×

zbMATH — the first resource for mathematics

A linearization technique for solving general 3-D shape optimization problems in spherical coordinates. (English) Zbl 1410.49049
Summary: Regarding the some useful advantages of spherical coordinates for some special problems, in this paper, based on Radon measure properties, we present a new and basic solution method for general shape optimization problems defined in spherical coordinates. Indeed, our goal is to determine a bounded shape located over the \((x,y)\)-plane, such that its projection in the \((x,y)\)-plane and its volume is given and also it minimizes some given surface integral. To solve these kinds of problems, we somehow extend the embedding process in Radon measures space. First, the problem is converted into an infinite-dimensional linear programming one. Then, using approximation scheme and a special way for discretization in spherical region, this problem is reduced to a finite-dimensional linear programming one. Finally, the solution of this new problem is used to construct a nearly optimal smooth surface by applying an outlier detection algorithm and curve fitting. More than reducing the complexity, this approach in comparison with the other methods has some other advantages: linear treatment for even nonlinear problems, and the minimization is global and does not depend on initial shape and mesh design. Numerical examples are also given to demonstrate the effectiveness of the new method, especially for classical and obstacle problems.
MSC:
49Q10 Optimization of shapes other than minimal surfaces
93B51 Design techniques (robust design, computer-aided design, etc.)
90C05 Linear programming
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Alimorad, H.: Introducing shape-measure method for designing 3-D optimal shapes. Ph. D Thesis mathematics, Shiraz University of Technology, Shiraz, Iran (2015)
[2] Allaire, G.; Jouve, F., A level-set method for variation and multiple loads structural optimization, Comput. Methods Appl. Mech. Eng., 194, 3269-3290, (2005) · Zbl 1091.74038
[3] Allalrf, G.; Jouv, F., A level-set method for variation and multiple loads structural optimization, Comput. Methods Appl. Mech. Eng., 194, 3269-3290, (2005) · Zbl 1091.74038
[4] Antonietti, P.F., Bigoni, N., Verani, M.: Mimetic finite difference method for shape optimization problems. In: Barth, T.J., Griebel, M., Keyes, D.E., Nieminen, R.M., Roose, D., Schlick, T. (eds.) Numerical Mathematics and Advanced Applications, pp. 125-132 (2013) · Zbl 1320.74111
[5] Burman, E.; Elfverson, D.; Hansbo, P.; Larson, MG; Larsson, K., Shape optimization using the cut finite element method, Comput. Methods Appl. Mech. Eng., 328, 242-261, (2018)
[6] Canelas, A.; Herskovits, J.; Telles, JCF, Shape optimization using the boundary element method and a SAND interior point algorithm for constrained optimization, Comput. Struct., 86, 1517-1526, (2008)
[7] Cheng, D.K.: Field and Wave Electromagnetics, 2nd edn. Addison-Wesley, Pearson (1989)
[8] Fakharzadeh, A.: Shapes, measure and elliptic equations. Ph.D Thesis Mathematics, University of Leeds, Leeds, England (1996)
[9] Fakharzadeh, A.; Rubio, JE, Best domain for an elliptic problem in cartesian coordinates by means of shape-measure, Asian J. Control, 11, 536-547, (2009)
[10] Farahi, MH; Borzabadi, AH; Mehneh, HH; Kamyad, AV, Measure theoretical approach for optimal shape design of a nozzle, J. Appl. Math. Comput., 17, 315-328, (2005) · Zbl 1060.49028
[11] Farahi, MH; Mehne, HH; Borzabadi, AH, Wing drag minimization by using measure theory, Optim. Methods Softw., 21, 169-177, (2006) · Zbl 1091.49033
[12] Friedman, A.: Variational Principles and Free Boundary Problems. Wiley, New York (1982) · Zbl 0564.49002
[13] Haslinger, J., Neittaanmaki, P.: Finite Element Approximation for Optimal Shape Design: Theory and Application. Wiley, New York (1988) · Zbl 0713.73062
[14] Haslinger, J., Neitaanmaki, P.: Finite Element Approximation for Optimal Shape, Material and Topology Design, 2nd edn. Wiley, Chichester (1996) · Zbl 0659.49005
[15] Hawkins, D.: Identication of Outliers. Chapman and Hall, London (1980)
[16] Khludnev, A.M., Sokolowski, J.: Modelling and Control in Solid Mechanics. Birkhauser, Basel (1997) · Zbl 0865.73003
[17] Kim, NH; Choi, KK; Botkin, ME, Numerical method for shape optimization using meshfree method, Struct. Multidiscip. Optim., 24, 418-429, (2002)
[18] Kriegel, H.-P., Kroger, P., Schubert, E., Zimek, A.: LoOP: local outlier probabilities. In: Proceedings of the ACM Conference on Information and knowledge Management (CIKM), Hong Kong, China (2009)
[19] Majava, K.; Tai, XC, A level set method for solving free boundary problems associated with obstacles, Int. J. Numer. Anal. Model., 1, 157-171, (2004) · Zbl 1082.65067
[20] Murat, F., Simon, J.: Optimal control with respect to the domain. Thesis (in French), University of Paris (1977)
[21] Murat, F., Simon, J.: Studies in Optimal Shape Design. Lecture Notes in Computer Science, vol. 41. Springer, Berlin (1976)
[22] Nazemi, AR; Farahi, MH, Shape optimization of an arterial bypass in cardiovascular systems, Iran. J. Oper. Res., 4, 127-145, (2013)
[23] Nitsche, J.C.C.: Introduction to Minimal Surfaces. Cambridge University Press, Cambridge (1989)
[24] Osserman, R.: Minimal Surfaces. Springer, Berlin (1997) · Zbl 0918.53003
[25] Pironneau, O., On optimal design in fluid mechanics, J. Fluid Mech., 64, 97-110, (1974) · Zbl 0281.76020
[26] Pironneau, O.: Optimal Shape Design for Elliptic Systems. Springer, New York (1983) · Zbl 0534.49001
[27] Rodrigues, J.F.: Obstacle Problems in Mathematical Physics. North-Holland, Amsterdam (1989)
[28] Royden, H.L.: Real Analysis, 3rd edn. Macmillan, New York (1988) · Zbl 0704.26006
[29] Rubio, J.E.: Control and Optimization; the Linear Treatment of Non-linear Problems. Manchester University Press, Manchester (1986) · Zbl 1095.49500
[30] Rudin, W.: Real and Complex Analysis, 2nd edn. Tata McGraw-Hill Publishing Co Ltd., New Dehli (1983) · Zbl 0142.01701
[31] Sokolowski, J., Zolesio, J.-P.: Introduction to Shape Optimization. Springer, Berlin (1992) · Zbl 0761.73003
[32] Thomas, G.B., Finney, R.L.: Calculus and Analytic Geometry, 9th edn. Addison-Wesley, Boston (1998) · Zbl 0766.26001
[33] Wang, D.; Sun, S.; Chen, Xi; Yu, Z., A 3D shape descriptor based on spherical harmonics through evolutionary optimization, Neurocomputing, 194, 183-191, (2016)
[34] Wilmott, P., Howison, S., Dewynne, J.: The Mathematics of Financial Derivative. Cambridge University Press, Cambridge (1995) · Zbl 0842.90008
[35] Wilson, DA; Rubio, JE, Existence of optimal controls for the diffusion equation, J. Optim. Theory Appl., 22, 91-101, (1977) · Zbl 0336.93046
[36] Young, L.C.: Calculus of Variations and Optimal Control Theory. AMS Chelsea Publishing, Philadelphia (1969) · Zbl 0177.37801
[37] Zhang, X.; Rayasam, M.; Subbarayan, G., A meshless, compositional approach to shape optimal design, Comput. Methods Appl. Mech. Eng., 196, 2130-2146, (2007) · Zbl 1173.74371
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.