Projection-based Model Order Reduction Research Articles

A reduced-order model for nonlinear radiative transfer problems based on moment equations and POD-Petrov-Galerkin projection of the normalized Boltzmann transport equation

A data-driven projection-based reduced-order model (ROM) for nonlinear thermal radiative transfer (TRT) problems is presented. The TRT ROM is formulated by (i) a hierarchy of low-order quasidiffusion (aka variable Eddington factor) equations for moments of the radiation intensity and (ii) the normalized Boltzmann transport equation (BTE). The multilevel system of moment equations is derived by projection of the BTE onto a sequence of subspaces which represent elements of the phase space of the problem. Exact closure for the moment equations is provided by the Eddington tensor. A Petrov-Galerkin (PG) projection of the normalized BTE is formulated using a proper orthogonal decomposition (POD) basis representing the normalized radiation intensity over the whole phase space and time. The Eddington tensor linearly depends on the solution of the normalized BTE. By linear superposition of the POD basis functions, a low-rank expansion of the Eddington tensor is constructed with coefficients defined by the PG projected normalized BTE. The material energy balance (MEB) equation is coupled with the effective grey low-order equations which exist on the same dimensional scale as the MEB equation. The resulting TRT ROM is structure and asymptotic preserving. A detailed analysis of the ROM is performed on the classical Fleck-Cummings (F-C) TRT multigroup test problem in 2D geometry. Numerical results are presented to demonstrate the ROM's effectiveness in the simulation of radiation wave phenomena. The ROM is shown to produce solutions with sufficiently high accuracy while using low-rank approximation of the normalized BTE solution. Essential physical characteristics of supersonic radiation wave are preserved in the ROM solutions.

STOCHASTIC GALERKIN METHOD AND PORT-HAMILTONIAN FORM FOR LINEAR FIRST-ORDER ORDINARY DIFFERENTIAL EQUATIONS

We consider linear first-order systems of ordinary differential equations (ODEs) in port-Hamiltonian (pH) form. Physical parameters are remodeled as random variables to conduct an uncertainty quantification. A stochastic Galerkin projection yields a larger deterministic system of ODEs, which does not exhibit a pH form in general. We apply transformations of the original systems such that the stochastic Galerkin projection becomes structure-preserving. Furthermore, we investigate meaning and properties of the Hamiltonian function belonging to the stochastic Galerkin system. A large number of random variables implies a high-dimensional stochastic Galerkin system, which suggests itself to apply model order reduction (MOR) generating a low-dimensional system of ODEs. We discuss structure preservation in projection-based MOR, where the smaller systems of ODEs feature pH form again. Results of numerical computations are presented using two test examples.

An improved linear parameter-varying modeling, model order reduction, and control design process for flexible aircraft

This paper presents an improved process for the modeling, model order reduction (MOR), and control design of a flexible flying wing unmanned aerial vehicle based on the linear parameter-varying (LPV) technology. By applying the Lagrangian formulation and doublet lattice method, the aeroservoelastic (ASE) model of the flexible aircraft is first derived and then rewritten as an airspeed-dependent LPV system, which consists of rigid-body states, structural modes, aerodynamic states, etc. This leads to a high-order LPV model, and the stability of the system varies from stable to unstable when the speed of the aircraft exceeds the critical flutter speed. To facilitate the control design, an oblique projection-based MOR method is then adopted to reduce the order of the plant. Different from previous studies, the MOR method is improved to systematically generate a reduced-order LPV model with consistent states for both the stable and unstable dynamics parts. Finally, an output-feedback LPV controller is designed to simultaneously achieve the body freedom flutter suppression and longitudinal attitude control. Numerical simulation results show the effectiveness of the improved LPV-based ASE modeling, order reduction, and control design process for the flexible aircraft.

Estimating flow fields with reduced order models

The estimation of fluid flows inside a centrifugal pump in realtime is a challenging task that cannot be achieved with long-established methods like CFD due to their computational demands. We use a projection-based reduced order model (ROM) instead. Based on this ROM, a realtime observer can be devised that estimates the temporally and spatially resolved velocity and pressure fields inside the pump. The entire fluid-solid domain is treated as a fluid in order to be able to consider moving rigid bodies in the reduction method. A greedy algorithm is introduced for finding suitable and as few measurement locations as possible. Robust observability is ensured with an extended Kalman filter, which is based on a time-variant observability matrix obtained from the nonlinear velocity ROM. We present the results of the velocity and pressure ROMs based on a unsteady Reynolds-averaged Navier-Stokes CFD simulation of a 2D centrifugal pump, as well as the results for the extended Kalman filter.

Neural-network-augmented projection-based model order reduction for mitigating the Kolmogorov barrier to reducibility

Inspired by our previous work on a quadratic approximation manifold [1], we propose in this paper a computationally tractable approach for combining a projection-based reduced-order model (PROM) and an artificial neural network (ANN) to mitigate the Kolmogorov barrier to reducibility of parametric and/or highly nonlinear, high-dimensional, physics-based models. The main objective of our PROM-ANN concept is to reduce the dimensionality of the online approximation of the solution beyond what is achievable using affine and quadratic approximation manifolds, while maintaining accuracy. In contrast to previous approaches that exploited one form or another of an ANN, the training of the ANN part of our PROM-ANN does not involve data whose dimension scales with that of the high-dimensional model; and the resulting PROM-ANN can be efficiently hyperreduced using any well-established hyperreduction method. Hence, unlike many other ANN-based model order reduction approaches, the PROM-ANN concept we propose in this paper should be practical for large-scale and industry-relevant computational problems. We demonstrate the computational tractability of its offline stage and the superior wall clock time performance of its online stage for a large-scale, parametric, two-dimensional, model problem that is representative of shock-dominated unsteady flow problems.

Reduced order model approaches for predicting the magnetic polarizability tensor for multiple parameters of interest

The magnetic polarizability tensor (MPT) is an economical characterisation of a conducting magnetic object, which can assist with identifying hidden targets in metal detection. The MPT’s coefficients depend on multiple parameters of interest including the object shape, size, electrical conductivity, magnetic permeability, and the frequency of excitation. The computation of the coefficients follow from post-processing an eddy current transmission problem solved numerically using high-order finite elements. To reduce the computational cost of constructing these characterisations for multiple different parameters, we compare three methods by which the MPT can be efficiently calculated for two-dimensional parameter sets, with different levels of code invasiveness. We compare, with numerical examples, a neural network regression of MPT eigenvalues with a projection-based reduced order model (ROM) and a neural network enhanced ROM (POD–NN) for predicting MPT coefficients.

Predictive reduced order modeling of chaotic multi-scale problems using adaptively sampled projections

An adaptive projection-based reduced-order model (ROM) formulation is presented for model-order reduction of problems featuring chaotic and convection-dominant physics. An efficient method is formulated to adapt the basis at every time-step of the on-line execution to account for the unresolved dynamics. The adaptive ROM is formulated in a Least-Squares setting using a variable transformation to promote stability and robustness. An efficient strategy is developed to incorporate non-local information in the basis adaptation, significantly enhancing the predictive capabilities of the resulting ROMs. A detailed analysis of the computational complexity is presented, and validated. The adaptive ROM formulation is shown to require negligible offline training and naturally enables both future-state and parametric predictions. The formulation is evaluated on representative reacting flow benchmark problems, demonstrating that the ROMs are capable of providing efficient and accurate predictions including those involving significant changes in dynamics due to parametric variations, and transient phenomena. A key contribution of this work is the development and demonstration of a comprehensive ROM formulation that targets predictive capability in chaotic, multi-scale, and transport-dominated problems.

Model Reduction of Linear Differential–Algebraic Equation Systems Using Constructed Spectral Projectors

This contribution introduces linear differential-algebraic equation (DAE) systems and provides the explicit construction steps of spectral projectors for DAEs with index 1 or 2. Further, an interpolatory projection-based model order reduction technique using spectral projectors is applied to an eddy current problem as well as an electrical circuit to compute a reduced order model.

Space‐local reduced‐order bases for accelerating reduced‐order models through sparsity

AbstractProjection‐based model order reduction (PMOR) methods based on linear or affine approximation subspaces accelerate numerical predictions by reducing the dimensionality of the underlying computational models. The state of the art of PMOR includes approximation methods based on state‐local subspaces—that is, subspaces associated with different regions of the solution manifold—and methods based on adaptive reduced‐order bases. For challenging applications such as those associated with highly nonlinear problems, such methods accelerate traditional PMOR by controlling the dimension of the reduced‐order basis (ROB) and associated projection‐based reduced‐order model (PROM). This article proposes an alternative as well as complementary approach for accelerating PMOR based on introducing sparsity into the ROB, in order to enhance the computational efficiency of the associated PROM. Specifically, the proposed approach introduces sparsity in PMOR by partitioning the computational domain rather than, or in addition to, the solution manifold, and therefore leads to the concept of a space‐local ROB. This concept is compatible with both concepts of a state‐local ROB and hyperreduction. It is demonstrated for two computational fluid dynamics problems in turbulent flow applications. Acceleration factors of the order of 1.5 relative to traditional PMOR and CPU time speedup factors of several orders of magnitude relative to high‐dimensional models are reported.

An accelerated subspaces recycling strategy for the deflation of parametric linear systems based on model order reduction

Krylov subspace recycling has been extensively used to facilitate the solution of sequences of linear systems by constructing a deflation subspace and accelerating the convergence of a corresponding iterative solver. However, most existing techniques update the recycled subspace sequentially for each system, thus inducing a potentially high computational cost. In that context, this work proposes a method to accelerate the above procedure for the case of multiresolution analyses of affine parametric systems, by decoupling the solution of the system from the construction of the recycled subspace. The proposed method follows the projection based Model Order Reduction (MOR) logic that splits operations into an offline and an online phase and therefore proves particularly beneficial in case of affine parametrizations that involve multiple affine coefficients. In the offline phase of the method an especially tailored version of the Automatic Krylov subspaces Recycling (AKR) algorithm (Panagiotopoulos et al., 2021), proposed within this work and denoted as AKR-D, is employed. In brief, AKR-D constructs a high quality recycling basis W for a predefined parameter interval Ψ by targeting a desirable convergence rate at an automatically generated set of parameter values Ω⊂Ψ. The upfront construction of W enables an a-priori Galerkin projection of the affinely described full parametric system to yield a reduced order model (ROM). This ROM can be leveraged in the online phase of the proposed recycling method to accelerate the construction of the corresponding projector P onto W=span{W}, whose implicit assembly is required in most recycling based deflation schemes. The effectiveness of the proposed strategy is investigated in terms of algorithmic complexity and is demonstrated on a densely parametrized system arising within the boundary integral solver of the Helmholtz equation and a random sparse parametrization example.

Reprint of: Robust and globally efficient reduction of parametric, highly nonlinear computational models and real time online performance

This paper considers the problem of parametric, nonlinear, projection-based model order reduction (PMOR) and makes three contributions to advance its state of the art. First, it presents a computational strategy for reducing the total offline cost of nonlinear PMOR by enabling hyperreduction to be performed adaptively within the typical greedy sampling procedure, rather than uniformly at each of its iteration. Then, it presents a more accurate and computationally efficient approach for training hyperreduction based on residual Jacobians rather than residuals. Finally, it demonstrates for a large-scale, industrial-grade, parametric, nonlinear application, that the two aforementioned contributions make both offline and online stages of nonlinear PMOR practical in today’s engineering environments. The paper also shows that despite the so-called Kolmogorov n-width barrier for the model reduction of convection-dominated transport problems, the current state of the art of nonlinear, Petrov–Galerkin PMOR equipped with the advances proposed in this paper is capable of accelerating by orders of magnitude the accurate solution of shape-parametric, high Reynolds number, viscous flow problems represented by the Reynolds-averaged Navier–Stokes equations augmented with turbulence modeling. All three contributions are presented and discussed in the contexts of steady-state CFD problems; however, they are equally applicable to unsteady CFD problems as well as problems in solid mechanics and structural dynamics.

Accelerating boundary element methods in wideband frequency sweep analysis by matrix-free model order reduction

In frequency-domain acoustic simulations, a wideband frequency sweep analysis is often required. This becomes particularly challenging for boundary element methods (BEM), as the system matrix has a complex dependency on frequency. Previous attempts have been made to accelerate the frequency sweep analysis for the conventional BEM i.e., without fast multipole or ℋ-matrices accelerations, which restrains the applications to limited scale problems due to the dense system matrix. This paper investigates and incorporates a rational Krylov projection based model order reduction (MOR) technology i.e., matrix-free method into various BEM formulations to address the critical challenge of accelerating frequency sweep analysis. Operating on the transfer functions of the BEM system’s input and output, the matrix-free MOR avoids the direct interference with the BEM kernels, which makes it feasible and attractive to work with existing fast BEM techniques such as ℋ-matrices and fast multipole method. The combined holistic BEM solution retains the merits of the acceleration from both MOR and fast BEM techniques. Significant speed-ups have been achieved by the matrix-free MOR for wideband BEM simulations. Numerical examples are presented to demonstrate the capability of the presented holistic approach in simulating various acoustic problems, including interior–exterior models, complex geometries and frequency dependent boundary conditions.

Acceleration of a Physics-Based Machine Learning Approach for Modeling and Quantifying Model-Form Uncertainties and Performing Model Updating

Abstract The nonparametric probabilistic method (NPM) for modeling and quantifying model-form uncertainties is a physics-based, computationally tractable, machine learning method for performing uncertainty quantification and model updating. It extracts from data information not captured by a deterministic, high-dimensional model (HDM) of dimension N and infuses it into a counterpart stochastic, hyperreduced, projection-based reduced-order model (SHPROM) of dimension n ≪ N. Here, the robustness and performance of NPM are improved using a two-pronged approach. First, the sensitivities of its stochastic loss function with respect to the hyperparameters are computed analytically, by tracking the complex web of operations underlying the construction of that function. Next, the theoretical number of hyperparameters is reduced from O(n2) to O(n), by developing a network of autoencoders that provides a nonlinear approximation of the dependence of the SHPROM on the hyperparameters. The robustness and performance of the enhanced NPM are demonstrated using two nonlinear, realistic, structural dynamics applications.

Minimally-invasive parametric model-order reduction for sweep-based radiation transport

We present a parametric reduced-order model for the neutral particle radiation transport equation. The approach devised is a minimally-intrusive, projection-based reduced-order model using global modes obtained via Proper Orthogonal Decomposition. The reduced-order model is specifically designed to work in a matrix-free fashion with radiation transport solvers relying on transport sweeps. The advantages and disadvantages of this model-order reduction approach are discussed and tested on two fixed-source radiation-transport benchmark problems, as well as a k-eigenvalue benchmark. The performance of the reduced-order model in terms of speedup and accuracy are found to be problem-dependent with speedup factors of around 2 for the fixed-source benchmarks and 43 for the k-eigenvalue benchmark. The corresponding reduced solution relative error for these speedups is 1% for the fixed-source benchmarks and 4 pcm for the k-eigenvalue benchmark.

Robust and globally efficient reduction of parametric, highly nonlinear computational models and real time online performance

This paper considers the problem of parametric, nonlinear, projection-based model order reduction (PMOR) and makes three contributions to advance its state of the art. First, it presents a computational strategy for reducing the total offline cost of nonlinear PMOR by enabling hyperreduction to be performed adaptively within the typical greedy sampling procedure, rather than uniformly at each of its iteration. Then, it presents a more accurate and computationally efficient approach for training hyperreduction based on residual Jacobians rather than residuals. Finally, it demonstrates for a large-scale, industrial-grade, parametric, nonlinear application, that the two aforementioned contributions make both offline and online stages of nonlinear PMOR practical in today’s engineering environments. The paper also shows that despite the so-called Kolmogorov n-width barrier for the model reduction of convection-dominated transport problems, the current state of the art of nonlinear, Petrov–Galerkin PMOR equipped with the advances proposed in this paper is capable of accelerating by orders of magnitude the accurate solution of shape-parametric, high Reynolds number, viscous flow problems represented by the Reynolds-averaged Navier–Stokes equations augmented with turbulence modeling. All three contributions are presented and discussed in the contexts of steady-state CFD problems; however, they are equally applicable to unsteady CFD problems as well as problems in solid mechanics and structural dynamics.

Quadratic approximation manifold for mitigating the Kolmogorov barrier in nonlinear projection-based model order reduction

A quadratic approximation manifold is presented for performing nonlinear, projection-based, model order reduction (PMOR). It constitutes a departure from the traditional affine subspace approximation that is aimed at mitigating the Kolmogorov barrier for nonlinear PMOR, particularly for convection-dominated transport problems. It builds on the data-driven approach underlying the traditional construction of projection-based reduced-order models (PROMs); is application-independent; is linearization-free; and therefore is robust for highly nonlinear problems. Most importantly, this approximation leads to quadratic PROMs that deliver the same accuracy as their traditional counterparts using however a much smaller dimension – typically, n2∼n1, where n2 and n1 denote the dimensions of the quadratic and traditional PROMs, respectively. The computational advantages of the proposed high-order approach to nonlinear PMOR over the traditional approach are highlighted for the detached-eddy simulation-based prediction of the Ahmed body turbulent wake flow, which is a popular CFD benchmark problem in the automotive industry. For a fixed accuracy level, these advantages include: a reduction of the total offline computational cost by a factor greater than five; a reduction of its online wall clock time by a factor greater than 32; and a reduction of the wall clock time of the underlying high-dimensional model by a factor greater than two orders of magnitude.

Projection-Based Model Reduction for Coupled Conduction—Enclosure Radiation Systems

Abstract A projection-based reduced order model (pROM) methodology has been developed for transient heat transfer problems involving coupled conduction and enclosure radiation. The approach was demonstrated on two test problems of varying complexity. The reduced order models demonstrated substantial speedups (up to 185×) relative to the full order model with good accuracy (less than 3% L∞ error). An attractive feature of pROMs is that there is a natural error indicator for the ROM solution: the final residual norm at each time-step of the converged ROM solution. Using example test cases, we discuss how to interpret this error indicator to assess the accuracy of the ROM solution. The approach shows promise for many-query applications, such as uncertainty quantification and optimization. The reduced computational cost of the ROM relative to the full-order model (FOM) can enable the analysis of larger and more complex systems as well as the exploration of larger parameter spaces.

Flutter Prediction Using Reduced-Order Modeling with Error Estimation

This paper presents a methodology for dynamic aeroelastic analysis of aircraft based on model order reduction with error estimation. A projection-based model order reduction approach is used to create an aerodynamic reduced-order model (ROM), which is coupled to a structural model to create an aeroelastic ROM. The governing aerodynamic equations are the linearized semidiscrete Euler equations. Flutter analysis is conducted by analyzing the eigenvalues of the aeroelastic ROM. A dual-weighted residual-based error estimator is presented which approximates the error in the eigenvalues obtained from the reduced eigenproblem relative to the eigenvalues from the high-dimensional aeroelastic model. The error estimator thus allows for the construction of aeroelastic ROMs with select eigenvalues that satisfy a user-prescribed accuracy. The aerodynamic ROM is constructed using approximate high-dimensional aeroelastic eigenvectors computed using the two-sided Jacobi–Davidson algorithm. Dynamic aeroelastic analyses are presented for the NACA 64A010 Isogai case, the AGARD 445.6 wing model, and a NACA 0012 benchmark case. The error estimator is shown to have good agreement with the exact error. For the test cases presented in this work, the cost of computing the flutter point at a given Mach number is equivalent to the cost of approximately 5 to 12 steady nonlinear flow evaluations of the high-dimensional Euler equations.

A hierarchical kriging approach for multi-fidelity optimization of automotive crashworthiness problems

Multi-fidelity optimization schemes enriching expensive high-fidelity functions with cheap-to-evaluate low-fidelity functions have gained popularity in recent years. In the present work, an optimization scheme based on a hierarchical kriging is proposed for large-scale and highly non-linear crashworthiness problems. After comparison to other multi-fidelity techniques an infill criterion called variable-fidelity expected improvement is applied and evaluated. This is complemented by two innovative techniques, a new approach regarding initial sampling and a novel way to generate the low-fidelity model for crash problems are suggested. For the former, a modified Latin hypercube sampling, pushing samples more towards design space boundaries, increases the quality of sampling selection. For the latter, a projection-based non-intrusive model order reduction technique accelerates and simplifies the low-fidelity model evaluation. The proposed techniques are investigated with two application problems from the field of automotive crashworthiness—a size optimization problem for lateral impact and a shape optimization problem for frontal impact. The use of a multi-fidelity scheme compared to baseline single-fidelity optimization saves computational effort while keeping an acceptable level of accuracy. Both suggested modifications, independently and especially combined, increase computational performance and result quality in the presented examples.

Efficient semi-implicit coupling fluid-structure interaction analysis via model-order reduction of dynamic grids

In this paper, a fluid-structure interaction (FSI) using the model-order reduction (MOR) of dynamic grids is proposed. Within the interface problem between the fluid and the structure, the grid deformation is based on the spring analogy, and it is further extended in order for the efficient computation. In this procedure, the projection-based MOR technique is employed. The relevant MOR technique is realized by using the proper orthogonal decomposition and the discrete empirical interpolation method (POD/DEIM). The resulting FSI framework is based on a semi-implicit coupling approach. In order to achieve this, a characteristic-based split (CBS) scheme is used for an arbitrary Lagrangian–Eulerian (ALE) formulation of the two-dimensional Navier-Stokes equation. Also, a structural analysis is based on the co-rotational (CR) formulation in order to simulate the geometrically nonlinear behavior. Both the fluid and structural analyses are combined with the semi-implicit coupling approach, and the relevant algorithm is embedded in the ALE/CBS scheme. Then, the present FSI analysis is verified using three examples, and the computational efficiency is evaluated. The resulting numerical data demonstrated the validity of the present analysis, and it is found that a significant reduction in the computational cost is possibly achieved with the proposed approach.