What's the shape optimization

Let $\Omega $ be a bounded domain with Lipschitz boundary in $\mathbb{R}^d, d=2,3$ and $u(\Omega )$ a minimizer of an energy functional $\mathcal{E}$ over a subset $V(\Omega )$ in the function space $W^{1,p}(\Omega ;\mathbb{R}^m),m\ge 1$, Consider a cost functional $F(u(\Omega ))$. The shape optimization of boundary is a problem of finding a method of deforming a given set $\Omega $ so as to lower the cost function, and optimized shape $\Omega ^O$ is given by $$ F(u(\Omega ^O)) \lt F(u(\Omega ))$$

A mathematician asked me whether $\Omega ^O$ gives the minimum or a local minimum? In practical problems, it is not important to find the minimizer of the cost functions. Consider windows as an example. The minimizer of energy functional of Poisson problem $-\Delta u=1$ with Dirichlet boudary is the circle if the area is same. This indicates that the influence of external forces is uniform in round windows. However, many windows are square shaped because they provide a wider field of view, are easier to make and have fewer losses to cut out. On the other hand, windows on ships that require strength are round windows.

In practical problems it is based on multiple evaluations and is not created by a single evaluation. Here, the optimisation problem is the problem to find lowering the cost functional from mathematical view. With advances in computers, the study of shape optimization has become interesting. Recent mathematical research is dominated by topological optimisation, level set methods, etc., but here the traditional methods based on shape sensitivity analysis are used as a basis. It differs from traditional research that we obtain interesting results due to the singularity of the solution.


$\Omega $:
Bounded domain with Lipschitz boundaryu,
Variational problems:
"$u\in \arg \min _{v\in V(\Omega )}\int _{\Omega }\{\widehat{W}(v)-fv\} -\int _{\Gamma _N}gv$" means that $u$ is the minimizer of the functional $\mathcal{E}(v;f,g,\Omega ) :=\int _{\Omega }\{\widehat{W}(v)-fv\} -\int _{\Gamma _N}gv$ over the set $V(\Omega )$ of functions, where $\Gamma _N$ is the part of $\partial \Omega $, $f,g$ are given functions.
Shape sensitivity:
Let $[t\mapsto \varphi _t ]\in C^2([0,\epsilon _0 );C^{0,1}(\mathbb{R}^d;\mathbb{R}^d))$, such that $\varphi _0(x)=x$.
Under construction


[Adams] R.A.Adams, Sobolev spaces, Academic Press, 1975.
[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) %\url
[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.