Abstract
We propose a topology optimization method for a flow field using transient information. The optimization algorithm of many conventional methods use the fully converged information of a flow field. In contrast, our approach uses the transient information of an unsteady flow field and update the design domain while solving the unsteady flow field, thereby greatly reducing the computational cost. The fluid and solid regions are clearly distinguished by a level-set function. Consequently, the boundary is concretely represented, and precise boundary conditions are applied on the wall boundary. The lattice Boltzmann method is employed as a fluid computation method. To implement the non-slip boundary conditions at the fluid-solid boundary, we apply bounce-back conditions. We update the domain according to a sensitivity analysis. A sensitivity is formulated based on the lattice Boltzmann equations without adjoint equations for self-adjoint flow. We approximately use the sensitivity for non-self-adjoint equations, i.e. lattice Boltzmann equations, and discuss the optimality and limitations. The approximated sensitivity also considers the bounce-back boundary conditions at the wall separating the fluid and solid regions.
Similar content being viewed by others
References
Aage, N., Poulsen, T.H., Gersborg-Hansen, A., Sigmund, O.: Topology optimization of large scale Stokes flow problems. Struct. Multidiscip. Optim. 35, 175–180 (2008)
Allaire, G., Jouve, F., Toader, A.-M.: Structural optimization using sensitivity analysis and a level-set method. J. Comput. Phys. 194, 363–393 (2004)
Andreasen, C.S., Sigmund, O.: Topology optimization of fluid-structure-interaction problemsin poroelasticity. Comput. Methods Appl. Mech. Eng. 258, 55–62 (2013)
Ayachit, U.: The ParaView Guide: A Parallel Visualization Application. Kitware Incorporated, USA (2015)
Bajaj, N., Subbarayan, G., Garimella, S.V.: Topological design of channels for squeeze flow optimization of thermal interface materials. Int. J. Heat Mass Transf. 55, 3560–3575 (2012)
Bendsøe, M.P., Kikuchi, N.: Generating optimal topologies in structural design using a homogenization method. Comput. Methods Appl. Mech. Eng. 71, 197–224 (1988)
Bhatnager, P., Gross, E., Krook, M.: A model for collision processes in gases, I: small amplitude processes in charged and neutral one-component system. Phys. Rev. 94, 511–525 (1954)
Borrval, T., Peterson, J.: Topology optimization of fluids in Stokes flow. Int. J. Numer. Methods Fluids 41, 77–107 (2003)
Bouzidi, M., Firdaouss, M., Lallemand, P.: Momentum transfer of a Boltzmann-lattice fluid with boundaries. Phys. Fluids 13, 3452–3459 (2001)
Brandenburg, C., Lindemann, F., Ulbrich, M., Ulbrich, S.: A continuous adjoint approach to shape optimization for Navier–Stokes flow. In: Künisch, K., Sprekels, J., Leugering, G., Tröltzsch, F. (Eds) Optimal Control of Coupled Systems of Partial Differential Equations, vol. 158 of International Series of Numerical Mathematics, pp. 35–56 . Birkhäuser, Basel (2009)
Challis, V.J., Guest, J.K.: Level set topology optimization of fluids in Stokes flow. Int. J. Numer. Methods Eng. 79, 1284–1308 (2009)
Deng, Y., Liu, Z., Wu, Y.: Topology optimization of steady and unsteady incompressible Navier–Stokes flows driven by body forces. Struct. Multidiscip. Optim. 47, 555–570 (2013)
Deng, Y., Liu, Z., Zhang, P., Liu, Y., Wu, Y.: Topology optimization of unsteady incompressible Navier–Stokes flows. J. Comput. Phys. 230, 6688–6708 (2011)
d’Humières, D., Ginzburg, I., Krafczyk, M., Lallemand, P., Luo, L.: Multiple-relaxation-time lattice Boltzmann models in three dimensions. Philos. Trans. R. Soc. A 360, 437–451 (2002)
d’Humis̀eres, D., Lallemand, P.: Numerical simulations of hydrodynamics with lattice gas automata in two dimensions. Complex Syst 1, 599–632 (1987)
Duan, X.-B., Ma, Y.-C., Zhang, R.: Optimal shape control of fluid flow using variational level set method. Phys. Lett. A 372, 1374–1379 (2008)
Geller, S., Krafczyk, M., Tölke, J., Turek, S., Hron, J.: Benchmark computations based on lattice-Boltzmann, finite element and finite volume methods for laminar flows. Comput. Fluids 35, 888–897 (2006)
Gersborg-Hansen, A., Sigmund, O., Haber, R.: Topology optimization of channel flow problem. Struct. Multidiscip. Optim. 30, 181–192 (2005)
Guest, J.K., Précost, J.H.: Topology optimization of creeping fluid flows using a Darcy-Stokes finite element. Int. J. Numer. Methods Eng. 66, 461–484 (2006)
Ho, C.-F., Chang, C., Lin, K.-H., Lin, C.-A.: Consistent boundary conditions for 2D and 3D lattice Boltzmann simulations. Comput. Model. Eng. Sci. 44, 137–155 (2009)
Kontoleontos, E., Papoutisis-Kiachagias, E., Zymaris, A., Papadinitriou, D., Giannakoglou, K.: Adjoint-based constrained topology optimization for viscous flows, including heat transfer. Eng. Optim. 45, 941–961 (2013)
Krafczyk, M., Tølke, J., Rank, E., Schulz, M.: Two-dimensional simulation of fluid-structure interaction using lattice-Boltzmann methods. Comput. Struct. 79, 2031–2037 (2001)
Krause, M., Thater, G., Heuveline, V.: Adjoint-based fluid flow control and optimisation with lattice Boltzmann methods. Comput. Math. Appl. 65, 945–960 (2013)
Kreissl, S., Maute, K.: Levelset based fluid topology optimization using the extended finite element method. Struct. Multidiscip. Optim. 46, 311–326 (2012)
Kreissl, S., Pingen, G., Evgrafov, A., Maute, K.: Topology optimization of flexible micro-fluidic devices. Struct. Multidiscip. Optim. 42, 495–516 (2010)
Kreissl, S., Pingen, G., Maute, K.: An explicit level set approach for generalized shape optimization of fluids with the lattice Boltzmann method. Int. J. Numer. Methods Eng. 65, 496–519 (2011)
Kreissl, S., Pingen, G., Maute, K.: Topology optimization for unsteady flow. Int. J. Numer. Methods Eng. 87, 1229–1253 (2011)
Lallemand, P., Luo, L.-S.: Theory of the lattice Boltzmann method: dispersion, dissipation, isotropy, Galilean invariance, and stability. Phys. Rev. E 61, 6546–6562 (2000)
Maute, K., Allen, M.: Conceptual design of aeroelastic structures by topology optimization. Struct. Multidiscip. Optim. 27, 27–42 (2004)
McNamara, G.R., Zanetti, G.: Use of Boltzmann equation to simulate lattice-gas automata. Phys. Rev. Lett. 61, 2332–2335 (1988)
Moos, O., Klimetzek, F., Rossmann, R.: Bionic optimization of air-guiding system. In: Proceedings of SAE 2004 World Congress and Exhibition, 2004-01-1377, pp. 95–100 (2004)
Özkaya, E., Gauger, N.: Single-step one-shot aerodynamic shape optimization. In: Kunisch, K., Sprekels, J., Leugering, G., Troltzsch, F. (Eds.) Optimal control of coupled systems of partial differential equations, Vol. 158 of International series of numerical mathematics, pp. 191–204. Birkhauser, Basel (2009)
Pingen, G., Evgrafov, A., Maute, K.: Topology optimization of flow domains using the lattice Boltzmann method. Struct. Multidiscip. Optim. 34, 507–524 (2007)
Pingen, G., Maute, K.: Optimal design for non-Newtonian flows using a topology optimization approach. Comput. Math. Appl. 59, 2340–2350 (2010)
Pingen, G., Waidmann, M., Evgrafov, A., Maute, K.: Adjoint parameter sensitivity analysis for the hydrodynamic lattice Boltzmann method with applications to design optimization. Comput. Fluids 38, 910–923 (2009)
Pingen, G., Waidmann, M., Evgrafov, A., Maute, K.: A parametric level-set approach for topology optimization of flow domains. Struct. Multidiscip. Optim. 41, 117–131 (2010)
Pironneau, O.: On optimum profiles in Stokes flow. J. Fluid Mech. 59, 117–128 (1973)
Steven, G., Li, Q., Xie, Y.: Evolutionary topology and shape design for general physical field problems. Comput. Mech. 26, 129–139 (2000)
Succi, S.: The lattice Boltzmann equation: for fluid dynamics and beyond. Oxford University Press, Oxford (2001)
Vincente, W., Picelli, R., Pavanello, R., Xie, Y.: Topology optimization of frequency responses of fluid-structure interaction system. Finite Elements Anal. Des. 98, 1–13 (2015)
Yaji, K., Yamada, T., Kubo, S., Izui, K., Nishiwaki, S.: A topology optimization method for a coupled thermal-fluid problem using level set boundary expressions. Int. J. Heat Mass Transf. 81, 878–888 (2015)
Yaji, K., Yamada, T., Yoshino, M., Matsumoto, T., Izui, K., Nishiwaki, S.: Topology optimization using the lattice Boltzmann method incorporating level set boundary expressions. J. Comput. Phys. 274, 158–181 (2014)
Yamada, T., Izui, K., Nishiwaki, S., Takezawa, A.: A topology optimization method based on the level set method incorporating a fictitious interface energy. Comput. Methods Appl. Mech. Eng. 199, 2876–2891 (2010)
Yonekura, K., Kanno, Y.: A flow topology optimization method for steady state flow using transient information of flow field solved by lattice Boltzmann method. Struct. Multidiscip. Optim. 51, 159–172 (2015)
Yonekura, K., Kanno, Y.: Erratum to: A flow topology optimization method for steady state flow using transient information of flow field solved by lattice Boltzmann method. Struct. Multidiscip. Optim. 54, 193–195 (2016)
Zhang, T., Shi, B., Guo, Z., Chai, Z., Lu, J.: General bounce-back scheme for concentration boundary condition in the lattice-Boltzmann method. Phys. Rev. E 85, 016701 (2012)
Ziegler, D.P.: Boundary conditions for lattice Bolzmann simulations. J. Stat. Phys. 71, 1171–1177 (1993)
Zou, Q., He, X.: On pressure and velocity boundary conditions for the lattice Boltzmann BGK model. J. Phys. Fluids 9, 1591–1598 (1997)
Author information
Authors and Affiliations
Corresponding author
About this article
Cite this article
Yonekura, K., Kanno, Y. Topology optimization method for interior flow based on transient information of the lattice Boltzmann method with a level-set function. Japan J. Indust. Appl. Math. 34, 611–632 (2017). https://doi.org/10.1007/s13160-017-0257-9
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s13160-017-0257-9
Keywords
- Flow optimization
- Lattice Boltzmann method
- Multiple relaxation time
- Level-set function
- Sensitivity analysis