Full-order Model Research Articles

Adjacency‐based, non‐intrusive reduced‐order modeling for fluid‐structure interactions

AbstractNon‐intrusive model reduction is a promising solution to system dynamics prediction, especially in cases where data are collected from experimental campaigns or proprietary software simulations. In this work, we present a method for non‐intrusive model reduction applied to Fluid‐Structure Interaction (FSI) problems. The approach is based on the a priori known sparsity of the full‐order system operators, which is dictated by grid adjacency information. In order to enforce this type of sparsity, we solve a “local”, regularized least‐squares problem for each degree of freedom on a grid, considering only the training data from adjacent degrees of freedom (DoFs), thus making computation and storage of the inferred full‐order operators feasible. After constructing the non‐intrusive, sparse full‐order model (FOM), Proper Orthogonal Decomposition (POD) is used for its projection to a reduced dimension subspace and thus the construction of a reduced‐order model (ROM). The methodology is applied to the challenging Hron‐Turek benchmark FSI3, for Re = 200. A physics‐informed, non‐intrusive ROM is constructed to predict the two‐way coupled dynamics of a solid with a deformable, slender tail, subject to an incompressible, laminar flow. Results considering the accuracy and predictive capabilities of the inferred reduced models are discussed.

Model Order Reduction with Dynamically Transformed Modes for Electrophysiological Simulations

The numerical simulation of neuromuscular processes in skeletal muscles typically imposes high computational costs, resulting in the need for techniques to reduce these costs. One submodel in these types of multiphysics multiscale simulations is the electrophysiological model, describing the propagation of an action potential (AP) in a muscle fiber. Involved in the simulation of the propagation are two traveling waves, resulting in difficulties for classical model order reduction (MOR) techniques based on linear subspaces.
 Instead, we apply the nonlinear MOR method shifted Proper Orthogonal Decomposition (sPOD) to construct dynamically transformed reduced basis functions depending on time-dependent paths spanning the adaptive reduced ansatz space.
 Our numerical experiments demonstrate that the constructed reduced ansatz space can accurately capture the dynamics of two fully separated wavefronts and reduce the degrees of freedom of the whole simulation. However, it cannot represent the activation of the AP in the center of the fiber and overlapping wave parts. The constructed reduced order model outperforms the high-dimensional full order model in terms of the computational costs while the accuracy is maintained and reaches speedup factors between 2 and 73 depending on the time discretization.

Data-driven reduced-order simulation of dam-break flows in a wetted channel with obstacles

Accurate and timely information on dam-break waves is essential for risk assessment and disaster mitigation. The unsteady flow interacting with in-channel obstacles renders numerical simulations computationally costly. This study establishes a machine learning (ML)-enhanced reduced-order model (ROM), which provides accelerated and accurate flow predictions. The model consists of three phases: dimensionality reduction, long short-term memory (LSTM) optimization and forecasting, and flow field reconstruction. The proper orthogonal decomposition (POD) first reduces the complexity of the physical system while maintaining the dominant flow dynamics. Subsequently, an LSTM fine-tuned by the grey wolf optimization (GWO) predicts the evolution of the POD coefficients in the reduced-order space. Lastly, the flow field is reconstructed using the high-energy POD modes and the estimated amplitudes. The proposed GWO-LSTM-ROM is evaluated for time-dependent dam-break flows in a wetted channel with obstacles. Based on the comparison of millions of data samples, the approach is highly consistent with the high-fidelity full-order model, with a coefficient of determination over 0.99. Meanwhile, the average computational efficiency is improved by 86%. The main contribution of this work is to develop an improved method for fast and accurate modeling of complex flows, benefiting a wide range of applications, e.g., multiphase flows and fluid-structure interactions.

An adaptive partitioned reduced order model of peridynamics for efficient static fracture simulation

In this paper, an adaptive partitioning strategy and the proper orthogonal decomposition–Galerkin method are combined to formulate an adaptive partitioned reduced order model (APROM) for fast solution of peridynamic (PD) models involving fractures. A boundary layer separation approach is presented to tackle the inaccurate reproduction of nonlocal boundary conditions in the reduced order model (ROM). Cracks invariably make ROM results more sensitive to the snapshots, i.e. a set of high-fidelity solutions simulated by the full order model (FOM), and this challenge is overcome by an adaptive partitioning strategy. The internal computational domain is divided into two parts: the crack region is modeled by full PD model, and the remaining region is dimensionally reduced. The partitioned configuration is automatically updated with the damage evolution. Several numerical tests are executed to verify the performance of the APROM, which shows that the APROM is able to successfully simulate various fracture phenomena. A significant improvement in computational efficiency is found: the number of degrees of freedom in the full PD model is greatly reduced, leading to an approximately 10 times improvement in CPU time without loss of accuracy. This paper provides strong theoretical support for accelerating PD solutions, thereby promoting the practical application of PD theory in large-scale engineering projects.

Reduced order modeling and model order reduction for continuum manipulators: an overview.

Soft robot's natural dynamics calls for the development of tailored modeling techniques for control. However, the high-dimensional configuration space of the geometrically exact modeling approaches for soft robots, i.e., Cosserat rod and Finite Element Methods (FEM), has been identified as a key obstacle in controller design. To address this challenge, Reduced Order Modeling (ROM), i.e., the approximation of the full-order models, and Model Order Reduction (MOR), i.e., reducing the state space dimension of a high fidelity FEM-based model, are enjoying extensive research. Although both techniques serve a similar purpose and their terms have been used interchangeably in the literature, they are different in their assumptions and implementation. This review paper provides the first in-depth survey of ROM and MOR techniques in the continuum and soft robotics landscape to aid Soft Robotics researchers in selecting computationally efficient models for their specific tasks.

A Fast Reduced-Order Model for Radial Integration Boundary Element Method Based on Proper Orthogonal Decomposition in the Non-Uniform Coupled Thermoelastic Problems

To efficiently address the challenge of thermoelastic coupling in functionally graded materials, we propose an approach that combines the radial integral boundary element method (RIBEM) with proper orthogonal decomposition (POD). This integration establishes a swift reduced-order model to transform the high-dimensional system of equations into a more manageable, low-dimensional counterpart. The implementation of this reduced-order model offers the potential for rapid numerical simulations of functionally graded materials (FGMs) under thermal shock loading. Initially, the RIBEM is utilized to resolve the thermal coupling issue within the FGMs. From these solutions, a snapshot matrix is constructed, capturing the solved temperature and displacement fields. Subsequently, the POD modes are established and a POD reduced-order model is constructed for the boundary element format of the thermally coupled problem. Finally, a system of low-order discrete differential equations is solved. Numerical experiments demonstrate that the results obtained from the reduced-order model closely align with those of the full-order model, even when considering variations in structural parameters or impact loads. Thus, the introduction of the reduced-order model not only guarantees solution accuracy but also significantly enhances computational efficiency.

Space-time adaptive model order reduction utilizing local low-dimensionality of flow field

The cost of unsteady flow simulations, such as large eddy simulations, has been increasing because of the large number of mesh cells and the use of high-order numerical schemes. In contrast, recent data-driven reduced-order models (ROMs) predict a flow field quickly; however, their application is often limited to relatively simple flow problems. In this study, we investigate space-time adaptive model order reduction for unsteady flow simulation, where full-order Navier-Stokes equations and a proper orthogonal decomposition (POD)-based reduced-order model are adaptively switched in a block-by-block manner on a multi-block Cartesian mesh. The full-order model (FOM) is switched to the ROM when POD modes effectively represent the unsteady flow field in each block. Owing to the local low-dimensionality of a flow field, the degree of freedom of fluid flow in each block can be smaller than that of the entire flow field; hence, the flow field is effectively represented by a small number of POD modes. First, the hybrid FOM/ROM approach is tested with a low-Reynolds-number flow around a circular cylinder, where the reduction in computational time and the accuracy of the simulated flow field are evaluated. We then consider a flow around a circular cylinder with a Reynolds number of 1000, where the flow field is reduced except for the highly unsteady wake. The potential computational advantage of this method is demonstrated by load balancing on a distributed memory computer. In addition, we utilize bred vector dimensions to quantify the degree of freedom of the flow field in each block, which is closely related to whether ROMs effectively reduce the flow field.

Regression-based identification and order reduction method for nonlinear dynamic structural models

In this paper, we propose a regression-based nonlinear reduced-order model for nonlinear structural dynamics problems, called the Nonlinear Identification and Dimension-Order Reduction (NLIDOR) algorithm. We evaluate the algorithm using a simple toy model, a chain of coupled oscillators and an actual three-dimensional flat plate. The results show that NLIDOR can accurately identify the natural frequencies and modes of the system and capture the nonlinear dynamical features, while the linear Dynamic Mode Decomposition (DMD) method can only capture linear features and is influenced by nonlinear terms. Compared with the full-order model (FOM), NLIDOR can effectively reduce computational cost, while compared with DMD, NLIDOR significantly improves computational accuracy. The results demonstrate the effectiveness and potential of NLIDOR for solving nonlinear dynamic problems in various applications.

Hyper-reduction for Petrov–Galerkin reduced order models

Projection-based Reduced Order Models are based on the idea of minimizing the discrete residual of a “full order model” (FOM) while at the same time constraining the unknowns to live in a space of reduced dimension. For problems with symmetric positive definite (SPD) Jacobians, this minimization can be achieved optimally by projecting the full order residual onto the approximation basis (Galerkin Projection). This approach is sub-optimal for problems with non-SPD Jacobians since it only guarantees that the projection of the residual onto the chosen basis is minimized and not the residual itself. One possible alternative in such cases is to directly minimize the 2-norm of the residual. This minimization can be achieved either by using QR factorization to solve the resulting least-squares problem or by employing the method of the normal equations (LSPG) to the same end. The first approach involves constructing and factorizing a rectangular tall and skinny matrix of size proportional to the number of unknowns of the FOM. The LSPG method avoids the use of the large matrix by directly constructing the product by its transpose. Unfortunately, constructing such a product element by element is not feasible and requires the use of a complementary mesh, which adds an extra layer of complexity to the hyper-reduction process when performing mesh sampling. The main idea of this work is to propose an alternative technique based on the idea of Petrov–Galerkin minimization. Essentially, we choose a left basis so that a least-squares minimization procedure can be carried out on a reduced problem while guaranteeing that the discrete full order residual is minimized. The resulting procedure is applicable to problems with both SPD and non-SPD Jacobians. Additionally, the resulting minimization problem can be assembled element by element, avoiding the use of the complementary mesh and simplifying implementation in the context of finite elements. The resulting technique is amenable to hyper-reduction by the use of the Empirical Cubature Method and can be readily applied in the context of nonlinear reduction procedures.

Hyperreduced-order modeling of thermally coupled flows

This article presents a full and reduced-order methodology based on the finite element method that allows the characterization of convective-dominant conjugate heat transfer flows. Reduced-order modeling (ROM) allows the description of several degrees of freedom problems, referred to as full-order modeling (FOM), through a reduced-order surrogate representation based on a Petrov-Galerkin projection onto a reduced subspace. The FOM and ROM problems are stabilized through the Variational Multi-Scale (VMS) method, which ensures stability in dominant convective problems and allows equal interpolation spaces for the pressure, velocity, and temperature. The proposed methodology is validated solving natural heat convection in a differentially heated square cavity, considering air as the working fluid and Rayleigh numbers in the interval 104≤Ra≤108. In this problem, the benefit of using a ROM approximation for the three unknown fields, i.e., velocity, pressure, and temperature, is compared against ROM approximation only in the velocity and pressure fields and a FOM type approximation for temperature. The benefit of using mesh-based hyperreduction and its computational performance and precision are also analyzed. Subsequently, the method simulates a conjugate heat transfer problem that includes natural heat convection, allowing the inclusion of different rheologies between the working fluids. The results verify the precision and speed up of the calculations by more than three hundred times with respect to the time necessary to solve a full-order formulation, emerging it as a great potential tool in the resolution of thermally coupled flows.

Scalable Projection-Based Reduced-Order Models for Large Multiscale Fluid Systems

Although projection-based reduced-order models (PROMs) have existed for decades, they have rarely been applied to large, nonlinear, multiscale, and multi-physics systems due to the complexity of effectively implementing such methods. Advances in hyper-reduction have enabled the scalable computation of PROMs for general nonlinear dynamical systems. Further, the recent model-form-preserving least squares with variable transformation method has proven capable of generating stable PROMs for extremely stiff multiphysics problems. In this work, we formulate a PROM framework combining these methodologies and demonstrate that robust, accurate, and cost-effective PROMs can be realized for complex nonreacting and reacting compressible flows. Along with an open-source toolchain for hyper-reduction sample mesh generation from extremely large data sets, this represents an end-to-end effort to assess the applicability of PROMs to large-scale, multiphysics problems of engineering interest. We examine practical considerations for implementing hyper-reduction methods and their effect on memory consumption, load balancing, and interprocessor communications. These considerations produce accurate PROMs that are three to four orders of magnitude more computationally efficient than the full-order model in recreating transonic flow over a cavity and reacting flow in a rocket combustor. Guidelines for data preparation, sample mesh construction, and online PROM solution which promote robust simulations are also provided.

Control oriented model order reduction for the hydraulic system of an autonomous wheel loader

Control design for an autonomous wheel loader requires a high-fidelity and low-order dynamics model. The objective of order reduction is to decrease the number of states in the model while maintaining performance comparable to the original model. Of all the power components in a wheel loader, the model order reduction for steering and working hydraulic systems is most demanding due to its nonlinear and stiff dynamical nature. This paper describes a physics-inspired model order reduction method that can reduce the model order by close to thirty percent. It is achieved by discarding orders with fast dynamics and consolidating multiple states into less new states. This method can also be extended to other types of off-road vehicles such as excavators, bulldozers, etc. Simulation results demonstrate that the reduced-order model exhibits very similar performance to the full-order model under typical operating conditions of wheel loaders, with an output error of less than six percent.

Model order reduction for deforming domain problems in a time‐continuous space‐time setting

AbstractIn the context of simulation‐based methods, multiple challenges arise, two of which are considered in this work. As a first challenge, problems including time‐dependent phenomena with complex domain deformations, potentially even with changes in the domain topology, need to be tackled appropriately. The second challenge arises when computational resources and the time for evaluating the model become critical in so‐called many query scenarios for parametric problems. For example, these problems occur in optimization, uncertainty quantification (UQ), or automatic control, and using highly resolved full‐order models (FOMs) may become impractical. To address both types of complexity, we present a novel projection‐based model order reduction (MOR) approach for deforming domain problems that takes advantage of the time‐continuous space‐time formulation. We apply it to two examples that are relevant to engineering or biomedical applications and conduct an error and performance analysis. In both cases, we are able to drastically reduce the computational expense for a model evaluation and, at the same time, maintain an adequate accuracy level, each compared to the original time‐continuous space‐time full‐order model (FOM). All in all, this work indicates the effectiveness of the presented MOR approach for deforming domain problems by taking advantage of a time‐continuous space‐time setting.

A reduced-order modeling for thermo-mechanical coupling analyses by using radial integration boundary element method

In order to efficiently solve thermo-mechanical coupling problems, this work combines the radial integration boundary element method (RIBEM) and the proper orthogonal decomposition (POD) to construct a fast reduced-order model. This model transforms high-dimensional systems into low-dimensional ones, enabling faster computation of thermo-mechanical coupling problems under thermal loads. Firstly, RIBEM is employed to solve coupled thermoelastic dynamic problems under thermal shock loads. Domain integrals are converted to boundary integrals using the radial integration method, establishing a purely boundary element algorithm without internal cells. Subsequently, finite difference techniques are applied to calculate the discretized algebraic equations, and the obtained field variables are used as a snapshot matrix to establish POD modes and construct the POD reduced-order model. Numerical examples demonstrate that the results of the reduced-order model are essentially consistent with the full-order model across various physical parameters and boundary conditions. Therefore, this reduced-order model significantly improves computational efficiency while maintaining high accuracy, especially when solving large-scale problems.

A reduced order model based on adaptive proper orthogonal decomposition incorporated with modal coefficient learning for digital twin in process industry

The digital twin (DT) technology provides a viable and promising direction for improving the level of the production status monitoring and the overall product quality in various fields. However, the accuracy of working condition identification, the timeliness of process adjustment, and the stability of product quality are put forward higher requirements in the process industry, which is characterized by nonlinear, large-scale, and dynamic complex systems. Therefore, it still remains a tricky challenge to construct and maintain an effective and accurate DT model in the process industry. A reduced order model (ROM) with the adaptive updating ability is proposed. The adaptive proper orthogonal decomposition (APOD) is adopted to achieve the continuous iteration and the adaptive optimization of the reduced basis set. Correspondingly, an adaptive learning algorithm based on the least squares support vector regression (LS-SVR) is developed to quickly obtain the modal coefficients and effectively circumvent the prohibitively high computational cost. In this way, the physical field of interest is expressed in a low-dimensional approximation with a high accuracy. The effectiveness of the method is verified by a case study in the process industry. Results show that the proposed model displays a high-precision fitting and a significant time saving for the full order model (FOM).

Towards Reduced Order Models via Robust Proper Orthogonal Decomposition to capture personalised aortic haemodynamics

Data driven, reduced order modelling has shown promise in tackling the challenges associated with computational and experimental haemodynamic models. In this work, we focus on the use of Reduced Order Models (ROMs) to reconstruct velocity fields in a patient-specific dissected aorta, with the objective being to compare the ROMs obtained from Robust Proper Orthogonal Decomposition (RPOD) to those obtained from the traditional Proper Orthogonal Decomposition (POD). POD and RPOD are applied to in vitro, haemodynamic data acquired by Particle Image Velocimetry and compare the decomposed flows to those derived from Computational Fluid Dynamics (CFD) data for the same geometry and flow conditions. In this work, PIV and CFD results act as surrogates for clinical haemodynamic data e.g. MR, helping to demonstrate the potential use of ROMS in real clinical scenarios. The flow is reconstructed using different numbers of POD modes and the flow features obtained throughout the cardiac cycle are compared to the original Full Order Models (FOMs).Robust Principal Component Analysis (RPCA), the first step of RPOD, has been found to enhance the quality of PIV data, allowing POD to capture most of the kinetic energy of the flow in just two modes similar to the numerical data that are free from measurement noise. The reconstruction errors differ along the cardiac cycle with diastolic flows requiring more modes for accurate reconstruction. In general, modes 1–10 are found sufficient to represent the flow field. The results demonstrate that the coherent structures that characterise this aortic dissection flow are described by the first few POD modes suggesting that it is possible to represent the macroscale behaviour of aortic flow in a low-dimensional space; thus significantly simplifying the problem, and allowing for more computationally efficient flow simulations or machine learning based flow predictions that can pave the way for translation of such models to the clinic.

Data-Driven Balanced Truncation for Predictive Model Order Reduction of Aeroacoustic Response

Rapid prediction of the aeroacoustic response is a key component in the design of aircraft and turbomachinery. In this work, we propose a technique for highly accelerated prediction of aeroacoustic response using a data-driven model reduction approach based on the eigensystem realization algorithm (ERA). Specifically, we create and compare ERA reduced-order models (ROMs) based on the training data generated by solving the linearized and nonlinear Euler equations, and we use them to predict the aeroacoustic response of an airfoil in a purely predictive setting subject to different types of gust loading. Activating each input channel separately in the full-order model (FOM) to generate the Markov sequence for training makes it computationally challenging to use the ERA in aeroacoustics applications. We address this bottleneck by first proposing a multifidelity gappy POD method to reduce the computation cost on the FOM and ROM levels by querying the high-resolution FOM only for the input channels identified by gappy POD. Second, we use tangential interpolation at the ROM level to reduce the size of the Hankel matrix. The proposed methods enable application of the ERA for accurate online acoustic response prediction, and they reduce the offline computation cost of ROMs.

Reduced Order Modelling of Shigesada-Kawasaki-Teramoto Cross-Diffusion Systems

Shigesada-Kawasaki-Teramoto (SKT) is the most known equation in population ecology for nonlinear cross-diffusion systems. The full order model (FOM) of the SKT system is constructed using symmetric interior penalty discontinuous Galerkin method (SIPG) in space and the semi-implicit Euler method in time. The reduced order models (ROMs) are solved using proper orthogonal decomposition (POD) Galerkin projection. Discrete empirical interpolation method (DEIM) is used to solve the nonlinearities of the SKT system. Numerical simulations show the accuracy and efficiency of the POD and POD-DEIM reduced solutions for the SKT system.

Transient Stability Analysis for Grid-Forming VSCs Based on Nonlinear Decoupling Method

With the increasing integration of renewable energy into the power grid, there is a growing demand for converters that not only provide stable power, but also support auxiliary functions such as grid-voltage regulation. Consequently, grid-forming strategies have attracted significant attention. However, due to the complexities of analyzing nonlinear coupling systems, a comprehensive transient stability analysis of grid-forming converters is still being explored. Conventional analysis methods rely on a simplified quasi-steady-state model for grid-forming voltage source converters (VSCs) and focus on analyzing the transient instability phenomenon caused by the outer power loop. However, this oversimplified model may yield incorrect conclusions. To address this limitation, this paper develops a full-order model that includes quadratic nonlinear terms to accurately represent the system’s nonlinear characteristics. The developed model is then decoupled into multiple low-order modes using a nonlinear decoupling method. These low-order modes can be analyzed using the mature inversing trajectory method, indirectly reflecting the transient stability of grid-forming VSCs under large disturbances. Through varying the inner and outer parameters, the transient stability of grid-forming VSCs is analyzed in detail. Furthermore, the analysis results are verified through hardware-in-loop (HIL) experiments.

Comparing different stabilization strategies for reduced order modeling of viscoelastic fluid flow problems

This paper presents a computational framework for model order reduction of viscoelastic fluid flows, with a particular focus on the stabilization of the reduced order model (ROM) and the efficient approximation of non-linear terms appearing in the governing equations. We compare three different stabilization approaches, two of which are based on offline stabilization only and one that uses both offline and online stabilization. Two main techniques for the treatment of non-linear terms are examined and compared: the discrete empirical interpolation method, and a reduced quadrature method based on a sparse recovery technique. Numerical experiments are conducted on two benchmark flows such as the flow past a sphere and the 4:1 contraction flow problem. We show that significant speedups can be achieved using offline streamline upwind Petrov–Galerkin stabilization together with hyper-reduction. We also discuss the potential accuracy losses of the ROM that could result from mesh-related issues of the full-order model (FOM) and propose possible remedies.