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Continuous and Discrete Adjoint Approach Based on Lattice Boltzmann Method in Aerodynamic Optimization Part I: Mathematical Derivation of Adjoint Lattice Boltzmann Equations

  • Mohamad Hamed Hekmat (a1) and Masoud Mirzaei (a1)
Abstract
Abstract

The significance of flow optimization utilizing the lattice Boltzmann (LB) method becomes obvious regarding its advantages as a novel flow field solution method compared to the other conventional computational fluid dynamics techniques. These unique characteristics of the LB method form the main idea of its application to optimization problems. In this research, for the first time, both continuous and discrete adjoint equations were extracted based on the LB method using a general procedure with low implementation cost. The proposed approach could be performed similarly for any optimization problem with the corresponding cost function and design variables vector. Moreover, this approach was not limited to flow fields and could be employed for steady as well as unsteady flows. Initially, the continuous and discrete adjoint LB equations and the cost function gradient vector were derived mathematically in detail using the continuous and discrete LB equations in space and time, respectively. Meanwhile, new adjoint concepts in lattice space were introduced. Finally, the analytical evaluation of the adjoint distribution functions and the cost function gradients was carried out.

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Corresponding author
Corresponding author. Email: mhekmat@dena.kntu.ac.ir
Email: mirzaei@kntu.ac.ir
References
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[1]Hekmat M. H., Aerodynamic Optimization of an Airfoil Using Adjoint Equations Approach, M. S. Thesis, K. N. Toosi University of Technology, 2009.
[2]Nadarajah S., A comparison of the discrete and continuous adjoint approach to automatic aero-dynamic optimization, AIAA paper 2000-0667, 38th Aerospace Sciences Meeting and Exhibit, Reno, Nevada, January 2000.
[3]Mittal R. and Iaccarino G., Immersed boundary methods, Ann. Rev. Fluid Mech., 37 (2005), pp. 239261.
[4]Jameson A., Aerodynamic design via control theory, J. Sci. Comput., also ICASE Report No. 88-64, 3 (1988), pp. 233260.
[5]Jameson A., Computational aerodynamics for aircraft design, Science, 245 (1989), pp. 361371.
[6]Jameson A., Automatic design of transonic airfoils to reduce the shock induced pressure drag, In Proceedings of the 31st Annual Conference on Aviation and Aeronautics, 517, 1990.
[7]Elliot J. and Peraire J., Aerodynamic design using unstructured meshes, AIAA paper 961941, 1996.
[8]Nadarajah S., The Discrete Adjoint Approach to Aerodynamic Shape Optimization, Ph.D. Thesis, Standford University, 2003.
[9]Hekmat M. H., Mirzaei M. and Izadpanah E., Constrained and non-constrained aerodynamic optimization using the adjoint equations approach, J. Mech. Sci. Tech., 23 (2009), pp. 19111923.
[10]Hekmat M. H., Mirzaei M. and Izadpanah E., Numerical investigation of adjoint method in aerodynamic optimization, Mech. Aerospace Eng. J., 5(1) (2009), pp. 7586.
[11]Tonomura O., Kano M. AND Hasebe S., Shape optimization of microchannels using CFD and adjoint method, 20th European Symposium on Computer Aided Process Engineering-ESCAPE20, 2010.
[12]Jacques , Peter E. V. AND Dwight R. P., Numerical sensitivity analysis for aerodynamic optimization: a survey of approaches, J. Comput. Fluids, 39 (2010), pp. 373391.
[13]Hicken J. E. and Zingg D. W., Aerodynamic optimization algorithm with integrated geometry parameterization and mesh movement, AlAA Journal, 48(2) (2010), pp. 400414.
[14]Freund J. B., Adjointt-based optimization for understanding and suppressing jet noise, Procedia IUTAM J., 1 (2010), pp. 5463.
[15]Tekitek M. M., Bouzidi M., Dubois F. and Lallemand P., Adjoint lattice Boltzmann equation for parameter identification, Comput. Fluids, 35 (2006), pp. 805813.
[16]Pingen G., Evgrafov A. and Maute K., Topology optimization of flow domains using the lattice Boltzmann method, Struct Multidisc Optim J., 34 (2007), pp. 507524.
[17]Pingen G., Evgrafov A. and Maute K., Parameter Sensitivity analysis for the hydrodynamic lattice Boltzmann method with application, J. Comput. Fluids, 38 (2009), pp. 910923.
[18]Krause M. J., Thater G. and Heuveline V., Adjoint-based fluid flow control and optimization with lattice Boltzmann methods, Comput. Math. Appl., 65 (2013), pp. 945960.
[19]Gladrow W. AND Dieter A., Lattice-Gas Cellular Automata and Lattice Boltzmann Models: an Introduction, Heidelberg, Springer, 2000.
[20]Bhatnagar P. L., Gross E. P. and Krook M., A model for collision processes in gases I. small amplitude processes in charged and neutral one-component systems, J. Phys. Rev., 94(3) (1954), pp. 511525.
[21]Sh. Luo L., The Lattice-Gas and Lattice Boltzmann Methods: Past, Present, and Future, in Wu J.-H. and Zhu Z.-J., editors, International Conference on Applied Computational Fluid Dynamics, 5283, 2000.
[22]Guo Z., Zheng C. and Shi B., Discrete lattice effects on the forcing term in the lattice Boltzmann method, Phys. Rev. E, 65(4) (2002), pp. 16.
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Advances in Applied Mathematics and Mechanics
  • ISSN: 2070-0733
  • EISSN: 2075-1354
  • URL: /core/journals/advances-in-applied-mathematics-and-mechanics
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