Model Order Reduction Methods for Discrete Systems via Discrete Pulse Orthogonal Functions

This paper explores model order reduction(MOR)methods for discrete linear and discrete bilinear systems via discrete pulse orthogonal functions(DPOFs).Firstly,the discrete linear systems and the discrete bilinear systems are expanded in the space spanned by DPOFs,and two recurrence formulas for the expansion coefficients of the system’s state variables are obtained.Then,a modified Arnoldi process is applied to both recurrence formulas to construct the orthogonal projection matrices,by which the reduced-order systems are obtained.Theoretical analysis shows that the output variables of the reducedorder systems can match a certain number of the expansion coefficients of the original system’s output variables.Finally,two numerical examples illustrate the feasibility and effectiveness of the proposed methods.

Routh Order Reduction Method of Relativistic Birkhoffian Systems

Routh order reduction method of the relativistic Birkhoffian equations is studied. For a relativistic Birkhoffian system, the cyclic integrals can be found by using the perfect differential method. Through these cyclic integrals, the order of the system can be reduced. If the relativistic Birkhoffian system has a cyclic integral, then the Birkhoffian equations can be reduced at least by two degrees and the Birkhoffian form can be kept. The relations among the relativistic Birkhoffian mechanics, the relativistic Hamiltonian mechanics, and the relativistic Lagrangian mechanics are discussed, and the Routh order reduction method of the relativistic Lagrangian system is obtained. And an example is given to illustrate the application of the result.

A Real-time Cutting Model Based on Finite Element and Order Reduction

Telemedicine plays an important role in Corona Virus Disease 2019(COVID-19).The virtual surgery simulation system,as a key component in telemedicine,requires to compute in real-time.Therefore,this paper proposes a realtime cutting model based on finite element and order reduction method,which improves the computational speed and ensure the real-time performance.The proposed model uses the finite element model to construct a deformation model of the virtual lung.Meanwhile,a model order reduction method combining proper orthogonal decomposition and Galerkin projection is employed to reduce the amount of deformation computation.In addition,the cutting path is formed according to the collision intersection position of the surgical instrument and the lesion area of the virtual lung.Then,the Bezier curve is adopted to draw the incision outline after the virtual lung has been cut.Finally,the simulation system is set up on the PHANTOM OMNI force haptic feedback device to realize the cutting simulation of the virtual lung.Experimental results show that the proposed model can enhance the real-time performance of telemedicine,reduce the complexity of the cutting simulation and make the incision smoother and more natural.

An Adaptive Substructure-Based Model Order Reduction Method for Nonlinear Seismic Analysis in OpenSees

Structural components may enter an initial-elastic state,a plastic-hardening state and a residual-elastic state during strong seismic excitations.In the residual-elastic state,structural components keep in an unloading/reloading stage that is dominated by a tangent stiffness,thus structural components remain residual deformations but behave in an elastic manner.It has a great potential to make model order reduction for such structural components using the tangent-stiffness-based vibration modes as a reduced order basis.In this paper,an adaptive substructure-based model order reduction method is developed to perform nonlinear seismic analysis for structures that have a priori unknown damage distribution.This method is able to generate time-varying substructures and make nonlinear model order reduction for substructures in the residual-elastic phase.The finite element program OpenSees has been extended to provide the adaptive substructure-based nonlinear seismic analysis.At the low level of OpenSees framework,a new abstract layer is created to represent the time-varying substructures and implement the modeling process of substructures.At the high level of OpenSees framework,a new transient analysis class is created to implement the solving process of substructure-based governing equations.Compared with the conventional time step integration method,the adaptive substructure-based model order reduction method can yield comparative results with a higher computational efficiency.

Advances of Model Order Reduction Research in Large-scale System Simulation

Model Order Reduction (MOR) plays more and more imp or tant role in complex system simulation, design and control recently. For example , for the large-size space structures, VLSI and MEMS (Micro-ElectroMechanical Systems) etc., in order to shorten the development cost, increase the system co ntrolling accuracy and reduce the complexity of controllers, the reduced order model must be constructed. Even in Virtual Reality (VR), the simulation and d isplay must be in real-time, the model order must be reduced too. The recent advances of MOR research are overviewed in the article. The MOR theor y and methods may be classified as Singular Value decomposition (SVD) based, the Krylov subspace based and others. The merits and demerits of the different meth ods are analyzed, and the existed problems are pointed out. Moreover, the applic ation’s fields are overviewed, and the potential applications are forecaste d. After the existed problems analyzed, the future work is described. There are som e problems in the traditional methods such as SVD and Krylov subspace, they are that it’s difficult to (1)guarantee the stability of the original system, (2) b e adaptive to nonlinear system, and (3) control the modeling accuracy. The f uture works may be solving the above problems on the foundation of the tradition al methods, and applying other methods such as wavelet or signal compression.

Model Order Reduction for Coupled Dynamic Characterization of Torsional Micromirrors

Numerical solutions could not perform rapid system-level simulation of the behavior of micro-electro-mechanical systems（MEMS） and analytic solutions for the describing partial differential equations are only available for simple geometries.Model order reduction（MOR） can extract approximate low-order model from the original large scale system.Conventional model order reduction algorithm is based on first-order system model,however,most structure mechanical MEMS systems are naturally second-order in time.For the purpose of solving the above problem,a direct second-order system model order reduction approach based on Krylov subspace projection for the coupled dynamic study of electrostatic torsional micromirrors is presented.The block Arnoldi process is applied to create the orthonormal vectors to construct the projection matrix,which enables the extraction of the low order model from the discretized system assembled through finite element analysis.The transfer functions of the reduced order model and the original model are expanded to demonstrate the moment-matching property of the second-order model reduction algorithm.The torsion and bending effect are included in the finite element model,and the squeeze film damping effect is considered as well.An empirical method considering relative error convergence is adopted to obtain the optimal choice of the order for the reduced model.A comparison research between the full model and the reduced model is carried out.The modeling accuracy and computation efficiency of the presented second-order model reduction method are confirmed by the comparison research results.The research provides references for MOR of MEMS.

State space solution to 3D multilayered elastic soils based on order reduction method

Starting with the governing equations in terms of displacements of 3D elastic media, the solutions to displacement components and their first derivatives are obtained by the application of a double Fourier transform and an order reduction method based on the Cayley-Hamilton theorem. Combining the solutions and the constitutive equations which connect the displacements and stresses, the transfer matrix of a single soil layer is acquired. Then, the state space solution to multilayered elastic soils is further obtained by introducing the boundary conditions and continuity conditions between adjacent soil layers. The numerical analysis based on the present theory is carried out, and the vertical displacements of multilayered foundation with a weak and a hard underlying stratums are compared and discussed.

Model Order Reduction of Complex Airframes Using Component Mode Synthesis for Dynamic Aeroelasticity Load Analysis

Airframe structural optimization at different design stages results in new mass and stiffness distributions which modify the critical design loads envelop. Determination of aircraft critical loads is an extensive analysis procedure which involves simulating the aircraft at thousands of load cases as defmed in the certification requirements. It is computationally prohibitive to use a GFEM （Global Finite Element Model） for the load analysis, hence reduced order structural models are required which closely represent the dynamic characteristics of the GFEM. This paper presents the implementation of CMS （Component Mode Synthesis） method for the generation of high fidelity ROM （Reduced Order Model） of complex airframes. Here, sub-structuring technique is used to divide the complex higher order airframe dynamical system into a set of subsystems. Each subsystem is reduced to fewer degrees of freedom using matrix projection onto a carefully chosen reduced order basis subspace. The reduced structural matrices are assembled for all the subsystems through interface coupling and the dynamic response of the total system is solved. The CMS method is employed to develop the ROM of a Bombardier Aerospace business jet which is coupled with aerodynamic model for dynamic aeroelasticity loads analysis under gust turbulence. Another set of dynamic aeroelastic loads is also generated employing a stick model of same aircraft. Stick model is the reduced order modelling methodology commonly used in the aerospace industry based on stiffness generation by unitary loading application. The extracted aeroelastic loads from both models are compared against those generated employing the GFEM. Critical loads modal participation factors and modal characteristics of the different ROMs are investigated and compared against those of the GFEM. Results obtained show that the ROM generated using Craig Bampton CMS reduction process has a superior dynamic characteristics compared to the stick model.

Projection-Based Dimensional Reduction of Adaptively Refined Nonlinear Models

Adaptive mesh refinement (AMR) is fairly practiced in the context of high-dimensional, mesh-based computational models. However, it is in its infancy in that of low-dimensional, generalized-coordinate-based computational models such as projection-based reduced-order models. This paper presents a complete framework for projection-based model order reduction (PMOR) of nonlinear problems in the presence of AMR that builds on elements from existing methods and augments them with critical new contributions. In particular, it proposes an analytical algorithm for computing a pseudo-meshless inner product between adapted solution snapshots for the purpose of clustering and PMOR. It exploits hyperreduction—specifically, the energy-conserving sampling and weighting hyperreduction method—to deliver for nonlinear and/or parametric problems the desired computational gains. Most importantly, the proposed framework for PMOR in the presence of AMR capitalizes on the concept of state-local reduced-order bases to make the most of the notion of a supermesh, while achieving computational tractability. Its features are illustrated with CFD applications grounded in AMR and its significance is demonstrated by the reported wall-clock speedup factors.

Second-order difference scheme for a nonlinear model of wood drying process

A numerical simulation for a model of wood drying process is considered. The model is given by a couple of nonlinear differential equations. One is a nonlinear parabolic equation and the other one is a nonlinear ordinary equation. A difference scheme is derived by the method of reduction of order. First, a new variable is introduced and the original problem is rewritten into a system of the first-order differential equations. Secondly, a difference scheme is constructed for the later problem. The solvability, stability and convergence of the difference scheme are proved by the energy method. The convergence order of the difference scheme is secondorder both in time and in space. A prior error estimate is put forward. The new variable is put aside to reduce the computational cost. A numerical example testifies the theoretical result.

Sliding Mode Control Design via Reduced Order Model Approach

This paper presents a design of continuous-time sliding mode control for the higher order systems via reduced order model. It is shown that a continuous-time sliding mode control designed for the reduced order model gives similar performance for thc higher order system. The method is illustrated by numerical examples. The paper also introduces a technique for design of a sliding surface such that the system satisfies a cost-optimality condition when on the sliding surface.

市场主体登记秩序法益的刑法保护

市场主体登记制度兼具公私法属性,所蕴含的秩序法益存在一元论与二元论之争。市场主体登记秩序相对于个人法益而言具有独立性,秩序法益保护具有效率、自由和安全的多元价值。秩序法益具有复杂性和抽象性,存在与个人法益并存或单独存在的类型形态,有必要运用法益还原论予以还原。基于类型化思维,根据秩序法益能否还原、是否必须还原、是否已还原为个人法益的不同情况,判断其应否成为刑法保护的法益、是否将个人法益所受的实际侵害作为具体定罪标准。市场主体虚假登记主要涉及“两虚一逃”犯罪,在认缴登记制改革下,“两虚一逃”呈现出罪化趋势,但仍存在入罪的现实必要性;实践中应针对其所侵犯的秩序法益的不同类型形态予以刑事责任认定。

还原论抑或加工论:深度学习的根基反思与理论走向

深度学习是学习者透过表象、超越知识、潜入心灵、全面领悟的一种知识学习现象、目标与状态,还原论与加工论是深度学习研究的两个理论原生点。问题、思维与意义是深度学习的三个关节点,回植情景、思维升级是深度学习创建的两大基本路径,贯通型深度学习理论是有机整合还原论与加工论的理念范型。面向未来,深度学习的理论走向是:构筑知识学习的立体世界,构建知识学习的生发链、生态环,建立学习力的系统提升机制。

Fast Model-based Design of High Performance Permanent Magnet Machine for Next Generation Electric Propulsion for Urban Aerial Vehicle Application

Model order reduction(MOR)is considered as a good alternative to reduce the computational scale for electro-magnetic problems.The aim of this work is to introduce the use of dynamic mode decomposition(DMD)as a promising tool for MOR to analyze its effectiveness in creating a fast model-based design platform for the permanent magnet motor design for ur-ban aerial vehicles(UAVs).Using a singular value decomposition(SVD)based DMD,the design process is constructed and verified against different scenarios.

Efficient numerical analysis of guided wave structures by compact FDFD with PVL method

An efficient numerical simulation technique is introduced to extract the propagation characteristics of a millimeter guided wave structure. The method is based on the application of the Krylov subspace model order reduction technique (Padé via Lanczos) to the compact finite difference frequency domain (FDFD) method. This new technique speeds up the solution by decreasing the originally larger system matrix into one lower order system matrix. Numerical experiments from several millimeter guided wave structures demonstrate the efficiency and accuracy of this algorithm.

Recent Advance in Non-Krylov Subspace Model Order Reduction of Interconnect Circuits

Model order reduction of interconnect circuits is an important technique to reduce the circuit complexity and improve the efficiency of post-layout verification process in the nanometer VLSI design. Existing works using the Krylov subspace method are very efficient, but the resulting models are less compact and lack global accuracy. Also, existing methods cannot handle interconnect circuits with large input and output ports. Recent advances in reduction techniques using non-Krylov subspace techniques such as truncated balanced realization （TBR） hold some promise to solve these problems. In this paper, we first review the classic TBR-based reduction methods and then present the recent developments in fast TBR-based reduction and techniques such as PMTBR, SBPOR, and ETBR methods. These newly proposed methods try to avoid the expensive computing steps in traditional TBR methods at some cost to accuracy to boost efficiency and scalability, which is critical to reduce large interconnect parasitics modeled as RLCK circuits. The ETBR method can also reduce circuits with massive ports by considering the input signals. We show the pros and cons of each method and compare them on a set of large interconnect circuits, and finally point to some new research directions for this area.

AN ACCELERATED WAVEFORM RELAXATION APPROACH BASED ON MODEL ORDER REDUCTION FOR LARGE COUPLING SYSTEMS

In this paper, we present an accelerated simulation approach on waveform relaxation using Krylov subspace for a large time-dependent system composed of some subsystems. This approach first allows these subsystems to be decoupled by waveform relaxation. Then the Arnoldi procedure based on Krylov subspace is provided to accelerate the simulation of the decoupled subsystems independently. For the new approach, the convergent conditions on waveform relaxation are derived. The robust behavior is also successfully illustrated via numerical examples.

Quadratic investigation of geochemical distribution by backward elimination approach at Glojeh epithermal Au(Ag)-polymetallic mineralization, NW Iran

The correspondence analysis will describe elemental association accompanying an indicator samples.This analysis indicates strong mineralization of Ag,As,Pb,Te,Mo,Au,Zn and to a lesser extent S,W,Cu at Glojeh polymetallic mineralization,NW Iran.This work proposes a backward elimination approach(BEA)that quantitatively predicts the Au concentration from main effects(X),quadratic terms(X2)and the first order interaction(Xi×Xj)of Ag,Cu,Pb,and Zn by initialization,order reduction and validation of model.BEA is done based on the quadratic model(QM),and it was eliminated to reduced quadratic model(RQM)by removing insignificant predictors.During the QM optimization process,overall convergence trend of R2,R2(adj)and R2(pred)is obvious,corresponding to increase in the R2(pred)and decrease of R2.The RQM consisted of(threshold value,Cu,Ag×Cu,Pb×Zn,and Ag2-Pb2)and(Pb,Ag×Cu,Ag×Pb,Cu×Zn,Pb×Zn,and Ag2)as main predictors of optimized model according to288and679litho-samples in trenches and boreholes,respectively.Due to the strong genetic effects with Au mineralization,Pb,Ag2,and Ag×Pb are important predictors in boreholes RQM,while the threshold value is known as an important predictor in the trenches model.The RQMs R2(pred)equal74.90%and60.62%which are verified by R2equal to73.9%and60.9%in the trenches and boreholes validation group,respectively.

A single-manufacturer multi-retailer sustainable reworking model for green and environmental sensitive demand under discrete ordering cost reduction

This paper develops an economic production quantity(EPQ)model for a singlemanufacturer multi-retailer(SMMR)production and reworking system with green and environmental sensitive customer demand.The minimum cost of the manufacturer has obtained under carbon emissions(CE)policies and discrete ordering cost reduction.The model is used to optimize the total number of shipments,greening investment level,environmental measure,and lot size for productions and rework.This research work determines that the manufacturer’s and retailer’s profits will be increased after considering the environmental and green dependent demand of customers.Further,the development of green and environmental demand is proposed to minimize the CE and maximize the demand for the customers.In the existing literature,no discrete investment is developed for reducing the cost of ordering for the retailer/buyer.However,in this paper,we have introduced it.We provide numerical examples to explain the models and determine the significance of model parameters.

A Local Deep Learning Method for Solving High Order Partial Differential Equations

At present, deep learning based methods are being employed to resolvethe computational challenges of high-dimensional partial differential equations(PDEs). But the computation of the high order derivatives of neural networks iscostly, and high order derivatives lack robustness for training purposes. We proposea novel approach to solving PDEs with high order derivatives by simultaneously approximating the function value and derivatives. We introduce intermediate variablesto rewrite the PDEs into a system of low order differential equations as what is donein the local discontinuous Galerkin method. The intermediate variables and the solutions to the PDEs are simultaneously approximated by a multi-output deep neuralnetwork. By taking the residual of the system as a loss function, we can optimizethe network parameters to approximate the solution. The whole process relies onlow order derivatives. Numerous numerical examples are carried out to demonstrate that our local deep learning is efficient, robust, flexible, and is particularlywell-suited for high-dimensional PDEs with high order derivatives.