Efficient Model Order Reduction Research Articles

Modified Iterative-Order-Reduction Substructuring Method with Interface Boundary Reduction

The iterative-order-reduction (IOR) substructuring method is an efficient model order reduction (MOR) technique for estimating eigenvalues and eigenvectors in large-scale structural systems. However, it can be computationally expensive when applied to models with highly refined meshes and a substantial number of substructures because it retains all the interface degrees of freedom (DOF), which increase the size of the reduced structural matrices. Moreover, the reduced structural matrices may suffer from ill-conditioning after a certain number of iterations, potentially yielding inaccurate results. This paper presents a modified IOR substructuring method that integrates interface boundary reduction to reduce the size of the original IOR substructuring model by decreasing the number of interface DOF. The performance of the proposed method is demonstrated through a comprehensive numerical comparison with the original IOR substructuring and other widely used MOR techniques, including the Craig–Bampton, enhanced Craig–Bampton, and iterated improved reduced system with a substructuring scheme. The results show that the proposed method achieves comparable accuracy while significantly reducing the computational time and memory usage.

Derivation of a superelement with deformable interfaces – applied to model flexure joint

Design and optimization, as well as real time control, of flexure mechanisms require efficient but accurate models. The flexures can be modelled using beam elements and the frame parts can be modelled using superelements. Such a superelement efficiently models arbitrarily shaped bodies by few coordinates, using models obtained by model order reduction. The interfaces between the frame parts and the flexures often experience considerable deformation which affects the stiffness. To define the interface deformation in a reduced order model, this paper derives a multipoint constraint formulation, which relates the nodes on the deformable interface surface of a finite element model to a few coordinates. The multipoint constraints are imposed using a combination of the Lagrange multiplier method and master–slave elimination for efficient model order reduction. The resulting reduced order models are used in the generalized-strain multi-node superelement (GMS) that was defined in (Dwarshuis et al. in Multibody Syst. Dyn. 56(4):367–399, 2022). The interface deformations can be coupled to the cross-sectional deformation of higher order beam elements (i.e. beam elements of which the deformation of the cross-sections is explicitly taken into account).This paper applies this technique to model flexure joints, where the flexures are modelled with beam elements, and the frame components and critical connections using the GMS. This approach gives generally over 94% accurate stiffness, compared to nonlinear finite element models. The errors were often more than 50% lower than errors of models which only contain beam elements.

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.

Robust error assessment for reduced order vibro-acoustic problems

Robust and efficient automatic model order reduction is important in the design process of vibro-acoustic structures. For correct assessment of the reduction error without requiring the solution of the full order model, reliable error estimation techniques are required. Such error estimators can be used in iterative algorithms to automatically determine a reasonable order and expansion point distribution for reduced-order models. An accurate estimation of the approximation error is important in order to design an efficient adaptive model order reduction process requiring as little a priori knowledge of the system response as possible. In this contribution, different techniques for estimating the approximation error of reduced-order models are compared and employed in adaptive algorithms aiming at generating reduced-order models of vibro-acoustic systems. For many applications, vibro-acoustic systems need to be evaluated in a specific frequency range, as certain system characteristics are only observed in this region. Therefore, the employed error estimation methods are assessed regarding their effectiveness estimating the error in a particular frequency region.

Model order reduction using hyperreduction methods (DEIM, ECSW) for magnetodynamic FEM problems

Efficient Model Order Reduction Methods for the calculation of non-linear electromagnetic systems can be realized using hyperreduction strategies that are known from structural dynamics. Precalculated full field solution snapshots are used to build a reduction basis using a singular value decomposition. The non-linear behavior is further reduced by a weighting function on selected elements. This paper presents the application of the discrete empirical interpolation method (DEIM) and the energy-conserving sampling and weighting (ECSW) method to magnetodynamic FEM problems. The magnetodynamic FEM formulation is based on the modified vector-potential in combination with edge-elements. Applying the hyperreduction methods, a significant reduction of online computational time was achieved while preserving very good accuracy of the simulation.

An efficient model order reduction scheme for dynamic contact in linear elasticity

The paper proposes an approach for the efficient model order reduction of dynamic contact problems in linear elasticity. Instead of the augmented Lagrangian method that is widely used for mechanical contact problems, we prefer here the Linear Complementarity Programming (LCP) method as basic methodology. It has the advantage of resulting in the much smaller dual problem that is associated with the governing variational principle and that turns out to be beneficial for the model order reduction. Since the shape of the contact zone depends strongly on the acting outer forces, the LCP for the Lagrange multipliers has to be solved in each time step. The model order reduction scheme, on the other hand, is applied to the large linear system for the displacements and computed in advance by means of an Arnoldi process. In terms of computational effort the reduction scheme is very appealing because the contact constraints are fully satisfied while the reduction acts only on the displacements. As an extension of our approach, we furthermore take up the idea of the Craig-Bampton method in order to distinguish between interior nodes and the nodes in the contact zone. A careful performance analysis closes the paper.

Efficient Estimation of Electrical Machine Behavior by Model Order Reduction

The proper orthogonal decomposition (POD) is an efficient model order reduction method, which is frequently coupled with the discrete empirical interpolation method (DEIM) to solve nonlinear electromagnetic problems. A drawback of this method is that instabilities can occur related to the reduction operator of the nonlinear part. In this contribution, different DEIMs and the Gappy POD are employed and analyzed. Consecutively, the methods are employed to efficiently estimate the behavior of a permanent magnet synchronous machine in terms of global quantities, such as torque and iron losses.

Improving the generalized Bloch mode synthesis method using algebraic condensation

The generalized Bloch mode synthesis (GBMS) is an efficient model-order reduction technique to reduce the computational cost for band-structure calculations of periodic media. In this article, an attempt is made to improve the computational performance of the GBMS method by employing algebraic condensation (AC). The interior component of the unit cell is firstly partitioned into a set of small substructures automatically in the algebraic domain, and these substructures are then absorbed into boundary sets by using the improved reduced system (IRS) method. The boundary degrees of freedom (DOFs) are subsequently reduced via the enhanced Craig–Bampton (ECB) method and the local-level characteristic constraint mode reduction, respectively. The coupling residual modes of IRS and ECB processes are then compensated after imposing the Bloch boundary conditions. Through several numerical examples, the performance of the proposed method is demonstrated in terms of accuracy and computational cost. The influence of partitioned numbers on the computational efficiency of the proposed method is finally discussed. The numerical results show that the computational efficiency of the GBMS method can be greatly enhanced by using AC for the unit cells with relatively large DOFs or with dense meshes of the exterior boundary, which hence extends the application scope of the GBMS method.

Multiport Model Order Reduction With Multiple Expansion Points

Multiple expansion points are introduced to the multiport Cauer ladder network method to realize an efficient model order reduction of an eddy-current field. The proposed method can set multiple target frequencies to cover a wide frequency range with a small number of multiport electric network elements. A time-dependent response to pulsewidth-modulation (PWM) excitation is accurately reconstructed by setting the target frequencies at both fundamental and carrier frequencies.

Multi-Port Model Order Reduction Using a Matrix Cauer Ladder Network

To achieve efficient multi-port model order reduction, a multi-port Cauer ladder network (CLN) method is formulated that directly yields resistance and inductance matrices that constitute the network elements in the matrix Cauer form. The eddy-current field driven by multiple power sources is accurately reconstructed using a small number of network elements. The matrix Cauer form achieves faster convergence of the transfer function than a single-port CLN method and almost the same convergence as a block Pade via Lanczos (PVL) method.

A Reduced-Order Model for the Vibration Analysis of Mistuned Blade–Disc–Shaft Assembly

An effective reduced-order model is presented in this paper for the vibration analysis of a mistuned blade–disc–shaft assembly considering the flexibility of the shaft and the rotordynamic effects. For the sake of accurate modeling and quantitative analysis, three-dimensional (3D) finite element models were employed in obtaining the governing equations of motion with the Coriolis force, centrifugal stiffening, and spin softening effects taken into account. Then, an efficient model order reduction technique based on the coordinate projection by normal modes of tuned assembly and cyclic symmetry analysis was developed for mistuned blade–disc–shaft assembly. The criterion of whether one matrix could be incorporated in cyclic symmetry analysis is presented. During the modeling, the mistuning in blade and disc was taken into account and dealt with independently. In mistuning projection, the blade and disc parts were both projected onto their tuned counterparts of the sector model, where the boundary conditions were set to be fixed and free, respectively. Finally, an example of a blade–disc–shaft assembly was employed to validate the effectiveness of the presented method in free and forced vibration analysis.

An efficient model order reduction of frequency-dependent double panel systems

This paper treats a typical industrial vibroacoustic problem, consisting of a double panel under forced vibration which radiates to the far-field. A finite element model is proposed, where both structural links and porous material are represented as equivalent models. The addition of frequency-dependent elements, as mechanical links and porous material in the cavity, leads to a balance between airborne and structural-borne transmission through panels in the mid-frequency range, which demands a suitable model to represent these phenomena. In order to cope with such large-scale models, a dedicated parametric model order reduction is proposed. The frequency-dependent terms are separated from the system matrices, allowing an offline reduction of the system. The Second-Order Arnoldi procedure is employed to calculate the reduced bases at multiple frequencies, chosen by a rational interpolation scheme. The expansion frequencies are selected by a novel adaptive windowing algorithm, which aims at minimizing the number of frequencies within the frequency range. The proposed approach has shown to be effective in reducing significantly the computational time while maintaining precision compared to the direct method.

Efficient Time-Domain Sensitivity Analysis of Active Networks

The conventional sensitivity analysis based on model-order reduction (MOR) techniques guarantees the passivity and, consequently, the stability of the reduced sensitivity circuit provided that the original circuit is passive. This excludes a large class of circuits that are stable but not necessarily passive. In this article, an efficient MOR method is presented for the sensitivity analysis of active stable circuits. The proposed algorithm preserves the stability of both the original and associated sensitivity equations, and is based on reducing the first-order variational equations in the form of a set of stable multi-input differential equations. The sensitivity equations are decomposed into several subsystems of equations where each subsystem contains a dedicated cluster of inputs, thereby avoiding the significant increase in the size of the reduced model due to increasing the number of inputs.

Hybrid reduction of mistuned bladed disks for nonlinear forced response analysis with dry friction

An efficient model order reduction technique for forced vibration analysis of a mistuned bladed disk with friction contact is presented. The present method applicable for small blade-to-blade mistuning, provides significant reduction in time without loss of accuracy for the nonlinear statistical analysis of mistuned bladed disks. This is achieved by obtaining a reduced order model of the mistuned bladed disks without generating its stiffness and mass matrices and considering the effect of mistuning on the static modes used in the reduction. This new approach can be interpreted as a hybrid of two existing techniques: Component mode mistuning (CMM) and a Dual model order reduction technique. The Dual method is a free interface based Component Mode Synthesis (CMS) method whose unique formulation makes it appropriate for mistuning analysis. The reduction basis in Dual approach includes free interface normal modes and residual flexibility attachment modes of mistuned bladed disk. These modes are computed using CMM and a proposed approximate approach, respectively. The resulting governing equation are solved in the frequency domain using Harmonic Balance method while contact forces are computed by an Alternating Frequency–Time method and a node-to-node 3D coupled contact model. The method is applied to a simplified shrouded turbine bladed disk, the results demonstrate the accuracy of the proposed method and the importance of tackling friction nonlinearity and mistuning simultaneously.

Cauer Ladder Network With Multiple Expansion Points for Efficient Model Order Reduction of Eddy-Current Field

The Cauer ladder network (CLN) method provides an efficient representation of eddy-current fields. This paper proposes a reformulation of the CLN method that considers expansion points using the magnetic/current vector potential. To cover a wide range of frequencies, the expansion point may be changed at a later stage of the network. Only a few stages of the ladder network are required for the eddy-current field to be reconstructed accurately around the target frequency using the expansion points.

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

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

Stabilized principal interval decomposition method for model reduction of nonlinear convective systems with moving shocks

An efficient and reliable model order reduction of nonlinear systems poses a challenge for nonstationary problems with convective, non-periodic, and non-equilibrium dynamics. To that end, we put forth a localized basis selection strategy based on the proper orthogonal decomposition (POD) and principal interval decomposition (PID) to construct a stable reduced-order modeling framework to capture the unsteady dynamics of nonlinear systems effectively. The implementation of an eddy viscosity (EV) based closure model in POD–PID approach yields the proposed POD–PID–EV projection-based reduced-order modeling approach for nonlinear partial differential equations. Solving the nonlinear Burgers’ equation with various spatio-temporal dynamical complexities, it is shown that the present approach yields significant improvements in accuracy over the standard POD–Galerkin model with a negligibly small computational overhead. Furthermore, we show that strong moving discontinuities can be effectively captured in the low-dimensional space with the proposed approach.

Nonlinear dynamical behaviors of a complicated dual-rotor aero-engine with rub-impact

In this paper, the nonlinear dynamical behaviors of a complicated dual-rotor aero-engine with rub-impact are investigated. A novel framework is proposed, in which the sophisticated geometrical structure is considered by finite solid element method and efficient model order reduction is applied to the model. The validity and efficiency of the reduced-order model are verified through critical speed and eigen problems. Its stable and unstable solutions are calculated by means of the assembly technique and the multiple harmonic balance method combined with the alternating frequency/time domain technique (MHB–AFT). The accurate frequency amplitudes are obtained accordingly for each harmonic component. The stabilities of the solutions are checked by the Floquet theory. Through the numerical computations, some complex nonlinear phenomena, such as combined frequency vibration, hysteresis, and resonant peak shifting, are discovered when the rub-impact occurs. The results also show that the control parameters of mass eccentricity, rub-impact stiffness, and rotational speed ratio make significant but different influences on the dynamical characteristics of the system. Therefore, a key innovation of this paper is the marriage of a hybrid modeling method—accurate modeling technique combined with model order reduction and solution method—highly efficient semi-analytic method of MHB–AFT. The proposed framework is benefit for parametric study and provides a better understanding of the nonlinear dynamical behaviors of the real complicated dual-rotor aero-engine with rub-impact.

Model-order reduction of lumped parameter systems via fractional calculus

This study investigates the use of fractional order differential models to simulate the dynamic response of non-homogeneous discrete systems and to achieve efficient and accurate model order reduction. The traditional integer order approach to the simulation of non-homogeneous systems dictates the use of numerical solutions and often imposes stringent compromises between accuracy and computational performance. Fractional calculus provides an alternative approach where complex dynamical systems can be modeled with compact fractional equations that not only can still guarantee analytical solutions, but can also enable high levels of order reduction without compromising on accuracy. Different approaches are explored in order to transform the integer order model into a reduced order fractional model able to match the dynamic response of the initial system. Analytical and numerical results show that, under certain conditions, an exact match is possible and the resulting fractional differential models have both a complex and frequency-dependent order of the differential operator. The implications of this type of approach for both model order reduction and model synthesis are discussed.

Understanding and reducing complex systems pharmacology models based on a novel input-response index.

A growing understanding of complex processes in biology has led to large-scale mechanistic models of pharmacologically relevant processes. These models are increasingly used to study the response of the system to a given input or stimulus, e.g., after drug administration. Understanding the input-response relationship, however, is often a challenging task due to the complexity of the interactions between its constituents as well as the size of the models. An approach that quantifies the importance of the different constituents for a given input-output relationship and allows to reduce the dynamics to its essential features is therefore highly desirable. In this article, we present a novel state- and time-dependent quantity called the input-response index that quantifies the importance of state variables for a given input-response relationship at a particular time. It is based on the concept of time-bounded controllability and observability, and defined with respect to a reference dynamics. In application to the brown snake venom-fibrinogen (Fg) network, the input-response indices give insight into the coordinated action of specific coagulation factors and about those factors that contribute only little to the response. We demonstrate how the indices can be used to reduce large-scale models in a two-step procedure: (i) elimination of states whose dynamics have only minor impact on the input-response relationship, and (ii) proper lumping of the remaining (lower order) model. In application to the brown snake venom-fibrinogen network, this resulted in a reduction from 62 to 8 state variables in the first step, and a further reduction to 5 state variables in the second step. We further illustrate that the sequence, in which a recursive algorithm eliminates and/or lumps state variables, has an impact on the final reduced model. The input-response indices are particularly suited to determine an informed sequence, since they are based on the dynamics of the original system. In summary, the novel measure of importance provides a powerful tool for analysing the complex dynamics of large-scale systems and a means for very efficient model order reduction of nonlinear systems.