An explicit level set approach for generalized shape optimization of fluids with the lattice Boltzmann method

International Journal for Numerical Methods in Fluids

Volumn 65, Issue 5, 2011, Pages 496-519

  Kreissl, Sebastian ^a   Pingen, Georg ^b   Maute, Kurt ^a  

^a University of Colorado at Boulder   (United States)

^b University of Colorado at Colorado Springs   (United States)

Author keywords

Adjoint sensitivity analysis; Immersed boundary technique; Interpolation boundary condition; Nonlinear programming; Regularity control

Indexed keywords

ADJOINT SENSITIVITY ANALYSIS; BOX CONSTRAINTS; DESIGN VARIABLES; FILTERING METHOD; FLUID-SOLID INTERFACES; FLUID-STRUCTURE INTERFACES; HAMILTON JACOBI EQUATIONS; IMMERSED BOUNDARY TECHNIQUE; INTERPOLATION SCHEMES; LATTICE BOLTZMANN METHOD; LEVEL SET; LEVEL SET APPROACH; LEVEL SET METHOD; LEVEL SET REPRESENTATION; NONLINEAR PROGRAMMING METHODS; OPTIMIZATION APPROACH; PENALIZATION METHOD; REGULARITY CONTROL; SECOND ORDERS; SMOOTH BOUNDARY; STEADY-STATE FLOWS; TOPOLOGY OPTIMIZATION;

BOUNDARY CONDITIONS; DROP FORMATION; FLUIDS; INTERPOLATION; NONLINEAR EQUATIONS; NONLINEAR PROGRAMMING; NUMERICAL METHODS; SENSITIVITY ANALYSIS;

SHAPE OPTIMIZATION;

EID: 78651243524     PISSN: 02712091     EISSN: 10970363     Source Type: Journal    
DOI: 10.1002/fld.2193     Document Type: Article

References (57)

1. Bendsøe M, Sigmund O. Topology Optimization: Theory, Methods and Applications. Springer: Berlin, 2003.
2. Bruns T, Sigmund O, Tortorelli D. Numerical methods for the topology optimization of structures that exhibit snap-through. International Journal for Numerical Methods in Engineering 2002; 55:1215-1237. DOI: 10.1002/nme.544.
3. Haber R, Bendsøe M. Problem formulation, solution procedures and geometric modeling: key issues in variable-topology optimization. AIAA/USAF/NASA/ISSMO Symposium on Multidisciplinary Analysis and Optimization, No. AIAA-1998-4948 in Collection of Technical Papers. Pt. 3. AIAA: New York, 1998.
4. Ramm E, Maute K, Schwarz S. Adaptive topology and shape optimization. WCCM IV, Buenos Aires, Argentina, 29 June-2 July 1998; 19-38.
5. Ramm E, Maute K, Schwarz S. Conceptual design by structural optimization. EURO-C, Badgastein, Austria, 31 March-3 April 1998; 896-897.
6. Borrvall T, Petersson J. Topology optimization of fluids in stokes flow. International Journal for Numerical Methods in Fluids 2003; 41(1):77-107. DOI: 10.1002/fld.426.
7. Gersborg-Hansen A, Sigmund O, Haber R. Topology optimization of channel flow problems. Structural and Multidisciplinary Optimization 2005; 30(3):181-192. DOI: 10.1007/s00158-004-0508-7.
8. Guest JK, Prévost JH. Topology optimization of creeping fluid flows using a Darcy-Stokes finite element. International Journal for Numerical Methods in Engineering 2006; 66:461-484. DOI: 10.1002/nme.1560.
9. Othmer C. CFD topology and shape optimization with adjoint methods. VDI Fahrzeug- und Verkehrstechnik, 13. Internationaler Kongress, Berechnung und Simulation im Fahrzeugbau, Würzburg, 2006. Available from: .
10. Othmer C, de Villiers E, Weller HG. Implementation of a continuous adjoint for topology optimization of ducted flows. Eighteenth AIAA Computational Fluid Dynamics Conference. AIAA: Miami, FL, 2007.
11. Pingen G, Evgrafov A, Maute K. Topology optimization of flow domains using the lattice Boltzmann method. Structural and Multidisciplinary Optimization 2007; 36(6):507-524. DOI: 10.1007/s00158-007-0105-7.
12. Brinkman H. A calculation of the viscous force exerted by a flowing fluid on a dense swarm of particles. Applied Scientific Research, Section A 1947; 1:27.
13. De Ruiter M, van Keulen F. Topology optimization using a topology description function. Structural and Multidisciplinary Optimization 2004; 26(6):406-416. DOI: 10.1007/s00158-003-0375-7.
14. Raulli M, Maute K. Optimization of fully coupled electrostatic-fluid-structure interaction problems. Computers and Structures 2005; 83:221-233.
15. Schwarz S, Maute K, Ramm E. Topology and shape optimization for elastoplastic structural response. Computer Methods in Applied Mechanics and Engineering 2001; 190:2135-2155.
16. Succi S. The Lattice Boltzmann Equation: For Fluid Dynamics and Beyond. Oxford University Press: Oxford, 2001.
17. Hammer V, Olhoff N. Topology optimization of continuum structures subjected to pressure loading. Structural and Multidisciplinary Optimization 2000; 19:85-92.
18. Maute K, Ramm E. Adaptive topology optimization. Structural Optimization 1995; 10:100-112.
19. Pingen G. Optimal design for fluidic systems: topology and shape optimization with the lattice Boltzmann method. Ph.D. Thesis, University of Colorado at Boulder, 2008.
20. Allaire G, Jouve F, Toader AM. Structural optimization using sensitivity analysis and a level-set method. Journal of Computational Physics 2004; 194:363-393. DOI: 10.1016/j.jcp.2003.09.032.
21. Wang SY, Wang MY. A moving superimposed finite element method for structural topology optimization. International Journal for Numerical Methods in Engineering 2006; 65(11):1892-1922. DOI: 10.1002/nme.1527.
22. Mittal R, Gianluca I. Immersed boundary methods. Annual Review of Fluid Mechanics 2005; 37:239-261. DOI: 10.1146/annurev.fluid.37.061903.175743.
23. Gerstenberger A, Wall W. An extended finite element method/Lagrange multiplier based approach for fluid-structure interaction. Computer Methods in Applied Mechanics and Engineering 2008; 197:1699-1714.
24. Bouzidi M, Firdaouss M, Lallemand P. Momentum transfer of a lattice Boltzmann fluid with boundaries. Physics of Fluids 2001; 13(11):3452-3459.
25. Sethian J, Wiegmann A. Structural boundary design via level set and immersed interface methods. Journal of Computational Physics 2000; 163(2):489-528. DOI: 10.1006/jcph.2000.6581.
26. Wang MY, Wang X, Guo D. A level set method for structural topology optimization. Computer Methods in Applied Mechanics and Engineering 2003; 192(1-2):227-246. DOI: 10.1016/S0045-7825(02)00559-5.
27. Xia Q, Wang MY, Wang S, Chen S. Semi-Lagrange method for level-set-based structural topology and shape optimization. Structural and Multidisciplinary Optimization, vol. 31. Springer: Berlin, Heidelberg, 2006; 419-429. DOI: 10.1007/s00158-005-0597-y.
28. Wei P, Wang MY. Parametric structural shape and topology optimization method with radial basis functions and level-set method. Proceedings of IDETC/CIE 2006, ASME 2006 International Design Engineering Technical Conferences, Philadelphia, U.S.A., 2006.
29. Luo Z, Tong L. A level set method for shape and topology optimization of large-displacement compliant mechanisms. International Journal for Numerical Methods in Engineering 2008; 76(6):862-892. DOI: 10.1002/nme.2352.
30. De Ruiter M. Topology optimization using a topology description function approach. Ph.D. Thesis, Technische Universiteit Delft, 2005.
31. van Miegroet L, Duysinx P. Stress concentration minimization of 2d filets using x-fem and level set description. Structural and Multidisciplinary Optimization 2007; 33(4-5):425-438. DOI: 10.1007/s00158-006-0091-1.
32. Cunha AL. A fully Eulerian method for shape optimization with application to Navier-Stokes flows. Ph.D. Thesis, Carnegie Mellon University, Pittsburgh, PA, September 2004.
33. Duan X, Ma Y, Zhang R. Optimal shape control of fluid flow using variational level set method. Physics Letters A 2008; 372(9):1374-1379. DOI: 10.1016/j.physleta.2007.09.070.
34. Mohammadi B, Pironneau O. Theory and practice of optimal shape design. European Journal of Computational Mechanics 2008; 17(1-2):13-30.
35. Pingen G, Waidmann M, Evgrafov A, Maute K. Application of a parametric level-set approach to topology optimization fluid with the Navier-Stokes and lattice Boltzmann equations. WCSMO07. ISSMO: Seoul, Korea, 21-25 May 2007.
36. Yu D, Mei R, Shyy W. A unified boundary treatment in lattice Boltzmann method. APS Meeting Abstracts, November 2002. J11+.
37. Chen S, Doolen GD. Lattice Boltzmann method for fluid flows. Annual Review of Fluid Mechanics 1998; 30:329-364.
38. Pingen G, Evgrafov A, Maute K. Adjoint parameter sensitivity analysis for the hydrodynamic lattice Boltzmann method with applications to design optimization. Computers and Fluids 2009; 38(4):910-923. DOI: 10.1016/j.compfluid.2008.10.002.
39. Pingen G, Waidmann M, Evgrafov A, Maute K. A parametric level-set approach for topology optimization of flow domains. Structural and Multidisciplinary Optimization 2009; DOI: 10.1007/s00/58-009-0405-1.
40. Geier M, Liu Z, Greiner A, Korvink J. Lattice Boltzmann based structural topology optimization of micro channels. Second International Conference on Transport Phenomena in Micro and Nanodevices, Barga, Italy, 2006.
41. Yang Z. Analysis of lattice Boltzmann boundary conditions. Ph.D. Thesis, Universität Konstanz, 2007.
42. Bouzidi M, Firdaouss M, Lallemand P. Momentum transfer of a Boltzmann-lattice fluid with boundaries. Physics of Fluids 2001; 13(11):3452-3459.
43. Osher S, Sethian J. Fronts propagating with curvature dependent speed: algorithms based on Hamilton-Jacobi formulations. Journal of Computational Physics 1988; 79:12-49.
44. Wang MY, Wang X. PDE-driven level sets, shape sensitivity and curvature flow for structural topology optimization. Computer Modeling in Engineering and Sciences 2004; 6(4):373-395.
45. Burger M, Osher S. A survey on level set methods for inverse problems and optimal design. European Journal of Applied Mathematics 2005; 16:263-301. DOI: 10.1017/S0956792505006182.
46. Sethian J, Smereka P. Level set methods for fluid interfaces. Annual Review of Fluid Mechanics 2003; 35:341-372. DOI: 10.1146/annurev.fluid.35.101101.161105.
47. Wang MY, Wang S. Parametric shape and topology optimization with radial basis functions. IUTAM Symposium on Topological Design Optimization of Structures, Machines and Materials, Aalborg and Lyngby, Denmark, 2005; 13-22.
48. Buhmann M. Radial Basis Functions: Theory and Implementations. Cambridge University Press: Cambridge, 2003.
49. Mohammadi B. Global optimization, level set dynamics, incomplete sensitivity and regularity control. International Journal of Computational Fluid Dynamics 2007; 21(2):61-68.
50. Sethian J. Level Set Methods. Cambridge University Press: Cambridge, 1996.
51. Jameson A, Martinelli L, Pierce NA. Optimum aerodynamic design using the Navier-Stokes equations. Theoretical and Computational Fluid Dynamics 1998; 10:213-237.
52. Fung Y, Tong P. Classical and Computational Solid Mechanics. World Scientific Publishing: Singapore, 2001.
53. Kreisselmeier G, Steinhauser R. Systematic control design by optimizing a vector performance index. International Federation of Active Controls Symposium on Computer Aided Design of Control Systems, Zurich, Switzerland, 1979.
54. Sigmund O, Petersson J. Numerical instabilities in topology optimization: a survey on procedures dealing with checkerboards, mesh-dependencies and local minima. Structural and Multidisciplinary Optimization 1998; 16(1):168-175.
55. Daoud F, Firl M, Bletzinger KU. Filter techniques in shape optimization with CAD-free parametrization. Sixth World Congresses of Structural and Multidisciplinary Optimization, Rio de Janeiro, Brazil, 2005.
56. Mei R, Yu D, Shyy W, Luo LS. Force evaluation in the lattice Boltzmann method involving curved geometry. Physical Review E 2002; 65(4):041 203:1-041 203:14.
57. Svanberg K. A globally convergent version of MMA without linesearch. Proceedings of the First World Congress of Structural and Multidisciplinary Optimization, 28 May-2 June 1995; 9-16.