Contents

# Shape optimization

## Finite element analysis by the GJ-integral

In this subsection, to simplify the explanation, $\Omega$ is a polygon whose finite elements are triangles in 2-dimensional domain. Essentially the same results hold in the 3-dimensional case. In our problem, we cannot expect the smoothness of the solution $u$, but it will be able to assume the existance of a number $s, 1 \lt s \lt 2$ such that $u\in H^s(\Omega ;\mathbb{R}^m)$ (see e.g., [Gr85, Gr92, Kne05]).
Triangulation
Let $\mathcal{T}_h(\Omega )=\{T_k,\, k=1,…,n_t\}$ be a triangulation of $\overline{\Omega }=\cup _{k=1}^{n_t}T_k$ with the maximum diameter $h=\max \{\textrm{diam}(T_k):\, T_k\in \mathcal{T}_h(\Omega )\}$. Let us denote by $n_v$ the number of vertices $q^l, l=1,…,n_v$ of the triangles in $\mathcal{T}_h(\Omega )$
FE-spaces
We now introduce the basis function $\phi _l,l=1,…,n_v$ such that $\phi _k(q^l)=\delta ^l_k$, that is, linear polynomial ($P_1$) on each triangle, and the P1-finite element space \begin{eqnarray*} V_h(\mathcal{T}_h(\Omega ),P_1)&=&\{c_1\phi _1+…+c_{n_v}\phi _{n_v}\\ &&~~:\, c_i\in \mathbb{R},\, \phi _i |_K\in P_1,\, ∀K\in \mathcal{T}_h(\Omega ),\, 1\le i\le n_v\} \end{eqnarray*} For a continuous function $v(x),x\in \overline{\Omega }$, the $P_1$-approximation $v_h\in V_h(\mathcal{T}_h(\Omega ),P_1)$ is given by $c_i=v(q^i),\, i=1,…,n_v$. Let $\Pi _hv$ be the projection of $v$ into $\Pi _hv\in V_h(\mathcal{T}_h(\Omega ),P_1)$ defined by $c_i=v(q^i),\, i=1,…,n_v$.
Error estimation
Then by Taylor expansion, we can get the estimation for a smooth function $v$ (see e.g., [Coro.1.128, E-G04]) $$\|\Pi _hv-v\|_{1,\Omega }\le Ch\|v\|_{2,\Omega } \tag{3.25}$$ The operator $v\mapsto \Pi _hv\in H^1(\Omega ;\mathbb{R}^m)$ is orthogonal projection, so the Pythagoras identity yields (see e,g,, [Lemma 1.131, E-G04] $$\|(I-\Pi _h)v\|_{1,\Omega }\le \|v\|_{1,\Omega } \tag{3.26}$$ Using the interpolation theory of operators [7.17,7.43, Adams], we derive from (3.25) and (3.26) $$\|(I-\Pi _h)u\|_{1,\Omega }\le C^{s-1}h^{s-1}\|u\|_{s,\Omega },$$ The error between $u$ and $u_h$ is estimated by Céa's Lemma(see e.g., [Lemma 2.28, E-G04]), that is, $\|u-u_h\|_{1,\Omega }\le Ch^{s-1}$ with a constant $C \gt 0$ independent of $h$. If the problem (2.1) is linear, then by $\widehat{W}(x,u,\nabla u)=\delta \widehat{W}(x,u,\nabla u)[u]/2$, we can prove $$|R_{\Omega }(u,\mu )-R_{\Omega }(u_h,\mu )|\le C\left (\|f\|_{L^2(\Omega )}+\|u\|_{1,\Omega }+\|u_h\|_{1,\Omega }\right )\|u-u_h\|_{1,\Omega }\|\mu \|_{1,\infty ,\Omega }$$ with a constant $C \gt 0$ independent of $h$.\c
Error estimation of GJ-integral
So that $$|R_{\Omega }(u,\mu )-R_{\Omega }(u_h,\mu )|\le Ch^{s-1} \tag{3.27}$$ for $u\in H^s(\Omega ;\mathbb{R}^m), 1 \lt s \lt 2$. If $u\in H^s(\Omega ;\mathbb{R}^d), s \gt 3/2$, we can use the Hadamard formula (2.37) , but the FE analysis in (2.37) require an estimation of the FE-approximation like that $\|u-u_h\|_{s,\Omega }, s \gt 3/2$ by the Trace theorem (see e.g., [Theorem 5.5, Nec67]). However, even with the higher order FEM utilizing Lagrangian interpolation polynomials, we only evaluate the approximation in $H^1(\Omega ;\mathbb{R}^m)$, i.e., $\|u-u_h\|_{1,\Omega }$ (see e.g., [Coro.3.29, E-G04]). So, even if $u$ is smooth, it is difficult to improve the calculation accuracy of $P_{\Omega }(u_h,\mu )$ because $\nabla u_h$ is discontinuous on the interface $\overline{T_k}\cap \overline{T_l}$ of different meshs. For the reason just above, if we try to calculate J-integral (2.16a) with high accuracy we encounter same problem.
Crack problem and Mixed boundary condition
In crack problems (see FEM of J-integral , Section 2.2 ) and mixed boundary conditions (Section 2.5.2 ), the integral $J_{\omega }(u,\mu |\partial \omega )$ appear with $\omega$ containing the edge of crack or bound parts of different boundary conditions. From Corollary 2.4 , it is possible to change $J_{\omega }(u,\mu |\partial \omega )$ to $R_{\omega _0}(u,\beta _{\omega }\mu )$ only using a cut-off function $\beta _{\omega }$ of $\omega$ such as $\beta _{\omega }=1$ on $\omega$ and $\textrm{supp}\beta _{\omega }\subset \omega _0$. Here, we notice that $J_{\omega _0}(u,\mu |\partial \omega _0)-J_{\omega }(u,\mu |\partial \omega ) ≈P_{\omega _0\setminus \overline{\omega }}(u,\mu |(\partial \Omega \cap (\omega _0\setminus \overline{\omega }))$. In a crack problem, $P_{\omega _0\setminus \overline{\omega }}(u,\mu |(\partial \Omega \cap (\omega _0\setminus \overline{\omega }))$, $\partial \Omega =\partial D\cup \Sigma$, is a integral over a crack surface $\Sigma$ except the edge and $\mu \perp n$ for the normal vector $n$ of a crack surface, then $P_{\omega _0\setminus \overline{\omega }}(u,\mu |(\partial \Omega \cap (\omega _0\setminus \overline{\omega }))=0$. However $P_{\omega _0\setminus \overline{\omega }}(u,\mu |(\partial \Omega \cap (\omega _0\setminus \overline{\omega }))$ remains in mixed boundary condition. Since the constant $C$ in (3.27) is affected by $\nabla \beta _{\omega }$, it is difficult to bring $\omega$ closer to singular points .

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[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.
[Li85] F. Z. Li, C. F. Shih and A. Needleman, A comparison of methods for calculating energy release rates, Eng. Frac. Mech. 21(1985), 405--421.
[Lo82] H. G. deLorenzi, On the energy release rate and the J-integral for 3-D crack configurations. Int.. J. Fracture 19, 183-193 (1982).
[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.