The new Think-Discrete Do-Continuous (TDDC) adjoint has been developed during the DiStrAto project (full title: "Discretization Schemes for the Continuous Adjoint Equations Consistent with Discrete Adjoint, with Applications in Fluid Mechanics"), funded by the Hellenic Foundation for Research and Innovation (HFRI)
TDDC bridges the gap between the discrete and continuous variants of the adjoint method by proposing consistent discretization schemes (inspired by discrete adjoint) for the continuous adjoint equations and their boundary conditions, while maintaining the physical insight to the above offered by continuous adjoint.
The sensitivity derivatives computed by the TDDC adjoint are as accurate as in discrete adjoint, obtained though with a much lower memory footprint.
The new TDDC adjoint outperforms the continuous adjoint suing "standard" discretization schemes, and the so-computed more accurate sensitivity derivatives result in improved optimization behavior.
Highligths of the scientific achievements during the project follow.
Scientific Achievements
Consistent discretization schemes
Consistent discretization schemes for the continuous adjoint equations, inspired by the discrete adjoint equations which are derived first, with a clear physical meaning, are proposed and used.
The interested reader may find the detailed development of these schemes in the PhD texts of Dr. M.
Kontou
and Dr. N.
Galanos.
The former performed the development for the in-house GPU-enabled solver PUMA (compressible fluid, vertex-centered finite volume definition) while the latter in the open-source CFD toolbox OpenFOAM (incompressible fluid, cell-centered).
Accurate Sensitivity Derivatives
The accuracy of the TDDC adjoint in computing sensitivity derivatives (SDs) is compared with Finite Differences as well as with continuous adjoint with "standard" discretization schemes widely used in the literature, in a wide number of applications.
An example of the obtained accuracy on computational grids of different size (from coarse to fine) is shown in the figures below.
Reference SDs are computed by Finite Differences (FDs).
The TDDC adjoint is capable or reproducing the SDs of FDs with high accuracy (at least 6 significant digits) on any grid, while discrepancies appear on the SDs computed based on the Standard adjoint discretization.
Improved Optimization Behavior
The consistency and accuracy of the SDs with the TDDC adjoint leads to improved optimization behavior.
Comparison with the "standard" continuous adjoint reveales the benefits of the TDDC adjoint in terms of optimization convergence rate and quality of the optimized solution.
In other words, the TDDC adjoint (compared with "standard" continuous adjoint) can obtain a solution of same quality in less or equal optimization cycles, and very often the TDDC adjoint can lead to a better design.
Dissemination Activities
The outcomes of the DiStrAto project have been disseminated through publications in scientific journals and international conferences.
A detailed list with all the relevant publications follows.
Scientific Journals:
X.S. Trompoukis, K.T. Tsiakas, V.G. Asouti, K.C. Giannakoglou.
Continuous adjoint-based shape optimization of a turbomachinery stage using a 3D volumetric parameterization.
International Journal for Numerical Methods in Fluids 2023; 95(7):1054-1075; https://doi.org/10.1002/fld.5187
N. Galanos, E.M. Papoutsis-Kiachagias and K.C. Giannakoglou.
The Cut-Cell Method for the Conjugate Heat Transfer Topology Optimization of Turbulent Flows Using the "Think Discrete-Do Continuous" Adjoint.
Energies 2024; 17(8):1817; https://doi.org/10.3390/en17081817
M.G. Kontou, X.S. Trompoukis, V.G. Asouti, K.C. Giannakoglou.
The Continuous Adjoint Method to the γ-Reθt Transition Model Coupled with the Spalart-Allmaras Model for Compressible Flows.
International Journal for Numerical Methods in Fluids 2024; https://doi.org/10.1002/fld.5319
A.-S.I. Margetis, E.M. Papoutsis-Kiachagias and K.C. Giannakoglou.
The continuous adjoint to the incompressible (D)DES Spalart-Allmaras turbulence models.
Computers & Fluids 2024; 284: 106439; https://doi.org/10.1016/j.compfluid.2024.106439
N. Galanos, E.M. Papoutsis-Kiachagias and K.C. Giannakoglou.
A Continuous Adjoint Cut-Cell Formulation for Toppology Optimization of bi-fluid Heat Exchangers.
International Journal of Numerical Methods in Heat & Fluid Flow 2025; https://doi.org/10.1108/HFF-08-2024-0642
International Conferences:
K.C. Giannakoglou, V.G. Asouti, E.M. Papoutsis-Kiachagias, N. Galanos, M.G. Kontou, X.S. Trompoukis.
The Think Discrete-Do Continuous Adjoint in Aerodynamic Shape Optimization.
EUROGEN 2023, 15th International Conference on Evolutionary and Deterministic Methods for Design, Optimization and Control, Chania, Greece, June 1-3 2023.
M.G. Kontou, X.S. Trompoukis, V.G. Asouti, K.C. Giannakoglou.
On the discretization of the continuous adjoint to the Euler equations in shape optimization.
ADMOS 2023, International Conference on Adaptive Modeling and Simulation, Gothenburg, Sweden, June 19-21 2023.
N. Galanos, E.M. Papoutsis-Kiachagias, K.C. Giannakoglou.
A Continuous Adjoint Cut-Cell formulation for Topology Optimization with one or two Fluids and Conjugate Heat Transfer.
ICCHMT 2023, 14th International Conference on Computational Heat and Mass Transfer, Dusseldorf, Germany, September 4-8, 2023.
M.G. Kontou, X.S. Trompoukis, V.G. Asouti, K.C. Giannakoglou.
Consistent Discretization Schemes for The Continuous Adjoint Equations in Aerodynamic Shape Optimization for Turbulent/Transitional Flows.
The 9th European Congress on Computational Methods in Applied Sciences and Engineering, ECCOMAS Congress 2024, Lisbon, Portugal, June 3-7 2024.
Researchers' Night:
The research team of DiStrAto participated in the European Researchers' Night events of 2023 and 2024 (29/09/2023 and 27/09/2024)
at the National Technical University of Athens.
Acknowledgment
This work was supported by the Hellenic Foundation for Research and Innovation (H.F.R.I.) under the
"2nd Call for H.F.R.I. Research Projects to support Faculty Members & Researchers" (Project Number: 3821).
