Model Order Reduction Framework Research Articles

Structure-preserving interpolation for model reduction of parametric bilinear systems

In this paper, we present a projection-based interpolation framework for structure-preserving model order reduction of parametric bilinear dynamical systems. We introduce a general setting, covering a broad variety of different structures for parametric bilinear systems, and then provide conditions on projection spaces for the interpolation of structured subsystem transfer functions such that the system structure and parameter dependencies are preserved in the reduced-order model. Two benchmark examples with different parameter dependencies are used to demonstrate the theoretical analysis.

Non-intrusive data-driven ROM framework for hemodynamics problems

Reduced order modeling (ROM) techniques are numerical methods that approximate the solution of parametric partial differential equation (PED) by properly combining the high-fidelity solutions of the problem obtained for several configurations, i.e. for several properly chosen values of the physical/geometrical parameters characterizing the problem. By starting from a database of high-fidelity solutions related to a certain values of the parameters, we apply the proper orthogonal decomposition with interpolation (PODI) and then reconstruct the variables of interest for new values of the parameters, i.e. different values from the ones included in the database. Furthermore, we present a preliminary web application through which one can run the ROM with a very user-friendly approach, without the need of having expertise in the numerical analysis and scientific computing field. The case study we have chosen to test the efficiency of our algorithm is represented by the aortic blood flow pattern in presence of a left ventricular (LVAD) assist device when varying the pump flow rate.

Learning reduced‐order dynamics for parametrized shallow water equations from data

AbstractThis paper discusses a non‐intrusive data‐driven model order reduction method that learns low‐dimensional dynamical models for a parametrized shallow water equation. We consider the shallow water equation in non‐traditional form (NTSWE). We focus on learning low‐dimensional models in a non‐intrusive way. That means, we assume not to have access to a discretized form of the NTSWE in any form. Instead, we have snapshots that can be obtained using a black‐box solver. Consequently, we aim at learning reduced‐order models only from the snapshots. Precisely, a reduced‐order model is learnt by solving an appropriate least‐squares optimization problem in a low‐dimensional subspace. Furthermore, we discuss computational challenges that particularly arise from the optimization problem being ill‐conditioned. Moreover, we extend the non‐intrusive model order reduction framework to a parametric case, where we make use of the parameter dependency at the level of the partial differential equation. We illustrate the efficiency of the proposed non‐intrusive method to construct reduced‐order models for NTSWE and compare it with an intrusive method (proper orthogonal decomposition). We furthermore discuss the predictive capabilities of both models outside the range of the training data.

Generalization of the Neville–Aitken interpolation algorithm on Grassmann manifolds: Applications to reduced order model

AbstractAn extension of the well‐known Neville–Aitken's algorithm for interpolation on the Grassmann manifold in the framework of parametric model order reduction is presented. Interpolation points on are the subspaces spanned by bases obtained by Proper Orthogonal Decomposition of available solutions associated with the chosen parameter sampling. The Neville–Aitken's algorithm is performed recursively via the geodesic barycenter of two points. Three CFD applications are presented: (i) the Von Karman vortex shedding street, (ii) the lid‐driven cavity with inflow and (iii) the flow induced by a rotating solid. Numerical results are relevant with respect to accuracy while the asymptotic complexity is comparable to the state of the art.

Efficient Flutter Prediction Using Reduced-Order Modeling

Typical aerodynamic shape optimization and multidisciplinary optimization algorithms omit high-fidelity flutter predictions due to the associated computational costs. This paper presents a model order reduction framework as a step toward flutter-constrained aircraft optimization. The Euler equations linearized about a steady-state solution of the nonlinear Euler equations are used as the governing unsteady flow equations. Using a proper orthogonal decomposition approach, a reduced basis is constructed onto which the governing equations are projected. The result is a linear reduced-order model (ROM) with significantly fewer degrees of freedom capable of rapidly approximating aerodynamic forces. This ROM is coupled to a linear structural model to create a single monolithic aeroelastic system. The eigenvalues of the resulting system are analyzed for various flow conditions to determine the onset of flutter in the system. To ensure the stability of the ROM, the use of a stabilizing inner product is demonstrated. The flutter boundaries obtained for both a two-degree-of-freedom airfoil structure and the AGARD 445.6 wing model show good agreement with the full-order model and with the literature.

A hyper-reduction computational method for accelerated modeling of thermal cycling-induced plastic deformations

For materials under cyclic thermal loadings, temperature and strain rate-dependent creep deformation can occur due to the thermal expansion mismatch near material interfaces, leading to deterioration of fatigue life. Simulation of the nonlinear mechanical deformation processes of materials subjected to thermal cycling with high-fidelity numerical models often consume high computational costs due to material nonlinearities and long thermal loading period. To accelerate such thermal cycling simulation, an efficient reduced-order modeling framework is proposed. We adopt the reproducing kernel particle method (RKPM) to generate offline high-fidelity model snapshots, which are utilized to create low-dimensional surrogate models based on the proper orthogonal decomposition under Galerkin projection. Based on the stabilized conforming nodal integration technique, gradient smoothing at single integration point per integration cell is employed in conjunction with a least-squares stabilization for high computational efficiency. To further speed up the simulation, a hyper-reduction method is introduced to avoid the full domain integration during the elastoplastic online simulation. Numerical examples on modeling of thermal cycling-induced plastic deformation and thermal fatigue life prediction of a flip chip assembly are analyzed to demonstrate the effectiveness of the present hyper-reduced-order model in significant reduction of computational time while preserving desired accuracy.

Data‐Driven Learning of Reduced‐Order Dynamics for a Parametrized Shallow Water Equation

AbstractA non‐intrusive data‐driven model order reduction method is introduced that learns low‐dimensional dynamical models for a parametrized non‐traditional shallow water equation (NTSWE) [1]. The reduced‐order model is learnt by setting an appropriate least‐squares optimization problem [2] in a low‐dimensional subspace. The non‐intrusive model order reduction framework is extended to a parametric case using the parameter dependency at the level of the partial differential equation. The efficiency of the proposed non‐intrusive method is illustrated to construct reduced‐order models for NTSWE and compared with an intrusive method, proper orthogonal decomposition with Galerkin projection. Furthermore the predictability of both models outside the range of the training data is discussed.

A GFEM-based reduced-order homogenization model for heterogeneous materials under volumetric and interfacial damage

This manuscript presents a multiscale reduced-order modeling framework for heterogeneous materials that accounts for both cohesive interface failure and continuum damage. The model builds on the eigendeformation-based reduced-order homogenization model (EHM), which relies on the pre-calculation of a set of coefficient tensors that account for the effects of linear and nonlinear material behavior between regions of the domain known as parts. A k-means clustering algorithm is used to optimally construct these parts and a new formulation for the partitioning of interfaces using this method is proposed. The extraction of the volumetric and interfacial influence functions is performed using the Interface-enriched Generalized Finite Element Method (IGFEM), which relies on a finite element discretization that does not conform to the material phase boundaries. A Lagrange multiplier method is used in this preprocessing phase, allowing for the reuse of the matrix factorization for different influence function problems and hence leading to efficiency improvement. A newly proposed traction calculation for the interface partition is also adopted to alleviate the instability caused by traction calculations along interfaces. The accuracy and efficiency of the IGFEM–EHM method is assessed through comparison with reference IGFEM simulations. The method is then used to extract the nonlinear multiscale response of particulate, unidirectional fiber-reinforced, and woven composites.

Learning stable reduced-order models for hybrid twins

AbstractThe concept of “hybrid twin” (HT) has recently received a growing interest thanks to the availability of powerful machine learning techniques. This twin concept combines physics-based models within a model order reduction framework—to obtain real-time feedback rates—and data science. Thus, the main idea of the HT is to develop on-the-fly data-driven models to correct possible deviations between measurements and physics-based model predictions. This paper is focused on the computation of stable, fast, and accurate corrections in the HT framework. Furthermore, regarding the delicate and important problem of stability, a new approach is proposed, introducing several subvariants and guaranteeing a low computational cost as well as the achievement of a stable time-integration.

Latent-space time evolution of non-intrusive reduced-order models using Gaussian process emulation

Non-intrusive reduced-order models (ROMs) have recently generated considerable interest for constructing computationally efficient counterparts of nonlinear dynamical systems emerging from various domain sciences. They provide a low-dimensional emulation framework for systems that may be intrinsically high-dimensional. This is accomplished by utilizing a construction algorithm that is purely data-driven. It is no surprise, therefore, that the algorithmic advances of machine learning have led to non-intrusive ROMs with greater accuracy and computational gains. However, in bypassing the utilization of an equation-based evolution, it is often seen that the interpretability of the ROM framework suffers. This becomes more problematic when black-box deep learning methods are used which are notorious for lacking robustness outside the physical regime of the observed data. In this article, we propose the use of a novel latent-space interpolation algorithm based on Gaussian process regression. Notably, this reduced-order evolution of the system is parameterized by control parameters to allow for interpolation in space. The use of this procedure also allows for a continuous interpretation of time which allows for temporal interpolation. The latter aspect provides information, with quantified uncertainty, about full-state evolution at a finer resolution than that utilized for training the ROMs. We assess the viability of this algorithm for an advection-dominated system given by the inviscid shallow water equations.

Non-intrusive reduced order model of urban airflow with dynamic boundary conditions

To address the problem that urban wind field simulations are limited by high computational requirements, a supervised machine learning framework for the non-intrusive model order reduction of urban airflow is proposed to provide fast predictions with dynamic boundary conditions. An integrated atmospheric modeling system is established by coupling the Parallelized LES Model (PALM) with the Urban Canopy Model (UCM) which embedded in the Weather Research and Forecasting (WRF) model to simulate the high-resolution urban wind field. Within this machine learning framework, encoders and decoders are trained to compress and reproduce the full-order airflow data. The eXtreme Gradient Boost regression models (XGBoost) are trained to map the boundary conditions to the latent vectors in the encoders and decoders. Thus, high resolution urban airflow can be predicted rapidly by the XGBoost-decoder models with the input of boundary conditions. By comparing four types of encoding-decoding models (i.e., the proper orthogonal decomposition models, autoencoders with linear/nonlinear fully connected neural networks and autoencoders with Convolutional Neural Network (AE-CNN)), we find that the AE-CNN model with advantages of portable model size and convenient updating procedure is capable for high resolution simulations of urban airflow with dynamic boundary conditions at a low computational cost.

A framework for parametric reduction in large-scale nonlinear dynamical systems

This manuscript presents a general parametric model order reduction (pMOR) framework for nonlinear dynamical systems. At first, a family of local nonlinear reduced order models (ROMs) is generated for different parametric space vectors using the nonlinear moment matching (NLMM) scheme along-with the discrete empirical interpolation method (DEIM). Then, the nonlinear reduced order model for any new parameter is obtained by interpolating the neighbouring reduced order models after projecting them onto a universal subspace. The advantage of such a scheme is that the parametric dependency is maintained in the reduced nonlinear models. Finally, we substantiate our observations by a suite of numerical tests.

Reduced order framework for convection dominant and pure diffusive problems based on combination of deep long short‐term memory and proper orthogonal decomposition/dynamic mode decomposition methods

AbstractIn many real‐world applications, mathematical models are highly complex, and numerical simulations in high‐dimensional systems are challenging. Model order reduction is a useful method to obtain a reasonable approximation by significantly reducing the computational cost of such problems. Deep learning technology is a recent improvement in artificial neural networks that can find more hidden information from the data. Deep learning has the advantage of processing data in its raw form and trains the nonlinear system with different levels of representation and predicts the data. In this article, a reduced order model framework based on a combination of deep learning [long short‐term memory (LSTM)] and proper orthogonal decomposition/dynamic mode decomposition (POD/DMD) modes is presented. Due to the robustness and stability of the LSTM recurrent neural network in predicting chaotic dynamical systems, we consider LSTM architecture to develop our data‐driven reduced order modeling (ROM). We investigate the proposed method performance by solving two well‐known canonical cases: a steady shear flow exhibiting the Kelvin‐Helmholtz instability, and two‐dimensional and unsteady mass diffusion equation. The focus of this article is to use LSTM deep recursive neural network to learn the time dynamics and POD/DMD to generate the order reduction model. The results show that the proposed method is very accurate in predicting time dynamics and input reconstruction.

Fast interactive CFD evaluation of hemodynamics assisted by RBF mesh morphing and reduced order models: the case of aTAA modelling

The medical digital twin is emerging as a viable opportunity to provide patient-specific information useful for treatment, prevention and surgical planning. A bottleneck toward its effective use when computational fluid dynamics (CFD) techniques and tools are adopted for the high fidelity prediction of blood flow, is the significant computing cost required. Reduced order models (ROM) looks to be a promising solution for facing the aforementioned limit. In fact, once ROM data processing is accomplished, the consumption stage can be performed outside the computer-aided engineering software adopted for simulation and, in addition, it could be also implemented on interactive software visualization interfaces that are commonly employed in the medical context. In this paper we demonstrate the soundness of such a concept by numerically investigating the effect of the bulge shape for the ascending thoracic aorta aneurysm case. Radial basis functions (RBF) based mesh morphing enables the implementation of a parametric shape, which is used to build up the ROM framework and data. The final result is an inspection tool capable to visualize, interactively and almost in real-time, the effect of shape parameters on the entire flow field. The approach is first verified considering a morphing action representing the progression from an average healthy patient to an average aneurismatic one (Capellini et al. in Proceedings VII Meeting Italian Chapter of the European Society of Biomechanics (ESB-ITA 2017), 2017; Capellini et al. in J. Biomech. Eng. 140(11):111007-1–111007-10, 2018). Then, a set of shape parameters, suitable to consistently represent a widespread number of possible bulge configurations, are defined and accordingly generated. The concept is showcased taking into account the steady flow field at systolic peak conditions, using ANSYS®Fluent®and its ROM environment for CFD and ROM calculations respectively, and the RBF MorphTM software for shape parametrization.

Multi-fidelity bayesian optimization using model-order reduction for viscoplastic structures

One of the main issues when dealing with the numerical optimization of mechanical structures is the balance between computation time and model accuracy. The work presented herein aims at accelerating global optimization by using the framework of Bayesian optimization on a quantity of interest together with multiple levels of fidelity. These multi-fidelity data are generated from a model-order reduction framework: the LATIN Proper Generalized Decomposition. Within this framework, a reduced-order basis is generated on-the-fly and re-exploited to reduce the computational cost of observations. This strategy is illustrated on two elasto-viscoplastic test-cases for which significant speedups can be observed.

A POD-DEIM reduced model for compressible gas reservoir flow based on the Peng-Robinson equation of state

The efficient simulation of gas flow in porous media is highly required in petroleum engineering, CO2 sequestration, etc. However, it still remains a great challenge attributing to the gas compressibility compared with incompressible fluid flows. The commonly-used equation of states (EOS) of the gas are cubic equation and need to be updated in each iteration, and it also leads to the nonlinearity in flow equations, thus the computational cost mainly stems from the solution of gas EOS. In this paper, a hybrid reduced order model (ROM) coupling the proper orthogonal decomposition (POD) and the discrete empirical interpolation method (DEIM) is presented to accelerate the calculation of compressible single-phase gas reservoir flow, in which the Peng-Robinson EOS (P-R EOS) is considered to describe the gas states. To this end, in the hybrid ROM framework POD is applied to solve the flow equation and DEIM is used to solve the P-R EOS, respectively. The selection of POD modes and DEIM interpolation points, which plays a crucial role in the hybrid ROM, is discussed and carried out carefully. Performances of the proposed POD-DEIM-ROM are evaluated and demonstrated by two numerical cases. Simulation results illustrate that the proposed hybrid ROM displays a satisfactory computational speed-up (two orders of magnitudes faster) without sacrificing numerical accuracy significantly compared with the standard finite difference method. In addition, DEIM shows excellent acceleration and it is a perfect choice for solving the cubic gas EOS.

Stable compact modeling of piezoelectric energy harvester devices

Purpose Based on the framework of Krylov subspace-based model order reduction (MOR), compact models of the piezoelectric energy harvester devices can be generated. However, the stability of reduced piezoelectric model often cannot be preserved. In previous research studies, “MOR after Schur,” “Schur after MOR” and “multiphysics structure preserving MOR” methods have proven successful in obtaining stable reduced piezoelectric energy harvester models. Though the stability preservation of “MOR after Schur” and “Schur after MOR” methods has already been mathematically proven, the “multiphysics structure preserving MOR” method was not. This paper aims to provide the missing mathematical proof of “multiphysics structure preserving MOR.” Design/methodology/approach Piezoelectric energy harvesters can be represented by system of differential-algebraic equations obtained by the finite element method. According to the block structure of its system matrices, “MOR after Schur” and “Schur after MOR” both perform Schur complement transformations either before or after the MOR process. For the “multiphysics structure preserving MOR” method, the original block structure of the system matrices is preserved during MOR. Findings This contribution shows that, in comparison to “MOR after Schur” and “Schur after MOR” methods, “multiphysics structure preserving MOR” method performs the Schur complement transformation implicitly, and therefore, stabilizes the reduced piezoelectric model. Originality/value The stability preservation of the reduced piezoelectric energy harvester model obtained through “multiphysics structure preserving MOR” method is proven mathematically and further validated by numerical experiments on two different piezoelectric energy harvester devices.

Model order reduction assisted by deep neural networks (ROM-net)

In this paper, we propose a general framework for projection-based model order reduction assisted by deep neural networks. The proposed methodology, called ROM-net, consists in using deep learning techniques to adapt the reduced-order model to a stochastic input tensor whose nonparametrized variabilities strongly influence the quantities of interest for a given physics problem. In particular, we introduce the concept of dictionary-based ROM-nets, where deep neural networks recommend a suitable local reduced-order model from a dictionary. The dictionary of local reduced-order models is constructed from a clustering of simplified simulations enabling the identification of the subspaces in which the solutions evolve for different input tensors. The training examples are represented by points on a Grassmann manifold, on which distances are computed for clustering. This methodology is applied to an anisothermal elastoplastic problem in structural mechanics, where the damage field depends on a random temperature field. When using deep neural networks, the selection of the best reduced-order model for a given thermal loading is 60 times faster than when following the clustering procedure used in the training phase.

A Rigorous Model Order Reduction Framework for Waste Heat Recovery Systems Based on Proper Orthogonal Decomposition and Galerkin Projection

A proper orthogonal decomposition (POD) and Galerkin projection-based model order reduction framework is developed for the evaporator heat exchanger in heavy-duty diesel engine organic Rankine cycle waste heat recovery system. The dynamics in the heat exchanger are first modeled by a finite-volume model, composed of highly nonlinear, coupled, partial differential equations, and then used to generate snapshots from which basis functions are defined. Reduced order models (ROMs) are then derived using the Galerkin projection approach. The accuracy and the execution time of different POD ROMs are evaluated against the high-fidelity finite-volume model. The results show that the POD ROM dimension can be selected based on the specific requirements of accuracy and computation time demanded by the intended model utilization. The proposed ROM framework can be utilized to generate ROMs with various dimensions for different purposes, such as estimator design, model-based control, and benchmark generation.

Data-driven recovery of hidden physics in reduced order modeling of fluid flows

In this article, we introduce a modular hybrid analysis and modeling (HAM) approach to account for hidden physics in reduced order modeling (ROM) of parameterized systems relevant to fluid dynamics. The hybrid ROM framework is based on using first principles to model the known physics in conjunction with utilizing the data-driven machine learning tools to model the remaining residual that is hidden in data. This framework employs proper orthogonal decomposition as a compression tool to construct orthonormal bases and a Galerkin projection (GP) as a model to build the dynamical core of the system. Our proposed methodology, hence, compensates structural or epistemic uncertainties in models and utilizes the observed data snapshots to compute true modal coefficients spanned by these bases. The GP model is then corrected at every time step with a data-driven rectification using a long short-term memory (LSTM) neural network architecture to incorporate hidden physics. A Grassmann manifold approach is also adopted for interpolating basis functions to unseen parametric conditions. The control parameter governing the system’s behavior is, thus, implicitly considered through true modal coefficients as input features to the LSTM network. The effectiveness of the HAM approach is then discussed through illustrative examples that are generated synthetically to take hidden physics into account. Our approach, thus, provides insights addressing a fundamental limitation of the physics-based models when the governing equations are incomplete to represent underlying physical processes.