Level set topology optimization of stationary fluid-structure interaction problems

  • Nicholas Jenkins
  • Kurt MauteEmail author


This paper introduces a topology optimization approach that combines an explicit level set method (LSM) and the extended finite element method (XFEM) for designing the internal structural layout of fluid-structure interaction (FSI) problems. The FSI response is predicted by a monolithic solver that couples an incompressible Navier-Stokes flow model with a small-deformation solid model. The fluid mesh is modeled as an elastic continuum that deforms with the structure. The fluid model is discretized with stabilized finite elements and the structural model by a generalized formulation of the XFEM. The nodal parameters of the discretized level set field are defined as explicit functions of the optimization variables. The optimization problem is solved by a nonlinear programming method. The LSM-XFEM approach is studied for two- and three-dimensional FSI problems at steady-state and compared against a density topology optimization approach. The numerical examples illustrate that the LSM-XFEM approach convergences to well-defined geometries even on coarse meshes, regardless of the choice of objective and constraints. In contrast, the density method requires refined grids and a mass penalization to yield smooth and crisp designs. The numerical studies show that the LSM-XFEM approach can suffer from a discontinuous evolution of the design in the optimization process as thin structural members disconnect. This issue is caused by the interpolation of the level set field and the inability to represent particular geometric configurations in the XFEM model. While this deficiency is generic to the LSM-XFEM approach used here, it is pronounced by the nonlinear response of FSI problems.


Topology optimization Extended finite Element method Level set method Fluid-structure interaction Hydroelasticity Monolithic solver 



The authors acknowledge the support of the National Science Foundation under grant CMMI 1235532. The opinions and conclusions presented in this paper are those of the authors and do not necessarily reflect the views of the sponsoring organization.


  1. Allen M, Maute K (2002) Reliability based optimization of aeroelastic structures. In: Proceedings of the 9th AIAA/ISSMO Symposium on Multidisciplinary Analysis and Optimization, Atlanta, GA, September 4 – 6Google Scholar
  2. Allen M, Maute K (2004) Reliability based optimization of aeroelastic structures. Struct Multidiscip Optim 24:228–242CrossRefGoogle Scholar
  3. Amestoy P, Duff I, L’Excellent JY (2000) Multifrontal parallel distributed symmetric and unsymmetric solvers. Comput Methods Appl Mech Eng 184(24):501–520CrossRefzbMATHGoogle Scholar
  4. Angot P, Bruneau CH, Fabrie P (1999) A penalization method to take into account obstacles in incompressible viscous flows. Numer Math 81(4):497–520MathSciNetCrossRefGoogle Scholar
  5. Bendsøe M, Sigmund O (1999) Material interpolation schemes in topology optimization. Arch Appl Mech 69(9-10):635–654CrossRefGoogle Scholar
  6. Bourdin B (2001) Filters in topology optimization. Int J Numer Methods Eng 50(9):2143–2158MathSciNetCrossRefGoogle Scholar
  7. Brampton C, Kim HA, Cuningham J (2012) Level set topology optimisation of aircraft wing considering aerostructural interaction. In: 12th AIAA Aviation Technology, Integration, and Opterations (ATIO) Conference and 14th AIAA/ISSM Multidisciplinary Analysis and Optimization Conference, Indianapolis, IN, September 17–19Google Scholar
  8. COMSOL (2008) Mems module model library: Fluid-structure interaction Tech. rep. COMSOLGoogle Scholar
  9. van Dijk N, Maute K, Langelaar M, Keulen F (2013) Level-set methods for structural topology optimization: a review. Struct Multidiscip Optim pp 1–36Google Scholar
  10. Dunning P, Stanford B, Kim H (2014) Coupled aerostructural topology optimization using a level set method for 3d aircraft wings. Structural and Multidisciplinary Optimization. doi: 10.1007/s00158-014-1200-1
  11. Gerstenberger A, Wall WA (2008) An extended finite element method/Lagrange multiplier based approach for fluid-structure interaction. Comput Methods Appl Mech Eng 197:1699–1714MathSciNetCrossRefzbMATHGoogle Scholar
  12. Ghazlane I, Carrier G, Dumont A, Marcelet M, Désidéri J A (2011) Aerostructural optimization with the adjoint method. In: EUROGEN 2011, Capua, Italy, September 14 - 15Google Scholar
  13. Gomes A A, Suleman A (2008) Topology optimization of a reinforced wing box for enhanced roll maneuvers. AIAA J 46:548– 556CrossRefGoogle Scholar
  14. Guest J (2009) Topology optimization with multiple phase projection. Comput Methods Appl Mech Eng 199 (1-4):123–135MathSciNetCrossRefzbMATHGoogle Scholar
  15. Guo S (2007) Aeroelastic optimization of an aerobatic aircraft wing structure. Aerosp Sci Technol 11(5):396–404CrossRefzbMATHGoogle Scholar
  16. Guo S, Cheng W, Cui D (2005) Optimization of composite wing structures for maximum flutter speed. In: 46th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics & Materials Conference, Austin, TX, April 21 - 25Google Scholar
  17. Hubner B, Walhorn E, Dinkler D (2004) A monolithic approach to fluid-structure interaction using space-time finite elements. Comput Methods Appl Mech Eng 193(23-26):2087–2104CrossRefGoogle Scholar
  18. Kennedy G, Kenway G, Martins J (2014) Towards gradient-based design optimization of flexible transport aircraft with flutter constraints. In: 15th AIAA/ISSMO Multidisciplinary Analysis and Optimization Conference, Atlanta,Georgia, June 16-20Google Scholar
  19. Kreissl S, Maute K (2011) Topology optimization for unsteady flow. Int J Numer Methods Eng 87:1229–1253MathSciNetzbMATHGoogle Scholar
  20. Kreissl S, Maute K (2012) Levelset based fluid topology optimization using the extended finite element method. Struct Multidiscip Optim:1–16Google Scholar
  21. Lang C, Makhija D, Doostan A, Maute K (2014) A simple and efficient preconditioning scheme for heaviside enriched xfem. Comput Mech 54(5):1357–1374MathSciNetCrossRefzbMATHGoogle Scholar
  22. Makhija D, Maute K (2014a) Level set topology optimization of scalar transport problems. Struct Multidiscip Optim. pp 1–9Google Scholar
  23. Makhija D, Maute K (2014b) Numerical instabilities in level set topology optimization with the extended finite element method. Struct Multidiscip Optim 49(2):185–197MathSciNetCrossRefGoogle Scholar
  24. Martins J, Alonso J, Reuther J (2005) A coupled-adjoint sensitivity analysis method for high-fidelity aero-structural design. Optim Eng 6(1):33–62CrossRefzbMATHGoogle Scholar
  25. Maute K, Reich G (2006) Integrated multidisciplinary topology optimization approach to adaptive wing design. AIAA Journal of Aircraft 43(1):253–263CrossRefGoogle Scholar
  26. Maute K, Nikbay M, Farhat C (2003) Sensitivity analysis and design optimization of three-dimensional nonlinear aeroelastic systems by the adjoint method. Int J Numer Methods Eng 56(6):911–933CrossRefzbMATHGoogle Scholar
  27. Petersson J, Sigmund O (1998) Slope constrained topology optimization. Int J Numer Methods Eng 41 (8):1417–1434MathSciNetCrossRefzbMATHGoogle Scholar
  28. Saad Y (1994) Ilut: A dual threshold incomplete lu factorization. Numerical Linear Algebra Appl 1(4):387–402MathSciNetCrossRefzbMATHGoogle Scholar
  29. Sala M, Stanley KS, Heroux MA (2008) On the design of interfaces to sparse direct solvers. ACM Trans Math Softw 34(2):9MathSciNetGoogle Scholar
  30. Sigmund O (2007) Morphology-based black and white filters for topology optimization. Struct Multidiscip Optim 33(4-5):401–424CrossRefGoogle Scholar
  31. Sigmund O, Maute K (2012) Sensitivity filtering from a continuum mechanics perspective. Struct Multidiscip Optim 46(4):471–475MathSciNetCrossRefzbMATHGoogle Scholar
  32. Sigmund O, Maute K (2013) Topology optimization approaches: A comparative review. Struct Multidiscip Optim 48(6):1031–1055MathSciNetCrossRefGoogle Scholar
  33. Stanford B (2008) Aeroelastic analysis and optimization of membrane micro air vehicle wings. PhD thesis, University of FloridaGoogle Scholar
  34. Stanford B, Beran P (2013) Aerothermoelastic topology optimization with flutter and buckling metrics. Struct Multidiscip Optim 48(1):149–171MathSciNetCrossRefzbMATHGoogle Scholar
  35. Stanford B, Ifju P (2009) Aeroelastic topology optimization of membrane structures for micro air vehicles. Struct Multidiscip Optim 38(3):301–316CrossRefGoogle Scholar
  36. Stein K, Tezduyar TE, Benney R (2004) Automatic mesh update with the solid-extension mesh moving technique. Comput Methods Appl Mech Eng 193(21-22):2019–2032CrossRefzbMATHGoogle Scholar
  37. Svanberg K (2002) A class of globally convergent optimization methods based on conservative convex separable approximations. SIAM J on Optim 12(2):555–573MathSciNetCrossRefzbMATHGoogle Scholar
  38. 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):1321–1346CrossRefGoogle Scholar
  39. Tezduyar TE, Mittal S, Ray SE, Shih R (1992) Incompressible flow computations with stabilized bilinear and linear equal-order-interpolation velocity-pressure elements. Comput Methods Appl Mech Eng 95:221–242CrossRefzbMATHGoogle Scholar
  40. Tran AB, Yvonnet J, He QC, Toulemonde C, Sanahuja J (2011) A multiple level set approach to prevent numerical artifacts in complex microstructures with nearby inclusions within xfem. Int J Numer Methods Eng 85(11):1436–1459CrossRefzbMATHGoogle Scholar
  41. Villanueva CH, Maute K (2014) Density and level set-xfem schemes for topology optimization of 3-d structures. Comput Mech 54(1):133–150MathSciNetCrossRefzbMATHGoogle Scholar
  42. Yoon GH (2009) Topology optimization for stationary fluid-structure interaction problems using a new monolithic formulation. Int J Numer Meth Engng 82:591–616Google Scholar
  43. Yoon GH (2014) Stress-based topology optimization method for steady-state fluidstructure interaction problems. Comput Methods Appl Mech Eng 278:499–523CrossRefGoogle Scholar
  44. Zhou M, Rozvany G (1991) The COC algorithm, part II: topological, geometry and generalized shape optimization. Comput Methods Appl Mech Eng 89(1–3):309–336CrossRefGoogle Scholar

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© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  1. 1.Department of Aerospace Engineering SciencesUniversity of Colorado at BoulderBoulderUSA

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