Model Order Reduction Method Research Articles

Reduced-order state-space models for two-dimensional discrete systems via bivariate discrete orthogonal polynomials

This paper investigates the new model order reduction (MOR) methods via bivariate discrete orthogonal polynomials for two-dimensional (2-D) discrete systems. The 2-D discrete system is described by the Kurek model. First, we deduce algebraically the shift-transformation matrix of the classical discrete orthogonal polynomials of one variable. By means of the shift-transformation matrices, 2-D discrete systems are expanded in the spaces spanned by bivariate discrete orthogonal polynomials. The coefficient matrices are calculated from matrix equations. Then the reduced-order systems are produced by the orthogonal projection matrices defined by the coefficient matrices. Theoretical analysis shows that the reduced-order systems can match a certain number of coefficient vectors of the original outputs. Finally, one numerical example is simulated to demonstrate the feasibility and effectiveness of the proposed methods.

Model order reduction via substructuring for a nonlinear, differential‐algebraic machine tool model with moving loads

AbstractMachine tools are permanently exposed to complex static, dynamic and thermic loads. This often results in an undesired displacement of the tool center point (TCP), causing errors in the production process and thus limiting the achievable workpiece quality. Recent research efforts try to counter these effects by process‐parallel solutions utilizing machine internal data. Therefore, fast simulation models are a fundamental requirement. Here, model order reduction (MOR) becomes crucial. The resulting low‐dimensional models are required for various applications, e.g., in digital twins, the correction of thermally induced errors at the TCP during the production process as well as for lifetime calculations in predictive maintenance. In this contribution, we present a strategy of MOR for a coupled thermo‐mechanical model with a nonlinear subsystem and moving loads, using the example of a feed axis with nonlinear machine components. Applying tailored substructuring techniques, we are able to separate the linear and nonlinear system components. This allows to apply classic linear MOR methods to the much larger linear part, whereas the small nonlinear part is kept, and thus enables drastically reduced computing times. The relative movement is considered by a switched system approach. Transient thermo‐mechanical interactions of the feed axis are calculated in a final investigation, comparing the performance of the resulting reduced‐order model and the original one.

Data-driven reduced order modeling for mechanical oscillators using Koopman approaches

Data-driven reduced order modeling methods that aim at extracting physically meaningful governing equations directly from measurement data are facing a growing interest in recent years. The HAVOK-algorithm is a Koopman-based method that distills a forced, low-dimensional state-space model for a given dynamical system from a univariate measurement time series. This article studies the potential of HAVOK for application to mechanical oscillators by investigating which information of the underlying system can be extracted from the state-space model generated by HAVOK. Extensive parameter studies are performed to point out the strengths and pitfalls of the algorithm and ultimately yield recommendations for choosing tuning parameters. The application of the algorithm to real-world friction brake system measurements concludes this study.

SOBMOR: Structured Optimization-Based Model Order Reduction

Model order reduction (MOR) methods that are designed to preserve structural features of a given full order model (FOM) often suffer from a lower accuracy when compared to their non-structure-preserving counterparts. In this paper, we present a framework for structure-preserving MOR, which allows us to compute structured reduced order models (ROMs) with a much higher accuracy. The framework is based on parameter optimization, i.e., the elements of the system matrices of the ROM are iteratively varied to minimize an objective functional that measures the difference between the FOM and the ROM. The structural constraints can be encoded in the parametrization of the ROM. The method only depends on frequency response data and can thus be applied to a wide range of dynamical systems. We illustrate the effectiveness of our method on a port-Hamiltonian and on a symmetric second-order system in a comparison with other structure-preserving MOR algorithms.

Reduced-order discrete modeling method and nonlinear analysis of a discontinuous conduction mode buck converter with a constant power load

Discontinuous conduction mode (DCM) DC–DC converters with constant power loads (CPLs) are the widely used power electronics cascade topologies of distributed renewable energy power systems. There exists rich nonlinear dynamics of these cascade topologies, which tend to induce unstable operation of the distributed renewable energy systems. However, the difficulties of output regulation and discrete modeling for DCM DC–DC converters with CPLs decelerate the progress of the study of nonlinear dynamics in these systems. To study the mechanism of these nonlinear dynamics for DCM DC–DC converters with CPLs, this paper proposes a reduced-order discrete modeling method for the system by considering a current-controlled discontinuous conduction mode (CC-DCM) Buck converter with a CPL as example. The nonlinear dynamics of CC-DCM Buck converter with a CPL are studied by using the proposed reduced-order discrete model. The theoretical results display complex nonlinear behaviors of this system, such as multicycle orbits, states jumping between different periodic orbits or between periodic orbits and chaos, chaotic states, and orbits interleaving with the variation of the power of CPL. Both simulation and experimental results illustrate the effectiveness of the proposed reduced-order discrete modeling method and the correctness of nonlinear characteristic analysis of the system. Based on the proposed reduced-order discrete modeling method, the problems of discrete modeling DCM DC–DC converters with CPLs in DCM states using the stroboscopic mapping method are resolved. This study is expected to provide reference for the further study and application of these power electronic cascaded systems.

An improved model order reduction method for dynamic analysis of large-scale structures with local nonlinearities

In this study, an improved model order reduction method is developed for the dynamic analysis of large-scale structures with local nonlinearities. In this method, the reduced system associated with the elements in nonlinear state is reconstructed adaptively based on the expressions of the element generalized deformation, which is defined by the eigen-decomposition of the element stiffness matrix. With the introduction of a simplified pre-conditioning method, a novel Newton iteration approach with Krylov subspace methods is proposed to solve the reduced system efficiently. Thus, dynamic analysis can be conducted efficiently for the entire structure. To expand the scope of application, an alternating iterative solution algorithm based on the model order reduction method is further developed for the structural systems simulated with both pseudo-static models and parameterized dynamic models. According to the results of numerical investigation, the novel Newton iteration approach significantly improves the solution efficiency of dynamic analysis for the large-scale structures with local nonlinearities. Meanwhile, the implementation of the alternating iterative solution algorithm contributes a new idea to the fast solution of structural systems with various types of nonlinear models.

Model order reduction for parameterized electromagnetic problems using matrix decomposition and deep neural networks

A non-intrusive model order reduction (MOR) method for solving parameterized electromagnetic scattering problems is proposed in this paper. A database collecting snapshots of high-fidelity solutions is built by solving the parameterized time-domain Maxwell equations for some values of the material parameters using a fullwave solver based on a high order discontinuous Galerkin time-domain (DGTD) method. To perform a prior dimensionality reduction, a set of reduced basis (RB) functions are extracted from the database via a two-step proper orthogonal decomposition (POD) method. Intrinsic coordinates of the high-fidelity solutions are further compressed through a convolutional autoencoder (CAE) network. Singular value decomposition (SVD) is then used to extract the principal components of the low dimensional coding matrices generated by CAE, and a cubic spline interpolation-based (CSI) approach is employed for approximating the dominating time- and parameter-modes of these matrices. The generation of the reduced basis and the training of the CAE and CSI are accomplished in the offline stage, thus the RB solution for given time/parameter values can be quickly recovered via outputs of the interpolation model and decoder network. In particular, the offline and online stages of the proposed RB method are completely decoupled, which ensures the validity of the method. The performance of the proposed CAE-CSI ROM is illustrated with numerical experiments for scattering of a plane wave by a 2-D dielectric disk and a multi-layer heterogeneous medium.

Review of vortex lattice method for supersonic aircraft design

AbstractThere has been a renewed interest in developing environmentally friendly, economically viable, and technologically feasible supersonic transport aircraft and reduced order modeling methods can play an important contribution in accelerating the design process of these future aircraft. This paper reviews the use of the vortex lattice method (VLM) in modeling the general aerodynamics of subsonic and supersonic aircraft. The historical overview of the vortex lattice method is reviewed which indicates the use of this method for over a century for development and advancements in the aerodynamic analysis of subsonic and supersonic aircraft. The preference of VLM over other potential flow-solvers is because of its low order highly efficient computational analysis which is quick and efficient. Developments in VLM covering steady, unsteady state, linear and non-linear aerodynamic characteristics for different wing planform for the purpose of several different types of design optimisation is reviewed. For over a decade classical vortex lattice method has been used for multi-objective optimisation studies for commercial aircraft and unmanned aerial vehicle’s aerodynamic performance optimisation. VLM was one of the major potential flow solvers for studying the aerodynamic and aeroelastic characteristics of many wings and aircraft for NASA’s supersonic transport mission (SST). VLM is a preferred means for solving large numbers of computational design parameters in less time, more efficiently, and cheaper when compared to conventional CFD analysis which lends itself more to detailed study and solving the more challenging configuration and aerodynamic features of civil supersonic transport.

Comparison of reduced basis construction methods for Model Order Reduction, with application to non-linear low frequency electromagnetics

Numerical simulation is more and more used during the design stage of a manufactured product in order to optimize its performances. However, it is often too time consuming, particularly when it is used to solve optimization problems, preventing an intensive usage. A-posteriori Model Order Reduction methods can be very effective to shorten the computational time. An approximated solution is then sought in a space of small dimension defined by a reduced basis. The accuracy of such methods is highly dependent on the choice of the reduced basis, extracted from preliminary numerical simulation. The method usually applied to construct such reduced basis is based on the Singular Value Decomposition (SVD), which can be time consuming, and is not adapted when a large collection of preliminary numerical simulations must be used to construct the basis. An alternative to this approach has been proposed recently with the Maximum Entropy Snapshot Sampling (MESS) method. In this paper, we propose to compare these methods with other approaches usually used for clustering or data classification based on vectors distance calculation, like the Centroidal Voronoi Tessellation (CVT), Density Based Spatial Clustering of Applications with Noise (DBSCAN), and Clustering Using Representatives (CURE). The methods are compared on a complex and realistic nonlinear problem in low frequency electromagnetics. The quality of the reduced bases obtained by the different methods are compared. Then, field distributions and global quantities, like eddy current losses and magnetic energy, are computed from the reconstructed results, to further analyze the quality of the reduced bases.

Crash topology optimization for front-end safety parts of battery electric vehicle using an improved equivalent static loads method

Crash topology optimization is a typical nonlinear dynamic response structural topology optimization problem, which is one of the most difficult problem in the structural design field. The equivalent static load method (ESLM) provides a well-defined pattern to solve such difficult problems, which can convert a nonlinear dynamic response optimization into multi-load steps optimization problem with the equivalent static loads (ESLs). However, due to the large deformation in the crash condition and nodal characteristics of the ESLs, it is hard to solve the crash topology optimization directly using the standard ESLM. To expand the application scope of the ESLM, an improved ESLs calculation method is proposed by using the model order reduction method and energy principle, which only acting on some nodes and can be scaled adaptively. Correspondingly, to enrich the connotation of topology optimization, a crash topology optimization method is proposed by using the improved ESLs, which can solve the numerical problems in the design domain by guaranteeing the topology optimization with the improved ESLs perform in the linear rang. First, the principle of the standard ESLM is introduced, and corresponding problems and deficiencies in solving the crash topology optimization are summarized. Then, to solve the above problems, the improved ESLs calculation method is proposed. Meanwhile, the corresponding crash topology strategy is proposed based on the improved ESLs. Finally, to verify the effectiveness and engineering application value of the proposed crash topology optimization method, a test-verified frontal crash simulation model of the BEV front-end is established, and their safety parts are redesigned by using the proposed method. The results show that, the proposed method can effectively solve the crash topology optimization of thin-walled structures under large deformation crash condition. This method also provides a new idea and practical method for the crashworthiness and lightweight design of automobile structures.

A CFD-Based Data-Driven Reduced Order Modeling Method for Damaged Ship Motion in Waves

A simple CFD-based data-driven reduced order modeling method was proposed for the study of damaged ship motion in waves. It consists of low-order modeling of the whole concerned parameter range and high-order modeling for selected key scenarios identified with the help of low-order results. The difference between the low and high-order results for the whole parameter range, where the main trend of the physics behind the problem is expected to be captured, is then modeled by some commonly used machine learning or data regression methods based on the data from key scenarios which is chosen as Piecewise Cubic Hermite Interpolating Polynomial (PCHIP) in this study. The final prediction is obtained by adding the results from the low-order model and the difference. The low and high-order modeling were conducted through computational fluid dynamics (CFD) simulations with coarse and refined meshes. Taking the roll Response Amplitude Operator (RAO) of a DTMB-5415 ship model with a damaged cabin as an example, the proposed physics-informed data-driven model was shown to have the same level of accuracy as pure high-order modeling, whilst the computational time can be reduced by 22~55% for the studied cases. This simple reduced order modeling approach is also expected to be applicable to other ship hydrodynamic problems.

Methods of Model Order Reduction for Coupled Systems Applied to a Brake Disc‐Wheel Composite

AbstractIn this contribution, investigations on model order reduction for coupled systems composed from components of a passenger car are shown. In today's development processes, the simulation of mechanical components is indispensable and large Finite Element models are often used for this purpose. For the calculation of time‐domain or frequency‐domain analyses, for example, a lot of computing power is required. However, with the application of model order reduction methods, this effort can be reduced, but this results in a trade‐off between the reduction error and the computational time. Since the computation of reduction bases for complete systems can be computationally expensive, it is of interest to be able to reduce components individually and then assemble them into a reduced overall model. This can result in both, a saving of computational effort when creating the bases, as well as a saving of the required memory space. Furthermore, there are many possible combinations of components in the modular systems of today's automotive industry, which emphasizes the model order reduction by parts and not by assemblies. In this work, methods of model order reduction for coupled systems are presented and will be tested on components in the chassis of a sports car. Therefore, an assembly consisting of a brake disc and wheel rim together with the wheel hub are investigated. For this purpose, the eigenmodes and transfer functions of the overall model, the reduced overall model and the assembly built from individual reduced bodies are compared.

Adaptive methods for MOR in non‐linear FE simulations of unit cells

AbstractIn this contribution, an adaptive method for Model Order Reduction (MOR) is presented for simulating physically non‐linear material behavior of micro‐heterogeneous structures. The method is purely online and includes a Galerkin projection, which is performed using a projection operator initially gained by a Proper Orthogonal Decomposition (POD). By this projection, the number of degrees of freedom to solve for is decreased drastically and can be projected back to the vector space of the high fidelity problem after the solution process. The composition of the projection operator follows an adaptive algorithm, which evaluates an update of the operator if convergence is not ensured during the Newton Raphson scheme. Later on, during the simulation, the number of degrees of freedom to solve is controlled using a POD‐based control criterion for sorting out non‐relevant basis vectors from previous updates. This results in an efficient method and leads to a satisfying speedup of the simulation process. The performance and efficiency of the presented adaptive MOR scheme is investigated in numerical simulations of a hexagon‐shaped unit cell.

Interpolation‐based parametric model order reduction of automotive brake systems for frequency‐domain analyses

Brake squeal describes noise with different frequencies that can be emitted during the braking process. Typically, the frequencies are in the range of 1 to 16 kHz. Although the noise has virtually no effect on braking performance, strong attempts are made to identify and eliminate the noise as it can be very unpleasant and annoying. In the field of numerical simulation, the brake is typically modeled using the Finite Element method, and this results in a high-dimensional equation of motion. For the analysis of brake squeal, gyroscopic and circulatory effects, as well as damping and friction, must be considered correctly. For the subsequent analysis, the high-dimensional damped nonlinear equation system is linearized. This results in terms that are non-symmetric and dependent on the rotational frequency of the brake rotor. Many parameter points to be evaluated implies many evaluations to determine the relevant parameters of the unstable system. In order to increase the efficiency of the process, the system is typically reduced with a truncated modal transformation. However, with this method the damping and the velocity-dependent terms, which have a significant influence on the system, are neglected for the calculation of the eigenmodes, and this can lead to inaccurate reduced models. In this paper, we present results of other methods of model order reduction applied on an industrial high-dimensional brake model. Using moment matching methods combined with parametric model order reduction, both the damping and the various parameter-dependent terms of the brake model can be taken into account in the reduction step. Thus, better results in the frequency domain can be obtained. On the one hand, as usual in brake analysis, the complex eigenvalues are evaluated, but on the other hand also the transfer behavior in terms of the frequency response. In each case, the classical and the new reduction method are compared with each other.

Parametric model order reduction and vibration analysis of pipeline system based on adaptive dynamic substructure method

In the dynamics topology optimization process of the pipeline system, the parametric model order reduction (PMOR) method is needed to improve the efficiency of model reconstruction and vibration analysis. In this paper, the general PMOR and vibration analysis methods of pipeline system based on adaptive dynamic substructure method are innovatively proposed. Based on the spatial 8-node conforming solid element, a spatial 8-node nonconforming solid (Solid-NC) element with high accuracy is obtained by introducing the node-free displacement items. The node coordinates of the finite element models corresponding to the straight-line and curved arc segments of the pipeline system are obtained based on the direction vector method and the vector decomposition method respectively, thus the parametric finite element model of the pipeline system is established. The pipeline body is divided into single interface substructures and double interface substructures based on the dynamic substructure method, thus the PMOR method that can change the number of substructures or elements contained in the substructure with the pipeline shape or mesh density is obtained, and then the natural characteristic and vibration response solutions of the pipeline system are performed. Finally, the rationality and efficiency of the proposed PMOR method are verified by an example.

A computationally effective time-restricted stability preserving [formula omitted]-optimal model order reduction approach

Several approaches for reducing model order on the definite time segments have become the topic of investigation in a series of papers that bring challenges during application in a large-scale setting. The subject of discussion of this paper is the computationally efficient time-restricted H2-optimal model order reduction method of higher dimensional sparse systems that requires the solutions of time-restricted Lyapunov and Sylvester equations. Our discussion is on developing the algorithms to solve these matrix equations that face difficulty when calculating the matrix exponential of the large-scale matrices. As a result, an efficient remedy is also proposed to compute the matrix exponential. Our ideas are also evaluated for index-1 descriptor systems apart from the generalized structure. Numerical analyses are conducted on several benchmark examples to illustrate how accurate and efficient our suggested approaches are by comparing them with the existing methods.

Nonlinear model order reduction and vibration control of a membrane antenna structure

In this paper, model order reduction (MOR) of the nonlinear dynamic model of a membrane antenna structure is studied. Nonlinear vibration control of the membrane antenna structure by using cable actuators has also been investigated. First, the nonlinear dynamic model of the membrane antenna structure is established by using the nonlinear finite element method. Then, MOR of this nonlinear model is carried out. The vibration modes obtained by the linearized model are used as transformation bases, and the method of polynomial fitting is applied to approximate the nonlinear terms. Based on the reduced nonlinear dynamic model, by using the tension cables as actuators, a vibration controller is proposed to suppress the nonlinear vibration of the structure. The nonlinear vibration of the membrane antenna structure is suppressed by adjusting the tension force in the cable actuators. Numerical simulations show that the reduced dynamic model obtained by the presented MOR method can approximate the original nonlinear model very well meanwhile reduce the calculation cost significantly, and the nonlinear vibration of the structure can be suppressed effectively by the cable actuators with the proposed control law.

A model order reduction of hysteretic eddy-current fields in electric machines using Cauer ladder network method

This study aims to establish a hysteretic model order reduction method for electric machines using the Cauer ladder network (CLN) method. The first inductor of the CLN represents the hysteresis effect, which is computed using the scalar play model identified from the static I–ϕ (current–magnetic flux) loops. An iron-core inductor was analyzed, and pulse-width modulation (PWM) voltage input was applied to the coil. The proposed method yielded results that agreed with the transient finite element (FE) analysis and effectively reduced the computation time.

Parameter Estimation for a Kinetic Model of a Cellular System Using Model Order Reduction Method

Order reduction methods are important tools for systems engineering and can be used, for example, for parameter estimation of kinetic models for systems biology applications. In particular, the Proper Orthogonal Decomposition (POD) method produces a reduced-order model of a system that is used for solving inverse problems (parameter estimation). POD is an intrusive model order reduction method that is aimed to obtain a lower-dimensional system for a high-dimensional system while preserving the main features of the original system. We use a singular value decomposition (SVD) to compute a reduced basis as it is usually numerically more robust to compute the singular values of the snapshot matrix instead of the eigenvalues of the corresponding correlation matrix. The reduced basis functions are then used to construct a data-fitting function that fits a known experimental data set of system substance concentrations. The method is applied to calibrate a kinetic model of carbon catabolite repression (CCR) in Escherichia coli, where the regulatory mechanisms on the molecular side are well understood and experimental data for a number of state variables is available. In particular, we show that the method can be used to estimate the uptake rate constants and other kinetic parameters of the CCR model.

Parallel model order reduction methods based on structured matrix analysis for discrete-time systems with parametric uncertainty

This paper explores two novel parallel parametric model order reduction methods for discrete-time systems with parametric uncertainty. The proposed methods allow that the parametric dependence is non-affine. With the shift-transformation matrix of the discrete orthogonal polynomials, including Charlier polynomials and Krawtchouk polynomials, the expansion coefficients of state variable in discrete orthogonal polynomials space can be obtained by solving the system of linear algebraic equations (SLAE). By applying the structured matrix analysis of the SLAE, two parallel strategies based on the equivalent transformation of block bi-diagonal matrices and the block discrete Fourier transform of block ϵ-circulant matrices are proposed to solve the SLAE. Then, the projection matrix is constructed to reduce discrete-time parametric systems. Moreover, we analyze the properties of the proposed parallel methods, including the invariable coefficients, the invertibility, and the error analysis. Finally, the effectiveness of the proposed methods is illustrated by the numerical experiments.