Reduced Basis Space Research Articles

An LSTM-enhanced surrogate model to simulate the dynamics of particle-laden fluid systems

The numerical treatment of fluid–particle systems is a very challenging problem because of the complex coupling phenomena occurring between the two phases. Although an accurate mathematical modelling is available to address this kind of applications, the computational cost of the numerical simulations is very expensive. The use of the most modern high performance computing infrastructures could help to mitigate such an issue but not completely to fix it. In this work we develop a non intrusive data-driven reduced order model (ROM) for Computational Fluid Dynamics (CFD) - Discrete Element Method (DEM) simulations. The ROM is built using the proper orthogonal decomposition (POD) for the computation of the reduced basis space and the Long Short-Term Memory (LSTM) network for the computation of the reduced coefficients. We are interested to deal both with system identification and prediction. The most relevant novelties rely on (i) a filtering procedure of the full order snapshots to reduce the dimensionality of the reduced problem and (ii) a preliminary treatment of the particle phase. The accuracy of our ROM approach is assessed against the classic Goldschmidt fluidized bed benchmark problem. Finally, we also provide some insights about the efficiency of our ROM approach.

Projection-based reduced order modeling of an iterative scheme for linear thermo-poroelasticity

This paper explores an iterative approach to solve linear thermo-poroelasticity problems, with its application as a high-fidelity discretization utilizing finite elements during the training of projection-based reduced order models. One of the main challenges in addressing coupled multi-physics problems is the complexity and computational expenses involved. In this study, we introduce a decoupled iterative solution approach, integrated with reduced order modeling, aimed at augmenting the efficiency of the computational algorithm. The iterative technique we employ builds upon the established fixed-stress splitting scheme that has been extensively investigated for Biot’s poroelasticity. By leveraging solutions derived from this coupled iterative scheme, the reduced order model employs an additional Galerkin projection onto a reduced basis space formed by a small number of modes obtained through proper orthogonal decomposition. The effectiveness of the proposed algorithm is demonstrated through numerical experiments, showcasing its computational prowess.

Data-Driven Reduced Order Modelling for Patient-Specific Hemodynamics of Coronary Artery Bypass Grafts with Physical and Geometrical Parameters

In this work the development of a machine learning-based Reduced Order Model (ROM) for the investigation of hemodynamics in a patient-specific configuration of Coronary Artery Bypass Graft (CABG) is proposed. The computational domain is referred to left branches of coronary arteries when a stenosis of the Left Main Coronary Artery (LMCA) occurs. The method extracts a reduced basis space from a collection of high-fidelity solutions via a Proper Orthogonal Decomposition (POD) algorithm and employs Artificial Neural Networks (ANNs) for the computation of the modal coefficients. The Full Order Model (FOM) is represented by the incompressible Navier-Stokes equations discretized using a Finite Volume (FV) technique. Both physical and geometrical parametrization are taken into account, the former one related to the inlet flow rate and the latter one related to the stenosis severity. With respect to the previous works focused on the development of a ROM framework for the evaluation of coronary artery disease, the novelties of our study include the use of the FV method in a patient-specific configuration, the use of a data-driven ROM technique and the mesh deformation strategy based on a Free Form Deformation (FFD) technique. The performance of our ROM approach is analyzed in terms of the error between full order and reduced order solutions as well as the speed-up achieved at the online stage.

Finite element based Model Order Reduction for parametrized one-way coupled steady state linear thermo-mechanical problems

This contribution focuses on the development of Model Order Reduction (MOR) for one-way coupled steady state linear thermo-mechanical problems in a finite element setting. We apply Proper Orthogonal Decomposition (POD) for the computation of reduced basis space. On the other hand, for the evaluation of the modal coefficients, we use two different methodologies: the one based on the Galerkin projection (G) and the other one based on Artificial Neural Network (ANN). We aim to compare POD-G and POD-ANN in terms of relevant features including errors and computational efficiency. In this context, both physical and geometrical parametrization are considered. We also carry out a validation of the Full Order Model (FOM) based on customized benchmarks in order to provide a complete computational pipeline. The framework proposed is applied to a relevant industrial problem related to the investigation of thermo-mechanical phenomena arising in blast furnace hearth walls. • Computational pipeline for fast and accurate thermo-mechanical FEM simulations. • Comparison between POD-Galerkin and POD-ANN model order reduction. • Parametric setting involving both physical properties and geometrical features. • Assessment of MOR approach with an industrial problem related to ironmaking process.

Physics-Based Greedy Algorithm for Reliable Fast Frequency Sweep in Electromagnetics via the Reduced-Basis Method

A reliable fast frequency sweep model order reduction (MOR) process is detailed. A reduced-basis approximation is taken into account to accurately and efficiently describe the electromagnetic field in a frequency band of analysis. The reduced-basis space is made up of snapshots of the electric field adaptively chosen on a physics-based greedy algorithm. Following a theoretical analysis starting from time-harmonic Maxwell’s equations, in- band eigenresonances are shown to significantly contribute to the electric field in the band of analysis. An impedance matrix transfer function description of the microwave circuit is used and a pole-residue expansion of this matrix-valued impedance transfer function is detailed. A fundamental physical property is identified, from which a straightforward a posteriori error estimator arises, namely, matrix residues in the pole-residue expansion of the impedance transfer function must be rank-1. This cheap computation is well-suited for fast greedy algorithm strategies in adaptive construction of the reduced-basis approximation. Contrary to what has been previously done, a straightforward physics-based greedy algorithm that does not need to scan the whole frequency band of interest to enrich the reduced-basis approximation is proposed. Finally, actual microwave applications illustrate the capabilities and efficiency of the new physics-based a posteriori error estimator in the proposed methodology.

Gaussian Stochastic Process Modeling of Blend Repaired Airfoil Modal Response Using Reduced Basis Mode Shape Approach

Abstract A machine learning (ML) approach is developed to predict the effect of blend repairs on airfoil frequency, modal assurance criterion (MAC), and modal displacement vectors. The method is demonstrated on a transonic research rig compressor rotor airfoil. A parametric definition of blend geometry is developed and shown to be capable of encompassing a large range of blend geometry. This blend repair geometry is used to modify the airfoil surface definition and a mesh morphing process transforms the nominal finite element model (FEM) to the repaired configuration. A multilevel full factorial sampling of the blend repair design space provides training data to a Gaussian stochastic process (GSP) regressor. The frequency and MAC results create a vector of training data for GSP calibration, but the airfoil mode shapes require further mathematical manipulation to avoid creating GSP models for each nodal displacement. This paper develops a method to significantly reduce blended airfoil mode shape emulation cost by transforming the mode shape training data into a reduced basis space using principal component analysis (PCA). The coefficients of this reduced basis are used to train a GSP that can then predict the values for new blended airfoils. The emulated coefficients are used with the reduced basis vectors in a reconstruction of blended airfoil mode shape. Validation data is computed at a full-factorial design that maximizes the distance from training points. It is found that large variations in modal properties from large blend repairs can be accurately emulated with a reasonable number of training points. The reduced basis approach of mode shape variation is shown to more accurately predict MAC variation when compared to direct MAC emulation. The added benefit of having the full modal displacement field also allows determination of other influences such as tip-timing limits and modal force values.

Using Reduced Basis Approximation for Efficient Surrogate-Based Inverse Identification of Thermo-Hydro-Mechanical Parameters from an In Situ Heating Test

In this paper, a reduced-order model built from the reduced basis (RB) method is used as a surrogate for an inverse identification problem related to coupled thermo-hydro-mechanical (THM) processes. The RB space—spanned by solutions of the governing THM equations—is constructed using a greedy adaptive procedure guided by an a posteriori error estimator that selects the optimal snapshot points in a given parametric space. The RB model is assessed in terms of accuracy and computational cost reduction for the three-dimensional transient coupled problem described by the ATLAS III small-scale in situ heating test. The substantial system size reduction and the associated significant computational gain result in a surrogate model suitable for parameter identification procedures in the Boom Clay material. The effectiveness of the proposed strategy is demonstrated by performing inverse analysis based on direct-search and genetic algorithm (GA) optimization supported by real sensor measurement data where 800 times faster computational speed-up was achieved.

A two-level parameterized Model-Order Reduction approach for time-domain elastodynamics

We present a two-level parameterized Model Order Reduction (pMOR) technique for the linear hyperbolic Partial Differential Equation (PDE) of time-domain elastodynamics. In order to approximate the frequency-domain PDE, we take advantage of the Port-Reduced Reduced-Basis Component (PR-RBC) method to develop (in the offline stage) reduced bases for subdomains; the latter are then assembled (in the online stage) to form the global domains of interest. The PR-RBC approach reduces the effective dimensionality of the parameter space and also provides flexibility in topology and geometry. In the online stage, for each query, we consider a given parameter value and associated global domain. In the first level of reduction, the PR-RBC reduced bases are used to approximate the frequency-domain solution at selected frequencies. In the second level of reduction, these instantiated PR-RBC approximations are used as surrogate truth solutions in a Strong Greedy approach to identify a reduced basis space; the PDE of time-domain elastodynamics is then projected on this reduced space. We provide a numerical example to demonstrate the computational capability and assess the performance of the proposed two-level approach.

A space-time certified reduced basis method for quasilinear parabolic partial differential equations

In this paper, we propose a certified reduced basis (RB) method for quasilinear parabolic problems with strongly monotone spatial differential operator. We provide a residual-based a posteriori error estimate for a space-time formulation and the corresponding efficiently computable bound for the certification of the method. We introduce a Petrov-Galerkin finite element discretization of the continuous space-time problem and use it as our reference in a posteriori error control. The Petrov-Galerkin discretization is further approximated by the Crank-Nicolson time-marching problem. It allows to use a POD-Greedy approach to construct the reduced-basis spaces of small dimensions and to apply the Empirical Interpolation Method (EIM) to guarantee the efficient offline-online computational procedure. In our approach, we compute the reduced basis solution in a time-marching framework while the RB approximation error in a space-time norm is controlled by our computable bound. Therefore, we combine a POD-Greedy approximation with a space-time Galerkin method.

A Novel Iterative Penalty Method to Enforce Boundary Conditions in Finite Volume POD-Galerkin Reduced Order Models for Fluid Dynamics Problems

A Finite-Volume based POD-Galerkin reduced order model is developed for fluid dynamics problems where the (time-dependent) boundary conditions are controlled using two different boundary control strategies: the lifting function method, whose aim is to obtain homogeneous basis functions for the reduced basis space and the penalty method where the boundary conditions are enforced in the reduced order model using a penalty factor. The penalty method is improved by using an iterative solver for the determination of the penalty factor rather than tuning the factor with a sensitivity analysis or numerical experimentation. The boundary control methods are compared and tested for two cases: the classical lid driven cavity benchmark problem and a Y-junction flow case with two inlet channels and one outlet channel. The results show that the boundaries of the reduced order model can be controlled with the boundary control methods and the same order of accuracy is achieved for the velocity and pressure fields. Finally, the reduced order models are 270-308 times faster than the full order models for the lid driven cavity test case and 13-24 times for the Y-junction test case.

Electric Field Integral Equation Fast Frequency Sweep for Scattering of Nonpenetrable Objects via the Reduced-Basis Method

A reduced-basis approximation to solve for the electromagnetic scattering from nonpenetrable targets in a frequency band of interest is studied in detail. Perfect electric scatterers in homogeneous media are considered. Even though the frequency behavior can be complicated, the electromagnetic scattering does not arbitrarily change in the frequency band of interest. Thus, instead of using computationally inefficient, large-dimension numerical approximation scenarios, such as the method of moments, to solve for integral equations in the band of interest, a low-dimension reduced-basis space is identified to approximate the smooth evolution of the electromagnetic scattering itself as frequency changes. As a result, the frequency-parameter scattering problem is solved within a reduced-order model approach. The electric field integral equation formulation is addressed and properly selected electric currents are considered to span the reduced-basis space in the frequency band of interest. The involved frequency dependence in the integral equation due to the wave propagation phenomena described in the free-space Green’s function has traditionally limited the applicability of reduced-basis approximations for fast frequency sweep in integral equation methods in electromagnetics. In this article, we focus on this difficulty and propose a physics-based approach where the frequency-parameter dependence shows an affine behavior in the integral equation, without relying on rather complicated interpolation schemes. As a result, a fast frequency sweep for integral equation problems in electromagnetics is proposed. Finally, real-life applications will illustrate the capabilities of this methodology.

Reduced Basis Greedy Selection Using Random Training Sets

Reduced bases have been introduced for the approximation of parametrized PDEs in applications where many online queries are required. Their numerical efficiency for such problems has been theoretically confirmed in Binev et al. (SIAM J. Math. Anal. 43 (2011) 1457–1472) and DeVore et al. (Constructive Approximation 37 (2013) 455–466), where it is shown that the reduced basis space Vn of dimension n, constructed by a certain greedy strategy, has approximation error similar to that of the optimal space associated to the Kolmogorov n-width of the solution manifold. The greedy construction of the reduced basis space is performed in an offline stage which requires at each step a maximization of the current error over the parameter space. For the purpose of numerical computation, this maximization is performed over a finite training set obtained through a discretization of the parameter domain. To guarantee a final approximation error ε for the space generated by the greedy algorithm requires in principle that the snapshots associated to this training set constitute an approximation net for the solution manifold with accuracy of order ε. Hence, the size of the training set is the ε covering number for M and this covering number typically behaves like exp(Cε−1/s) for some C > 0 when the solution manifold has n-width decay O(n−s). Thus, the shear size of the training set prohibits implementation of the algorithm when ε is small. The main result of this paper shows that, if one is willing to accept results which hold with high probability, rather than with certainty, then for a large class of relevant problems one may replace the fine discretization by a random training set of size polynomial in ε−1. Our proof of this fact is established by using inverse inequalities for polynomials in high dimensions.

A reduced-order shifted boundary method for parametrized incompressible Navier–Stokes equations

We investigate a projection-based reduced order model of the steady incompressible Navier–Stokes equations for moderate Reynolds numbers. In particular, we construct an “embedded” reduced basis space, by applying proper orthogonal decomposition to the Shifted Boundary Method, a high-fidelity embedded method recently developed. We focus on the geometrical parametrization through level-set geometries, using a fixed Cartesian background geometry and the associated mesh. This approach avoids both remeshing and the development of a reference domain formulation, as typically done in fitted mesh finite element formulations. Two-dimensional computational examples for one and three parameter dimensions are presented to validate the convergence and the efficacy of the proposed approach.

Stabilized reduced basis methods for parametrized steady Stokes and Navier–Stokes equations

It is well known in the Reduced Basis approximation of saddle point problems that the Galerkin projection on the reduced space does not guarantee the inf–sup approximation stability even if a stable high fidelity method was used to generate snapshots. For problems in computational fluid dynamics, the lack of inf–sup stability is reflected by the inability to accurately approximate the pressure field. In this context, inf–sup stability is usually recovered through the enrichment of the velocity space with suitable supremizer functions. The main goal of this work is to propose an alternative approach, which relies on the residual based stabilization techniques customarily employed in the Finite Element literature, such as Brezzi–Pitkaranta, Franca–Hughes, streamline upwind Petrov–Galerkin, Galerkin Least Square. In the spirit of offline–online reduced basis computational splitting, two such options are proposed, namely offline-only stabilization and offline–online stabilization. These approaches are then compared to (and combined with) the state of the art supremizer enrichment approach. Numerical results are discussed, highlighting that the proposed methodology allows to obtain smaller reduced basis spaces (i.e., neglecting supremizer enrichment) for which a modified inf–sup stability is still preserved at the reduced order level.

3D-VAR for parameterized partial differential equations: a certified reduced basis approach

In this paper, we propose a reduced order approach for 3D variational data assimilation governed by parameterized partial differential equations. In contrast to the classical 3D-VAR formulation that penalizes the measurement error directly, we present a modified formulation that penalizes the experimentally observable misfit in the measurement space. Furthermore, we include a model correction term that allows to obtain an improved state estimate. We begin by discussing the influence of the measurement space on the amplification of noise and prove a necessary and sufficient condition for the identification of a “good” measurement space. We then propose a certified reduced basis (RB) method for the estimation of the model correction, the state prediction, the adjoint solution, and the observable misfit with respect to the true state for real-time and many-query applications. A posteriori bounds are proposed for the error in each of these approximations. Finally, we introduce different approaches for the generation of the reduced basis spaces and the stability-based selection of measurement functionals. The 3D-VAR method and the associated certified reduced basis approximation are tested in a parameter and state estimation problem for a steady-state thermal conduction problem with unknown parameters and unknown Neumann boundary conditions.

POD–Galerkin reduced order methods for combined Navier–Stokes transport equations based on a hybrid FV-FE solver

The purpose of this work is to introduce a novel POD–Galerkin strategy for the semi-implicit hybrid high order finite volume/finite element solver introduced in Bermúdez et al. (2014) and Busto et al. (2018). The interest is into the incompressible Navier–Stokes equations coupled with an additional transport equation. The full order model employed in this article makes use of staggered meshes. This feature will be conveyed to the reduced order model leading to the definition of reduced basis spaces in both meshes. The reduced order model presented herein accounts for velocity, pressure, and a transport-related variable. The pressure term at both the full order and the reduced order level is reconstructed making use of a projection method. More precisely, a Poisson equation for pressure is considered within the reduced order model. Results are verified against three-dimensional manufactured test cases. Moreover a modified version of the classical cavity test benchmark including the transport of a species is analysed.

Fast solution methods of parameterized high-fidelity models

High-fidelity simulation based on solving partial differential equations has become an indispensable approach in innovative research of nuclear energy. However, due to the prohibitively high computation cost, it is yet unfeasible to run simulations on the whole reactor core level with all the physics resolved on fine meshes, especially in the analysis which need carry out many times of calculations. Model order reduction strategies are promising ways to achieve significant speedups by replacing the original large-scale model by a reduced order model of substantially smaller scale. Two effective model order reduction techniques, i.e., the reduced basis finite element method and the POD-Galerkin method, are studied and applied in the present study to solve the transient thermo-elastic problem and the fluid dynamics problem, respectively. For the former, the boundary conditions, the physical parameters and the body heat source are treated as parameters. The greedy-POD algorithm is applied to construct reduced basis space and the for the latter, the Reynolds number is treated as a parameter. The pure POD method is applied to construct the basis space. During on the online stage, three to five orders of computation speed up has been achieved by using the built reduced order models compared with that by the full order models while the accuracy of the results is well ensured. It is demonstrated that model order reduction techniques will be a practical way to deploy numerical reactors.

Multi space reduced basis preconditioners for parametrized Stokes equations

We introduce a two-level preconditioner for the efficient solution of large scale saddle-point linear systems arising from the finite element (FE) discretization of parametrized Stokes equations. This preconditioner extends the Multi Space Reduced Basis (MSRB) preconditioning method proposed in Dal Santo et al. (2018); it combines an approximated block (fine grid) preconditioner with a reduced basis (RB) solver which plays the role of coarse component. A sequence of RB spaces, constructed either with an enriched velocity formulation or a Petrov–Galerkin projection, is built. Each RB coarse component is defined to perform a single iteration of the iterative method at hand. The flexible GMRES (FGMRES) algorithm is employed to solve the resulting preconditioned system and targets small tolerances with a very small iteration count and in a very short time. Numerical test cases for Stokes flows in three dimensional parameter-dependent geometries are considered to assess the numerical properties of the proposed technique in different large scale computational settings.

A non-intrusive reduced basis approach for parametrized heat transfer problems

Computation Fluid Dynamics (CFD) simulation has become a routine design tool for i) predicting accurately the thermal performances of electronics set ups and devices such as cooling system and ii) optimizing configurations. Although CFD simulations using discretization methods such as finite volume or finite element can be performed at different scales, from component/board levels to larger system, these classical discretization techniques can prove to be too costly and time consuming, especially in the case of optimization purposes where similar systems, with different design parameters have to be solved sequentially. The design parameters can be of geometric nature or related to the boundary conditions. This motivates our interest on model reduction and particularly on reduced basis methods. As is well documented in the literature, the offline/online implementation of the standard RB method (a Galerkin approach within the reduced basis space) requires to modify the original CFD calculation code, which for a commercial one may be problematic even impossible. For this reason, we have proposed in a previous paper, with an application to a simple scalar convection diffusion problem, an alternative non-intrusive reduced basis approach (NIRB) based on a two-grid finite element discretization. Here also the process is two stages: offline, the construction of the reduced basis is performed on a fine mesh; online a new configuration is simulated using a coarse mesh. While such a coarse solution, can be computed quickly enough to be used in a rapid decision process, it is generally not accurate enough for practical use. In order to retrieve accuracy, we first project every such coarse solution into the reduced space, and then further improve them via a rectification technique. The purpose of this paper is to generalize the approach to a CFD configuration.

A Compact Basis for Reliable Fast Frequency Sweep via the Reduced-Basis Method

A reliable reduced-order model (ROM) for fast frequency sweep in time-harmonic Maxwell’s equations by means of the reduced-basis method is detailed. Taking frequency as a parameter, the electromagnetic field in microwave circuits does not arbitrarily vary as frequency changes, but evolves on a very low-dimensional manifold. Approximating this low-dimensional manifold by a low dimension subspace, namely, reduced-basis space, gives rise to an ROM for fast frequency sweep in microwave circuits. This avoids carrying out time-consuming finite-element analysis for each frequency in the band of interest. The behavior of the solutions to Maxwell’s equations as a function of the frequency parameter is studied and highlighted. As a result, a compact reduced-basis space for efficient model-order reduction is proposed. In this paper, the reduced-basis space is composed of two parts: 1) eigenmodes hit in the frequency band of interest, which form an orthogonal, fundamental set that describes the natural oscillating dynamics of the electromagnetic field and 2) whatever else electromagnetic fields, sampled in the frequency band of interest, that are needed to achieve convergence in the reduced-basis approximation. The reduced-basis method aims not only to find out a reduced-basis space in an efficient way, but also to certify the reliability of the approximation carried out. Emphasis is placed on a fast evaluation of the ROM error measure and on providing a reliable convergence criterion. This approach is applied to both narrowband resonating structures and wideband nonresonanting devices in order to show the capabilities of the method in real-life applications.