Contents

# Shape optimization

## Optimization

We now consider problems of finding the shape of singular points from shape sensitivity, that is, for a given cost function $F$, find a mapping $\varphi ^O\in W^{1,\infty }(\Omega ;\mathbb{R}^d)$ such that $F(u^{\varphi ^O}) \lt F(u)$ where $u$ and $u^{\varphi ^O}$ are solutions of problems (2.1) and (2.25a) with $\varphi _t=x+t\mu ^O, \varphi ^O=\varphi _{\epsilon ^O}$ with $\mu ^O\in W^{1,\infty }(\Omega ;\mathbb{R}^d)$ and a number $\epsilon ^O \gt 0$. In our shape optimization there is the case that $\varphi ^O(\Omega )=\Omega$ such as mixed boundary conditions ($\varphi ^O(\Gamma _D)\neq \Gamma _D$). Expanding the cost function $F$ of $u^{\varphi _t}, \varphi _t=x+t\mu , \mu \in W^{1,\infty }(\Omega ;\mathbb{R}^d)$ with respect to $t$, we derive $$F(u^{\varphi _t})=F(u)+tdF(u)[\mu ]+o(t), dF(u)[\mu ] =\left .\frac{d}{dt}F(u^{\varphi _t})\right |_{t=0}$$ In the usual shape shape derivative $dF(u)[\mu ]$ is represented ordinary \begin{equation} dF(u)[\mu ]=\int _{\partial \Omega }g_{\Gamma }(\mu \cdot n)\, ds \tag{3.13} \end{equation} with an integrable function $g_{\Gamma }$ defined on $\Gamma$ if the boundary $\partial \Omega$ is smooth [Theorem 2.27, Sok92]. The advantage of the boundary expression (3.13) is that a descent direction $-g_{\Gamma }n$ is readily available. However, numerical instability phenomenon appears by the method of directly moving the node on the boundary $\partial \Omega$ using the shape gradient $-g_{\Gamma }n$. There are two ways to eliminate such instability; one is to consider it as a problem of numerical stability with a finite dimensional design space, called parametric method (see e.g.[Sa99]). Another is to find general shapes from the sensitivity of a shape function with respect to an arbitrary perturbation of these shapes, called nonparametric method where (before discretization) the design space is infinite-dimensional. In Japan there is the famous nonparametric method called the H1-gradient method (originally called the traction method'') [Az94, A-W96, Az20]. Find $\mu ^O$ by solving the additional auxiliary variational problem \begin{equation} b(\mu ^O,\eta ) = -dF(u)[\eta ] ~~ ∀\eta \in M(\Omega ) \tag{3.14} \end{equation} where $b(\cdot ,\cdot )$ is a coersive bilinear form \begin{equation} b(\eta ,\eta )\ge \alpha _b\|\eta \|_{1,\Omega }^2 ~~ ∀\eta \in M(\Omega ) \tag{3.15} \end{equation} with a constant $\alpha _b \gt 0$, and $M(\Omega )$ is a suitable subspace of $H^1(\Omega ;\mathbb{R}^d)$. Azegami [Az94, A-W96] chose for the bilinear form $b(\cdot ,\cdot )$ from linear elasticity. There is similar nonparametric method in French researcher, for example, Allaire[A-P06, Al07] use the bilinear form $b(\mu ,\eta ) =\int _{\Omega }\{\nabla \mu :\nabla \eta +\mu \cdot \eta \} dx$ to find shape optimization, and we also call it H1-gradient method here. By replacing $dF(u)[\mu ]$ in (3.14) with the shape sensitivities by GJ-integral, we have for any $\eta \in M(\Omega )$ when the conditions in Corollary 2.13 are satisfied, \begin{equation} b(\mu ^O,\eta )= \left \lbrace \array{R_{\Omega }(u,\eta )+\int _{\partial \Omega }fu(\eta \cdot n)ds& \textrm{(energy)}\\ -2\left \{R_{\Omega }(u,\eta )+\int _{\partial \Omega }fu(\eta \cdot n)ds\right \} & \textrm{(mean compliance)}\\ -\delta R_{\Omega }(u,\eta )[u_g]&\\ -\int _{\partial \Omega } (fu_g+\widehat{g}(u)-\nabla _z\widehat{g}(u)u)(\eta \cdot n)ds\\ (F(u)=\int _{\Omega }\widehat{g}(u)dx)&\\ }\right . \tag{3.16} \end{equation} where $u_g$ is the solution of the problem (3.8) . Here we notice that the right-hand side of (3.16) take the finite value for all weak solution of (2.1) . The solution $\mu ^O$ of (3.16) is unique. Putting $\varphi ^O_t=x+t\mu ^O$, we have \begin{eqnarray} F(u^{\varphi ^O_t})&=&F(u)+\left .t\frac{d}{dt}F(u^{\varphi ^O_t})\right |_{t=0}+o(t)\\ &=&F(u)-tb(\mu ^O,\mu ^O)+o(t)\notag \\ &\le &F(u)-t\alpha _b\|\mu ^O\|_{1,2,\Omega }^2+o(t)\notag \tag{3.17} \end{eqnarray} If $\mu ^O\neq 0$, we can take an appropriate number $\epsilon ^O$ such that $F(u^{\varphi _t^O}) \lt F(u), 0\le t\le \epsilon ^O$, so that $\varphi ^O=x+\epsilon ^O\mu ^O$ gives the optimization of the singular points with respect to the cost function $F$. We already indicated that $[\eta \mapsto R_{\Omega }(u,\eta )]$ is continuous on $W^{1,\infty }(\Omega ,\mathbb{R}^d)$, but in the H1-gradient method $[\eta \mapsto R_{\Omega }(u,\eta )]$ needs to be continuous on $H^1(\Omega ;\mathbb{R}^d)$. If there is some regularity such as $u\in H^s(\Omega ;\mathbb{R}^m),s \gt 1$, then we will extend $[\eta \mapsto R_{\Omega }(u,\eta )]$ to the continious functional on $H^1(\Omega ;\mathbb{R}^m)$. In numerical calculation of (3.16) , for example FEM, (3.16) is well-defined if $[u\mapsto R_{\Omega }(u_h,\mu )]$ give the good approximation of $[u\mapsto R_{\Omega }(u,\mu )]$ for a FE-approximation $u_h$ of $u$. In the interface problem (2.51) , H1-gradient method becomes as follows when the conditions in Corollary 2.18 are satisfied: \begin{equation} b(\mu ^O,\eta )= \left \lbrace \array{\sum _{\kappa =1}^K R_{\Omega _{\kappa }}(u,\eta )+\int _{\partial \Omega }fu(\eta \cdot n)ds& \textrm{(energy)}\\ -2\left \{\sum _{\kappa =1}^KR_{\Omega _{\kappa }}(u_{\kappa },\eta )+\int _{\partial \Omega }fu(\eta \cdot n)ds\right \} &\textrm{(mean compliance)}\\ -\sum _{\kappa =1}^K\delta R_{\Omega _{\kappa }}(u,\eta )[u_g]&\\     -\int _{\partial \Omega } (fu_g+\widehat{g}(u)-\nabla _z\widehat{g}(u)u)(\eta \cdot n)ds &(F=\int _{\Omega }\widehat{g}(u)dx) }\right . \tag{3.18} \end{equation} The H1-gradient of eigenvalue, when satisfying the conditions in Theorem 3.3 , is \begin{equation} b(\mu ^O,\eta )=2R_{\Omega }^E(u_{\lambda },\eta )+\lambda \int _{\partial \Omega }u_{\lambda }^2(\eta \cdot n)ds \tag{3.19} \end{equation} For the shape optimization of energy and compliance under Stokes problem, H1-gradient become \begin{equation} b(\mu ^O,\eta )= \left \lbrace \array{R_{\Omega }^S((u,p),\eta )+\int _{\partial \Omega }fu(\eta \cdot n)ds& \textrm{(energy)}\\ -2\left \{R_{\Omega }^S((u,p),\eta )+\int _{\partial \Omega }fu(\eta \cdot n)ds\right \} & \textrm{(mean compliance)} }\right . \tag{3.20} \end{equation}

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. [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.