Model Order Reduction Techniques Research Articles

Nonlinear model order reduction of resonant piezoelectric micro-actuators: An invariant manifold approach

This paper presents a novel derivation of the direct parametrisation method for invariant manifolds able to build simulation-free reduced-order models for nonlinear piezoelectric structures, with a particular emphasis on applications to Micro Electro Mechanical Systems. The constitutive model adopted accounts for the hysteretic and electrostrictive response of the piezoelectric material by resorting to the Landau-Devonshire theory of ferroelectrics. Results are validated with full-order simulations operated with a harmonic balance finite element method to highlight the reliability of the proposed reduction procedure. Numerical results show a remarkable gain in terms of computing time as a result of the dimensionality reduction process over low dimensional invariant sets. Results are also compared with experimental data to highlight the remarkable benefits of the proposed model order reduction technique.

Adaptive MOR techniques for non‐linear FE simulations of unit cells

AbstractThis work shows an adaptive Model Order Reduction (MOR) scheme for the computation of unit cells with non‐linear material behavior. A reduction of dimensionality and computational cost is achieved by projecting the high‐fidelity system of equations (SOE) from the full vector space to a lower dimensional subspace. An adaptive algorithm updates the projection operator, which is initially gained from a Proper Orthogonal Decomposition (POD), to ensure convergence and optimal size control of the reduced problem. Numerical examples show a comparison between the discretization of full displacements and fluctuations of a hexagonal unit cell, which show the independence of the MOR basis from the macroscopic displacements.

Efficient thermomechanically coupled FE‐FFT‐based multiscale simulation of polycrystals

AbstractIn general, the overall macroscopic material behavior of any structural component is directly dependent on its underlying microstructure. For metal components, the associated microstructure is given in terms of a polycrystal. To enable the simulation of the related microstructural and overall elasto‐viscoplastic material behavior, a two‐scale simulation approach can be used. In this context, we use a FE‐FFT‐based two‐scale method, which is an efficient alternative to the classical FE2 method for the simulation of periodic microstructures. In addition, we consider a thermomechanically coupled framework to account for both thermal and mechanical loads. Finally, we incorporate a model order reduction technique based on a coarsely discretized microstructure to develop an efficient two‐scale simulation technique. As a demonstration of the feasibility of the proposed simulation framework, a numerical example will be investigated.

Concise and Compatible MOR-Based Self-Adjoint EM Sensitivity Analysis for Fast Frequency Sweep

It is of great significance to perform adjoint electromagnetic (EM) sensitivity analysis when designing or optimizing a microwave component. Recent advances in adjoint EM sensitivity analysis using a fast frequency sweep have gained a considerable speedup over that using discrete frequency sweep. However, it needs complex computational loops for calculating the adjoint vectors, resulting in extra time for an increased number of forward/backward (F/B) substitutions of the original system matrix along with additional large matrix multiplications. The proposed method addresses this situation and proposes a novel model order reduction (MOR)-based self-adjoint EM sensitivity analysis for a fast frequency sweep. This method derives more concise and more compatible formulations for calculating adjoint sensitivities to decrease the number of F/B substitutions and matrix multiplications. The proposed method subsequently develops a new self-adjoint EM sensitivity analysis algorithm for a fast frequency sweep which further decreases half of the number of F/B substitutions. Compared with the existing adjoint sensitivity method for a fast frequency sweep, the technique we proposed can take less time due to the more concise sensitivity formulations while additionally obtaining better sensitivity accuracy in a relatively wide frequency range, especially when a higher MOR order is needed. Besides, the self-adjoint EM sensitivity analysis formulation can be compatible with various MOR techniques. Two EM examples of microwave components are used to demonstrate the proposed technique.

A reduced basis ensemble Kalman method

In the process of reproducing the state dynamics of parameter dependent distributed systems, data from physical measurements can be incorporated into the mathematical model to reduce the parameter uncertainty and, consequently, improve the state prediction. Such a data assimilation process must deal with the data and model misfit arising from experimental noise as well as model inaccuracies and uncertainties. In this work, we focus on the ensemble Kalman method (EnKM), a particle-based iterative regularization method designed for a posteriori analysis of time series. The method is gradient free and, like the ensemble Kalman filter (EnKF), relies on a sample of parameters or particle ensemble to identify the state that better reproduces the physical observations, while preserving the physics of the system as described by the best knowledge model. We consider systems described by parameterized parabolic partial differential equations and employ model order reduction techniques to generate surrogate models of different accuracy with uncertain parameters. Their use in combination with the EnKM involves the introduction of the model bias which constitutes a new source of systematic error. To mitigate its impact, an algorithm adjustment is proposed accounting for a prior estimation of the bias in the data. The resulting RB-EnKM is tested in different conditions, including different ensemble sizes and increasing levels of experimental noise. The results are compared to those obtained with the standard EnKF and with the unadjusted algorithm.

Model order reduction of time-domain vibro-acoustic finite element simulations with non-locally reacting absorbers

Time-domain vibro-acoustic finite element simulation gained considerable interest in recent years because of the rise of some applications such as auralization and virtual sensing. Efficiency and stability are of great importance for this topic. The number of elements needed per wavelength to reach acceptable accuracy often results in a large model size, which requires lots of computational resources and cannot run efficiently. Model order reduction can significantly alleviate this problem by reducing the size of these models, while maintaining the high-fidelity property. However, many model order reduction techniques fail to preserve the stability for non-locally reacting absorbers, which are often accounted for by using the Helmholtz equation with frequency-dependent density and bulk modulus, known as the equivalent fluid method. When discretized into a finite element model, frequency-dependent mass and stiffness matrices are introduced, which hinder the preservation of stability in the context of model order reduction. This paper presents a method that enables the creation of a stable reduced order model of a vibro-acoustic system with non-locally absorbers. This method reforms the Helmholtz equation and considers it as the interconnection of several passive systems. Stacking the states of these subsystems gives the final descriptor representation of the non-locally reacting absorbers. Furthermore, a possible second-order representation of the descriptor model is chosen such that it satisfies the stability-preserving condition under model order reduction. The resulting second-order model is coupled with a vibro-acoustic system in a velocity potential-displacement formulation, leading to a second-order model satisfying the aforementioned stability-preserving condition. The proposed method is successfully verified by several numerical simulations.

An operator inference oriented approach for linear mechanical systems

Model order reduction techniques allow the construction of low-dimensional surrogate models that can accelerate engineering design processes. Often, these techniques are intrusive, meaning that they require direct access to underlying high-fidelity models. Accessing these models is laborious or may not even be possible in some cases. Therefore, there is an interest in developing non-intrusive model order reduction techniques to construct low-dimensional models directly from simulated or experimental data. In this work, we focus on a recent data-driven methodology, namely operator inference, that aims at inferring the reduced operators using only trajectories of high-fidelity models. We present an extension of operator inference for linear mechanical systems, preserving the underlying second-order structure. Furthermore, we study a particular case in which complete information about the external forces is available. In this formulation, the reduced operators, having certain properties inspired by the original system matrices, are enforced by adding constraints to the inference problem. We illustrate the proposed methodologies using three numerical examples.

A novel multistage damage detection method for trusses using time-history data based on model order reduction and deep neural network

This article first proposes a multistage damage identification approach for trusses using time-series data relied upon model order reduction (MOR) and deep neural network (DNN). In the first step, an acceleration–displacement-based strain energy indicator (ADSEI), which is computed from the acceleration and displacement data incompletely measured at limited sensors and unmeasured ones inferred from a second-order Neumann series expansion (SNSE)-relied MOR technique, is utilized to eliminate low-risk damage candidates, aiming to only keep high-potential ones. This can dramatically reduce the number of output neurons of the DNN model in the second step. The input data employed to build such a DNN are the finite element method (FEM)-simulated acceleration and displacement signals corresponding to measured degrees of freedom (DOFs). Low-risk flawed candidates predicted by this DNN are then excluded via a suggested damage threshold. By repeating such a manner, the accuracy of the DNN models constructed in the subsequent stages is therefore enhanced continuously, although these upgraded DNNs only require a moderate dataset, simple architectures, and much less computational cost for their training and testing processes. Accordingly, both the location and severity of damaged members can be reliably and precisely diagnosed by time-series data measured in a fairly short interval at a few sensors, even with high noises. Several numerical examples of 3D trusses are tested to affirm the efficiency and feasibility of the current approach.

A Rapid Surrogate Model for Estimating Aviation Noise Impact across Various Departure Profiles and Operating Conditions

Aviation noise remains a key barrier to the sustainable growth of commercial aviation. The advent of emerging technologies, such as urban air mobility, and the renewed interest in commercial supersonic transport aircraft, has only further raised concerns over the resultant community noise exposure. The foundation of any noise mitigation effort is the ability to accurately model noise metrics over a wide range of scenarios. Aviation noise is influenced by a wide variety of factors, including aircraft type, payload weight, thrust settings, airport elevation, ambient weather, and flight trajectory. Traditional noise modeling paradigms rely on physics-based and empirical calculations, which are computationally expensive. Attempts at speeding up the computations with alternate models could deliver on speed or accuracy, but not both. Recent research has indicated that model order reduction techniques hold promise for transforming and greatly reducing the number of quantities that need to be modeled. Paired with surrogate modeling techniques, a rapid and accurate noise model can be generated. The research presented in this manuscript expands on the model order reduction method and develops a rapid noise surrogate model, which can account for the piloting actions, the ambient temperature, and airport elevation. The presented results indicate that the method works well with minimal error for most modeling scenarios. The results also outline avenues for improvement, such as using a different class of surrogate models or modeling additional training cases. The model developed in this research has numerous applications for multi-query applications, such as parametric trade-off analyses and optimization studies. With the inclusion of airport and aircraft parameters, the model enables the development of frameworks that optimize piloting actions for noise mitigation on the ground.

A dimensionality reduction approach for convolutional neural networks

The focus of this work is on the application of classical Model Order Reduction techniques, such as Active Subspaces and Proper Orthogonal Decomposition, to Deep Neural Networks. We propose a generic methodology to reduce the number of layers in a pre-trained network by combining the aforementioned techniques for dimensionality reduction with input-output mappings, such as Polynomial Chaos Expansion and Feedforward Neural Networks. The motivation behind compressing the architecture of an existing Convolutional Neural Network arises from its usage in embedded systems with specific storage constraints. The conducted numerical tests demonstrate that the resulting reduced networks can achieve a level of accuracy comparable to the original Convolutional Neural Network being examined, while also saving memory allocation. Our primary emphasis lies in the field of image recognition, where we tested our methodology using VGG-16 and ResNet-110 architectures against three different datasets: CIFAR-10, CIFAR-100, and a custom dataset.

Real-time simulation of the transplanted tooth using model order reduction.

The biomechanics of transplanted teeth remain poorly understood due to a lack of models. In this context, finite element (FE) analysis has been used to evaluate the influence of occlusal morphology and root form on the biomechanical behavior of the transplanted tooth, but the construction of a FE model is extremely time-consuming. Model order reduction (MOR) techniques have been used in the medical field to reduce computing time, and the present study aimed to develop a reduced model of a transplanted tooth using the higher-order proper generalized decomposition method. The FE model of a previous study was used to learn von Mises root stress, and axial and lateral forces were used to simulate different occlusions between 75 and 175N. The error of the reduced model varied between 0.1% and 5.9% according to the subdomain, and was the highest for the highest lateral forces. The time for the FE simulation varied between 2.3 and 7.2h. In comparison, the reduced model was built in 17s and interpolation of new results took approximately 2.10-2s. The use of MOR reduced the time for delivering the root stresses by a mean 5.9h. The biomechanical behavior of a transplanted tooth simulated by FE models was accurately captured with a significant decrease of computing time. Future studies could include using jaw tracking devices for clinical use and the development of more realistic real-time simulations of tooth autotransplantation surgery.

POD-Based reduced order methods for optimal control problems governed by parametric partial differential equation with varying boundary control

In this work we propose tailored model order reduction for varying boundary optimal control problems governed by parametric partial differential equations. With varying boundary control, we mean that a specific parameter changes where the boundary control acts on the system. This peculiar formulation might benefit from model order reduction. Indeed, fast and reliable simulations of this model can be of utmost usefulness in many applied fields, such as geophysics and energy engineering. However, varying boundary control features very complicated and diversified parametric behaviour for the state and adjoint variables. The state solution, for example, changing the boundary control parameter, might feature transport phenomena. Moreover, the problem loses its affine structure. It is well known that classical model order reduction techniques fail in this setting, both in accuracy and in efficiency. Thus, we propose reduced approaches inspired by the ones used when dealing with wave-like phenomena. Indeed, we compare standard proper orthogonal decomposition with two tailored strategies: geometric recasting and local proper orthogonal decomposition. Geometric recasting solves the optimization system in a reference domain simplifying the problem at hand avoiding hyper-reduction, while local proper orthogonal decomposition builds local bases to increase the accuracy of the reduced solution in very general settings (where geometric recasting is unfeasible). We compare the various approaches on two different numerical experiments based on geometries of increasing complexity.

Comparative Study of Model Order Reduction Techniques

Comparative Study of Model Order Reduction Techniques

An improved whale optimization algorithm for the model order reduction of large-scale systems

An improved whale optimization algorithm (IWOA) is developed for the model order reduction (MOR) of large-scale systems (LSS) in this paper. An equivalent reduced order model (ROM) for the higher-order system (HOS) is derived by considering integral square error (ISE) as the objective function using IWOA. Many practical systems of single-input and single-output (SISO) and multi-input and multi-output (MIMO) systems are considered to examine the worth of the proposed technique. The powerfulness and robustness of the proposed design technique are tested on various typical examples. Several simulation results have been reported to demonstrate the efficacy of IWOA. To prove the potentiality of the suggested technique, the results have been compared with the familiar classical MOR techniques as well as other heuristic algorithms available in the literature.

Localized model order reduction and domain decomposition methods for coupled heterogeneous systems

SummaryWe propose a model order reduction technique to accurately approximate the behavior of multi‐component systems without any a‐priori knowledge of the coupled model. In the offline phase, we construct independent surrogate models by solving the local problems with parametrized interface boundary conditions, while we combine them using domain decomposition techniques during the online phase. We show the potential of the proposed approach in terms of accuracy, computational performance and robustness in a series of test cases, including nonlinear and multi‐physics problems.

Reduced-order digital twin and latent data assimilation for global wildfire prediction

Abstract. The occurrence of forest fires can impact vegetation in the ecosystem, property, and human health but also indirectly affect the climate. The Joint UK Land Environment Simulator – INteractive Fire and Emissions algorithm for Natural envirOnments (JULES-INFERNO) is a global land surface model, which simulates vegetation, soils, and fire occurrence driven by environmental factors. However, this model incurs substantial computational costs due to the high data dimensionality and the complexity of differential equations. Deep-learning-based digital twins have an advantage in handling large amounts of data. They can reduce the computational cost of subsequent predictive models by extracting data features through reduced-order modelling (ROM) and then compressing the data to a low-dimensional latent space. This study proposes a JULES-INFERNO-based digital twin fire model using ROM techniques and deep learning prediction networks to improve the efficiency of global wildfire predictions. The iterative prediction implemented in the proposed model can use current-year data to predict fires in subsequent years. To avoid the accumulation of errors from the iterative prediction, latent data assimilation (LA) is applied to the prediction process. LA manages to efficiently adjust the prediction results to ensure the stability and sustainability of the prediction. Numerical results show that the proposed model can effectively encode the original data and achieve accurate surrogate predictions. Furthermore, the application of LA can also effectively adjust the bias of the prediction results. The proposed digital twin also runs 500 times faster for online predictions than the original JULES-INFERNO model without requiring high-performance computing (HPC) clusters.

Hybrid Data Assimilation methods, Part II: Application to the DYNASTY experimental facility

Hybrid Data Assimilation (HDA) methods aim at combining the advantages of mathematical models and experimental observations by integrating Model Order Reduction techniques into a Data Assimilation framework, thus reducing the solution time whilst keeping the accuracy of the models to the desired level. HDA methods provide tools able to estimate the state of a system assuming to have measurements available per each field describing the system (e.g., neutron flux, temperature, velocity, …). However, it is not always possible to measure every field of interest for various reasons, thus these techniques should be adapted: in particular, it is legitimate to investigate the possibility of extracting some information from indirect measurement. Following the assessment of the two selected HDA methods (Generalised Empirical Interpolation Method and Parameterised Background Data-Weak) performed on a numerical benchmark test case discussed in the first part of this two-part work, the present paper now deals with their testing on a real-world experimental facility and their validation with experimental data. This work represents the first step for a deep validation phase to assess their efficiency and reliability, applying the Generalised Empirical Interpolation Method (GEIM), the Parameterised-Background Data-Weak (PBDW) formulation and the Indirect Reconstruction (IR) algorithm to DYNASTY, an experimental facility for studying natural circulation built at Politecnico di Milano. The robustness of these methods is high when the models are accurate, however when real system are analysed the model discrepancy can be present and these techniques may suffer.

Hybrid data assimilation methods, Part I: Numerical comparison between GEIM and PBDW

Hybrid Data Assimilation (HDA) methods are a class of numerical methods that aim at integrating Model Order Reduction (MOR) techniques into a Data Assimilation (DA) framework, thus combining mathematical models and experimental data. The objective is to reduce the solution time using MOR algorithms whilst keeping the accuracy of the models at the desired level using observations, which serve as an update to the a priori prediction of the model. This two-part work investigated HDA techniques by applying them to two classes of problems: numerical benchmark cases (part 1) and experimental facilities (part 2). In particular, this paper discusses the former, focusing on the numerical formulation of the methodologies and on the effect of noisy data. Indeed, real-world experimental data are always polluted by errors and uncertainties; therefore, it is critical to first assess the performance of these techniques on numerical benchmark cases with the artificial introduction of random noise before applying them to real-world experimental facilities. As such, this paper applies the Generalised Empirical Interpolation Method (GEIM) and the Parameterised-Background Data-Weak (PBDW) formulation to a non-adiabatic airflow over the classical computational fluid-dynamics benchmark of the 3D Backward Facing Step (BFS). Results show how both algorithms are valuable tools to reconstruct the state of the system when measurements are available, whilst assessing the effect of noise on the available data; in particular, the GEIM is a bit better than the PBDW since a lower reconstruction error is achieved with fewer sensors.

Error Estimation of the Cauer Ladder Network Method Based on Eigenfunction Expansion

The Cauer ladder network (CLN) method is a model order reduction technique used for quasi-static electromagnetic field analysis. In this study, we derive an explicit formula for the error of the truncated transfer function and its error estimators using the eigenvalues of a linear operator and the orthogonal polynomials associated with the basis functions of the CLN method. The effectiveness of the proposed method is clarified through the finite element analysis of a 2-D inductor model.

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.