POD Basis Research Articles

Efficient Cardiovascular Parameters Estimation for Fluid-Structure Simulations Using Gappy Proper Orthogonal Decomposition.

As full-scale detailed hemodynamic simulations of the entire vasculature are not feasible, numerical analysis should be focused on specific regions of the cardiovascular system, which requires the identification of lumped parameters to represent the patient behavior outside the simulated computational domain. We present a novel technique for estimating cardiovascular model parameters using gappy Proper Orthogonal Decomposition (g-POD). A POD basis is constructed with FSI simulations for different values of the lumped model parameters, and a linear operator is applied to retain information that can be compared to the available patient measurements. Then, the POD coefficients of the reconstructed solution are computed either by projecting patient measurements or by solving a minimization problem with constraints. The POD reconstruction is then used to estimate the model parameters. In the first test case, the parameter values of a 3-element Windkessel model are approximated using artificial patient measurements, obtaining a relative error of less than 4.2%. In the second case, 4 sets of 3-element Windkessel are approximated in a patient's aorta geometry, resulting in an error of less than 8% for the flow and less than 5% for the pressure. The method shows accurate results even with noisy patient data. It automatically calculates the delay between measurements and simulations and has flexibility in the types of patient measurements that can handle (at specific points, spatial or time averaged). The method is easy to implement and can be used in simulations performed in general-purpose FSI software.

Retrieval of initial condition for Burgers' equation using reduced-order EnKF via POD-based sparse observations

<p style='text-indent:20px;'>The data assimilation (DA) is a popular method to solve uncertainty quantification problem which has attracted more and more attention in disaster assessment and climate change research. However, the DA usually faces the computational problem associated with "the curse of dimensionality'' when the high-dimensional inverse problem is involved. For this, the usual strategy is to combine the dimensionality reduction method with the DA to mitigate the computational demand in practical application. In this paper, we develop a reduced-order model (ROM)-based EnKF used for retrieving the initial condition, where the prediction model and the observation model are all POD-dependent. In addition to the reduced-order model (ROM) being prediction model, the idea of compressed sensing motivates the acquisition of the observation model that allows an efficient combination of the EnKF with QD algorithm to obtain the optimal sparse observation locations. In this way, computational acceleration is gained in retrieving the initial condition. The effectiveness of this algorithm is demonstrated through retrieving the initial condition of an one-dimensional (1-D) Burgers' equation. Experimental results show that a satisfactory retrieval result is obtained using the current ROM-based EnKF with the computational (CPU) time, <inline-formula><tex-math id="M1">\begin{document}$ 42.64\rm{s} $\end{document}</tex-math></inline-formula>, nearly 25 times faster, and the error result <inline-formula><tex-math id="M2">\begin{document}$ 7.47\times 10^{-3} $\end{document}</tex-math></inline-formula>, two order of magnitude <inline-formula><tex-math id="M3">\begin{document}$ O(10^2) $\end{document}</tex-math></inline-formula> more accurate, than the implementation of the traditional full-order model (FOM)-based EnKF where the computational instability is observed at each iteration though in the same condition as set in the ROM-based EnKF, leading to a poor result (the CPU time <inline-formula><tex-math id="M4">\begin{document}$ 1057.97\rm{s} $\end{document}</tex-math></inline-formula>, and the error accuracy <inline-formula><tex-math id="M5">\begin{document}$ 4.06\times 10^{-1}) $\end{document}</tex-math></inline-formula>. The present study extends the application of ROM to efficiently dealing with ill-posed problem as a regularization strategy through replacing the FOM with the ROM such that the low-order truncation of the POD basis leads to the ROM with as few degrees of freedom as possible, which can efficiently inhibit the higher frequency errors, and hence achieve stability of the proposed approach.</p>

A reduced order model for the finite element approximation of eigenvalue problems

In this paper we consider a reduced order method for the approximation of the eigensolutions of the Laplace problem with Dirichlet boundary condition. We use a time continuation technique that consists in the introduction of a fictitious time parameter. We use a POD approach and we present some theoretical results showing how to choose the optimal dimension of the POD basis. The results of our computations confirm the optimal behavior of our approximate solution. We compute the first eigenvalue and discuss how to approximate the next eigenmodes.

Efficient Reduced Order Modeling of Large Data Sets Obtained from CFD Simulations

The ever-increasing computational power has shifted direct numerical simulations towards higher Reynolds numbers and large eddy simulations towards industrially-relevant flow scales. However, this increase in both temporal and spatial resolution has severely increased the computational cost of model order reduction techniques. Reducing the full data set to a smaller subset in order to perform reduced-order modeling (ROM) may be an interesting method to keep the computational effort reasonable. Moreover, non-tomographic particle image velocimetry measurements obtain a 2D data set of a 3D flow field and an interesting research question would be to quantify the difference between this 2D ROM compared to the 3D ROM of the full flow field. To provide an answer to both issues, the aim of this study was to test a new method for obtaining POD basis functions from a small subset of data initially and using them afterwards in the ROM of either the complete data set or the reduced data set. Hence, no new method of ROM is presented, but we demonstrate a procedure to significantly reduce the computational effort required for the ROM of very large data sets and a quantification of the error introduced by reducing the size of those data sets. The method applies eigenvalue decomposition on a small subset of data extracted from a full 3D simulation and the obtained temporal coefficients are projected back on the 3D velocity fields to obtain the 3D spatial modes. To test the method, an annular jet was chosen as a flow topology due to its simple geometry and the rich dynamical content of its flow field. First, a smaller data set is extracted from the 2D cross-sectional planes and ROM is performed on that data set. Secondly, the full 3D spatial structures are reconstructed by projecting the temporal coefficients back on the 3D velocity fields and the 2D spatial structures by projecting the temporal coefficients back on the 2D velocity fields. It is shown that two perpendicular lateral planes are sufficient to capture the relevant large-scale structures. As such, the total processing time can be reduced by a factor of 136 and up to 22 times less RAM is needed to complete the ROM processing.

Greedy Non-Intrusive Reduced-Order Model’s application in dynamic blowing and suction flow control to suppress the flow separation

The Greedy Non-Intrusive Reduced-Order Model (GNIROM) is applied in dynamic blowing and suction’s flow control to suppress the GAW1 airfoil’s flow separation. The effects of the uniform blowing and uniform suction on the flow fields and aerodynamic coefficients have been studied firstly. By comparing a uniform suction control with a dynamic blowing and suction control, it has been found that the dynamic blowing and suction can provide a higher lift coefficient and a lower energy loss. Finally, for the sake of achieving an optimal dynamic blowing and suction control, an optimization of the dynamic blowing and suction’s parameters is taken based on the GNIROM, consisting of a Non-Intrusive ROM (NIROM) built by the two-level POD and Radial Basis Function (RBF), and the relative greedy method. A cost function has been proposed by considering suppressing the flow separation, improving the lift-drag characteristics and reducing the energy loss together. The GNIROM has been built for unsteady vorticity fields with the parameters of the dynamic blowing and suction varying. Based on the GNIROM’s approximation, the optimal control parameters’ values have been determined quickly. Compared to the initial dynamic blowing and suction control, the optimal one has reduced the drag coefficient by 43.4 percents and the energy loss by 42 percents.

A hybrid partitioned deep learning methodology for moving interface and fluid–structure interaction

In this work, we present a hybrid partitioned deep learning framework for the reduced-order modeling of moving interfaces and predicting fluid–structure interaction. Using the discretized Navier–Stokes in the arbitrary Lagrangian–Eulerian reference frame, we generate the full-order flow snapshots and point cloud displacements as target physical data for the learning and inference of coupled fluid–structure dynamics. The hybrid operation of this methodology comes by combining two separate data-driven models for fluid and solid subdomains via deep learning-based reduced-order models (DL-ROMs). The proposed multi-level framework comprises the partitioned data-driven drivers for unsteady flow and the moving point cloud displacements. At the fluid–structure interface, the force information is exchanged synchronously between the two partitioned subdomain solvers. The first component of our proposed framework relies on the proper orthogonal decomposition-based recurrent neural network (POD-RNN) as a DL-ROM procedure to infer the point cloud with a moving interface. This model utilizes the POD basis modes to reduce dimensionality and evolve them in time via long short-term memory-based recurrent neural networks (LSTM-RNNs). The second component employs the convolution-based recurrent autoencoder network (CRAN) as a self-supervised DL-ROM procedure to infer the nonlinear flow dynamics at static Eulerian probes. We introduce these probes as spatially structured query nodes in the moving point cloud to treat the Lagrangian-to-Eulerian conflict together with convenience in training the CRAN driver. To determine these Eulerian probes, we construct a novel snapshot-field transfer and load recovery algorithm. They are chosen in such a way that the two components (i.e., POD-RNN and CRAN) are constrained at the interface to recover the bulk force quantities. These DL-ROM-based data-driven drivers rely on the LSTM-RNNs to evolve the low-dimensional states. A popular prototypical fluid–structure interaction problem of flow past a freely oscillating cylinder is considered to assess the efficacy of the proposed methodology for a different set of reduced velocities that lead to vortex-induced vibrations. The proposed framework tracks the interface description with acceptable accuracy and predicts the nonlinear wake dynamics over the chosen test data range. The proposed framework aligns with the development of partitioned digital twin of engineering systems, especially those involving moving boundaries and fluid–structure interactions.

An efficient PODI method for real-time simulation of indenter contact problems using RBF interpolation and contact domain decomposition

This paper proposes an efficient proper orthogonal decomposition with interpolation (PODI) method for real-time simulation of indenter contact problems. In the offline stage, POD models are constructed from displacement and von-Mises stress snapshots of full-order finite element solutions of indenter contact problems with training contact locations and indentation depths. Indentation depth and contact location are respectively used to interpolate POD basis vectors and POD coefficients by Lagrange interpolation with a congruence transformation and radial basis function (RBF) interpolation. In the online stage, displacements and von-Mises stress of solids under a new indentation loading are obtained very quickly by using the interpolated POD matrix and POD coefficients. A clustering-based contact domain decomposition (CDD) scheme is proposed to further improve the computational performance of the present PODI–RBF​ method. Numerical results show that the PODI–RBF method using the CDD scheme can be used to obtain finite element solutions of indenter contact problems at 400 ∼900 Hz on a personal computer for finite element models with 8000 ∼16000 nodes, which is promising to real-time simulation of indenter contact problems.

Pressure Stabilization Strategies for a LES Filtering Reduced Order Model

We present a stabilized POD–Galerkin reduced order method (ROM) for a Leray model. For the implementation of the model, we combine a two-step algorithm called Evolve-Filter (EF) with a computationally efficient finite volume method. In both steps of the EF algorithm, velocity and pressure fields are approximated using different POD basis and coefficients. To achieve pressure stabilization, we consider and compare two strategies: the pressure Poisson equation and the supremizer enrichment of the velocity space. We show that the evolve and filtered velocity spaces have to be enriched with the supremizer solutions related to both evolve and filter pressure fields in order to obtain stable and accurate solutions with the supremizer enrichment method. We test our ROM approach on a 2D unsteady flow past a cylinder at Reynolds number 0≤Re≤100. We find that both stabilization strategies produce comparable errors in the reconstruction of the lift and drag coefficients, with the pressure Poisson equation method being more computationally efficient.

Modeling neutronic transients with Galerkin projection onto a greedy-sampled, POD subspace

Presented here is an application of POD-Galerkin projection to develop a reduced-order-model (ROM) that approximates the solution of the multi-group, time-dependent, diffusion equation with a lower computational cost. Such reduced models are often sought in the context of parametric studies and uncertainty quantification, which require successive evaluation of the full-order model (FOM). To construct the ROM efficiently, greedy sampling was used to generate a reduced subspace that represents the entire parameter (here, cross-sections) domain. By using this subspace, the ROM is expected to yield accurate predictions when used in place of the FOM for uncertainty quantification conducted, e.g., with direct Monte-Carlo sampling. The TWIGL benchmark was used here for illustration. A ROM constructed with a POD basis of rank 10 was able to reproduce the core power with a maximum RMS error on the order of 10–6, and for the precursors concentration it was 10–9. Moreover, the statistical moments obtained from the ROM were in good agreement with those of the full-order-model. The mean relative error of the power sample mean prediction was about 4 ×10–6 for both the ramp and the step perturbations. In terms of the computational efficiency, the ROM time, including the projection, is approximately four-times faster than the full-order-model.

Parametric PGD model used with orthogonal polynomials to assess efficiently the building's envelope thermal performance

ABSTRACT Estimating the temperature field of a building envelope could be a time-consuming task. The use of a reduced-order method is then proposed: the Proper Generalized Decomposition method. The solution of the transient heat equation is then re-written as a function of its parameters: the boundary conditions, the initial condition, etc. To avoid a tremendous number of parameters, the initial condition is parameterized. This is usually done by using the Proper Orthogonal Decomposition method to provide an optimal basis. Building this basis requires data and a learning strategy. As an alternative, the use of orthogonal polynomials (Chebyshev, Legendre) is here proposed. Highlights Chebyshev and Legendre polynomials are used to approximate the initial condition Performance of Chebyshev and Legendre polynomials are compared to the POD basis Each basis combined with the PGD model is compared to laboratory measurements The influence of four different parameters on the accuracy of the basis is studied For each approximation basis, CPU calculation times are evaluated and compared

Nonlinear model reduction: A comparison between POD-Galerkin and POD-DEIM methods

Several nonlinear model reduction techniques are compared for the three cases of the non-parallel version of the Kuramoto-Sivashinsky equation, the transient regime of flow past a cylinder at Re=100 and fully developed flow past a cylinder at the same Reynolds number. The linear terms of the governing equations are reduced by Galerkin projection onto a POD basis of the flow state, while the reduced nonlinear convection terms are obtained either by a Galerkin projection onto the same state basis, by a Galerkin projection onto a POD basis representing the nonlinearities or by applying the Discrete Empirical Interpolation Method (DEIM) to a POD basis of the nonlinearities. The quality of the reduced order models is assessed as to their stability, accuracy and robustness, and appropriate quantitative measures are introduced and compared. In particular, the properties of the reduced linear terms are compared to those of the full-scale terms, and the structure of the nonlinear quadratic terms is analyzed as to the conservation of kinetic energy. It is shown that all three reduction techniques provide excellent and similar results for the cases of the Kuramoto-Sivashinsky equation and the limit-cycle cylinder flow. For the case of the transient regime of flow past a cylinder, only the pure Galerkin techniques are successful, while the DEIM technique produces reduced-order models that diverge in finite time.

Adaptive trust‐region POD for optimal control of the Cahn‐Hilliard equation

AbstractWe consider the optimal control problem of a phase field system with distributed control. A reduction of the computational complexity is achieved by a POD‐Galerkin approach where the expensive high‐fidelity PDE systems are replaced by low‐dimensional approximations. The POD basis functions are computed from snapshots which are space‐adapted finite element solutions of the governing equations. Within the trust‐region framework, the accuracy of the surrogate models is controlled in the course of the optimization. In the numerical example we compare the use of a relaxed double‐obstacle free energy with a smooth free energy concerning the influence on the accuracy of the solution and the optimization performance.

Analysis and Reconstruction of Air-cooled Condenser Based on Proper Orthogonal Decomposition Technique

Based on the previous experiments and numerical simulations, the low-order model of the air-cooled condenser was obtained by the orthogonal decomposition (POD) method, and the law of POD basis function and time coefficient distribution was obtained. Get the distribution of generalized “energy” between modes; The POD-based low-order model is used to reconstruct the pressure field and velocity field of the air-cooled condenser, and the relationship between the number of modes and the error of the result is obtained. The validity and rapidity of the reduced order method are proved by selecting the previous modal reconstruction flow fields.

POD basis interpolation via Inverse Distance Weighting on Grassmann manifolds

An adaptation of the Inverse Distance Weighting (IDW) method to the Grassmann manifold is carried out for interpolation of parametric POD bases. Our approach does not depend on the choice of a reference point on the Grassmann manifold to perform the interpolation, moreover our results are more accurate than those obtained in [ 7 ]. In return, our approach is not direct but iterative and its relevance depends on the choice of the weighting functions which are inversely proportional to the distance to the parameter. More judicious choices of such weighting functions can be carried out via kriging technics [ 23 ], this is the subject of a work in progress.

Parameterised non-intrusive reduced order methods for ensemble Kalman filter data assimilation

This paper presents a novel Ensemble Kalman Filter (EnKF) data assimilation method based on a parameterised non-intrusive reduced order model (P-NIROM) which is independent of the original computational code. EnKF techniques involve the expensive calculations of ensembles. In this work, the recently developed P-NIROM Xiao et al. [40] is incorporated into EnKF to speed up the ensemble simulations. A reduced order flow dynamical model is generated from the solution snapshots, which are obtained from a number of the high fidelity full simulations over the specific parametric space RP. The varying parameter is the background error covariance σ ∈ RP. Using the Smolyak sparse grid method, a set of parameters in the Gaussian probability density function is selected as the training points. The proposed method uses a two-level interpolation method for constructing the P-NIROM using a Radial Basis Function (RBF) interpolation method. The first level interpolation approach is used for generating the solution snapshots and POD basis functions for any given background error covariance while the second level interpolation approach for forming a set of hyper-surfaces representing the reduced system.The EnKF in combination with P-NIROM (P-NIROM-EnKF) has been implemented within an unstructured mesh finite element ocean model and applied to a three dimensional wind driven circulation gyre case. The numerical results show that the accuracy of ensembles and updated solutions using the P-NIROM-EnKF is maintained while the computational cost is significantly reduced by several orders of magnitude in comparison to the full-EnKF.

Global and local POD models for the prediction of compressible flows with DG methods

SummaryProper orthogonal decomposition (POD) allows to compress information by identifying the most energetic modes obtained from a database of snapshots. In this work, POD is used to predict the behavior of compressible flows by means of global and local approaches, which exploit some features of a discontinuous Galerkin spatial discretization. The presented global approach requires the definition of high‐order and low‐order POD bases, which are built from a database of high‐fidelity simulations. Predictions are obtained by performing a cheap low‐order simulation whose solution is projected on the low‐order basis. The projection coefficients are then used for the reconstruction with the high‐order basis. However, the nonlinear behavior related to the advection term of the governing equations makes the use of global POD bases quite problematic. For this reason, a second approach is presented in which an empirical POD basis is defined in each element of the mesh. This local approach is more intrusive with respect to the global approach but it is able to capture better the nonlinearities related to advection. The two approaches are tested and compared on the inviscid compressible flow around a gas‐turbine cascade and on the compressible turbulent flow around a wind turbine airfoil.

Benchmark aerodynamic shape optimization with the POD-based CST airfoil parametric method

This paper conducts a benchmark aerodynamic shape optimization with the developed POD-based CST airfoil parametric method. The POD analysis transforms the basis functions of the CST parametric method into reduced orthogonal POD basis functions which are used to describe the geometric variation of the aerodynamic shape. A benchmark aerodynamic shape optimization for drag minimization of RAE2822 airfoil in transonic viscous flow provided by AIAA aerodynamic design optimization discussion group is used to verify the optimization effect of the developed parametric method. The optimized results indicate that developed parametric method can maintain almost the same ability to cover potential airfoils with much fewer parameters as the original high-dimensional CST method does, which can overcome the contradiction of high-dimensionality of design parameters and increase of optimization difficulty well. Moreover, the benchmark optimized results verify the optimization effect of the developed method by comparing with benchmark results optimized by other international counterparts in the field of aerodynamic shape optimization.

Structure-preserving Galerkin POD reduced-order modeling of Hamiltonian systems

The proper orthogonal decomposition reduced-order model (POD-ROM) has been widely used as a computationally efficient surrogate model in large-scale numerical simulations of complex systems. However, when it is applied to a Hamiltonian system, a naive application of the POD method can destroy the Hamiltonian structure in the reduced-order model. In this paper, we develop a new reduced-order modeling approach for Hamiltonian systems, which modifies the Galerkin projection-based POD-ROM so that the appropriate Hamiltonian structure is preserved. Since the POD truncation can degrade the approximation of the Hamiltonian function, we propose to use a POD basis from shifted snapshots to improve the approximation to the Hamiltonian function. We further derive a rigorous a priori error estimate for the structure-preserving ROM and demonstrate its effectiveness in several numerical examples. This approach can be readily extended to dissipative Hamiltonian systems, port-Hamiltonian systems, etc.

Comparison of a separated flow response to localized and global-type disturbances

The flow structure and lift response of a separated flow over an airfoil that is subjected to an impulsive type of pitching motion are compared to the response produced by a localized pulse disturbance at the leading edge of an airfoil. Time-resolved PIV data are used to obtain the velocity field on the suction side of the airfoil. POD analysis shows that the majority of energy is contained within the first four modes. Strong similarities in the shapes of the POD basis functions are found, especially for the second mode, irrespective of the type of actuation (global or local). The time-varying coefficient of this second POD mode tracks the negative of the lift coefficient or circulation in each case. Basis functions from the localized actuation data were projected on the velocity field of the globally actuated flow to obtain a hybrid set of coefficients. The hybrid coefficients matched reasonably well with the coefficients obtained from the original POD analysis for the globally excited flow. Both types of actuation were found to generate very similar Lagrangian flow structures. The results suggest a certain degree of universality in the POD modes/flow structures for the separated flow over an airfoil, irrespective of the type of excitation.

A Dynamic Observer to Capture and Control Perturbation Energy in Noise Amplifier Flows

In this article, we present a technique to extract a reduced-order model of a transitional flat-plate boundary layer from simultaneous velocity snapshots and wall-shear stress measurements. The proposed approach combines a reduction of the degrees of freedom of the system by a projection of the velocity snapshots onto a POD basis together with a system-identification technique to obtain a state-space model of the flow. Such a model is then used in an optimal control framework to reduce the kinetic energy of the perturbation field and therefore delay transition.