scholarly journals A Block Arnoldi Algorithm Based Reduced-Order Model Applied to Large-Scale Algebraic Equations of a 3-D Field Problem

2021 ◽  
Vol 11 (20) ◽  
pp. 9435
Author(s):  
Ning Wang ◽  
Jiajia Chen ◽  
Huifang Wang ◽  
Shiyou Yang
Keyword(s):  
Temperature Field ◽  
Reduction Process ◽  
Reduced Order Model ◽  
Error Estimator ◽  
Transient Field ◽  
Order Model ◽  
Field Problem ◽  
Reduced Order ◽  
Arnoldi Algorithm ◽  
Block Arnoldi

In simulations of three-dimensional transient physics filled through a numerical approach, the order of the equation set of high-fidelity models is extremely high. To eliminate the large dimension of equations, a model order reduction (MOR) technique is introduced. In the existing MOR methods, the block Arnoldi algorithm-based MOR method is numerically stable, achieving a passively reduced order model. Nevertheless, this method performs poorly when it is applied to very wide-frequency transients. To eliminate this deficiency, multipoint MOR methods are emerging. However, it is hard to directly apply an existing multipoint MOR method to a 3-D transient field equation set. The implementation issues in a reduction process (such as the selection of expansion points, the number of moments matched at a point and the error bound) have not been explored in detail. In this respect, an adaptive multipoint model reduction model based on the Arnoldi algorithm is proposed to obtain the reduced-order models of a 3-D temperature field. The originality of this study is the proposal of a novel adaptive algorithm for selecting expansion points, matching moments automatically, using a posterior-error estimator based on temperature response coupled with a network topological method (NTM). The computational efficiency and accuracy of the proposed method are evaluated by the numerical results from solving the temperature field of a prototype insulated-gate bipolar transistor (IGBT).

A REDUCED ORDER MODEL BASED ON BLOCK ARNOLDI METHOD FOR AEROELASTIC SYSTEM

2014 ◽  
Vol 06 (06) ◽  
pp. 1450069 ◽  
Author(s):  
QIANG ZHOU ◽  
GANG CHEN ◽  
YUEMING LI
Keyword(s):  
Original System ◽  
Reduced Order Model ◽  
System Stability ◽  
Arnoldi Method ◽  
Order Model ◽  
Aeroelastic System ◽  
Reduced Order ◽  
Block Arnoldi ◽  
Flutter Boundary ◽  
Linearized Model

A reduced-order model (ROM) based on block Arnoldi algorithm to quickly predict flutter boundary of aeroelastic system is investigated. First, a mass–damper–spring dynamic system is tested, which shows that the low dimension system produced by the block Arnoldi method can keep a good dynamic property with the original system in low and high frequencies. Then a two-degree of freedom transonic nonlinear aerofoil aeroelastic system is used to validate the suitability of the block Arnoldi method in flutter prediction analysis. In the aerofoil case, the ROM based on a linearized model is obtained through a high-fidelity nonlinear computational fluid dynamics (CFD) calculation. The order of the reduced model is only 8 while it still has nearly the same accuracy as the full 9600-order model. Compared with the proper orthogonal decomposition (POD) method, the results show that, without snapshots the block Arnoldi/ROM has a unique superiority by maintaining the system stability aspect. The flutter boundary of the aeroelastic system predicted by the block Arnoldi/ROM agrees well with the CFD and reference results. The Arnoldi/ROM provides an efficient and convenient tool to quick analyze the system stability of nonlinear transonic aeroelastic systems.

scholarly journals A COMPARISON OF ONE- AND TWO-SIDED KRYLOV–ARNOLDI PROJECTION METHODS FOR FULLY COUPLED, DAMPED STRUCTURAL-ACOUSTIC ANALYSIS

2013 ◽  
Vol 21 (02) ◽  
pp. 1350004 ◽  
Author(s):  
R. SRINIVASAN PURI ◽  
DENISE MORREY
Keyword(s):  
Coupled System ◽  
Dimensional Subspace ◽  
Second Order ◽  
Reduced Order Model ◽  
Order Model ◽  
Reduced Order ◽  
Frequency Moments ◽  
Arnoldi Algorithm ◽  
Higher Dimensional ◽  
Fully Coupled

The two-sided second-order Arnoldi algorithm is used to generate a reduced order model of two test cases of fully coupled, acoustic interior cavities, backed by flexible structural systems with damping. The reduced order model is obtained by applying a Galerkin–Petrov projection of the coupled system matrices, from a higher dimensional subspace to a lower dimensional subspace, whilst preserving the low frequency moments of the coupled system. The basis vectors for projection are computed efficiently using a two-sided second-order Arnoldi algorithm, which generates an orthogonal basis for the second-order Krylov subspace containing moments of the original higher dimensional system. The first model is an ABAQUS benchmark problem: a 2D, point loaded, water filled cavity. The second model is a cylindrical air-filled cavity, with clamped ends and a load normal to its curved surface. The computational efficiency, error and convergence are analyzed, and the two-sided second-order Arnoldi method shows better efficiency and performance than the one-sided Arnoldi technique, whilst also preserving the second-order structure of the original problem.

scholarly journals Computer Vision with Error Estimation for Reduced Order Modeling of Macroscopic Mechanical Tests

Complexity ◽  
2018 ◽  
Vol 2018 ◽  
pp. 1-10 ◽  
Author(s):  
Franck Nguyen ◽  
Selim M. Barhli ◽  
Daniel Pino Muñoz ◽  
David Ryckelynck
Keyword(s):  
Neural Network ◽  
Computer Vision ◽  
Convolutional Neural Network ◽  
Digital Image ◽  
Mechanical Test ◽  
Reduced Order Model ◽  
Error Estimator ◽  
Reduced Order Models ◽  
Order Model ◽  
Reduced Order

In this paper, computer vision enables recommending a reduced order model for fast stress prediction according to various possible loading environments. This approach is applied on a macroscopic part by using a digital image of a mechanical test. We propose a hybrid approach that simultaneously exploits a data-driven model and a physics-based model, in mechanics of materials. During a machine learning stage, a classification of possible reduced order models is obtained through a clustering of loading environments by using simulation data. The recognition of the suitable reduced order model is performed via a convolutional neural network (CNN) applied to a digital image of the mechanical test. The CNN recommend a convenient mechanical model available in a dictionary of reduced order models. The output of the convolutional neural network being a model, an error estimator, is proposed to assess the accuracy of this output. This article details simple algorithmic choices that allowed a realistic mechanical modeling via computer vision.

PARTICLE SWARM OPTIMIZATION FOR MOR OF SINGULARLY PERTURBED SYSTEMS WITH CRITICAL FREQUENCY PRESERVATION AND APPLICATION TO POWER SYSTEMS SIMPLIFIED MODELING

2014 ◽  
Vol 23 (05) ◽  
pp. 1450073 ◽  
Author(s):  
OTHMAN M. K. ALSMADI ◽  
DIA I. ABU-AL-NADI ◽  
ZAER. S. ABO-HAMMOUR
Keyword(s):  
Particle Swarm Optimization ◽  
Critical Frequency ◽  
Mimo Systems ◽  
Particle Swarm ◽  
Reduction Process ◽  
Reduced Order Model ◽  
Reduced Model ◽  
Order Model ◽  
Swarm Optimization ◽  
Reduced Order

In this paper, a model-order reduction (MOR) technique with the advantage of critical frequency preservation capability is proposed using the particle swarm optimization (PSO). The new approach is capable of simplifying single-input single-output (SISO) systems as well as multi-input multi-output (MIMO) systems. If critical frequency preservation is desired, then this objective is achieved by retaining the exact critical frequencies of the full-order model as a subset in the reduced-order model. Otherwise, the reduction process is proceeded without such restriction. The reduction process is performed using the PSO technique to determine all of the necessary parameters in the reduced model. Determining the reduced-order model is performed based on minimizing the mean square error between the outputs of the original full-order model and the outputs of the reduced model. For method evaluation and validation, the proposed technique was applied to different models and compared with some of the well-known methods and recently published work for MOR. Results' comparison shows clearly the superiority of the proposed technique in terms of quality performance and accuracy of substructure preservation.

scholarly journals On error estimation for reduced-order modeling of linear non-parametric and parametric systems

2021 ◽  
Author(s):  
Lihong Feng ◽  
Peter Benner
Keyword(s):  
Reduced Order Modeling ◽  
Reduced Order Model ◽  
Error Estimator ◽  
Linear Dynamical System ◽  
Order Model ◽  
Reduced Order ◽  
Error Estimators ◽  
Theoretical Results ◽  
Parametric Systems ◽  
Better Than

Motivated by a recently proposed error estimator for the transfer function of the reduced-order model of a given linear dynamical system, we further develop more theoretical results in this work. Moreover, we propose several variants of the error estimator, and compare those variants with the existing ones both theoretically and numerically. It is shown that some of the proposed error estimators perform better than or equally well as the existing ones. All the error estimators considered can be easily extended to estimate the output error of reduced-order modeling for steady linear parametric systems.

scholarly journals Sampling-free parametric model reduction of systems with structured parameter variation

2018 ◽  
Author(s):  
Christopher Beattie ◽  
Serkan Gugercin ◽  
Zoran Tomljanović
Keyword(s):  
Model Reduction ◽  
Transfer Functions ◽  
Linear Time ◽  
Mass Matrix ◽  
Reduction Process ◽  
Parametric Model ◽  
Reduced Order Model ◽  
Computationally Efficient ◽  
Order Model ◽  
Reduced Order

We consider a parametric linear time invariant dynamical systems represented in state-space form as $$E \dot x(t) = A(p) x(t) + Bu(t), \\ y(t) = Cx(t),$$ where $E, A(p) \in \mathbb{R}^{n\times n}$, $B\in \mathbb{R}^{n\times m} $ and $C\in \mathbb{R}^{l\times n}$. Here $x(t)\in \mathbb{R}^{n} $ denotes the state variable, while $u(t)\in \mathbb{R}^{m}$ and $y(t)\in \mathbb{R}^{l}$ represent, respectively, the inputs and outputs of the system. We assume that $A(p)$ depends on $k\ll n$ parameters $p=(p_1, p_2, \ldots, p_k)$ such that we may write $$A(p)=A_0+U\,\diag (p_1, p_2, \ldots, p_k)V^T,$$ where $U, V \in \mathbb{R}^{n\times k}$ are given fixed matrices.We propose an approach for approximating the full-order transfer function $H(s;p)=C(s E -A(p))^{-1}B$ with a reduced-order model that retains the structure of parametric dependence and (typically) offers uniformly high fidelity across the full parameter range. Remarkably, the proposed reduction process removes the need for parameter sampling and thus does not depend on identifying particular parameter values of interest. Our approach is based on the classic Sherman-Morrison-Woodbury formula and allows us to construct a parameterized reduced order model from transfer functions of four subsystems that do not depend on parameters, allowing one to apply well-established model reduction techniques for non-parametric systems. The overall process is well suited for computationally efficient parameter optimization and the study of important system properties. One of the main applications of our approach is for damping optimization: we consider a vibrational system described by $$ \begin{equation}\label{MDK} \begin{array}{rl} M\ddot q(t)+(C_{int} + C_{ext})\dot q(t)+Kq(t)&=E w(t),\\ z(t)&=Hq(t), \end{array} \end{equation} $$ where the mass matrix, $M$, and stiffness matrix, $K$, are real, symmetric positive-definite matrices of order $n$. Here, $q(t)$ is a vector of displacements and rotations, while $ w(t) $ and $z(t) $ represent, respectively, the inputs (typically viewed as potentially disruptive) and outputs of the system. Damping in the structure is modeled as viscous damping determined by $C_{int} + C_{ext}$ where $C_{int}$ and $C_{ext}$ represent contributions from internal and external damping, respectively. Information regarding damper geometry and positioning as well as the corresponding damping viscosities are encoded in $C_{ext}= U\diag{(p_1, p_2, \ldots, p_k)} U^T$ where $U \in \mathbb{R}^{n\times k}$ determines the placement and geometry of the external dampers. The main problem is to determine the best damping matrix that is able to minimize influence of the disturbances, $w$, on the output of the system $z$. We use a minimization criteria based on the $\mathcal{H}_2$ system norm. In realistic settings, damping optimization is a very demanding problem. We find that the parametric model reduction approach described here offers a new tool with significant advantages for the efficient optimization of damping in such problems.

scholarly journals Adaptive POD-DEIM model reduction based on an improved error estimator

2018 ◽  
Author(s):  
Sridhar Chellappa ◽  
Lihong Feng ◽  
Peter Benner
Keyword(s):  
Dynamical Systems ◽  
Model Reduction ◽  
Nonlinear Model ◽  
Reduced Order Model ◽  
Error Estimator ◽  
Order Model ◽  
Reduced Order ◽  
Nonlinear Model Reduction ◽  
Nonlinear Approximations ◽  
Output Error

For reliable, efficient, rapid simulations of dynamical systems, a reduced order model (ROM) with certified accuracy is highly desirable. The ROM is derived from a full order model (FOM) through model reduction. In this work, we propose an adaptive approach for nonlinear model reduction by making use of a suitable output error estimator, thus enable the generation of a compact ROM, with appropriate balance between the state and nonlinear approximations.

A Refined Arnoldi Algorithm Based Krylov Subspace Technique for MEMS Model Order Reduction

2012 ◽  
Vol 503 ◽  
pp. 260-265
Author(s):  
Le Guan ◽  
Jia Li Gao ◽  
Zhi Wen Wang ◽  
Guo Qing Zhang ◽  
Jin Kui Chu
Keyword(s):  
Model Order Reduction ◽  
Krylov Subspace ◽  
Order Reduction ◽  
Original System ◽  
Reduced Order Model ◽  
System Matrix ◽  
Order Model ◽  
Model Order ◽  
Reduced Order ◽  
Arnoldi Algorithm

A refined approach producing MEMS numerical macromodels is proposed in this paper by generating the iterative Krylov subspace using a refined Arnoldi algorithm, which can reduce the degrees of freedom of the original system equations described by the state space method. Projection of the original system matrix onto the Krylov subspace which is spanned by a refined Arnoldi algorithm is still based on the transfer function moment matching principle. The idea of the iterative version is to expect that a new initial vector will contain more and more information on the required eigenvectors that is called refined vector. The refined approach improves approximation accuracy of the system matrix eigenvalues equivalent to a more accurate approximation to the poles of the system transfer function, obtaining a more accurate reduced-order model. The clamped beam model and the FOM model are reduced order by classical Arnoldi and refined Arnoldi algorithm in numerical experiments. From the computing result it is concluded that the refined Arnoldi algorithm based Krylov subspace technique for MEMS model order reduction has more accuracy and reaches lower order number of reduced order model than the classical Arnoldi process.

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