Parametric Model Order Reduction Method Research Articles

Towards System-Level Simulation of a Miniature Electromagnetic Energy Harvester Model

Energy harvesting, a solution to provide a lifetime power supply to wireless sensor nodes, has attracted widespread attention in the last two decades. An energy harvester collects ambient energy, e.g., solar, thermal, or vibration energy, and transforms it into electrical energy. In this work, we work on an electromagnetic energy harvester model, which is composed of four magnets oscillating along a coil. Such a device converts the vibrational energy into electrical energy. We reproduce the electromagnetic energy harvester model in finite element-based software. In order to include this model in a system-level simulation, the methodology of extracting a look-up table-based equivalent circuit model is presented. Such an equivalent circuit model enables the interaction of the electromagnetic energy harvester model with both electrical and mechanical compact models at the system-level. Furthermore, the matrix interpolation-based and algebraic parameterization-based parametric model order reduction methods are suggested for speeding up the generation of the equivalent circuit model and the design optimization process with respect to magnet dimensions. The efficiencies of these two methods are investigated and compared.

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.

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.

Stable Parametric Model Order Reduction of Piezoelectric Energy Harvester by Matrix Interpolation

In this work, we present a successful application of the matrix interpolation-based parametric model order reduction method for generating stable reduced-order models of a piezoelectric energy harvester device. In order to ensure the stability of these geometrically parameterized reduced order models, we have combined the above method with four stabilization approaches: MOR after Schur, MOR after implicit Schur, Schur after MOR, and multiphysics structure-preserving MOR. We apply the combined approaches directly to large-scale multi-physical finite element models enabling so, an efficient design optimization based on accurate numerical models.

Correction to: Semi-analytic contact technique in a non-linear parametric model order reduction method for gear simulations

Correction to: Semi-analytic contact technique in a non-linear parametric model order reduction method for gear simulations

Two-Step Predict and Correct Non-Intrusive Parametric Model Order Reduction for Changing Well Locations Using a Machine Learning Framework

The objective of this paper is to develop a two-step predict and correct non-intrusive Parametric Model Order Reduction (PMOR) methodology for the problem of changing well locations in an oil field that can eventually be used for well placement optimization to gain significant computational savings. In this work, we propose a two-step PMOR procedure, where, in the first step, a Proper Orthogonal Decomposition (POD)-based strategy that is non-intrusive to the simulator source code is introduced, as opposed to the convention of using POD as a simulator intrusive procedure. The non-intrusiveness of the proposed technique stems from formulating a novel Machine Learning (ML)-based framework used with POD. The features of the ML model (Random Forest was used here) are designed such that they take into consideration the temporal evolution of the state solutions and thereby avoid simulator access for the time dependency of the solutions. The proposed PMOR method is global, since a single reduced-order model can be used for all the well locations of interest in the reservoir. We address the major challenge of the explicit representation of the well location change as a parameter by introducing geometry-based features and flow diagnostics-inspired physics-based features. In the second step, an error correction model based on reduced model solutions is formulated to correct for discrepancies in the state solutions at well grid blocks expected from POD basis for new well locations. The error correction model proposed uses Artificial Neural Networks (ANNs) that consider the physics-based reduced model solutions as features, and is proved to reduce the error in QoI (Quantities of Interest), such as oil production rates and water cut, significantly. This workflow is applied to a simple homogeneous reservoir and a heterogeneous channelized reservoir using a section of SPE10 model that showed promising results in terms of model accuracy. Speed-ups of about 50×–100× were observed for different cases considered when running the test scenarios. The proposed workflow for Reduced-Order Modeling is “non-intrusive” and hence can increase its applicability to any simulator used. Additionally, the method is formulated such that all the simulation time steps are independent and hence can make use of parallel resources very efficiently and also avoid stability issues that can result from error accumulation over time steps.

Development of a parametric model order reduction method for laminated composite structures

This paper presents a parametric model order reduction (PMOR) method for laminated composite structures. The approach is applied to a lattice structure made of carbon fibre reinforced polymer (CFRP) beams. The global reduced order basis (ROB) used to apply the reduction is calculated with two different methods: the Modal-SVD (singular value decomposition) and the proper orthogonal decomposition (POD). Then, the reduced model is obtained by a Galerkin projection. The parametrization is carried out for the angle between the layers of the structure and for the thickness of the structure. The results show how the use of the reduced basis (RB) method to create reduced parametric models of composite structures is an efficient approach that allows to decrease the computational cost of the simulation.

Response statistics prediction for structures with uncertain attachments using a Krylov approach

This paper presents an approach to calculate the response statistics of a structure comprising a main, host structure with an attachment whose coupling location is not exactly known. The proposed approach is based on a Krylov model order reduction method, which reduces the order of a finite element model of the structure enabling an efficient way to estimate the statistics of the structural response. The multi-point constraint method is used to connect the finite element models of the attachment and the structure. One (or more) analysis based on Chebyshev nodes is performed for the attachment located at a given position(s) to obtain the local projection matrices. Afterwards, a Parametric Model Order Reduction method is applied to reduce the model order. Uncertainty in the location of the attachment is included in a Monte Carlo simulation. The method is illustrated using two numerical examples to demonstrate its accuracy and efficiency.

Fast Uncertainty Quantification in Low Frequency Electromagnetic Problems by an Integral Equation Method Based on Hierarchical Matrix Compression

A parametric model order reduction method combined with a polynomial spectral approximation is applied for the first time to a Volume Integral Equation method accelerated by a low-rank matrix compression technique. Such an approach allows for drastically reducing the computational cost required by uncertainty quantifications in electromagnetic problems. Moreover, the proposed numerical tool can be adopted for computing stochastic information (e.g. mean, variance, probability density function) of any electromagnetic quantity of interest, in order to test the reliability of industrial devices with uncertainties on the material parameters. Conductive, dielectric, and magnetic media which exhibit uncorrelated and correlated random material parameters are considered by the proposed method.

Interpolatory model reduction for quadratic-bilinear systems using error estimators

Purpose The purpose of this paper is to propose an interpolation-based projection framework for model reduction of quadratic-bilinear systems. The approach constructs projection matrices from the bilinear part of the original quadratic-bilinear descriptor system and uses these matrices to project the original system. Design/methodology/approach The projection matrices are constructed by viewing the bilinear system as a linear parametric system, where the input associated with the bilinear part is treated as a parameter. The advantage of this approach is that the projection matrices can be constructed reliably by using an a posteriori error bound for linear parametric systems. The use of the error bound allows us to select a good choice of interpolation points and parameter samples for the construction of the projection matrices by using a greedy-type framework. Findings The results are compared with the standard quadratic-bilinear projection methods and it is observed that the approximations through the proposed method are comparable to the standard method but at a lower computational cost (offline time). Originality/value In addition to the proposed model order reduction framework, the authors extend the one-sided moment matching parametric model order reduction (PMOR) method to a two-sided method that doubles the number of moments matched in the PMOR method.

Semi-analytic contact technique in a non-linear parametric model order reduction method for gear simulations

In this work we present a novel method for the solution of gear contact problems in flexible multi-body. These problems are characterized by significant variation in the location and size of the contact area, typically requiring a high number of degrees of freedom to correctly capture deformation and stress fields. Therefore fully dynamic simulation is computationally prohibitive. To overcome these limitations, we exploit a combined analytic-numerical contact model within a parametric model order reduction (PMOR) scheme. The reduction space consists of a truncated set of eigenvectors augmented with a parameter dependent set of residual static shape vectors. Each static shape is computed by interpolating among a set of displacement modes of the interacting bodies, obtained from a series of precomputed static contact analyses. During the contact analyses, an analytic model based on the Hertz theory describes the teeth local deformation. We implement the proposed method in an in-house code and we apply it to spur and helical gears dynamic contact analyses. We compare the results with classical PMOR schemes highlighting how the combined use of the semi-analytic contact model allows to decrease further the model complexity as well as the computational burden, for both static and dynamic cases. Finally, we validate the methodology by means of a comparison with experimental data found in literature, showing that the numerical method is able to capture quantitatively the static transmission error measurements in case of both helical and spur geared transmission for different torque levels.

An index-aware parametric model order reduction method for parameterized quadratic differential–algebraic equations

Modeling of sophisticated applications, such as coupled problems arising from nanoelectronics can lead to quadratic differential algebraic equations (DAEs). The quadratic DAEs may also be parameterized, due to variations in material properties, system configurations, etc., and they are usually subject to multi-query tasks, such as optimization, or uncertainty quantification. Model order reduction (MOR), specifically parametric model order reduction (pMOR), is known as a useful tool for accelerating the simulations in a multi-query context. However, pMOR dedicated to this particular structure, has not yet been systematically studied. Directly applying the existing pMOR methods may produce parametric reduced-order models (pROMs) which are less accurate, or may be very difficult to simulate. The same problem was already observed for linear DAEs, and could be eliminated by introducing splitting MOR techniques such as the index-aware MOR (IMOR) methods. We extend the IMOR methods to parameterized quadratic DAEs, thereby producing accurate and easy to simulate index-aware parametric reduced-order models (IpROMs). The proposed approach is so far limited to index-1 one-way coupled problems, but these often appear in computational nanoelectronics. We illustrate the performance of the new approach using industrial models for nanoelectronic structures.

Simulation of Moving Loads in Elastic Multibody Systems With Parametric Model Reduction Techniques

Abstract In elastic multibody systems, one considers large nonlinear rigid body motion and small elastic deformations. In a rising number of applications, e.g. automotive engineering, turning and milling processes, the position of acting forces on the elastic body varies. The necessary model order reduction to enable efficient simulations requires the determination of ansatz functions, which depend on the moving force position. For a large number of possible interaction points, the size of the reduced system would increase drastically in the classical Component Mode Synthesis framework. If many nodes are potentially loaded, or the contact area is not known a-priori and only a small number of nodes is loaded simultaneously, the system is described in this contribution with the parameter-dependent force position. This enables the application of parametric model order reduction methods. Here, two techniques based on matrix interpolation are described which transform individually reduced systems and allow the interpolation of the reduced system matrices to determine reduced systems for any force position. The online-offline decomposition and description of the force distribution onto the reduced elastic body are presented in this contribution. The proposed framework enables the simulation of elastic multibody systems with moving loads efficiently because it solely depends on the size of the reduced system. Results in frequency and time domain for the simulation of a thin-walled cylinder with a moving load illustrate the applicability of the proposed method.

Passivity-Preserving Parameterized Model Order Reduction Using Singular Values and Matrix Interpolation

We present a parameterized model order reduction method based on singular values and matrix interpolation. First, a fast technique using grammians is utilized to estimate the reduced order, and then common projection matrices are used to build parameterized reduced order models (ROMs). The design space is divided into cells, and a Krylov subspace is computed for each cell vertex model. The truncation of the singular values of the merged Krylov subspaces from the models located at the vertices of each cell yields a common projection matrix per design space cell. Finally, the reduced system matrices are interpolated using positive interpolation schemes to obtain a guaranteed passive parameterized ROM. Pertinent numerical results validate the proposed technique.

Interpolation-Based Parameterized Model Order Reduction of Delayed Systems

Three-dimensional electromagnetic methods are fundamental tools for the analysis and design of high-speed systems. These methods often generate large systems of equations, and model order reduction (MOR) methods are used to reduce such a high complexity. When the geometric dimensions become electrically large or signal waveform rise times decrease, time delays must be included in the modeling. Design space optimization and exploration are usually performed during a typical design process that consequently requires repeated simulations for different design parameter values. Efficient performing of these design activities calls for parameterized model order reduction (PMOR) methods, which are able to reduce large systems of equations with respect to frequency and other design parameters of the circuit, such as layout or substrate features. We propose a novel PMOR method for neutral delayed differential systems, which is based on an efficient and reliable combination of univariate model order reduction methods, a procedure to find scaling and frequency shifting coefficients and positive interpolation schemes. The proposed scaling and frequency shifting coefficients enhance and improve the modeling capability of standard positive interpolation schemes and allow accurate modeling of highly dynamic systems with a limited amount of initial univariate models in the design space. The proposed method is able to provide parameterized reduced order models passive by construction over the design space of interest. Pertinent numerical examples validate the proposed PMOR approach.

Robust Passivity Preserving Parametric Model Order Reduction using Matrix Interpolation

We present a novel robust parametric model order reduction method using matrix interpolation and common projection matrix. The design space is sampled over an estimation grid and transformation matrices are computed at the estimation points using a projection-based passivity preserving truncated balanced realization. A common projection matrix is obtained by the truncation of the singular values of the merged transformation matrices obtained from the estimation points in the design space. Finally, the reduced system matrices are interpolated using positive interpolation schemes to obtain a guaranteed passive parametric reduced model. The proposed technique is validated using pertinent numerical results.

EMPIRE: An Efficient and Compact Multiple-Parameterized Model-Order Reduction Method for Physical Optimization

Parameterized model-order reduction is useful for very large-scale integration VLSI physical design and optimization. In this paper, we propose an efficient yet accurate parameterized model-order reduction method EMPIRE for multiple parameters. It uses implicit moment matching to efficiently handle high-order moments of a large number of parameters. In addition, it can match the moments of different parameters with different accuracy according to their influence on the objective under study, and such influence is measured by the 2-norm of their coefficient matrix in the canonical form. It develops three algorithms to further suppress the size of the reduced model by finding a projection matrix that has a much smaller number of columns than the original one. Experimental results show that compared with the best existing algorithm CORE that uses explicit moment matching for the parameters, EMPIRE reduces waveform error by 47.8 × at a similar runtime.

Parametric model order reduction for squeeze film damping in perforated microstructures

In this paper, a parametric model order reduction (PMOR) method was presented to firstly extract the macromodels of squeeze film damping in perforated microstructures. The various design variables such as flow impedance of perforations, viscosity, ambient pressure, plate thickness and gap separation were preserved in the model order reduction (MOR) process. Thus when these parameters change, the extracted parametric macromodels could be reused parametrically instead of repeating the time-consuming MOR process in the traditional way. A micromachined capacitive accelerometer was used to demonstrate the feasibility and efficiency of the PMOR method for the squeeze film damping in perforated microstructures. The maximum relative error is less than 0.3% while the computational efficiency improves about 11 times compared with finite element method.