Methods and Algorithms for Scientific Computing

Shape Optimization Using the Finite Element Method on Multiple Meshes with Nitsche Coupling

Abstract

An important step in shape optimization with partial differential equation constraints is to adapt the geometry during each optimization iteration. Common strategies are to employ mesh deformation or remeshing, where one or the other typically lacks robustness or is computationally expensive. This paper proposes a different approach, in which the computational domain is represented by multiple, independent nonmatching meshes. The individual meshes can move independently, hence mesh deformation or remeshing is entirely avoided if the geometry can be parameterized by a combination of rigid motions and scaling. For general geometry changes, we present a deformation scheme tailored to nonmatching meshes. This deformation scheme is robust because the nonmatching mesh interfaces are free to move and computationally cheap because the scheme is applied only on a subset of the meshes. To solve the state and corresponding adjoint equations we use the multimesh finite element method. This method weakly enforces continuity over the nonmatching mesh interfaces by using Nitsche and additional stability terms. To obtain the shape derivatives we analyze both the strong formulation (Hadamard formulation) and weak formulation (method of mappings). We demonstrate the capabilities of our approach on the optimal placement of heat emitting wires in a cable to minimize the chance of overheating, the drag minimization in Stokes flow, and the orientation of nine objects in Stokes flow.

Keywords

  1. shape optimization
  2. multimesh finite element methods
  3. Hadamard representation

MSC codes

  1. 35Q93
  2. 49Q10
  3. 65M85
  4. 65N30
  5. 68N99

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References

1.
M. Aln\aes, J. Blechta, J. Hake, A. Johansson, B. Kehlet, A. Logg, C. Richardson, J. Ring, M. E. Rognes, and G. N. Wells, The fenics project version 1.5, Arch. Numer. Softw., 3 (2015), pp. 9--23.
2.
P. R. Amestoy, I. S. Duff, J. Koster, and J.-Y. L'Excellent, A fully asynchronous multifrontal solver using distributed dynamic scheduling, SIAM J. Matrix Anal. Appl., 23 (2001), pp. 15--41.
3.
K. Bandara, T. Rüberg, and F. Cirak, Shape optimisation with multiresolution subdivision surfaces and immersed finite elements, Comput. Meth. Appl. Mech. Engrg., 300 (2016), pp. 510--539.
4.
D. M. Belk, The role of overset grids in the development of the general purpose CFD code, in Surface Modeling, Grid Generation, and Related Issues in Computational Fluid Dynamic (CFD) Solutions, NASA Lewis Research Center, Cleveland, OH, 1995, pp. 193--204.
5.
J. Benek, J. Steger, and F. C. Dougherty, A flexible grid embedding technique with application to the euler equations, in Proceedings of the 6th Computational Fluid Dynamics Conference, Danvers, 1983, p. 1944.
6.
J. Benk, H.-J. Bungartz, M. Mehl, and M. Ulbrich, Immersed boundary methods for fluid-structure interaction and shape optimization within an FEM-based PDE toolbox, in Advanced Computing, Springer, New York, 2013, pp. 25--56.
7.
M. Berggren, A unified discrete--continuous sensitivity analysis method for shape optimization, in Applied and Numerical Partial Differential Equations, Springer, New York, 2010, pp. 25--39.
8.
A. Bernland, E. Wadbro, and M. Berggren, Acoustic shape optimization using cut finite elements, Int. J. Numer. Meth. Engrg., 113 (2018), pp. 432--449.
9.
P. T. Boggs, A. Althsuler, A. R. Larzelere, E. J. Walsh, R. L. Clay, and M. F. Hardwick, Dart system analysis, tech. report, Sandia National Laboratories, 2005.
10.
T. Borrvall and J. Petersson, Topology optimization of fluids in Stokes flow, Int. J. Numer. Methods Fluids, 41 (2003), pp. 77--107.
11.
F. Brezzi, J.-L. Lions, and O. Pironneau, Analysis of a Chimera method, Comptes Rendus de l'Academie des Sciences Series I Mathematics, 332 (2001), pp. 655--660.
12.
E. Burman, S. Claus, P. Hansbo, M. G. Larson, and A. Massing, CutFEM: Discretizing geometry and partial differential equations, Int. J. Numer. Meth. Engrg., 104 (2015), pp. 472--501.
13.
E. Burman, D. Elfverson, P. Hansbo, M. G. Larson, and K. Larsson, Shape optimization using the cut finite element method, Comput. Meth. Appl. Mech. Engrg., 328 (2018), pp. 242--261.
14.
W. Chan, R. Gomez, S. Rogers, and P. Buning, Best practices in overset grid generation, in Proceedings of the 32nd AIAA Fluid Dynamics Conference and Exhibit, 2002, p. 3191.
15.
M. C. Delfour and J.-P. Zolésio, Shapes and Geometries: Metrics, Analysis, Differential Calculus, and Optimization, SIAM, Philadelphia, 2011.
16.
S. W. Funke, P. E. Farrell, and M. Piggott, Tidal turbine array optimisation using the adjoint approach, Renewable Energy, 63 (2014), pp. 658--673.
17.
C. Geuzaine and J.-F. Remacle, Gmsh: A 3-d finite element mesh generator with built-in pre- and post-processing facilities, Int. J. Numer. Meth. Engrg., 79 (2009), pp. 1309--1331.
18.
C. V. Hansen, A. Logg, and C. Lundholm, Simulation of flow and view with applications in computational design of settlement layouts, preprint, arXiv:1610.02277, 2016, https://arxiv.org/abs/1610.02277.
19.
H. Harbrecht and F. Loos, Optimization of current carrying multicables, Comput. Optim. Appl., 63 (2016), pp. 237--271, https://doi.org/10.1007/s10589-015-9764-2.
20.
W. Henshaw, Overture: An object-oriented framework for overlapping grid applications, in Proceedings of the 32nd AIAA Fluid Dynamics Conference and Exhibit, 2002, p. 3189.
21.
M. Hinze, R. Pinnau, M. Ulbrich, and S. Ulbrich, Optimization with PDE constraints, vol. 23, Springer Science & Business Media, New York, 2008.
22.
A. Johansson, B. Kehlet, M. G. Larson, and A. Logg, MultiMesh finite element methods: Solving PDEs on multiple intersecting meshes, Comput. Meth. Appl. Mech. Engrg., 343 (2019), pp. 672--689.
23.
A. Johansson and M. G. Larson, A high order discontinuous Galerkin Nitsche method for elliptic problems with fictitious boundary, Numerische Mathematik, 123 (2013), pp. 607--628.
24.
A. Johansson, M. G. Larson, and A. Logg, High order cut finite element methods for the Stokes problem, Adv. Model. Simulation Engrg. Sci., 2 (2015), 24.
25.
W. Liao and H. M. Tsai, Aerodynamic design optimization by the adjoint equation method on overset grids, in Proceedings of the 44th AIAA Aerospace Sciences Meeting and Exhibit, 2006, p. 54.
26.
A. Logg and G. N. Wells, Dolfin: Automated finite element computing, ACM Trans. Math. Softw., 37 (2010), 20.
27.
F. Loos, Joule Heating in Connecting Structures of Automotive Electric Devices--Modelling, Simulation and Optimization, PhD thesis, München, Univ. der Bundeswehr, Diss., 2014, 2014.
28.
B. Mohammadi and O. Pironneau, Applied shape optimization for fluids, Oxford University Press, Oxford, 2010.
29.
G. Mosetti, C. Poloni, and B. Diviacco, Optimization of wind turbine positioning in large windfarms by means of a genetic algorithm, J. Wind Engrg. Industrial Aerodynamics, 51 (1994), pp. 105--116.
30.
F. Murat and J. Simon, Etude de problèmes d'optimal design, in IFIP Technical Conference on Optimization Techniques, Springer, New York, 1975, pp. 54--62.
31.
A. R. Najafi, M. Safdari, D. A. Tortorelli, and P. H. Geubelle, A gradient-based shape optimization scheme using an interface-enriched generalized FEM, Comput. Meth. Appl. Mech. Engrg., 296 (2015), pp. 1--17, https://doi.org/10.1016/j.cma.2015.07.024.
32.
E. J. Nielsen and M. A. Park, Using an adjoint approach to eliminate mesh sensitivities in computational design, AIAA J., 44 (2006), pp. 948--953.
33.
J. Nitsche, Über ein Variationsprinzip zur Lösung von Dirichlet-Problemen bei Verwendung von Teilräumen, die keinen Randbedingungen unterworfen sind, in Abhandlungen aus dem mathematischen Seminar der Universität Hamburg, vol. 36, Springer, New York, 1971, pp. 9--15.
34.
L. Noël and P. Duysinx, Shape optimization of microstructural designs subject to local stress constraints within an xfem-level set framework, Structural and Multidisciplinary Optimization, 55 (2017), pp. 2323--2338.
35.
T. E. Oliphant, Python for scientific computing, Comput. Sci. Engrg., 9 (2007), pp. 10--20.
36.
O. Pironneau, On optimum design in fluid mechanics, J. Fluid Mech., 64 (1974), pp. 97--110.
37.
O. Pironneau, Optimal shape design for elliptic systems, Springer Science & Business Media, New York, 2012.
38.
J. J. Reuther, A. Jameson, J. J. Alonso, M. J. Rimlinger, and D. Saunders, Constrained Multipoint Aerodynamic Shape Optimization Using an Adjoint Formulation and Parallel Computers, part 1, Journal of Aircraft, 36 (1999), pp. 51--60.
39.
N. Schlömer, G. Cervone, Antonio amd McBain, F. Gokstorp, M. Bussonnier, S. Chen, N. Wagner, and N. Sime, Pygmsh. Zenodo, 2018, https://doi.org/10.5281/zenodo.1173105.
40.
N. Schlömer, N. Wagner, L. Dalcin, T. Li, L. Schwarz, C. Coutinho, G. McBain, A. Cervone, T. Langlois, S. Peak, M. Bussonnier, G. Jacquenot, G. A. Vaillant, and A. Croucher, Meshio. Zenodo, 2018, https://doi.org/10.5281/zenodo.1173115.
41.
S. Schmidt, Efficient Large Scale Aerodynamic Design Based on Shape Calculus, Ph.D. thesis, University of Trier, Germany, 2010.
42.
S. Schmidt, A Two Stage CVT/Eikonal Convection Mesh Deformation Approach for Large Nodal Deformations, preprint, arXiv:1411.7663, 2014, https://arxiv.org/abs/1411.7663.
43.
S. Schmidt, C. Ilic, V. Schulz, and N. R. Gauger, Three-dimensional large-scale aerodynamic shape optimization based on shape calculus, AIAA J., 51 (2013), pp. 2615--2627.
44.
S. Schmidt, E. Wadbro, and M. Berggren, Large-scale three-dimensional acoustic horn optimization, SIAM J. Sci. Comput., 38 (2016), pp. B917--B940.
45.
V. Schulz and M. Siebenborn, Computational comparison of surface metrics for PDE constrained shape optimization, Comput. Meth. Appl. Math., 16 (2016), pp. 485--496.
46.
J. Sokolowski and J.-P. Zolesio, Introduction to shape optimization, in Introduction to Shape Optimization, Springer, New York, 1992, pp. 5--12.
47.
R. Stangl, Ein Euler-Verfahren zur Berechnung der Strömung um einen Hubschrauber im Vorwärtsflug, Herbert Utz Verlag, New York, 1996.
48.
G. Starius, Composite mesh difference methods for elliptic boundary value problems, Numerische Mathematik, 28 (1977), pp. 243--258.
49.
J. Steger, The Chimera method of flow simulation, in Workshop on applied CFD, Vol. 188, Univ of Tennessee Space Institute, 1991.
50.
J. L. Steger, F. C. Dougherty, and J. A. Benek, A Chimera grid scheme.[multiple overset body-conforming mesh system for finite difference adaptation to complex aircraft configurations], in Advances in Grid Generation: Proceedings of the Applied Mechanics, Bioengineering, and Fluids Engineering Conference, Houston, TX, June 20--22, 1983 (A84-11576 02-64), American Society of Mechanical Engineers, New York, 1983, pp 59--69.
51.
K. Stein, T. Tezduyar, and R. Benney, Mesh moving techniques for fluid-structure interactions with large displacements, J. Appl. Mech., 70 (2003), pp. 58--63.
52.
E. Tekin and I. Sabuncuoglu, Simulation optimization: A comprehensive review on theory and applications, IIE Trans., 36 (2004), pp. 1067--1081.
53.
L. Van Miegroet, N. Moës, C. Fleury, and P. Duysinx, Generalized shape optimization based on the level set method, in Proceedings of the 6th World Congress of Structural and Multidisciplinary Optimization (WCSMO6), 2005.
54.
C. H. Villanueva and K. Maute, Cutfem topology optimization of $3$d laminar incompressible flow problems, Comput. Meth. Appl. Mech. Engrg., 320 (2017), pp. 444--473.
55.
E. Volkov, The method of composite meshes, Automat. Program. Numer. Meth. Funct. Anal., 96 (1970), pp. 145--185.
56.
A. Wächter and L. T. Biegler, On the implementation of an interior-point filter line-search algorithm for large-scale nonlinear programming, Math. Program., 106 (2006), pp. 25--57.

Information & Authors

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Published In

cover image SIAM Journal on Scientific Computing
SIAM Journal on Scientific Computing
Pages: A1923 - A1948
ISSN (online): 1095-7197

History

Submitted: 23 May 2018
Accepted: 13 March 2019
Published online: 20 June 2019

Keywords

  1. shape optimization
  2. multimesh finite element methods
  3. Hadamard representation

MSC codes

  1. 35Q93
  2. 49Q10
  3. 65M85
  4. 65N30
  5. 68N99

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