Omega )}\mathcal{E}(v;f,g,\Omega ) \tag{2.1} \end{eqnarray} \begin{eqnarray} \mathcal{E}(v;f,g,\Omega )&:=&\int _{\Omega }\{\widehat{W}(v)-fv\} dx- \int _{\Gamma _N}gv\, ds \tag{2.2} \end{eqnarray} over a subset $V(\Omega )$ of $H^1(\Omega ;\mathbb{R}^m)$ where $\Gamma _N$ is the part of $\partial \Omega$, which leads $\mathcal{L}(u)=f$ given in ((2.4) ). For simplicity, we used $L^2(\Omega ;\mathbb{R}^m), H^1(\Omega ;\mathbb{R}^m)$, but we can formulate them for nonlinear equations by using $L^p(\Omega ;\mathbb{R}^m), W^{1,p}(\Omega ;\mathbb{R}^m)$ with $p\ge 1$. The following are examples of $\mathcal{L}, \widehat{W}$. Partial differential equations $(m=1)$:
Partial differential equations

Poisson's equation:
$-\Delta u=f,\mathcal{L}(u)=-\Delta u$;$\widehat{W}(\xi ,z,\zeta )=|\zeta |^2/2$, $\widehat{W}(v)=\frac{1}{2}|\nabla v |^2$.
If $V(\Omega )=H_0^1(\Omega ):=\{v\in H^1(\Omega ):v=0\textrm{ on }\partial \Omega \}$ then $$-\Delta u(x) = f(x)~~x\in \Omega ;~~u(x)=0~~x\in \partial \Omega$$

Linear elliptic equation:
$\mathcal{L}(u)=-\partial _i (a_{ij}(x)\partial _j u)+q(x)u, a_{ij}=a_{ji}$; $\widehat{W}(\xi ,z,\zeta )=(a_{ij}(\xi )\zeta _i\zeta _j+q(\xi )z^2)/2$, $\widehat{W}(v)=\frac{1}{2}(a_{ij}(x)\partial _iv\partial _jv+q(x)v^2)$ with ellipticity $a_{ij}\zeta _i\zeta _j \gt 0 ∀\zeta \in \mathbb{R}^d\setminus \{0\}$.
If $V(\Omega )=H_0^1(\Omega ,\Gamma _D):=\{v\in H^1(\Omega ):~u=0~\textrm{on }\Gamma _D\}$ then $$-\sum _{i,j=1}^d \partial _i a_{ij}\partial _j u + q(x)u(x)= f~~\textrm{in }\Omega ;~~u=0~~\textrm{on }\Gamma _D;~$$

Partial differential system $(m \gt 1)$

Vector Poisson:
$-\Delta u_i=f_i$ with $u=(u_1,…,u_m), \mathcal{L}=(\mathcal{L}_1,…,\mathcal{L}_m), m \gt 1$; $\mathcal{L}_i(u)= -\Delta u_i, \widehat{W}(\xi ,z,\zeta )=\frac{1}{2}\sum _{i=1}^m\sum _{j=1}^d\zeta _{ij}^2$.

Linear elasticity:
$\mathcal{L}=(\mathcal{L}_1,…,\mathcal{L}_d)$, $\mathcal{L}_i(u)=-\partial _j \sigma _{ij}(x,u), \sigma _{ij}(x,u)=C_{ijkl}e_{kl}(u)$; $\widehat{W}(\xi ,z,\zeta )=\frac{1}{2}C_{ijkl}(\xi )e_{kl}(\zeta )e_{ij}(\zeta )$, $e_{ij}(\zeta )=\frac{1}{2}(\zeta _{i,j}+\zeta _{j,i})$ with Hooke's tensor $C_{ijkl}(\xi )$ having symmetry $C_{ijkl}=C_{jilk}=C_{klij}$ satisfying ellipticity $C_{ijkl}e_{kl}e_{ij} \gt \alpha e_{ij}e_{ij}$ with a number $\alpha \gt 0$ for all non zero $(e_{ij})$. The material is called homogeneous, if Hooke's tensor $C_{ijkl}$ are constant, that is, $\widehat{W}(\xi ,z,\zeta )$ is indenpendent of $\xi$. When the material is isotropic, $C_{ijkl}=\lambda \delta _{ij}\delta _{kl}+ G(\delta _{ik}\delta _{jl}+\delta _{il}\delta _{jk})$ with Kronecker delta $\delta _{ij}$ and Lame's constants $\lambda ,G$ and $\widehat{W}(v)=\frac{1}{2}\lambda ({\rm div}v)^2+G e_{ij}e_{ij}$. The ellipticity hold if $G \gt 0, 3\lambda +2G \gt 0$.

Hyperelasticity:
$-\partial _j\sigma _{ij}(x,u)=f_i, \sigma _{ij}(x,u)=\partial \widehat{W}(u)/\partial e_{ij}$. Hyperelastic material contains nonlinearities (e.g. [Cir88]).

In the above examples, we assumed that $[(\xi ,z,\zeta )\mapsto \widehat{W}(\xi ,z,\zeta )]\in C^1(\mathbb{R}^d\times \mathbb{R}^m\times \mathbb{R}^{m\times d})$. Let us denote with $v\in V(\Omega )$ \begin{equation*} \delta \widehat{W}(u)[v]:=\lim _{\epsilon →0}\epsilon ^{-1} \left [\widehat{W}(u+\epsilon v)-\widehat{W}(u)\right ] \end{equation*} when $u+\epsilon v\in V(\Omega )$ for small $\epsilon$. Then we have $$\lim _{\epsilon →0}\epsilon ^{-1}\left [\mathcal{E}(u+\epsilon v;f,g,\Omega )- \mathcal{E}(u;f,g,\Omega )\right ] =\int _{\Omega }\{\delta \widehat{W}(u)[v]-fv\} dx-\int _{\Gamma _N}gv\, ds \tag{2.3}$$ The problem (2.1) is called linear if $V(\Omega )$ is the closed subspace of $H^1(\Omega ;\mathbb{R}^m)$ and $a(w,v)=\int _{\Omega }\delta \widehat{W}(w)[v]dx$ is bilinear form on $V(\Omega )\times V(\Omega )$. For a given open set $\omega \subset \mathbb{R}^d$, if $u |_{\omega \cap \Omega }\in H^2(\omega \cap \Omega ;\mathbb{R}^m), u±\epsilon v\in V(\Omega ) ∀v\in C_0^\infty (\omega \cap \Omega ;\mathbb{R}^m)$ with $|\epsilon |\le \epsilon _0, \textrm{small }\epsilon _0 \gt 0$, then we obtain the partial differential equation/system $\mathcal{L}(u)=f$ $$-\textrm{div}\left [\nabla _\zeta \widehat{W}(u)\right ] +\widehat{W}_z(u)=f \textrm{in }L^2(\omega \cap \Omega ;\mathbb{R}^m) \tag{2.4}$$ where \begin{eqnarray*} \nabla _{\zeta }\widehat{W}(x,z,\zeta )&:=&\left (\array{\partial \widehat{W}(x,z,\zeta )/\partial \zeta _{1,1}&…&\partial \widehat{W}(x,z,\zeta )/\partial \zeta _{1,d}\\ \vdots &\ddots &\vdots \\ \partial \widehat{W}(x,z,\zeta )/\partial \zeta _{m,1}&…&\partial \widehat{W}(x,z,\zeta )/\partial \zeta _{m,d} }\right )\\ \textrm{div}\left [\nabla _\zeta \widehat{W}(u)\right ]&=&\left (\array{\partial _1[\partial \widehat{W}(x,z,\zeta )/\partial \zeta _{1,1}]+&…&+\partial _d [\partial \widehat{W}(x,z,\zeta )/\partial \zeta _{1,d}]\\ &\vdots &\\ \partial _1[\partial \widehat{W}(x,z,\zeta )/\partial \zeta _{m,1}]+&…&+\partial _d [\partial \widehat{W}(x,z,\zeta )/\partial \zeta _{m,d}] }\right )\\ \widehat{W}_z(x,z,\zeta )&:=&\partial \widehat{W}/\partial z \textrm{if }m=1; :=(\partial \widehat{W}/\partial z_1,…,\partial \widehat{W}/\partial z_m)^T \textrm{if }m \gt 1 \end{eqnarray*}
The solution $u$ obtained by (2.1) is called regular inside $\omega$' if $u |_{\omega \cap \Omega }$ is in $H^2(\omega \cap \Omega ;\mathbb{R}^m)$ and satisfy (2.4) .

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.

# References

[A-P06] G. Allaire and O. Pantz, Structural optimization with FreeFem++, Struct. Multidiscip. Opt, 32 (2006), 173--181.
[Al07] G. Allaire, Conception optimale de structures, Springer, 2007.
[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.
[B-S04] M.P. Bends{\o }e and O. Sigmund, Topology optimization: theory, methods, and applications, Springer, 2004.
[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] R. Correa and A. Seeger, Directional derivative of a minimax function. Nonlinear Anal., 9(1985), 13--22.
[D-Z88] M.C. Delfour and J.-P. Zolésio, Shape sensitivity analysis via min max differentiability, SIAM J. Control and Optim., 26(1988), 834--862.
[D-D81] Ph. Destuynder and M. Djaoua, Sur une interprétation de l'intégrale de Rice en théorie de la rupture fragile. Math. Meth. in Appl. Sci., 3 (1981), 70--87.
[E-G04] A. Em and J.-L. Guermond, Theory and practice of finite elements, Springer, 2004.
[Es56] J.D. Eshelby, The Continuum theory of lattice defects, Solid State Physics, 3 (1956), 79--144.
[F-O78] D. Fujiwara and S. Ozawa, The Hadamard variational formula for the Green functions of some normal elliptic boundary value problems, Proc. Japan Acad., 54 (1978), 215--220.
[G-S52] P.R. Garabedian and M. Schiffer, Convexity of domain functionals, J.Anal.Math., 2 (1952), 281--368.
[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, Pitman, 1985.
[Gr92] P. Grisvard, Singularities in boundary value problems, Springer, 1992.
[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] E.J. Haug, K.K. Choi and V. Komkov, Design sensitivity analysis of structural systems, Academic Press, 1986.
[ffempp] F. Hecht, New development in freefem++. J. Numer. Math. 20 (2012), 251--265. 65Y15, (FreeFem++ URL:http://www.freefem.org)
[Kato] T. Kato, Perturbation theory for linear operators, Springer, 1980.
[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} M. Kimura and I. Wakano, Shape derivative of potential energy and energy release rate in rracture mechanics, J. Math-for-industry, 3A (2011), 21--31.
[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. http://elib.uni-stuttgart.de/opus/volltexte/2005/2191/.
[Ko06] V.A. Kovtunenko, Primal-dual methods of shape sensitivity analysis for curvilinear cracks with nonpenetration, IMA Jour. Appl. Math. 71 (2006), 635--657.
[K-O18] V.A. Kovtunenko and K. Ohtsuka, Shape differentiability of Lagrangians and application to stokes problem, SIAM J. Control Optim. 56 (2018), 3668--3684.
[M-P01] B. Mohammadi and O. Pironneau, Applied shape optimization for fluids. Oxford University Press, 2001.
[Na94] S.Nazarov and B.A.Plamenevsky, Elliptic problems in domains with piecewise smooth boundaries, de Gruyter Expositions in Mathematics 13. Walter de Gruyter \& Co., 1994.
[Nec67] J. Nečas, Direct methods in the theory of elliptic equations, Springer, 2012. Translated from `Méthodes directes en théorie des équations elliptiques, 1967, Masson''.
[Noe18] E. Noether, Invariante variationsprobleme, göttinger nachrichten, Mathematisch-Physikalische Klasse (1918), 235--257.
[N-S13] A.A. Novotny and J. Sokolowski, Topological derivatives in shape optimization, Springer, 2013.
[Oh81] K. Ohtsuka, Generalized J-integral and three dimensional fracture mechanics I, Hiroshima Math. J., 11(1981), 21--52.
[Oh85] K. Ohtsuka, Generalized J-integral and its applications. I. -- Basic theory, Japan J. Appl. Math., 2 (1985), 329--350.
[O-K00] K. Ohtsuka and A. Khludnev, Generalized J-integral method for sensitivity analysis of static shape design, Control \& Cybernetics, 29 (2000), 513--533.
[Oh02] K. Ohtsuka, Comparison of criteria on the direction of crack extension, J. Comput. Appl. Math., 149 (2002), 335--339.
[Oh02-2] K. Ohtsuka, Theoretical and numerical analysis on 3-dimensional brittle fracture, Mathematical Modeling and Numerical Simulation in Continuum Mechanics, Springer, 2002, 233--251.
[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, Japan J. Indust. Appl. Math., 29 (2012), 23--35.
[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] A.N. Pressley, Elementary differential geometry, Springer, 2010.
[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, North-Holland, 1991.
[Sok92] J. Sokolowski and J.-P. Zolesio, Introduction to shape optimization, Springer, 1992.
[St14] K. Sturm, On shape optimization with non-linear partial differential equations, Doctoral thesis, Technische Universiltät of Berlin, 2014. https://d-nb.info/106856959X/34
[Sumi] Y. Sumi, Mathematical and computational analyses of cracking formation, Springer, 2014.
[Zei/2B] E. Zeidler. Nonlinear functional analysis and its applications II/B, Springer, 1990.
[Z-S73] O.C. Zienkiewicz and J.S. Campbell, Shape optimization and sequential linear programming, Optimum Structural Design, Wiley, 1973, 109--126.