Approximating solutions of non-linear parametrized physical problems by interpolation presents a major challenge in terms of accuracy. In fact, pointwise interpolation of such solutions is rarely efficient and leads generally …
Approximating solutions of non-linear parametrized physical problems by interpolation presents a major challenge in terms of accuracy. In fact, pointwise interpolation of such solutions is rarely efficient and leads generally to incorrect results. However, instead of using a straight forward interpolation on solutions, reduced order models can be interpolated. More particularly, Amsallem and Farhat proposed an efficient POD reduced order model interpolation technique based on differential geometry tools. This approach, named in this paper ITSGM (Interpolation On a Tangent Space of the Grassmann Manifold), allows through the passage to the tangent space of the Grassmann manifold, to approximate accurately the reduced order basis associated to a new untrained parameter. This basis is used afterwards to build the interpolated ROM describing the temporal dynamics by performing the Galerkin projection on the high fidelity model. Such Galerkin ROMs require to access to the underlying high fidelity model, leading by that to intrusive ROMs. In this paper, and contrary to the ITSGM/Galerkin approach, we propose a non-intrusive reduced order modeling method which is independent of the governing equations. This method is named through this paper Bi-CITSGM (Bi-Calibrated ITSGM). It consists first to interpolate the spatial and temporal POD sampling bases considered as representatives of points on Grassmann manifolds, by the ITSGM method. Then, the resulting bases modes are reclassified by introducing two orthogonal matrices. These calibration matrices are determined as analytical solutions of two optimization problems. Results on the flow problem past a circular cylinder where the parameter of interpolation is the Reynolds number, show that for new untrained Reynolds number values, the developed approach produces satisfyingly accurate solutions in a real computational time.
Most genetic algorithms (GAs) used in literature to solve control problems are time consuming and involve important storage memory requirements. In fact, the search in GAs is iteratively performed on …
Most genetic algorithms (GAs) used in literature to solve control problems are time consuming and involve important storage memory requirements. In fact, the search in GAs is iteratively performed on a population of chromosomes (control parameters). As a result, the cost functional needs to be evaluated through solving the high fidelity model or by performing the experimental protocol for each chromosome and for many generations. To overcome this issue, a non intrusive reduced real coded genetic algorithm (RGA) for near real time optimal control is designed. This algorithm uses precalculated parametrized solution snapshots stored in the POD (Proper Orthogonal Decomposition) reduced form, to predict the solution snapshots for chromosomes over generations. The method used for this purpose is a hyper reduced version of the Bi-CITSGM method (Bi Calibrated Interpolation on the Tangent Space of the Grassmann Manifold) designed specially for non linear parametrized solution snapshots interpolation. This hyper reduced approach referred to as Hyper Bi-CITSGM, is proposed in such a way to accelerate the usual Bi-CITSGM process by bringing this last to a significantly low dimension. Thus, the whole optimization process by RGA can be performed in near real time. The potential of RGA in terms of accuracy and CPU time is demonstrated on control problems of the flow past a cylinder and flow in a lid driven cavity when the Reynolds number value varies.
A new application of subspaces interpolation for the construction of nonlinear Parametric Reduced Order Models (PROMs) is proposed. This approach is based upon the Riemannian geometry of the manifold formed …
A new application of subspaces interpolation for the construction of nonlinear Parametric Reduced Order Models (PROMs) is proposed. This approach is based upon the Riemannian geometry of the manifold formed by the quotient of the set of full-rank N-by-q matrices by the orthogonal group of dimension q. By using a set of untrained parametrized Proper Orthogonal Decomposition (POD) subspaces of dimension q, the subspace for a new untrained parameter is obtained as the generalized Karcher barycenter which solution is sought after by solving a simple fixed point problem. Contrary to existing PROM approaches, the proposed barycentric PROM is by construction easier to implement and more flexible with respect to change in parameter values. To assess the potential of the barycentric PROM, numerical experiments are conducted on the parametric flow past a circular cylinder and the flow in a lid driven cavity when the value of Reynolds number varies. It is demonstrated that the proposed barycentric PROM approach achieves competitive results with considerably reduced computational cost.
Optimal control of turbulent mixed-convection flows has attracted considerable attention from researchers. Numerical algorithms such as Genetic Algorithms (GAs) are powerful tools that allow to perform global optimization. These algorithms …
Optimal control of turbulent mixed-convection flows has attracted considerable attention from researchers. Numerical algorithms such as Genetic Algorithms (GAs) are powerful tools that allow to perform global optimization. These algorithms are particularly of great interest in complex optimization problems where cost functionals may lack smoothness and regularity. In turbulent flow optimization, the hybridization of GA with high fidelity Computational Fluid Dynamics (CFD) is extremely demanding in terms of computational time and memory storage. Thus, alternative approaches aiming to alleviate these requirements are of great interest. Nowadays, data driven approaches gained attention due to their potential in predicting flow solutions based only on preexisting data. In the present paper, we propose a near-real time data-driven genetic algorithm (DDGA) for inverse parameter identification problems involving turbulent flows. In this optimization framework, the parametrized flow data are used in their reduced form obtained by the POD (Proper Orthogonal Decomposition) and solutions prediction is made by interpolating the temporal and the spatial POD subspaces through a recently developed Riemannian barycentric interpolation. The validation of the proposed optimization approach is carried out in the parameter identification problem of the turbulent mixed-convection flow in a cavity. The objective is to determine the inflow temperature and inflow velocity corresponding to a given temperature distribution in a restricted area of the spatial domain. The results show that the proposed genetic programming optimization framework is able to deliver good approximations of the optimal solutions within less than two minutes.
This article presents error bounds for a velocity-pressure segregated POD reduced order model discretization of the Navier-Stokes equations. The stability is proven in L infinity (L 2 ) and energy …
This article presents error bounds for a velocity-pressure segregated POD reduced order model discretization of the Navier-Stokes equations. The stability is proven in L infinity (L 2 ) and energy norms for velocity, with bounds that do not depend on the viscosity, while for pressure it is proven in a semi-norm of the same asymptotic order as the L 2 norm with respect to the mesh size. The proposed estimates are calculated for the two flow problems, the flow past a cylinder and the lid-driven cavity flow. Their quality is then assessed in terms of the predicted logarithmic slope with respect to the velocity POD contribution ratio. We show that the proposed error estimates allow a good approximation of the real errors slopes and thus a good prediction of their rate of convergence.
Generally, reduced order models of fluid flows are obtained by projecting the Navier-Stokes equations onto a reduced subspace spanned by vector functions that carry the meaningful information of the dynamics. …
Generally, reduced order models of fluid flows are obtained by projecting the Navier-Stokes equations onto a reduced subspace spanned by vector functions that carry the meaningful information of the dynamics. A common method to generate such subspace is the Proper Orthogonal Decomposition. This projection strategy is intrusive since it assumes that the approximation model used to solve the high fidelity flow problem is known and accessible. In the present article, we propose a non-intrusive paradigm based on Machine Learning to build Closed Parametric Reduced Order Models (ML-PROM) relevant to fluid dynamics. With no prior knowledge requirement of the approximation model, this method is purely data-driven as it operates directly on data regardless their origin, DNS, RANS simulations or experiment. The key idea to build the ML-CPROM is to use the generally known form of Galerkin ROMs and assimilate the time variations of the temporal POD modes to a nonlinear quadratic form taking a closure right hand side member. This last term is introduced to account for uncaptured dynamics due to data noise, POD truncation errors and time integration schemes. For parameter variation, the ML-CPROM is updated by interpolation on the quotient manifold of the set of maximal rank matrices by the orthogonal group. The predicted closure term is afterwards calculated by a Long-Short-Term-Memory neural network. We assess the potential of the proposed approach on the parametric examples of the lid-driven cavity flow and the flow past a cylinder by varying the Reynolds number through the viscosity, and the Ahmed-Body flow by varying the geometry through the rear slant angle. We show on these examples that our method enables to recover the dynamics with a good accuracy, not only for the training parameter points, but also for unseen parameter values for which the model was not priory trained.
A machine learning-based methodology for blending data-driven turbulent closures for the Reynolds-Averaged Navier-Stokes (RANS) equations is proposed to improve the generalizability across different flow scenarios. Data-driven models based on sparse …
A machine learning-based methodology for blending data-driven turbulent closures for the Reynolds-Averaged Navier-Stokes (RANS) equations is proposed to improve the generalizability across different flow scenarios. Data-driven models based on sparse Bayesian learning and symbolic regression are pre-trained for specific flow classes, including turbulent channel flows, separated flows, and axisymmetric jets. The specialized models (called "\textit{experts}") are then blended using a set of weighting functions that reflect the local likelihood of a model to capture the reference data. The weighting functions are expressed as functions of a set of local flow features, and Random Forest regressors (RFR) are trained alongside the expert models to learn the mapping between the features and the weights. The training dataset includes additional data for flows ranging from shear flows at equilibrium to separated flows. The resulting blended model is then used to predict unseen flows, including a turbulent zero-pressure-gradient flat plate and a wall-mounted hump involving attached and separated boundary layer regions. The results show that the blending strategy adapts to the local flow conditions and effectively leverages the strengths of individual models, promoting the best-performing models locally, in a manner consistent with the training scenarios. The blended model provides a robust approximation for unseen flows that exhibit similar underlying physical characteristics to the training cases, while also successfully extrapolating to flows with different geometries and physical configurations.
Model reduction is a key technology for large-scale physical systems in science and engineering, as it brings behavior expressed in many degrees of freedom to a more manageable size that …
Model reduction is a key technology for large-scale physical systems in science and engineering, as it brings behavior expressed in many degrees of freedom to a more manageable size that subsequently allows control, optimization, and analysis with multi-query algorithms. We introduce an enhanced regression technique tailored to uncover quadratic parametric reduced-order dynamical systems from data. Our method, termed Improved Greedy Identification of Latent Dynamics (I-GILD), refines the learning phase of the original GILD approach. This refinement is achieved by reorganizing the quadratic model coefficients, allowing the minimum-residual problem to be reformulated using the Frobenius norm. Consequently, the optimality conditions lead to a generalized Sylvester equation, which is efficiently solved using the conjugate gradient method. Analysis of the convergence shows that I-GILD achieves superior convergence for quadratic model coefficients compared to GILD's steepest gradient descent, reducing both computational complexity and iteration count. Additionally, we derive an error bound for the model predictions, offering insights into error growth in time and ensuring controlled accuracy as long as the magnitudes of initial error is small and learning residuals are well minimized. The efficacy of I-GILD is demonstrated through its application to numerical and experimental tests, specifically the flow past Ahmed body with a variable rear slant angle, and the lid-driven cylindrical cavity problem with variable Reynolds numbers, utilizing particle-image velocimetry (PIV) data. These tests confirm I-GILD's ability to treat real-world dynamical system challenges and produce effective reduced-order models.
Model reduction is a key technology for large-scale physical systems in science and engineering, as it brings behavior expressed in many degrees of freedom to a more manageable size that …
Model reduction is a key technology for large-scale physical systems in science and engineering, as it brings behavior expressed in many degrees of freedom to a more manageable size that subsequently allows control, optimization, and analysis with multi-query algorithms. We introduce an enhanced regression technique tailored to uncover quadratic parametric reduced-order dynamical systems from data. Our method, termed Improved Greedy Identification of Latent Dynamics (I-GILD), refines the learning phase of the original GILD approach. This refinement is achieved by reorganizing the quadratic model coefficients, allowing the minimum-residual problem to be reformulated using the Frobenius norm. Consequently, the optimality conditions lead to a generalized Sylvester equation, which is efficiently solved using the conjugate gradient method. Analysis of the convergence shows that I-GILD achieves superior convergence for quadratic model coefficients compared to GILD's steepest gradient descent, reducing both computational complexity and iteration count. Additionally, we derive an error bound for the model predictions, offering insights into error growth in time and ensuring controlled accuracy as long as the magnitudes of initial error is small and learning residuals are well minimized. The efficacy of I-GILD is demonstrated through its application to numerical and experimental tests, specifically the flow past Ahmed body with a variable rear slant angle, and the lid-driven cylindrical cavity problem with variable Reynolds numbers, utilizing particle-image velocimetry (PIV) data. These tests confirm I-GILD's ability to treat real-world dynamical system challenges and produce effective reduced-order models.
A machine learning-based methodology for blending data-driven turbulent closures for the Reynolds-Averaged Navier-Stokes (RANS) equations is proposed to improve the generalizability across different flow scenarios. Data-driven models based on sparse …
A machine learning-based methodology for blending data-driven turbulent closures for the Reynolds-Averaged Navier-Stokes (RANS) equations is proposed to improve the generalizability across different flow scenarios. Data-driven models based on sparse Bayesian learning and symbolic regression are pre-trained for specific flow classes, including turbulent channel flows, separated flows, and axisymmetric jets. The specialized models (called "\textit{experts}") are then blended using a set of weighting functions that reflect the local likelihood of a model to capture the reference data. The weighting functions are expressed as functions of a set of local flow features, and Random Forest regressors (RFR) are trained alongside the expert models to learn the mapping between the features and the weights. The training dataset includes additional data for flows ranging from shear flows at equilibrium to separated flows. The resulting blended model is then used to predict unseen flows, including a turbulent zero-pressure-gradient flat plate and a wall-mounted hump involving attached and separated boundary layer regions. The results show that the blending strategy adapts to the local flow conditions and effectively leverages the strengths of individual models, promoting the best-performing models locally, in a manner consistent with the training scenarios. The blended model provides a robust approximation for unseen flows that exhibit similar underlying physical characteristics to the training cases, while also successfully extrapolating to flows with different geometries and physical configurations.
Generally, reduced order models of fluid flows are obtained by projecting the Navier-Stokes equations onto a reduced subspace spanned by vector functions that carry the meaningful information of the dynamics. …
Generally, reduced order models of fluid flows are obtained by projecting the Navier-Stokes equations onto a reduced subspace spanned by vector functions that carry the meaningful information of the dynamics. A common method to generate such subspace is the Proper Orthogonal Decomposition. This projection strategy is intrusive since it assumes that the approximation model used to solve the high fidelity flow problem is known and accessible. In the present article, we propose a non-intrusive paradigm based on Machine Learning to build Closed Parametric Reduced Order Models (ML-PROM) relevant to fluid dynamics. With no prior knowledge requirement of the approximation model, this method is purely data-driven as it operates directly on data regardless their origin, DNS, RANS simulations or experiment. The key idea to build the ML-CPROM is to use the generally known form of Galerkin ROMs and assimilate the time variations of the temporal POD modes to a nonlinear quadratic form taking a closure right hand side member. This last term is introduced to account for uncaptured dynamics due to data noise, POD truncation errors and time integration schemes. For parameter variation, the ML-CPROM is updated by interpolation on the quotient manifold of the set of maximal rank matrices by the orthogonal group. The predicted closure term is afterwards calculated by a Long-Short-Term-Memory neural network. We assess the potential of the proposed approach on the parametric examples of the lid-driven cavity flow and the flow past a cylinder by varying the Reynolds number through the viscosity, and the Ahmed-Body flow by varying the geometry through the rear slant angle. We show on these examples that our method enables to recover the dynamics with a good accuracy, not only for the training parameter points, but also for unseen parameter values for which the model was not priory trained.
This article presents error bounds for a velocity-pressure segregated POD reduced order model discretization of the Navier-Stokes equations. The stability is proven in L infinity (L 2 ) and energy …
This article presents error bounds for a velocity-pressure segregated POD reduced order model discretization of the Navier-Stokes equations. The stability is proven in L infinity (L 2 ) and energy norms for velocity, with bounds that do not depend on the viscosity, while for pressure it is proven in a semi-norm of the same asymptotic order as the L 2 norm with respect to the mesh size. The proposed estimates are calculated for the two flow problems, the flow past a cylinder and the lid-driven cavity flow. Their quality is then assessed in terms of the predicted logarithmic slope with respect to the velocity POD contribution ratio. We show that the proposed error estimates allow a good approximation of the real errors slopes and thus a good prediction of their rate of convergence.
Optimal control of turbulent mixed-convection flows has attracted considerable attention from researchers. Numerical algorithms such as Genetic Algorithms (GAs) are powerful tools that allow to perform global optimization. These algorithms …
Optimal control of turbulent mixed-convection flows has attracted considerable attention from researchers. Numerical algorithms such as Genetic Algorithms (GAs) are powerful tools that allow to perform global optimization. These algorithms are particularly of great interest in complex optimization problems where cost functionals may lack smoothness and regularity. In turbulent flow optimization, the hybridization of GA with high fidelity Computational Fluid Dynamics (CFD) is extremely demanding in terms of computational time and memory storage. Thus, alternative approaches aiming to alleviate these requirements are of great interest. Nowadays, data driven approaches gained attention due to their potential in predicting flow solutions based only on preexisting data. In the present paper, we propose a near-real time data-driven genetic algorithm (DDGA) for inverse parameter identification problems involving turbulent flows. In this optimization framework, the parametrized flow data are used in their reduced form obtained by the POD (Proper Orthogonal Decomposition) and solutions prediction is made by interpolating the temporal and the spatial POD subspaces through a recently developed Riemannian barycentric interpolation. The validation of the proposed optimization approach is carried out in the parameter identification problem of the turbulent mixed-convection flow in a cavity. The objective is to determine the inflow temperature and inflow velocity corresponding to a given temperature distribution in a restricted area of the spatial domain. The results show that the proposed genetic programming optimization framework is able to deliver good approximations of the optimal solutions within less than two minutes.
A new application of subspaces interpolation for the construction of nonlinear Parametric Reduced Order Models (PROMs) is proposed. This approach is based upon the Riemannian geometry of the manifold formed …
A new application of subspaces interpolation for the construction of nonlinear Parametric Reduced Order Models (PROMs) is proposed. This approach is based upon the Riemannian geometry of the manifold formed by the quotient of the set of full-rank N-by-q matrices by the orthogonal group of dimension q. By using a set of untrained parametrized Proper Orthogonal Decomposition (POD) subspaces of dimension q, the subspace for a new untrained parameter is obtained as the generalized Karcher barycenter which solution is sought after by solving a simple fixed point problem. Contrary to existing PROM approaches, the proposed barycentric PROM is by construction easier to implement and more flexible with respect to change in parameter values. To assess the potential of the barycentric PROM, numerical experiments are conducted on the parametric flow past a circular cylinder and the flow in a lid driven cavity when the value of Reynolds number varies. It is demonstrated that the proposed barycentric PROM approach achieves competitive results with considerably reduced computational cost.
Approximating solutions of non-linear parametrized physical problems by interpolation presents a major challenge in terms of accuracy. In fact, pointwise interpolation of such solutions is rarely efficient and leads generally …
Approximating solutions of non-linear parametrized physical problems by interpolation presents a major challenge in terms of accuracy. In fact, pointwise interpolation of such solutions is rarely efficient and leads generally to incorrect results. However, instead of using a straight forward interpolation on solutions, reduced order models can be interpolated. More particularly, Amsallem and Farhat proposed an efficient POD reduced order model interpolation technique based on differential geometry tools. This approach, named in this paper ITSGM (Interpolation On a Tangent Space of the Grassmann Manifold), allows through the passage to the tangent space of the Grassmann manifold, to approximate accurately the reduced order basis associated to a new untrained parameter. This basis is used afterwards to build the interpolated ROM describing the temporal dynamics by performing the Galerkin projection on the high fidelity model. Such Galerkin ROMs require to access to the underlying high fidelity model, leading by that to intrusive ROMs. In this paper, and contrary to the ITSGM/Galerkin approach, we propose a non-intrusive reduced order modeling method which is independent of the governing equations. This method is named through this paper Bi-CITSGM (Bi-Calibrated ITSGM). It consists first to interpolate the spatial and temporal POD sampling bases considered as representatives of points on Grassmann manifolds, by the ITSGM method. Then, the resulting bases modes are reclassified by introducing two orthogonal matrices. These calibration matrices are determined as analytical solutions of two optimization problems. Results on the flow problem past a circular cylinder where the parameter of interpolation is the Reynolds number, show that for new untrained Reynolds number values, the developed approach produces satisfyingly accurate solutions in a real computational time.
Most genetic algorithms (GAs) used in literature to solve control problems are time consuming and involve important storage memory requirements. In fact, the search in GAs is iteratively performed on …
Most genetic algorithms (GAs) used in literature to solve control problems are time consuming and involve important storage memory requirements. In fact, the search in GAs is iteratively performed on a population of chromosomes (control parameters). As a result, the cost functional needs to be evaluated through solving the high fidelity model or by performing the experimental protocol for each chromosome and for many generations. To overcome this issue, a non intrusive reduced real coded genetic algorithm (RGA) for near real time optimal control is designed. This algorithm uses precalculated parametrized solution snapshots stored in the POD (Proper Orthogonal Decomposition) reduced form, to predict the solution snapshots for chromosomes over generations. The method used for this purpose is a hyper reduced version of the Bi-CITSGM method (Bi Calibrated Interpolation on the Tangent Space of the Grassmann Manifold) designed specially for non linear parametrized solution snapshots interpolation. This hyper reduced approach referred to as Hyper Bi-CITSGM, is proposed in such a way to accelerate the usual Bi-CITSGM process by bringing this last to a significantly low dimension. Thus, the whole optimization process by RGA can be performed in near real time. The potential of RGA in terms of accuracy and CPU time is demonstrated on control problems of the flow past a cylinder and flow in a lid driven cavity when the Reynolds number value varies.