Contents

# Shape optimization

## Procedure of FE-calculation

Energy minimization.
Step1
Calculate the finite element approximation $u_h$ of ((2.1) ).
Step2
Set $M(\Omega )$, for example, determine the parts $\Gamma _F, \Gamma _V, \Gamma _H$ of $\partial \Omega$ where the shape is fixed on $\Gamma _F$, fixed vertical direction on $\Gamma _V$, and fixed the horizontal direction on $\Gamma _H$. In this case $M(\Omega )$ is defined by \begin{eqnarray*} \{\eta \in H^1(\mathbb{R}^d;\mathbb{R}^d):\, &\eta =0\textrm{ on }\Gamma _F,\, &\eta \cdot n=0\textrm{ on }\Gamma _V,\,\\ &&\eta -(\eta \cdot n)n=0\textrm{ on }\Gamma _H\} , \end{eqnarray*} Use the H1-gradient method; calculate the FE-approximation $\mu ^O_h$ of (3.16) by \begin{eqnarray} b_h(\mu ^O_h,\eta _h)&=& R_{\Omega }(u_h,\eta _h)+ \int _{\partial \Omega }fu_h(\eta _h\cdot n)ds ~~ ∀\eta _h\in M(\Omega ) \tag{3.28} \end{eqnarray} Here $b_h(\cdot ,\cdot )$ is an FE-approximation of $b_{E}$ or $b_{A}$ (e.g. [A-P06, Al07]) \begin{eqnarray*} b_{E}(\mu ,\eta )&=&\int _{\Omega }(\nabla \mu :\nabla \eta +\mu \cdot \eta )dx \\ b_{A}(\mu ,\eta )&=&\int _{\Omega }\left \{\frac{1}{2}\sigma _{ij}(\mu )\varepsilon _{ij}(\eta )+\mu \cdot \eta \right \} dx\notag \\ \sigma _{ij}(\mu )&=&\lambda \textrm{div}\mu \delta _{ij}+2G\varepsilon _{ij}(\mu ), \, \varepsilon _{ij}(\mu )=\frac{1}{2}\left (\frac{\partial \mu _{i}}{\partial x_j} +\frac{\partial \mu _{h,j}}{\partial x_i}\right )\notag \\ \lambda &=& \frac{E\nu }{(1+\nu )(1-2\nu )}, G=\frac{E}{2(1+\nu )}\\ &&(E:\textrm{Young's modulus},\, \nu :\textrm{Poisson ratio}) \end{eqnarray*} $b_A$ is complex, but we can control deformation by controlling the constants $\lambda$ and $G$.
Step3
Calculate the constraint by the H1-gradient method. For example, if the volume (area) is constant, calculate $\mu ^c_h$ by \begin{equation} b_h(\mu ^c_h,\eta _h) =-\int _{\Omega }\textrm{div}\eta _h\, dx ~~ ∀\eta _h\in M(\Omega ) \tag{3.29} \end{equation}
Step4
Incorporate constraint conditions using the Lagrangian multiplier method. If the volume (area) is constant, calculate $$\lambda =-\ell _o/\ell _c,~ \ell _o=\int _{\Omega }\textrm{div }\mu ^O_h\, dx,~ \ell _c=\int _{\Omega }\textrm{div }\mu ^c_h\, dx,$$ and set $\mu ^O_h=\mu ^O_h+\lambda \mu ^c_h$.
Step5
Properly take $\epsilon \gt 0$ and make a new mesh $\mathcal{T}_h(\Omega ^{\varphi _{\epsilon }})$ $$\Omega ^{\varphi _{\epsilon }}=\{\varphi _{\epsilon }(x):\, x\in \Omega \} ,\, \varphi _{\epsilon }(x)=x+\epsilon \mu ^O_h(x)$$ Here $\epsilon \gt 0$ must be taken to be that $\mathcal{E}(u^{\varphi _\epsilon }_h;f,g,\Omega ^{\varphi _\epsilon }) \le \mathcal{E}(u_h;f,g,\Omega )$. If the inequality does not satisfy, we need to take a smaller $\epsilon$.
Step6
By repeating (Step1)--(Step5), we can get shapes that are more optimized.
In the actual calculation, the mesh $\mathcal{T}_h (\Omega ^{\varphi _{\epsilon }})$ created by moving $q^l+\epsilon \mu ^O_h(q^l)$ for each vertex $q^l,l=1,…,n_v$ we often find triangle elements collapse. Therefore, it is necessary to remake the mesh (examples are in Section 4.1 ).
interface problems
For the interface problems (Section 2.5.3 , also called transmission problems), it is necessary to change (1)--(3) in just above as follows.
Step1
Calculate the finite element approximation $u_h$ of ((2.50) ).
Step2
Use the H1-gradient method; calculate the FE-approximation $\mu ^O_h$ by (3.18) \begin{eqnarray} b_h(\mu ^O_h,\eta _h)&=& \sum _{\kappa =1}^K R_{\Omega _{\kappa }}(u_h,\eta _h)+ \int _{\partial \Omega }fu_h(\eta _h\cdot n)ds ~~ ∀\eta _h\in M(\Omega ) \tag{3.30} \end{eqnarray}
Step3
Calculate the constraint by the H1-gradient method. In interface problems, we can consider the case that the shape of the boundary $\partial \Omega$ is fixed, but the interfaces $(\partial \Omega _{\kappa }\cap \partial \Omega _l)\setminus \partial \Omega , \kappa \neq l$ can move,

Mean compliance.
Only change (3.28) to \begin{equation} b_h(\mu _h^O,\eta _h)=-2\left \{R_{\Omega }(u_h,\eta _h)+\int _{\partial \Omega } fu_h(\eta _h\cdot n)ds\right \} \tag{3.31} \end{equation} Examples are in Section 4.3 , and the case of the interface is in Section 4.4 .

Least mean-square error, etc.
The shape derivative of the cost function $\int _{\Omega }\widehat{g}(u)dx$ is represented by $\delta R_{\Omega }(u,\mu _{\varphi })[u_g]$ (3.7) using the adjoint variable $u_g$.
\li Calculate $u_{h}$ of the problem (2.1) . \li Calculate FE-approximation $u_{g,h}$ of the problem (3.8) $$\mathcal{E}(u_{g,h};\nabla _z\widehat{g}(u_h),0,\Omega ) =\min _{v_h\in V_h(\Omega )} \mathcal{E}(v_h;\nabla _z\widehat{g}(u_h),0,\Omega )$$ \li Use H1-gradient method such as \begin{eqnarray} &&-b_h(\mu ^O_h,\eta _h)=\delta R_{\Omega }(u_h,\eta _h)[u_{g,h}]\\ &&~+\int _{\partial \Omega }(fu_{g,h}+\widehat{g}(u_h)-\nabla _z\widehat{g}(u_h)u_h)(\eta _h\cdot n)ds ~~ ∀\eta _h\in M(\Omega )\notag \tag{3.32} \end{eqnarray} \li Incorporate constraint conditions using Lagrangian multiplier method.
Examples are in Section 4.2 .

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.