Advertisement

Structural and Multidisciplinary Optimization

, Volume 51, Issue 2, pp 267–285 | Cite as

Level set topology optimization of scalar transport problems

  • David Makhija
  • Kurt Maute
RESEARCH PAPER

Abstract

This paper studies level set topology optimization of scalar transport problems, modeled by an advection-diffusion equation. Examples of such problems include the transport of energy or mass in a fluid. The geometry is defined via a level set method (LSM). The flow field is predicted by a hydrodynamic Boltzmann transport model and the scalar transport by a standard advection-diffusion model. Both models are discretized by the extended Finite Element Method (XFEM). The hydrodynamic Boltzmann equation is well suited for the XFEM as it allows for convenient enforcement of boundary conditions along immersed boundaries. In contrast, Navier Stokes models require more complex approaches to impose Dirichlet boundary conditions, such as stabilized Lagrange multiplier and Nitsche methods.

The combination of the LSM and the XFEM is an alternative to density-based topology optimization methods which have been applied previously to scalar transport problems. Density methods often suffer from a fuzzy description of boundaries, spurious diffusion through “void” regions, and the presence of fictitious material in the optimized design. This paper illustrates that the LSM/XFEM approach addresses these three concerns. The proposed approach is studied with two dimensional problems at steady state conditions. Both “fluid-void” and “fluid-solid” optimization problems are considered. For the “fluid-void” case, optimization results are obtained without spurious diffusion through “void” regions. For the “fluid-solid” case, the analysis recovers strong gradients of the flow and scalar fields at the fluid-solid interface, using moderately refined meshes.

Keywords

Topology optimization Level set method Extended finite element method Hydrodynamic Boltzmann transport equations Energy and mass transport Spurious diffusion 

Notes

Acknowledgements

The authors acknowledge the support of the National Science Foundation under grant EFRI-SEED 1038305 and CBET 1246854. The opinions and conclusions presented in this paper are those of the authors and do not necessarily reflect the views of the sponsoring organization.

References

  1. Allaire G, Jouve F, Toader AM (2004) Structural optimization using sensitivity analysis and a level-set method. J Comput Phys 194(1):363–393CrossRefzbMATHMathSciNetGoogle Scholar
  2. Andreasen CS, Gersborg AR, Sigmund O (2009) Topology optimization of microfluidic mixers. Int J Numer Methods Fluids 61(5):498–513CrossRefzbMATHMathSciNetGoogle Scholar
  3. Angot P, Bruneau CH, Fabrie P (1999) A penalization method to take into account obstacles in viscous flows. Numer Math 81:497–520CrossRefzbMATHMathSciNetGoogle Scholar
  4. Avila M, Codina R, Principe J (2011) Spatial approximation of the radiation transport equation using a subgrid-scale finite element method. Comput Methods Appl Mech Eng 200(5-8):425–438CrossRefzbMATHMathSciNetGoogle Scholar
  5. Bhatnagar PL, Gross EP, Krook M (1954) A model for collision processes in gases. I. small amplitude processes in charged and neutral one-component systems. Phys Rev 94(3):511–525CrossRefzbMATHGoogle Scholar
  6. Bijl H, Carpenter MH, Vatsa VN, Kennedy CA (2002) Implicit time integration schemes for the unsteady compressible Navier-Stokes equations: Laminar flow. J Comput Phys 179(1):313–329CrossRefzbMATHGoogle Scholar
  7. Borrvall T, Petersson J (2003) Topology optimization of fluids in stokes flow. Int J Numer Methods Fluids 41(1):77–107CrossRefzbMATHMathSciNetGoogle Scholar
  8. Brooks AN, Hughes TJ (1982) Streamline upwind/Petrov-Galerkin formulations for convection dominated flows with particular emphasis on the incompressible Navier-Stokes equations. Comput Methods Appl Mech Eng 32(1-3):199–259CrossRefzbMATHMathSciNetGoogle Scholar
  9. Cao N, Chen S, Jin S, Martínez D (1997) Physical symmetry and lattice symmetry in the lattice Boltzmann method. Phys Rev E 55:R21–R24CrossRefGoogle Scholar
  10. Chen H (1998) Volumetric formulation of the lattice Boltzmann method for fluid dynamics: Basic concept. Phys Rev E 58:3955–3963CrossRefGoogle Scholar
  11. Codina R (1998) Comparison of some finite element methods for solving the diffusion-convection-reaction equation. Comput Methods Appl Mech Eng 156(1-4):185–210CrossRefzbMATHMathSciNetGoogle Scholar
  12. Codina R (2001) A stabilized finite element method for generalized stationary incompressible flows. Comput Methods Appl Mech Eng 190(2021):2681–2706CrossRefzbMATHMathSciNetGoogle Scholar
  13. Codina R (2002) Stabilized finite element approximation of transient incompressible flows using orthogonal subscales. Comput Methods Appl Mech Eng 191(39-40):4295–4321CrossRefzbMATHMathSciNetGoogle Scholar
  14. Daux C, Moes N, Dolbow J, Sukumark N, Belytschko T (2000) Arbitrary branched and intersecting cracks with the extended nite element method. Int J Numer Meth Engng 48:1741–1760CrossRefzbMATHGoogle Scholar
  15. Dede E (2010) Multiphysics optimization, synthesis, and application of jet impingement target surfaces. In: Thermal and Thermomechanical Phenomena in Electronic Systems (ITherm), 2010 12th IEEE Intersociety Conference on, pp 1–7Google Scholar
  16. van Dijk N, Langelaar M, van Keulen F (2012) Explicit level-set-based topology optimization using an exact heaviside function and consistent sensitivity analysis. Int J Numer Meth Engng 91(1):67–97Google Scholar
  17. van Dijk N, Maute K, Langelaar M, van Keulen F (2013) Levelset methods for structural topology optimization A review. Struct Multidiscip Optim 48(3):437–472Google Scholar
  18. Dolbow J, Harari I (2009) An efficient finite element method for embedded interface problems. Int J Numer Meth Engng 78:229–252CrossRefzbMATHMathSciNetGoogle Scholar
  19. Düster A, Demkowicz L, Rank E (2006) High-order finite elements applied to the discrete Boltzmann equation. Int J Numer Methods Eng 67(8):1094–1121CrossRefzbMATHGoogle Scholar
  20. Duysinx P, Miegroet L, Jacobs T, Fleury C (2006) Generalized shape optimization using x-fem and level set methods. , In: IUTAM Symposium on Topological Design Optimization of Structures. Springer, Machines and Materials, pp 23–32Google Scholar
  21. Evans B, Morgan K, Hassan O (2011) A discontinuous finite element solution of the Boltzmann kinetic equation in collisionless and BGK forms for macroscopic gas flows. Appl Math Model 35(3):996–1015CrossRefzbMATHMathSciNetGoogle Scholar
  22. Fries T, Belytschko T (2010) The extended/generalized finite element method: an overview of the method and its applications. Int J Numer Methods Eng 84(3):253–304zbMATHMathSciNetGoogle Scholar
  23. Fries TP (2009) The intrinsic xfem for two-fluid flows. Int J Numer Meth Fluids 60(4):437–471CrossRefzbMATHMathSciNetGoogle Scholar
  24. Gersborg-Hansen A, Sigmund O, Haber RB (2005) Topology optimization of channel flow problems. Struct Multidiscip Optim 30(3):181–192CrossRefzbMATHMathSciNetGoogle Scholar
  25. Gersborg-Hansen A, Bendse MP, Sigmund O (2006) Topology optimization of heat conduction problems using the finite volume method. Struct Multidiscip Optim 31(4):251–259CrossRefzbMATHMathSciNetGoogle Scholar
  26. Gerstenberger A, Wall WA (2008) An extended finite element method/Lagrange multiplier based approach for fluid-structure interaction. Comput Methods Appl Mech Eng 197:1699–1714CrossRefzbMATHMathSciNetGoogle Scholar
  27. Grad H (1949) On the kinetic theory of rarefied gases. Commun Pur Appl Math 2(4):331–407CrossRefzbMATHMathSciNetGoogle Scholar
  28. Guest J, Prévost J, Belytschko T (2004) Achieving minimum length scale in topology optimization using nodal design variables and projection functions. Int J Numer Methods Eng 61(2):238– 254CrossRefzbMATHGoogle Scholar
  29. Guo X, Zhang W, Zhong W (2014) Explicit feature control in structural topology optimization via level set method. Comput Methods Appl Mech Eng 272:354–378CrossRefzbMATHMathSciNetGoogle Scholar
  30. Hansbo A, Hansbo P (2004) A finite element method for the simulation of strong and weak discontinuities in solid mechanics. Comput Methods Appl Mech Eng 193(3335):3523–3540CrossRefzbMATHMathSciNetGoogle Scholar
  31. Hauke G (2002) A simple subgrid scale stabilized method for the advection-diffusion-reaction equation. Comput Methods Appl Mechanics Eng 191(27–28):2925–2947CrossRefzbMATHMathSciNetGoogle Scholar
  32. Hughes TJR, Mallet M (1986) A new finite element formulation for computational fluid dynamics: III. The generalized streamline operator for multidimensional advective-diffusive systems. Comput Methods Appl Mechanics Eng 58(3):305–328CrossRefzbMATHMathSciNetGoogle Scholar
  33. Juntunen M, Stenberg R (2009) Nitsches method for general boundary conditions. Math Comput 78:1353–1374CrossRefzbMATHMathSciNetGoogle Scholar
  34. Kontoleontos EA, Papoutsis-Kiachagias EM, Zymaris AS, Papadimitriou DI, Giannakoglou KC (2013) Adjoint-based constrained topology optimization for viscous flows, including heat transfer. Eng Opt 45(8):941–961CrossRefMathSciNetGoogle Scholar
  35. Kreisselmeier G, Steinhauser R (1979) Systematic control design by optimizing a vector performance index. , In: International Federation of Active Contrals Symposium on Computer Aided Design of Control Systems. Zurich, SwitzerlandGoogle Scholar
  36. Kreissl S, Maute K (2011) Topology optimization for unsteady flow. Int J Numer Methods Eng 87:1229–1253zbMATHMathSciNetGoogle Scholar
  37. Kreissl S, Maute K (2012) Levelset based fluid topology optimization using the extended finite element method. Struct Multidiscip Optim 46(3):311–326CrossRefzbMATHMathSciNetGoogle Scholar
  38. Kreissl S, Pingen G, Evgrafov A, Maute K (2010) Topology optimization of flexible micro-fluidic devices. Struct Multidiscip Optim 42(4):495–516Google Scholar
  39. Kreissl S, Pingen G, Maute K (2011) An explicit level set approach for generalized shape optimization of fluids with the lattice boltzmann method. Int J Numer Methods Fluids 65(5):496–519CrossRefzbMATHGoogle Scholar
  40. Lang C, Makhija D, Doostan A,Maute K (2013) A simple and efficient preconditioning scheme for xfem with heaviside enrichments. http://arxiv.org/abs/1312.6092
  41. Lee T, Lin CL (2001) A characteristic Galerkin method for discrete Boltzmann equation. J Comput Phys 171(1):336–356CrossRefzbMATHMathSciNetGoogle Scholar
  42. Li Y, LeBoeuf EJ, Basu PK (2004) Least-squares finite-element lattice Boltzmann method. Phys Rev E 69(065):701Google Scholar
  43. Li Y, LeBoeuf E, Basu P (2005) Least-squares finite-element scheme for the lattice Boltzmann method on an unstructured mesh. Phys Rev E 72(4)(046):711Google Scholar
  44. Luo Z, Tong L, Wang MY, Wang S (2007) Shape and topology optimization of compliant mechanisms using a parameterization level set method. J Comput Phys 227(1):680–705CrossRefzbMATHMathSciNetGoogle Scholar
  45. Makhija D, Maute K (2014) Numerical instabilities in level set topology optimization with the extended finite element method. Struct Multidiscip Optim 49(2):185–197Google Scholar
  46. Makhija D, Pingen G, Yang R, Maute K (2012) Topology optimization of multi-component flows using a multi-relaxation time lattice Boltzmann method. Comput Fluids 67(0):104–114CrossRefMathSciNetGoogle Scholar
  47. Makhija D, Pingen G, Maute K (2014) An immersed boundary method for fluids using the xfem and the hydrodynamic boltzmann transport equation. Comput Methods Appl Mech Eng 273:37–55Google Scholar
  48. Matsumori T, Kondoh T, Kawamoto A, Nomura T (2013) Topology optimization for fluid-thermal interaction problems under constant input power. Struct Multidiscip Optim 47(4):571–581CrossRefzbMATHGoogle Scholar
  49. Maute K, Kreissl S, Makhija D, Yang R (2011) Topology optimization of heat conduction in nano-composites. Shizuoka, JapanGoogle Scholar
  50. Mei R, Shyy W (1998) On the finite difference-based lattice Boltzmann method in curvilinear coordinates. J Comput Phys 143(2):426–448CrossRefzbMATHMathSciNetGoogle Scholar
  51. van Miegroet L, Duysinx P (2007) Stress concentration minimization of 2d filets using x-fem and level set description. Struct Multidiscip Optim 33(4-5):425–438CrossRefGoogle Scholar
  52. van Miegroet L, Moës N Fleury C, Duysinx P (2005) Generalized shape optimization based on the level set method. In: 6 th World Congress of Structural and Multidisciplinary OptimizationGoogle Scholar
  53. Min M, Lee T (2011) A spectral-element discontinuous Galerkin lattice Boltzmann method for nearly incompressible flows. J Comput Phys 230(1):245–259CrossRefzbMATHMathSciNetGoogle Scholar
  54. Nannelli F, Succi S (1992) The lattice Boltzmann equation on irregular lattices. J Stat Phys 68:401–407CrossRefzbMATHMathSciNetGoogle Scholar
  55. Okkels F, Gregersen M, Bruus H (2009) Topology optimization of fully nonlinear lab-on-a-chip systems. In: Proceedings of 8th World Congress on Structural and Multidisciplinary Optimization, June 1–5, 2009. Lisbon, PortugalGoogle Scholar
  56. Othmer C (2006) CFD topology and shape optimization with adjoint methods. , In: VDI Fahrzeug- und Verkehrstechnik. Internationaler Kongress, Berechnung und Simulation im Fahrzeugbau, Würzburg, p 13Google Scholar
  57. Othmer C, de Villiers E, Weller HG (2007) Implementation of a continuous adjoint for topology optimization of ducted flows. In: Proceedings of the 18th AIAA Computational Fluid Dynamics Conference Miami. AIAA, FLGoogle Scholar
  58. Patil DV, Lakshmisha K (2009) Finite volume TVD formulation of lattice Boltzmann simulation on unstructured mesh. J Comput Phys 228(14):5262–5279CrossRefzbMATHMathSciNetGoogle Scholar
  59. Peng G, Xi H, Duncan C, Chou SH (1998) Lattice Boltzmann method on irregular meshes. Phys Rev E 58:R4124–R4127CrossRefGoogle Scholar
  60. Peng G, Xi H, Duncan C, Chou SH (1999) Finite volume scheme for the lattice Boltzmann method on unstructured meshes. Phys Rev E 59:4675–4682CrossRefGoogle Scholar
  61. Pingen G, Evgrafov A, Maute K (2009) Adjoint parameter sensitivity analysis for the hydrodynamic lattice Boltzmann method with applications to design optimization. Comput Fluids 38(4):910–923CrossRefzbMATHMathSciNetGoogle Scholar
  62. Pingen G, Waidmann M, Evgrafov A, Maute K (2010) A parametric level-set approach for topology optimization of flow domains. Struct Multidiscip Optim 41(1):117–131CrossRefzbMATHMathSciNetGoogle Scholar
  63. Shi X, Lin J, Yu Z (2003) Discontinuous Galerkin spectral element lattice Boltzmann method on triangular element. Int J Numer Methods Fluids 42(11):1249–1261CrossRefzbMATHGoogle Scholar
  64. Sigmund O, Maute K (2013) Topology optimization approaches: A comparative review. Struct Multidiscip Optim 48(6):1031–1055Google Scholar
  65. Stenberg R (1995) On some techniques for approximating boundary conditions in the finite element method. J Comput Appl Math 63(1-3):139–148CrossRefzbMATHMathSciNetGoogle Scholar
  66. Struchtrup H (2005) Macroscopic Transport Equations for Rarefied Gas Flows. SpringerGoogle Scholar
  67. Struchtrup H, Torrilhon M (2003) Regularization of Grad’s 13 moment equations: Derivation and linear analysis. Fluids 15:2668–2680CrossRefMathSciNetGoogle Scholar
  68. Svanberg K (2002) A class of globally convergent optimization methods based on conservative convex separable approximations. SIAM J Optim 12(2):555–573CrossRefzbMATHMathSciNetGoogle Scholar
  69. Terada K, Asai M, Yamagishi M (2003) Finite cover method for linear and non-linear analyses of heterogeneous solids. Int J Numer Methods Eng 58(9):1321–1346CrossRefzbMATHGoogle Scholar
  70. Tölke J, Krafczyk M, Schulz M, Rank E (2000) Discretization of the Boltzmann equation in velocity space using a Galerkin approach. Comput Phys Commun 129(13):91–99CrossRefzbMATHGoogle Scholar
  71. Tran AB, Yvonnet J, He QC, Toulemonde C, Sanahuja J (2011) A multiple level set approach to prevent numerical artefacts in complex microstructures with nearby inclusions within xfem. Int J Numer Methods Eng 85(11):1436–1459CrossRefzbMATHGoogle Scholar
  72. Ubertini S, Succi S (2005) Recent advances of lattice Boltzmann techniques on unstructured grids. Progress in Computational Fluid Dynamics, an International Journal 5(1):85– 96CrossRefMathSciNetGoogle Scholar
  73. Ubertini S, Bella G, Succi S (2003) Lattice Boltzmann method on unstructured grids: Further developments. Phys Rev E 68(016):701MathSciNetGoogle Scholar
  74. Villanueva C, Maute K (2014) Density and level set-xfem schemes for topology optimization of 3-D structures. Comput Mech 54(1):133–150Google Scholar
  75. Wang MY, Wang X, Guo D (2003) A level set method for structural topology optimization. Comput Methods Appl Mechanics Eng 192(1-2):227–246CrossRefzbMATHGoogle Scholar
  76. Wang S, Wang M (2006) Radial basis functions and level set method for structural topology optimization. Int J Numer Methods Eng 65(12):2060–2090CrossRefzbMATHGoogle Scholar
  77. Wei P, Wang M, Xing X (2010) A study on X-FEM in continuum structural optimization using a level set model. Comput Aided Des 42(8):708–719CrossRefGoogle Scholar
  78. Xi H, Peng G, Chou SH (1999) Finite-volume lattice Boltzmann method. Phys Rev E 59:6202–6205CrossRefGoogle Scholar
  79. Yang J, Huang J (1995) Rarefied flow computations using nonlinear model Boltzmann equations. J Comput Phys 120(2):323–339CrossRefzbMATHGoogle Scholar
  80. Yoon G (2009) Topology optimization for stationary fluid-structure interaction problems using a new monolithic formulation , In: Proceedings of 8th World Congress on Structural and Multidisciplinary Optimization. Lisbon, PortugalGoogle Scholar
  81. Yu D, Mei R, Luo L, Shyy W (2003) Viscous flow computations with the method of lattice Boltzmann equation. Prog Aerosp Sci 39(5):329–367CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  1. 1.Department of Mechanical EngineeringUniversity of Colorado at BoulderBoulderUSA
  2. 2.Department of Aerospace Engineering SciencesUniversity of Colorado at BoulderBoulderUSA

Personalised recommendations