143
Views
0
CrossRef citations to date
0
Altmetric
Articles

On the Contribution of Wall Distance Fields to the Adjoint of a RANS Model

, &
Pages 687-704 | Received 02 Aug 2022, Accepted 30 Jan 2023, Published online: 16 Feb 2023

References

  • Allmaras, S. R., F. T. Johnson, and P. R. Spalart. 2012. “Modifications and Clarifications for the Implementation of the Spalart-Allmaras Turbulence Model.” In 46th AIAA Fluid Dynamics Conference, Big Island, HI, USA: AIAA.
  • Anderson, W. K., and D. L. Bonhaus. 1997. “Aerodynamic Design on Unstructured Grids for Turbulent Flows.” Tech. Rep. NASA Technical Memorandum 112867. Hampton, Virginia: NASA.
  • Anderson, W. K., and D. L. Bonhaus. 1999. “Airfoil Design on Unstructured Grids for Turbulent Flows.” AIAA Journal 37 (2): 185–191. doi:10.2514/2.712.
  • Bassi, F., and S. Rebay. 1997. “A High-Order Accurate Discontinuous Finite Element Method for the Numerical Solution of the Compressible Navier–Stokes Equations.” Journal of Computational Physics 131 (2): 267–279. doi:10.1006/jcph.1996.5572.
  • Belyaev, A. G., and P.-A. Fayolle. 2015. “On Variational and PDE-Based Distance Function Approximations.” In Computer Graphics Forum, 104–118, Wiley Online Library. doi:10.1111/cgf.12611
  • Bischof, C., P. Khademi, A. Mauer, P. Hovland, and A. Carle. 1995. “Adifor 2.0 Users Guide (Revision B).” Tech. Rep., Argonne National Lab., IL (USA).
  • Boger, D. A.. 2001. “Efficient Method for Calculating Wall Proximity.” AIAA Journal 39 (12): 2404–2406. doi:10.2514/2.1251.
  • Bueno-Orovio, A., C. Castro, F. Palacios, and E. Zuazua. 2012. “Continuous Adjoint Approach for the Spalart-Allmaras Model in Aerodynamic Optimization.” AIAA Journal 50 (3): 631–646. doi:10.2514/1.J051307.
  • Castro-Díaz, M., F. Hecht, B. Mohammadi, and O. Pironneau. 1997. “Anisotropic Unstructured Mesh Adaption for Flow Simulations.” International Journal for Numerical Methods in Fluids 25 (4): 475–491. doi:10.1002/(ISSN)1097-0363.
  • Chien, K.-Y.. 1982. “Predictions of Channel and Boundary-Layer Flows with a Low-Reynolds-Number Turbulence Model.” AIAA Journal 20 (1): 33–38. doi:10.2514/3.51043.
  • Dhert, T., T. Ashuri, and J. R. Martins. 2017. “Aerodynamic Shape Optimization of Wind Turbine Blades Using a Reynolds-Averaged Navier–Stokes Model and an Adjoint Method.” Wind Energy 20 (5): 909–926. doi:10.1002/we.2070 .
  • Driver, J., and D. W. Zingg. 2007. “Numerical Aerodynamic Optimization Incorporating Laminar-Turbulent Transition Prediction.” AIAA Journal 45 (8): 1810–1818. doi:10.2514/1.23569.
  • Dwight, R. P., and J. Brezillon. 2006. “Effect of Approximations of the Discrete Adjoint on Gradient-Based Optimization.” AIAA Journal 44 (12): 3022–3031. doi:10.2514/1.21744.
  • Economon, T., F. Palacios, and J. Alonso. 2012. “A Coupled-Adjoint Method for Aerodynamic and Aeroacoustic Optimization.” In 12th AIAA Aviation Technology, Integration, and Operations (ATIO) Conference and 14th AIAA/ISSMO Multidisciplinary Analysis and Optimization Conference, p. 5598. Indiana, USA: AIAA. doi:10.2514/6.2012-5598
  • Fidkowski, K. J., and D. L. Darmofal. 2011. “Review of Output-Based Error Estimation and Mesh Adaptation in Computational Fluid Dynamics.” AIAA Journal 49 (4): 673–694. doi:10.2514/1.J050073.
  • Fraysse, F., E. Valero, and J. Ponsin. 2012. “Comparison of Mesh Adaptation Using the Adjoint Methodology and Truncation Error Estimates.” AIAA Journal 50 (9): 1920–1932. doi:10.2514/1.J051450.
  • Galbraith, M. C., S. R. Allmaras, and D. L. Darmofal. 2018. “Sans Rans Solutions for 3D Benchmark Configurations.” In 2018 AIAA Aerospace Sciences Meeting, 1570. Kissimmee, Florida, USA: AIAA. doi:10.2514/6.2018-1570
  • Giles, M. B., and N. A. Pierce. 2000. “An Introduction to the Adjoint Approach to Design.” Flow, Turbulence and Combustion 65 (3–4): 393–415. doi:10.1023/A:1011430410075.
  • Hamed, A.. 2022. “Multi-Objective Optimization Method of Trimaran Hull Form for Resistance Reduction and Propeller Intake Flow Improvement.” Ocean Engineering 244: 110352. doi:10.1016/j.oceaneng.2021.110352.
  • Hartmann, R., J. Held, and T. Leicht. 2011. “Adjoint-Based Error Estimation and Adaptive Mesh Refinement for the RANS and K–ω Turbulence Model Equations.” Journal of Computational Physics 230 (11): 4268–4284. doi:10.1016/j.jcp.2010.10.026.
  • Hascoët, L., and V. Pascual. 2013. “The Tapenade Automatic Differentiation Tool: Principles, Model, and Specification.” ACM Transactions on Mathematical Software 39 (3): 1–43. doi:10.1145/2450153.2450158 .
  • He, P., G. Filip, J. R. Martins, and K. J. Maki. 2019. “Design Optimization for Self-Propulsion of a Bulk Carrier Hull Using a Discrete Adjoint Method.” Computers & Fluids 192: 104259. doi:10.1016/j.compfluid.2019.104259.
  • Jameson, A.. 1988. “Aerodynamic Design Via Control Theory.” Journal of Scientific Computing 3 (3): 233–260. doi:10.1007/BF01061285.
  • Jameson, A., and L. Martinelli. 2000. “Aerodynamic Shape Optimization Techniques Based on Control Theory.” In Computational Mathematics Driven by Industrial Problems, 151–221. Martina Franca, Italy: Springer. doi:10.1007/BFb0103920
  • Jones, E., E. Oliphant, and P. Peterson. 2001. “Scipy Open Source Scientific Tools for Python.” https://www.scipy.org/.
  • Kühl, N., J. Kröger, M. Siebenborn, M. Hinze, and T. Rung. 2021. “Adjoint Complement to the Volume-of-Fluid Method for Immiscible Flows.” Journal of Computational Physics 440: 110411. doi:10.1016/j.jcp.2021.110411.
  • Kavvadias, I., E. Papoutsis-Kiachagias, G. Dimitrakopoulos, and K. Giannakoglou. 2015. “The Continuous Adjoint Approach to the K–ω SST Turbulence Model with Applications in Shape Optimization.” Engineering Optimization 47 (11): 1523–1542. doi:10.1080/0305215X.2014.979816.
  • Kenway, G. K., G. J. Kennedy, and J. R. Martins. 2014. “Scalable Parallel Approach for High-Fidelity Steady-State Aeroelastic Analysis and Adjoint Derivative Computations.” AIAA Journal 52 (5): 935–951. doi:10.2514/1.J052255.
  • Kenway, G. K., C. A. Mader, P. He, and J. R. Martins. 2019. “Effective Adjoint Approaches for Computational Fluid Dynamics.” Progress in Aerospace Sciences 110: 100542. doi:10.1016/j.paerosci.2019.05.002.
  • Kim, C. S., C. Kim, and O. H. Rho. 2001. “Sensitivity Analysis for the Navier–Stokes Equations with Two-Equation Turbulence Models.” AIAA Journal 39 (5): 838–845. doi:10.2514/2.1387.
  • Kim, C. S., C. Kim, and O. H. Rho. 2003. “Feasibility Study of Constant Eddy-Viscosity Assumption in Gradient-Based Design Optimization.” Journal of Aircraft 40 (6): 1168–1176. doi:10.2514/2.7206.
  • Kim, H.-J., C. Kim, O.-H. Rho, and K. Lee. 1999. “Aerodynamic Sensitivity Analysis for Navier–Stokes Equations.” In 37th Aerospace Sciences Meeting and Exhibit, 402. Reno, NV, USA: AIAA. doi:10.2514/6.1999-402
  • Korivi, V. M., A. C. I. Taylor, P. A. Newman, G. W. Hou, and H. E. Jones. 1992 Feb. “An Incremental Strategy for Calculating Consistent Discrete CFD Sensitivity Derivatives.” Tech. Rep. NASA-TM-104207, NASA Langley Research Center.
  • Kröger, J., N. Kühl, and T. Rung. 2018. “Adjoint Volume-of-Fluid Approaches for the Hydrodynamic Optimisation of Ships.” Ship Technology Research 65 (1): 47–68. doi:10.1080/09377255.2017.1411001.
  • Liem, R. P., G. K. Kenway, and J. R. Martins. 2015. “Multimission Aircraft Fuel-Burn Minimization Via Multipoint Aerostructural Optimization.” AIAA Journal 53 (1): 104–122. doi:10.2514/1.J052940.
  • Lyu, Z., G. K. Kenway, and J. R. Martins. 2015. “Aerodynamic Shape Optimization Investigations of the Common Research Model Wing Benchmark.” AIAA Journal 53 (4): 968–985. doi:10.2514/1.J053318.
  • Lyu, Z., G. K. Kenway, C. Paige, and J. Martins. 2013. “Automatic Differentiation Adjoint of the Reynolds-Averaged Navier–Stokes Equations with a Turbulence Model.” In 21st AIAA Computational Fluid Dynamics Conference, 2581. San Diego, CA: AIAA. doi:10.2514/6.2013-2581
  • Madsen, M. H. A., F. Zahle, N. N. Sørensen, and J. R. Martins. 2019. “Multipoint High-Fidelity CFD-Based Aerodynamic Shape Optimization of a 10 MW Wind Turbine.” Wind Energy Science 4 (2): 163–192. doi:10.5194/wes-4-163-2019.
  • Manservisi, S., and F. Menghini. 2016. “Numerical Simulations of Optimal Control Problems for the Reynolds Averaged Navier–Stokes System Closed with a Two-Equation Turbulence Model.” Computers & Fluids 125: 130–143. doi:10.1016/j.compfluid.2015.11.007.
  • Marta, A. C., and S. Shankaran. 2013. “On the Handling of Turbulence Equations in RANS Adjoint Solvers.” Computers & Fluids 74: 102–113. doi:10.1016/j.compfluid.2013.01.012.
  • Martins, J. R.. 2022. “Aerodynamic Design Optimization: Challenges and Perspectives.” Computers & Fluids 239: 105391. doi:10.1016/j.compfluid.2022.105391.
  • Martins, J. R., J. J. Alonso, and J. J. Reuther. 2004. “High-Fidelity Aerostructural Design Optimization of a Supersonic Business Jet.” Journal of Aircraft 41 (3): 523–530. doi:10.2514/1.11478.
  • Mavriplis, D. J.. 2007. “Discrete Adjoint-Based Approach for Optimization Problems on Three-Dimensional Unstructured Meshes.” AIAA Journal 45 (4): 741–750. doi:10.2514/1.22743.
  • Menter, F. R.. 1994. “Two-Equation Eddy-Viscosity Turbulence Models for Engineering Applications.” AIAA Journal 32 (8): 1598–1605. doi:10.2514/3.12149.
  • Mura, G. L.. 2017. “Mesh Sensitivity Investigation in the Discrete Adjoint Framework.” Ph.D. thesis, University of Sheffield.
  • Nadarajah, S., and A. Jameson. 2000. “A Comparison of the Continuous and Discrete Adjoint Approach to Automatic Aerodynamic Optimization.” In 38th Aerospace Sciences Meeting and Exhibit, 667. Reno, NV, USA: AIAA. doi:10.2514/6.2000-667
  • Nadarajah, S., and A. Jameson. 2001. “Studies of the Continuous and Discrete Adjoint Approaches to Viscous Automatic Aerodynamic Shape Optimization.” In 15th AIAA Computational Fluid Dynamics Conference, 2530. Anaheim,CA, USA: AIAA. doi:10.2514/6.2001-2530
  • Nazemian, A., and P. Ghadimi. 2022. “Multi-Objective Optimization of Ship Hull Modification Based on Resistance and Wake Field Improvement: Combination of Adjoint Solver and CAD-CFD-Based Approach.” Journal of the Brazilian Society of Mechanical Sciences and Engineering 44 (1): 1–27. doi:10.1007/s40430-021-03335-4.
  • Nemec, M., and M. Aftosmis. 2007. “Adjoint Error Estimation and Adaptive Refinement for Embedded-Boundary Cartesian Meshes.” 18th AIAA Computational Fluid Dynamics Conference 4187. doi:10.2514/6.2007-4187 .
  • Nemec, M., and D. Zingg. 2001. “Towards Efficient Aerodynamic Shape Optimization Based on the Navier–Stokes Equations.” 15th AIAA Computational Fluid Dynamics Conference, 2532. Anaheim, CA, USA: AIAA. doi:10.2514/6.2001-2532
  • Nemec, M., D. W. Zingg, and T. H. Pulliam. 2004. “Multipoint and Multi-Objective Aerodynamic Shape Optimization.” AIAA Journal 42 (6): 1057–1065. doi:10.2514/1.10415.
  • Newman, J. C., W. K. Anderson, and D. L. Whitfield. 1998. Multidisciplinary Sensitivity Derivatives Using Complex Variables. Mississippi: Mississippi State University Publication. MSSU-EIRS-ERC-98-08.
  • Nielsen, E. J.. 1998. “Aerodynamic Design Sensitivities on an Unstructured Mesh Using the Navier–Stokes Equations and a Discrete Adjoint Formulation.” Blacksburg, Virginia: Ph.D. thesis, Virginia Tech. https://hdl.handle.net/10919/29459.
  • Nielsen, E. J., J. Lu, M. A. Park, and D. L. Darmofal. 2004. “An Implicit, Exact Dual Adjoint Solution Method for Turbulent Flows on Unstructured Grids.” Computers & Fluids 33 (9): 1131–1155. doi:10.1016/j.compfluid.2003.09.005.
  • Nielsen, E. J., and M. A. Park. 2006. “Using an Adjoint Approach to Eliminate Mesh Sensitivities in Computational Design.” AIAA Journal 44 (5): 948–953. doi:10.2514/1.16052.
  • Ntanakas, G., M. Meyer, and K. C. Giannakoglou. 2018. “Employing the Time-Domain Unsteady Discrete Adjoint Method for Shape Optimization of Three-Dimensional Multirow Turbomachinery Configurations.” Journal of Turbomachinery 140 (8): 081006. doi:10.1115/1.4040564.
  • Othmer, C.. 2008. “A Continuous Adjoint Formulation for the Computation of Topological and Surface Sensitivities of Ducted Flows.” International Journal for Numerical Methods in Fluids 58 (8): 861–877. doi:10.1002/fld.v58:8.
  • Othmer, C.. 2014. “Adjoint Methods for Car Aerodynamics.” Journal of Mathematics in Industry 4 (1): 1–23. doi:10.1186/2190-5983-4-6.
  • Papoutsis-Kiachagias, E., V. Asouti, K. Giannakoglou, K. Gkagkas, S. Shimokawa, and E. Itakura. 2019. “Multi-Point Aerodynamic Shape Optimization of Cars Based on Continuous Adjoint.” Structural and Multidisciplinary Optimization 59 (2): 675–694. doi:10.1007/s00158-018-2091-3.
  • Papoutsis-Kiachagias, E. M., and K. C. Giannakoglou. 2016. “Continuous Adjoint Methods for Turbulent Flows, Applied to Shape and Topology Optimization: Industrial Applications.” Archives of Computational Methods in Engineering 23 (2): 255–299. doi:10.1007/s11831-014-9141-9.
  • Papoutsis-Kiachagias, E., A. Zymaris, I. Kavvadias, D. Papadimitriou, and K. Giannakoglou. 2015. “The Continuous Adjoint Approach to the K–ε Turbulence Model for Shape Optimization and Optimal Active Control of Turbulent Flows.” Engineering Optimization 47 (3): 370–389. doi:10.1080/0305215X.2014.892595.
  • Paszke, A., S. Gross, S. Chintala, G. Chanan, E. Yang, Z. DeVito, Z. Lin, et al. 2017. “Automatic Differentiation in Pytorch.” NIPS Autodiff Workshop. Long Beach, CA, USA: 31st Conference on Neural Information Processing Systems (NIPS 2017).
  • Peraire, J., J. Peiro, and K. Morgan. 1992. “Adaptive Remeshing for Three-Dimensional Compressible Flow Computations.” Journal of Computational Physics 103 (2): 269–285. doi:10.1016/0021-9991(92)90401-J.
  • Peter, J. E., and R. P. Dwight. 2010. “Numerical Sensitivity Analysis for Aerodynamic Optimization: A Survey of Approaches.” Computers & Fluids 39 (3): 373–391. doi:10.1016/j.compfluid.2009.09.013.
  • Pironneau, O.. 1974. “On Optimum Design in Fluid Mechanics.” Journal of Fluid Mechanics 64 (1): 97–110. doi:10.1017/S0022112074002023.
  • Pletcher, R. H., J. C. Tannehill, and D. Anderson. 2013. Computational Fluid Mechanics and Heat Transfer. CRC press.
  • Qin, N., and X. Liu. 2006. “Flow Feature Aligned Grid Adaptation.” International Journal for Numerical Methods in Engineering 67 (6): 787–814. doi:10.1002/nme.1648.
  • Roe, P. L.. 1989. “The Use of the Riemann Problem in Finite Difference Schemes.” In Seventh International Conference on Numerical Methods in Fluid Dynamics, 354–359, Stanford University, Stanford, California and NASA/Ames (USA): Springer.
  • Roget, B., and J. Sitaraman. 2013. “Wall Distance Search Algorithm Using Voxelized Marching Spheres.” Journal of Computational Physics 241: 76–94. doi:10.1016/j.jcp.2013.01.035.
  • Secco, N. R., and J. R. Martins. 2019. “RANS-Based Aerodynamic Shape Optimization of a Strut-Braced Wing with Overset Meshes.” Journal of Aircraft 56 (1): 217–227. doi:10.2514/1.C034934.
  • Sethian, J. A.. 1999. “Fast Marching Methods.” SIAM Review 41 (2): 199–235. doi:10.1137/S0036144598347059.
  • Sharma, M., N. A. Wukie, M. Ugolotti, and M. G. Turner. 2018. “Unsteady Turbomachinery Simulations Using Harmonic Balance on a Discontinuous Galerkin Discretization.” V02CT42A054, Oslo, Norway: ASME Turbo Expo 2018: Turbomachinery Technical Conference and Exposition. doi:10.1115/GT2018-77204.
  • Shi, L., and Z. J. Wang. 2015. “Adjoint-Based Error Estimation and Mesh Adaptation for the Correction Procedure Via Reconstruction Method.” Journal of Computational Physics 295: 261–284. doi:10.1016/j.jcp.2015.04.011.
  • Siddappaji, K.. 2012. “Parametric 3D Blade Geometry Modeling Tool for Turbomachinery Systems.” Master's thesis, University of Cincinnati. https://rave.ohiolink.edu/etdc/view?acc_num=ucin1337264652.
  • Soemarwoto, B. I.. 1996. “Multi-Point Aerodynamic Design by Optimization.” Ph.D. thesis, The Netherlands: TUDelft. https://resolver.tudelft.nl/uuid:5a126ea6-74fc-4678-b84c-18ca3fd962c5.
  • Spalart, P., and S. Allmaras. 1992. “A One-Equation Turbulence Model for Aerodynamic Flows.” In 30th Aerospace Sciences Meeting and Exhibit, 439. Reno, NV, US: AIAA. doi:10.2514/6.1992-439
  • Stück, A., and T. Rung. 2013. “Adjoint Complement to Viscous Finite-Volume Pressure-Correction Methods.” Journal of Computational Physics 248: 402–419. doi:10.1016/j.jcp.2013.01.002.
  • Trompoukis, X., K. Tsiakas, V. Asouti, M. Kontou, and K. Giannakoglou. 2021. “Continuous Adjoint-Based Optimization of an Internally Cooled Turbine Blade–Mathematical Development and Application.” International Journal of Turbomachinery, Propulsion and Power 6 (2): 20. doi:10.3390/ijtpp60-20020.
  • Tucker, P.. 2003. “Differential Equation-Based Wall Distance Computation for DES and RANS.” Journal of Computational Physics 190 (1): 229–248. doi:10.1016/S0021-9991(03)00272-9.
  • Tucker, P.. 2011. “Hybrid Hamilton–Jacobi–Poisson Wall Distance Function Model.” Computers & Fluids 44 (1): 130–142. doi:10.1016/j.compfluid.2010.12.021.
  • Tucker, P. G., C. L. Rumsey, P. R. Spalart, R. E. Bartels, and R. T. Biedron. 2005. “Computations of Wall Distances Based on Differential Equations.” AIAA Journal 43 (3): 539–549. doi:10.2514/1.8626.
  • Ugolotti, M., M. Turner, and P. D. Orkwis. 2019. “An Adjoint-Based Sensitivity Formulation Using the Discontinuous Galerkin Method.” In AIAA Aviation 2019 Forum, 3202. Dalla, TX, USA: AIAA. doi:10.2514/6.2019-3202
  • Ugolotti, M., N. A. Wukie, M. Turner, and P. D. Orkwis. 2018. “Discrete-Adjoint Solver Tests and Consistency Analysis for Discontinuous Galerkin Discretization.” In 2018 Fluid Dynamics Conference, 4154. Atlanta, GA, USA: AIAA. doi:10.2514/6.2018-4154
  • Vasilopoulos, I., V. G. Asouti, K. C. Giannakoglou, and M. Meyer. 2021. “Gradient-Based Pareto Front Approximation Applied to Turbomachinery Shape Optimization.” Engineering with Computers 37 (1): 449–459. doi:10.1007/s00366-019-00832-y.
  • Wang, D., and L. He. 2010. “Adjoint Aerodynamic Design Optimization for Blades in Multistage Turbomachines–Part I: Methodology and Verification.” Journal of Turbomachinery 132 (2): doi:10.1115/1.3103928.
  • Wolf, E. M., C. R. Schrock, and N. A. Wukie. 2018. “Implementation of an Arbitrary Lagrangian–Eulerian Moving Mesh Capability in the Chidg Discontinuous Galerkin Code with Applications to Fluid-Structure Interaction.” In 2018 Fluid Dynamics Conference, 4268. Atlanta, GA, USA: AIAA. doi:10.2514/6.2018-4268
  • Wukie, N. A.. 2018. “A Discontinuous Galerkin Method for Turbomachinery and Acoustics Applications.” Ph.D. thesis, University of Cincinnati. https://rave.ohiolink.edu/etdc/view?acc_num=ucin1543840344167045.
  • Wukie, N. A., and P. D. Orkwis. 2016. “On Fully-Implicit Solutions of the Time-Linearized Euler Equations in a DG/Chimera Solver.” In 22nd AIAA/CEAS Aeroacoustics Conference, 2975. Lyon, France: AIAA. doi:10.2514/6.2016-2975
  • Wukie, N. A., and P. D. Orkwis. 2016. “A Implicit, Discontinuous Galerkin Chimera Solver Using Automatic Differentiation.” 54th AIAA Aerospace Sciences Meeting, 2054. San Diego, CA, USA: AIAA. doi:10.2514/6.2016-2054
  • Wukie, N. A., and P. D. Orkwis. 2017. “A p-Poisson Wall Distance Approach for Turbulence Modeling.” In 23rd AIAA Computational Fluid Dynamics Conference, 3945. Denver, CO, USA: AIAA. doi:10.2514/6.2017-3945
  • Wukie, N. A., P. D. Orkwis, and C. R. Schrock. 2017. “A Chimera-Based, Zonal Discontinuous Galerkin Method.” In 23rd AIAA Computational Fluid Dynamics Conference, 3947. Denver, CO, USA: AIAA. doi:10.2514/6.2017-3947
  • Wukie, N. A., P. D. Orkwis, and M. G. Turner. 2016. “A Fully-Implicit, Giles-Type Nonreflecting Boundary Condition in a DG-Chimera Turbomachinery Solver.” In 46th AIAA Fluid Dynamics Conference, 3335. Washington, DC, USA: AIAA.
  • Xu, S., and S. Timme. 2017. “Robust and Efficient Adjoint Solver for Complex Flow Conditions.” Computers & Fluids 148: 26–38. doi:10.1016/j.compfluid.2017.02.012.
  • Yu, W., and M. Blair. 2013. “DNAD, a Simple Tool for Automatic Differentiation of Fortran Codes Using Dual Numbers.” Computer Physics Communications 184 (5): 1446–1452. doi:10.1016/j.cpc.2012.12.025.
  • Zhao, H.. 2005. “A Fast Sweeping Method for Eikonal Equations.” Mathematics of Computation 74 (250): 603–627. doi:10.1090/mcom/2005-74-250.https://www.jstor.org/stable/4100081.
  • Zymaris, A., D. Papadimitriou, K. Giannakoglou, and C. Othmer. 2009. “Continuous Adjoint Approach to the Spalart–Allmaras Turbulence Model for Incompressible Flows.” Computers & Fluids 38 (8): 1528–1538. doi:10.1016/j.compfluid.2008.12.006.
  • Zymaris, A., D. Papadimitriou, K. C. Giannakoglou, and C. Othmer. 2010. “Adjoint Wall Functions: A New Concept for Use in Aerodynamic Shape Optimization.” Journal of Computational Physics 229 (13): 5228–5245. doi:10.1016/j.jcp.2010.03.037.

Reprints and Corporate Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

To request a reprint or corporate permissions for this article, please click on the relevant link below:

Academic Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

Obtain permissions instantly via Rightslink by clicking on the button below:

If you are unable to obtain permissions via Rightslink, please complete and submit this Permissions form. For more information, please visit our Permissions help page.