Skip to main content
SpringerLink
Log in

Partitioned solution of an unsteady adjoint for strongly coupled fluid-structure interactions and application to parameter identification of a one-dimensional problem

  • Research Paper
  • Published:
Structural and Multidisciplinary Optimization Aims and scope Submit manuscript

Abstract

Unsteady fluid-structure interaction (FSI) simulations are generally time-consuming. Gradient-based methods are preferred to minimise the computational cost of parameter identification studies (and more in general optimisation) with a high number of parameters. However, calculating the cost function’s gradient using finite differences becomes prohibitively expensive for a high number of parameters. Therefore, the adjoint equations of the unsteady FSI problem are solved to obtain this gradient at a cost almost independent of the number of parameters. Here, both the forward and the adjoint problems are solved in a partitioned way, which means that the flow equations and the structural equations are solved separately. The application of interest is the identification of the arterial wall’s stiffness by comparing the motion of the arterial wall with a reference, possibly obtained from non-invasive imaging. Due to the strong interaction between the fluid and the structure, quasi-Newton coupling iterations are applied to stabilise the partitioned solution of both the forward and the adjoint problem.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6

Similar content being viewed by others

References

  • Balasubramanian R, Newman J (2006) Discrete direct and adjoint sensitivity analysis for arbitrary Mach number flows. Int J Numer Methods Eng 66(2):297

    Article  MathSciNet  MATH  Google Scholar 

  • Bertoglio C, Moireau P, Gerbeau JF (2012) Sequential parameter estimation for fluid-structure problems. Application to hemodynamics. Int J Numer Methods Biomed Eng 28:434–455. doi:10.1002/cnm.1476

    Article  MathSciNet  Google Scholar 

  • Bonnet M, Constantinescu A (2005) Inverse problems in elasticity. Inverse Probl 21(2):R1

    Article  MathSciNet  Google Scholar 

  • Causin P, Gerbeau JF, Nobile F (2005) Added-mass effect in the design of partitioned algorithms for fluid-structure problems. Comput Methods Appl Mech Eng 194(42–44):4506

    Article  MathSciNet  MATH  Google Scholar 

  • Cowper G (1966) The shear coefficient in Timoshenko’s beam theory. J Appl Mech 33(2):335

    Article  MATH  Google Scholar 

  • Degroote J, Bruggeman P, Haelterman R, Vierendeels J (2008) Stability of a coupling technique for partitioned solvers in FSI applications. Comput Struct 86(23–24):2224

    Article  Google Scholar 

  • Degroote J, Bathe KJ, Vierendeels J (2009) Partitioned simulation of the interaction between an elastic structure and free surface flow. Comput Struct 87(11–12):793

    Article  Google Scholar 

  • Degroote J, Annerel S, Vierendeels J (2010a) Stability analysis of Gauss-Seidel iterations in a partitioned simulation of fluid-structure interaction. Comput Struct 88(5–6):263

    Article  Google Scholar 

  • Degroote J, Haelterman R, Annerel S, Bruggeman P, Vierendeels J (2010b) Performance of partitioned procedures in fluid-structure interaction. Comput Struct 88(7–8):446

    Article  Google Scholar 

  • Eberhart R, Shi Y, Kennedy J (2001) Swarm intelligence. Morgan Kaufmann, San Mateo

    Google Scholar 

  • Farhat C, van der Zee K, Geuzaine P (2006) Provably second-order time-accurate loosely-coupled solution algorithms for transient nonlinear computational aeroelasticity. Comput Methods Appl Mech Eng 195(17–18):1973

    Article  MATH  Google Scholar 

  • Fazzolari A, Gauger N, Brezillon J (2007) Efficient aerodynamic shape optimization in MDO Context. J Comput Appl Math 203(2):548

    Article  MathSciNet  MATH  Google Scholar 

  • Felippa C, Park K, Farhat C (2001) Partitioned analysis of coupled mechanical systems. Comput Methods Appl Mech Eng 190(24–25):3247

    Article  MATH  Google Scholar 

  • Formaggia L, Lamponi D, Quarteroni A (2003) One-dimensional models for blood flow in arteries. J Eng Math 47(3–4):251

    Article  MathSciNet  MATH  Google Scholar 

  • Gauger N, Fazzolari A (2009) MegaDesign and MegaOpt—German initiatives for aerodynamic simulation and optimization in aircraft design, results of the closing symposium of the MegaDesign and MegaOpt projects, Braunschweig, Germany, 23–24 May 2007. Adjoint methods for coupled CFD–CSM optimization. In: Notes on numerical fluid mechanics and multidisciplinary design, vol 107. Springer, Berlin, pp 237–246

  • Gee M, Küttler U, Wall W (2011) Truly monolithic algebraic multigrid for fluid-structure interaction. Int J Numer Methods Eng 85(8):987

    Article  MATH  Google Scholar 

  • Gerbeau JF, Vidrascu M (2003) A quasi-Newton algorithm based on a reduced model for fluid-structure interaction problems in blood flows. ESAIM: Math Model Numer Anal 37(4):631

    Article  MathSciNet  MATH  Google Scholar 

  • Goldberg D (1989) Genetic algorithms in search optimization and machine learning. Addison Wesley, Reading

    MATH  Google Scholar 

  • Gorissen D, Couckuyt I, Demeester P, Dhaene T, Crombecq K (2010) A surrogate modeling and adaptive sampling toolbox for computer based design. J Mach Learn Res 11:2051

    Google Scholar 

  • Griewank A, Walther A (2000) Algorithm 799: revolve: an implementation of checkpointing for the reverse or adjoint mode of computational differtiation. ACM Trans Math Softw 26(1):19

    Article  MATH  Google Scholar 

  • Haelterman R, Degroote J, Van Heule D, Vierendeels J (2009) Analytical study of the least squares quasi-Newton method for interaction problems. SIAM J Numer Anal 47(3):2347

    Article  MathSciNet  MATH  Google Scholar 

  • Haelterman R, Degroote J, Van Heule D, Vierendeels J (2010) On the similarities between the quasi-Newton inverse least squares method and GMRes. SIAM J Numer Anal 47(6):4660

    Article  MathSciNet  MATH  Google Scholar 

  • Heil M (2004) An efficient solver for the fully coupled solution of large-displacement fluid-structure interaction problems. Comput Methods Appl Mech Eng 193(1–2):1

    Article  MathSciNet  MATH  Google Scholar 

  • Hojjat M, Stavropoulou E, Gallinger T, Israel U, Wüchner R, Bletzinger KU (2010) Fluid-structure interaction in the context of shape optimization and computational wind engineering. In: Bungartz HJ, Mehl M, SchäferM (eds) Fluid-structure interaction II: modelling, simulation, optimization. Lecture notes in computational science and engineering. Springer, Berlin, pp 351–381

    Google Scholar 

  • Hübner 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

    Article  MATH  Google Scholar 

  • Kennedy G, Hansen J, Martins J (2011) A Timoshenko beam theory with pressure corrections for layered orthotropic beams. Int J Solids Struct 48(16–17):2373

    Article  Google Scholar 

  • Kirkpatrick S, Gelatt C, Vecchi M (1983) Optimization by simulated annealing. Science 220(4598):671

    Article  MathSciNet  MATH  Google Scholar 

  • Küttler U, Wall W (2008) Fixed-point fluid-structure interaction solvers with dynamic relaxation. Comput Mech 43(1):61

    Article  MATH  Google Scholar 

  • Lagrée PY (2000) An inverse technique to deduce the elasticity of a large artery. Eur Phys J Appl Phys 9(2):153

    Article  Google Scholar 

  • Li F, Sun Z (2007) A finite difference scheme for solving the Timoshenko beam equations with boundary feedback. J Comput Appl Math 200(2):606

    Article  MathSciNet  MATH  Google Scholar 

  • Martin V, Clément F, Decoene A, Gerbeau JF (2005) Parameter identification for a one-dimensional blood flow model. ESAIM: Proc 14:174

    MATH  Google Scholar 

  • Martins J, Alonso J, Reuther J (2004) High-fidelity aerostructural design optimization of a supersonic business jet. J Aircr 41(3):523

    Article  Google Scholar 

  • Martins J, Alonso J, Reuther J (2005) A coupled-adjoint sensitivity analysis method for high-fidelity aero-structural design. Optim Eng 6(1):33

    Article  MATH  Google Scholar 

  • Maute K, Nikbay M, Farhat C (2001) Coupled analytical sensitivity analysis and optimization of three-dimensional nonlinear aeroelastic systems. Am Inst Aeron Astron J 39(11):2051

    Article  Google Scholar 

  • 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

    Article  MATH  Google Scholar 

  • Mok D, Wall W, Ramm E (2001) Accelerated iterative substructuring schemes for instationary fluid-structure interaction. In: KJ Bathe (ed) Computational fluid and solid mechanics. Elsevier, Amsterdam, pp 1325–1328

    Chapter  Google Scholar 

  • Michler C, van Brummelen E, de Borst R (2005) An interface Newton–Krylov solver for fluid-structure interaction. Int J Numer Methods Fluids 47(10–11):1189

    Article  MATH  Google Scholar 

  • Nielsen E, Diskin B, Yamaleev N (2009) Discrete adjoint-based design optimization of unsteady turbulent flows on dynamic unstructured grids. In: 19th AIAA computational fluid dynamics conference (San Antonio, TX, USA), pp 1–22. AIAA 2009-3802

  • Nocedal J, Wright S (1999) Numerical optimization. Springer series in operations research. Springer, New York

    Book  MATH  Google Scholar 

  • Palaniappan K, Sahu P, Alonso J, Jameson A (2006) Active flutter control using an adjoint method. In: 44th AIAA aerospace sciences meeting and exhibit (Reno, NV, USA), pp 1–11. AIAA 2006-844

  • Palaniappan K, Sahu P, Alonso J, Jameson A (2009) Design of adjoint based laws for wing flutter control. In: 47th AIAA aerospace sciences meeting (Orlando, FL, USA), pp 1–20. AIAA 2009-148

  • Papadimitriou D, Giannakoglou K (2008) Direct, adjoint and mixed approaches for the computation of Hessian in airfoil design problems. Int J Numer Methods Fluids 56(10):1929

    Article  MathSciNet  MATH  Google Scholar 

  • Quarteroni A, Tuveri M, Veneziani A (2000) Computational vascular fluid dynamics: problems, models and methods. Comput Visual Sci 2(4):163

    Article  MATH  Google Scholar 

  • Rumpfkeil M, Zingg D (2010) The optimal control of unsteady flows with a discrete adjoint method. Optim Eng 11(1):5

    Article  MathSciNet  Google Scholar 

  • Sternberg J, Griewank A (2006) Reduction of storage requirement by checkpointing for time-dependent optimal control problems in ODEs. In: Automatic differentiation: applications, theory, and implementations. Lecture notes in computational science and engineering, vol 50. Springer, Berlin, pp 99–110

    Chapter  Google Scholar 

  • Stück A, Camelli F, Löhner R (2010a) Adjoint-based design of shock mitigation devices. Int J Numer Methods Fluids 64(4):443

    Article  MATH  Google Scholar 

  • Stück A, Camelli F, Löhner R (2010b) Adjoint-based design of shock mitigation devices. In: 48th AIAA aerospace sciences meeting (Orlando, FL, USA), pp 1–30. AIAA 2010-1430

  • van Brummelen E (2009) Added mass effects of compressible and incompressible flows in fluid-structure interaction. J Appl Mech 76(2):021206

    Article  Google Scholar 

  • van Brummelen E (2011) Partitioned iterative solution methods for fluid-structure interaction. Int J Numer Methods Fluids 65(1–3):3

    Article  MATH  Google Scholar 

  • Vierendeels J, Riemslagh K, Dick E (1999) A multigrid semi-implicit line-method for viscous incompressible and low-Mach-number flows on high aspect ratio grids. J Comput Phys 154(2):310

    Article  MATH  Google Scholar 

  • Vierendeels J, Dumont K, Dick E, Verdonck P (2005) Analysis and stabilization of fluid-structure interaction algorithm for rigid-body motion. AIAA J 43(12):2549

    Article  Google Scholar 

  • Vignon-Clementel I, Figueroa C, Jansen K, Taylor C (2010) Outflow boundary conditions for 3D simulations of non-periodic blood flow and pressure fields in deformable arteries. Comput Methods Biomech Biomed Eng 13(5):625

    Article  Google Scholar 

  • Wang Q, Moin P, Iaccarino G (2009) Minimal repetition dynamic checkpointing algorithm for unsteady adjoint calculation. SIAM J Sci Comput 31(4):2549

    Article  MathSciNet  MATH  Google Scholar 

  • Wüchner R, Kupzok A, Bletzinger KU (2007) A framework for stabilized partitioned analysis of thin membrane-wind interaction. Int J Numer Methods Fluids 54(6–8):945

    Article  MATH  Google Scholar 

Download references

Acknowledgement

Joris Degroote gratefully acknowledges funding by a post-doctoral fellowship of the Research Foundation—Flanders (FWO).

Author information

Authors and Affiliations

  1. Department of Flow, Heat and Combustion Mechanics, Ghent University, Sint-Pietersnieuwstraat 41, 9000, Ghent, Belgium

    Joris Degroote

  2. Chair of Structural Analysis, Technische Universität München, Arcisstrasse 21, 80333, Munich, Germany

    Majid Hojjat, Electra Stavropoulou, Roland Wüchner & Kai-Uwe Bletzinger

Authors
  1. Joris Degroote
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Majid Hojjat
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Electra Stavropoulou
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Roland Wüchner
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Kai-Uwe Bletzinger
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Joris Degroote.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Degroote, J., Hojjat, M., Stavropoulou, E. et al. Partitioned solution of an unsteady adjoint for strongly coupled fluid-structure interactions and application to parameter identification of a one-dimensional problem. Struct Multidisc Optim 47, 77–94 (2013). https://doi.org/10.1007/s00158-012-0808-2

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00158-012-0808-2

Keywords

Search

Navigation