Two-dimensional shape optimization with nearly conformal transformations. (English) Zbl 1403.49043
Summary: In shape optimization it is desirable to obtain deformations of a given mesh without negative impact on the mesh quality. We propose a new algorithm using least square formulations of the Cauchy-Riemann equations. Our method allows us to deform meshes in a nearly conformal way and thus approximately preserves the angles of triangles during the optimization process. The performance of our methodology is shown by applying our method to some unconstrained shape functions and a constrained Stokes shape optimization problem.
MSC:
| 49Q10 | Optimization of shapes other than minimal surfaces |
| 93B40 | Computational methods in systems theory (MSC2010) |
| 65D99 | Numerical approximation and computational geometry (primarily algorithms) |
| 46E22 | Hilbert spaces with reproducing kernels (= (proper) functional Hilbert spaces, including de Branges-Rovnyak and other structured spaces) |
Keywords:
PDE constrained shape optimization; grid deformation; Cauchy-Riemann equations; conformal mappingsSoftware:
FiredrakeReferences:
| [1] | H. Attouch,, Viscosity solutions of minimization problems, SIAM J. Optim., 6 (1996), pp. 769–806, . · Zbl 0859.65065 |
| [2] | H. Begehr, Boundary value problems in complex analysis I, Bol. Asoc. Mat. Venez., 12 (2005), pp. 65–85. · Zbl 1260.30021 |
| [3] | J. Conway, Functions of One Complex Variable I, Springer, New York, Berlin, 1978. |
| [4] | M. C. Delfour and J.-P. Zolésio, Shapes and Geometries: Metrics, Analysis, Differential Calculus, and Optimization, 2nd ed., SIAM, Philadelphia, 2011, . · Zbl 1251.49001 |
| [5] | R. P. Dwight, Robust mesh deformation using the linear elasticity equations, in Computational Fluid Dynamics 2006, Springer, Berlin, Heidelberg, 2009, pp. 401–406. |
| [6] | M. Eigel and K. Sturm, Reproducing kernel Hilbert spaces and variable metric algorithms in PDE-constrained shape optimization, Optim. Methods Softw., (2018), pp. 268–296. · Zbl 1388.35036 |
| [7] | K. Eppler, S. Schmidt, V. Schulz, and C. Ilic, Preconditioning the pressure tracking in fluid dynamics by shape Hessian information, J. Optim. Theory Appl., 141 (2009), pp. 513–531. · Zbl 1178.49054 |
| [8] | T. Gamelin, Complex Analysis, Springer Science & Business Media, New York, 2003. |
| [9] | J. D. Gray and S. A. Morris, When is a function that satisfies the Cauchy-Riemann equations analytic?, Amer. Math. Monthly, 85 (1978), pp. 246–256. · Zbl 0416.30002 |
| [10] | X. Gu and S.-T. Yau, Global conformal surface parameterization, in Proceedings of the 2003 Eurographics/ACM SIGGRAPH Symposium on Geometry Processing, SGP ’03, Eurographics Association, Aire-la-Ville, Switzerland, 2003, pp. 127–137. |
| [11] | B. T. Helenbrook, Mesh deformation using the biharmonic operator, Internat. J. Numer. Methods Engrg., 56 (2003), pp. 1007–1021. · Zbl 1047.76044 |
| [12] | P. Hood and C. Taylor, Navier-Stokes equations using mixed interpolation, in Finite Element Methods in Flow Problems, UAH Press, Huntsville, AL, 1974, pp. 121–132. |
| [13] | K. Ito, K. Kunisch, and G. H. Peichl, Variational approach to shape derivatives, ESAIM Control Optim. Calc. Var., 14 (2008), pp. 517–539. · Zbl 1357.49148 |
| [14] | T. Iwaniec, L. V. Kovalev, and J. Onninen, The Nitsche conjecture, J. Amer. Math. Soc., 24 (2011), pp. 345–373. · Zbl 1214.31001 |
| [15] | T. Iwaniec and G. Martin, Geometric Function Theory and Non-linear Analysis, The Clarendon Press, Oxford University Press, 2001. · Zbl 1045.30011 |
| [16] | G. K. Karpouzas, E. M. Papoutsis-Kiachagias, T. Schumacher, E. de Villiers, K. C. Gianna Koglon, and C. Othmer, Adjoint optimization for vehicle external aerodynamics, Int. J. Automot. Eng., 7 (2016), pp. 1–7. |
| [17] | V. A. Kondrat’ev and O. A. Ole\uinik, Boundary value problems for a system in elasticity theory in unbounded domains. Korn inequalities, Russian Math. Surveys, 43 (1988), pp. 65–119, . · Zbl 0669.73005 |
| [18] | A. Laurain and K. Sturm, Distributed shape derivative via averaged adjoint method and applications, ESAIM Math. Model. Numer. Anal., 50 (2016), pp. 1241–1267. · Zbl 1347.49070 |
| [19] | B. Lévy, S. Petitjean, N. Ray, and J. Maillot, Least squares conformal maps for automatic texture atlas generation, ACM Trans. Graph., 21 (2002), pp. 362–371, . |
| [20] | B. Mohammadi and O. Pironneau, Applied Shape Optimization for Fluids, Oxford University Press, Oxford, UK, 2010. · Zbl 0970.76003 |
| [21] | F. Murat and J. Simon, Sur le contrôle par un domaine géométrique, Tech. report, 1976. |
| [22] | O. Pironneau, On optimum design in fluid mechanics, J. Fluid Mech., 64 (1974), pp. 97–110. · Zbl 0281.76020 |
| [23] | F. Rathgeber, D. A. Ham, L. Mitchell, M. Lange, F. Luporini, A. T. T. Mcrae, G.-T. Bercea, G. R. Markall, and P. H. J. Kelly, Firedrake: Automating the finite element method by composing abstractions, ACM Trans. Math. Software, 43 (2017), no. 3, 24. · Zbl 1396.65144 |
| [24] | D. Ridzal and D. P. Kouri, Rapid Optimization Library, Tech. report, Sandia National Laboratories (SNL-NM), Albuquerque, NM, 2014. |
| [25] | S. Schmidt, Efficient Large Scale Aerodynamic Design Based on Shape Calculus, Ph.D. thesis, University of Trier, Trier, Germany, 2010, . |
| [26] | S. Schmidt, A Two Stage CVT/Eikonal Convection Mesh Deformation Approach for Large Nodal Deformations, preprint, , 2014. |
| [27] | S. Schmidt, V. Schulz, C. Ilic, and N. Gauger, Large scale aerodynamic shape optimization based on shape calculus, in ECCOMAS-CFD&Optimization, I. H. Tuncer, ed., 2011. · Zbl 1284.76330 |
| [28] | F. Schottky, Ueber die conforme Abbildung mehrfach zusammenhängender ebener Flächen, J. Reine Angew. Math., 83 (1877), pp. 300–351. · JFM 09.0584.02 |
| [29] | V. Schulz and M. Siebenborn, Computational comparison of surface metrics for PDE constrained shape optimization, Comput. Methods Appl. Math., 16 (2016), pp. 485–496. · Zbl 1342.49065 |
| [30] | V. H. Schulz, A Riemannian view on shape optimization, Found. Comput. Math., 14 (2014), pp. 483–501. · Zbl 1296.49037 |
| [31] | A. Segall and M. Ben-Chen, Iterative closest conformal maps between planar domains, in Proceedings of the Symposium on Geometry Processing, SGP ’16, Eurographics Association, Goslar Germany, 2016, pp. 33–40, . |
| [32] | R. E. Smith and L.-E. Eriksson, Algebraic grid generation, Comput. Methods Appl. Mech. Engrg., 64 (1987), pp. 285–300. · Zbl 0609.65083 |
| [33] | J. Sokołowski and J.-P. Zolésio, Introduction to Shape Optimization. Shape Sensitivity Analysis, Springer, Berlin, 1992. · Zbl 0761.73003 |
| [34] | S. Steinberg, Fundamentals of Grid Generation, CRC Press, Boca Raton, FL, 1993. · Zbl 0855.65123 |
| [35] | K. Sturm, Minimax Lagrangian approach to the differentiability of nonlinear PDE constrained shape functions without saddle point assumption, SIAM J. Control Optim., 53 (2015), pp. 2017–2039, . · Zbl 1327.49073 |
| [36] | K. Sturm, Shape differentiability under non-linear PDE constraints, in New Trends in Shape Optimization, Internat. Ser. Numer. Math. 166, Birkhäuser/Springer, Cham, 2015, pp. 271–300. · Zbl 1329.49097 |
| [37] | A. Vaxman, C. Müller, and O. Weber, Conformal mesh deformations with Möbius transformations, ACM Trans. Graph., 34 (2015), 55, . · Zbl 1334.68271 |
| [38] | A. Zochowski and P. Holnicki, Interactive method of variational grid generation, J. Comput. Appl. Math., 26 (1989), pp. 281–287. · Zbl 0681.65088 |
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
