Reduced-order Approach Research Articles

Numerical identification of a viscoelastic substitute model for heterogeneous poroelastic media by a reduced order homogenization approach

The paper deals with the computational homogenization of pressure diffusion processes in a poroelastic medium. The underlying physical phenomena are of interest for the interpretation of seismic data with applications in hydrocarbon production and geothermal energy. Pressure diffusion is assumed to take place on a length scale much smaller than the observer scale. Thus, the macroscopic observer is not able to measure the properties of the poroelastic medium directly but notices an intrinsic viscous attenuation. Under these circumstances, the macro-scale can be interpreted as a single-phase solid with (apparent) viscoelastic properties. In this paper, we establish a numerical upscaling procedure based on a volume averaging concept. This enables us to identify the material properties of the viscoelastic substitute model in a numerically efficient manner. For this purpose, the poroelastic medium on the small scale is modelled in terms of the momentum balance of the biphasic mixture and a coupled diffusion equation. We approximate the poroelastic pressure field on the small scale by a linear combination of pressure modes forming a reduced orthogonal basis and being identified by a Proper Orthogonal Decomposition (POD) technique. From the superposition principle, the evaluation of the poroelastic continuity equation results in a proper identification of the evolution equations defining the apparent viscoelastic model. In comparison to the nested FE2 solution schemes, the reduced order approach only requires a small amount of off-line precomputations and, therefore, causes very low numerical costs. The proposed method is validated for the simple setup of a layered porous rock with alternating water-and gas-saturated zones.

Cooperative Output Regulation of Singular Heterogeneous Multiagent Systems.

This paper investigates the cooperative output regulation problem of singular heterogeneous multiagent systems. General distributed observers are proposed for every agent obtaining the estimated state of the exosystem. The feedforward control technique and reduced-order approach are used to design distributed singular output feedback controllers and distributed normal output feedback controllers. The proposed cooperative dynamic controller is dependent on the plant parameters and the interaction topologies. A simulation example is provided to demonstrate the effectiveness of the proposed design method.

Parameter identification for nonlinear elliptic-parabolic systems with application in lithium-ion battery modeling

In this paper the authors consider a parameter estimation problem for a nonlinear systems, which consists of one parabolic equation for the concentration and two elliptic equations for the potentials. The measurements are given as boundary values for one of the potentials. For its numerical solution the Gauss Newton method is applied. To speed up the solution process, a reduced-order approach based on proper orthogonal decomposition is utilized, where the accuracy is controlled by error estimators. Parameters, which can not be identified from the measurements, are identified by the subset selection method with $$QR$$QR pivoting. Numerical examples show the efficiency of the proposed approach.

Multisource statistically optimized near-field acoustical holography.

This paper presents a reduced-order approach to near-field acoustical holography (NAH) that allows the user to account for sound fields generated by multiple spatially separated sources. In this method, an equivalent wave model (EWM) of a given field is formulated to include combinations of planar, cylindrical, spherical, or other elementary wave functions in contrast to an EWM restricted to a single separable coordinate system. This can alleviate the need for higher-order functions, reduce the number of measurements, and decrease error. The statistically optimized near-field acoustical holography (SONAH) algorithm is utilized to perform the NAH projection after the formulation of the multisource EWM. The combined process is called multisource statistically optimized near-field acoustical holography (M-SONAH). This method is used to reconstruct simulated sound fields generated by combinations of a vibrating piston in a sphere and linear arrays of monopole sources. It is shown that M-SONAH can reconstruct near-field pressures in multisource environments with lower errors and fewer measurements than a strictly plane or cylindrical-wave formulation using the same simulated measurement.

Global–local nonlinear model reduction for flows in heterogeneous porous media

Abstract In this paper, we combine discrete empirical interpolation techniques, global mode decomposition methods, and local multiscale methods, such as the Generalized Multiscale Finite Element Method (GMsFEM), to reduce the computational complexity associated with nonlinear flows in highly-heterogeneous porous media. To solve the nonlinear governing equations, we employ the GMsFEM to represent the solution on a coarse grid with multiscale basis functions and apply proper orthogonal decomposition on a coarse grid. Computing the GMsFEM solution involves calculating the residual and the Jacobian on a fine grid. As such, we use local and global empirical interpolation concepts to circumvent performing these computations on the fine grid. The resulting reduced-order approach significantly reduces the flow problem size while accurately capturing the behavior of fully-resolved solutions. We consider several numerical examples of nonlinear multiscale partial differential equations that are numerically integrated using fully-implicit time marching schemes to demonstrate the capability of the proposed model reduction approach to speed up simulations of nonlinear flows in high-contrast porous media.

Real‐time monitoring of thermal processes by reduced‐order modeling

SummaryThis work presents a simple technique for real‐time monitoring of thermal processes. Real‐time simulation‐based control of thermal processes is a big challenge because high‐fidelity numerical simulations are costly and cannot be used, in general, for real‐time decision making. Very often, processes are monitored or controlled with a few measurements at some specific points. Thus, the strategy presented here is centered on fast evaluation of the response only where it is needed. To accomplish this, classical harmonic analysis is combined with recent model reduction techniques. This leads to an advanced harmonic methodology, which solves in real time the transient heat equation at the monitored point.In order to apply the reciprocity principle, harmonic analysis is used in the space‐frequency domain. Then, Proper Generalized Decomposition, a reduced order approach, pre‐computes a transfer function able to produce the output response for a given excitation. This transfer function is computed offline and only once. The response at the monitoring point can be recovered performing a computationally inexpensive post‐processing step. This last step can be performed online for real‐time monitoring of the thermal process. Examples show the applicability of this approach for a wide range of problems ranging from fast temperature evaluation to inverse problems. Copyright © 2014 John Wiley & Sons, Ltd.

A multisource-type representation statistically optimized near-field acoustical holography method

A reduced-order approach to near-field acoustical holography (NAH) that accounts for sound fields generated by multiple spatially separated sources of different types is presented. In this method, an equivalent wave model (EWM) of a given field is formulated based on rudimentary knowledge of source types and locations. The statistically optimized near-field acoustical holography (SONAH) algorithm is utilized to perform the NAH projection after the formulation of the multisource EWM. The combined process is called multisource-type representation SONAH (MSTR SONAH). This method is used to reconstruct simulated sound fields generated by combinations of multiple source types. It is shown that MSTR SONAH can successfully reconstruct the near field pressures in multisource environments where other NAH methods result in large errors. The MSTR SONAH technique can be extended to general sound fields where the shapes and locations of sources and scattering bodies are known.

A reduced-order nonlinear sliding mode observer for vehicle slip angle and tyre forces

In this paper, a reduced-order sliding mode observer (RO-SMO) is developed for vehicle state estimation. Several improvements are achieved in this paper. First, the reference model accuracy is improved by considering vehicle load transfers and using a precise nonlinear tyre model ‘UniTire’. Second, without the reference model accuracy degraded, the computing burden of the state observer is decreased by a reduced-order approach. Third, nonlinear system damping is integrated into the SMO to speed convergence and reduce chattering. The proposed RO-SMO is evaluated through simulation and experiments based on an in-wheel motor electric vehicle. The results show that the proposed observer accurately predicts the vehicle states.

Efficient Characterization of Uncertain Model Parameters with a Reduced-Order Ensemble Kalman Filter

Spatially variable model parameters are often highly uncertain and difficult to observe. This has prompted the widespread use of Bayesian characterization methods that can infer parameter values from measurements of related variables, while explicitly accounting for uncertainty. Ensemble versions of Bayesian characterization are particularly convenient when uncertain variables have complex spatial structures that do not conform to Gaussian descriptions. However, ensemble methods can be time consuming for high-dimensional problems. This paper describes a reduced-order approach to ensemble characterization that is particularly well suited for subsurface flow and transport problems. It uses a truncated discrete cosine transform to reduce the dimensionality of spatially variable time-invariant model parameters and a nonlinear extension of principle orthogonal decomposition to reduce the dimensionality of dynamic model states. The resulting nonlinear reduced-order model can be included in the forecast step of ...

A reduced-order model for Monte Carlo simulations of stochastic groundwater flow

We explore the ability of the greedy algorithm to serve as an effective tool for the construction of reduced-order models for the solution of fully saturated groundwater flow in the presence of randomly distributed transmissivities. The use of a reduced model is particularly appealing in the context of numerical Monte Carlo (MC) simulations that are typically performed, e.g., within environmental risk assessment protocols. In this context, model order reduction techniques enable one to construct a surrogate model to reduce the computational burden associated with the solution of the partial differential equation governing the evolution of the system. These techniques approximate the model solution with a linear combination of spatially distributed basis functions calculated from a small set of full model simulations. The number and the spatial behavior of these basis functions determine the computational efficiency of the reduced model and the accuracy of the approximated solution. The greedy algorithm provides a deterministic procedure to select the basis functions and build the reduced-order model. Starting from a single basis function, the algorithm enriches the set of basis functions until the largest error between the full and the reduced model solutions is lower than a predefined tolerance. The comparison between the standard MC and the reduced-order approach is performed through a two-dimensional steady-state groundwater flow scenario in the presence of a uniform (in the mean) hydraulic head gradient. The natural logarithm of the aquifer transmissivity is modeled as a second-order stationary Gaussian random field. The accuracy of the reduced basis model is assessed as a function of the correlation scale and variance of the log-transmissivity. We explore the performance of the reduced model in terms of the number of iterations of the greedy algorithm and selected metrics quantifying the discrepancy between the sample distributions of hydraulic heads computed with the full and the reduced model. Our results show that the reduced model is accurate and is highly efficient in the presence of a small variance and/or a large correlation length of the log-transmissivity field. The flow scenarios associated with large variances and small correlation lengths require an increased number of basis functions to accurately describe the collection of the MC solutions, thus reducing significantly the computational advantages associated with the reduced model.

Modeling and experimental verification of a flexible rotor/AMB system

PurposeIn this paper, the aim is to present a modeling strategy for a flexible rotor/active magnetic bearing (AMB) system with non‐collocation. Special attention is paid to the vibration reduction and the stable passage through the first critical speed.Design/methodology/approachThe finite element method based on Euller‐Bernoulli beam theory is applied in the formulation of the rotor model. Since rotor/AMB systems are complex mechatronic systems, reduced order approach is used in the control system design. This study applies the modal decomposition method and the modal truncation method, thus retaining the lower order bending modes. The obtained numerical results are compared with the measured open loop frequency responses and the existing differences are compensated in order to obtain accurate numerical model.FindingsFrequency response of the entire system model (flexible shaft, actuators, power amplifiers and sensors) with amplitudes expressed in rotor lateral displacements can be verified by the measured frequency responses. The deviations in the amplitude and phase diagrams are then successfully corrected using the appropriate model modifications.Practical implicationsThe results of this research find direct applications in flexible rotors supported by AMBs, e.g. high speed spindles, turbo molecular pumps, flywheel energy storage systems, etc. The presented procedure can be especially valuable in the design of model based controllers.Originality/valueAn AMB system model is developed and presented in this paper, in conjunction with a systematic description of an efficient procedure for the elimination of the typical mismatches between the simulation and experiment. Firstly, rotor/AMB test rig is stabilized with an appropriately tuned PID controller and an open loop frequency response is obtained for such a system. This response is then compared to corresponding simulation results for which mismatches are identified and eliminated thus yielding an accurate model of the system.

Mode decomposition methods for flows in high-contrast porous media. Global–local approach

In this paper, we combine concepts of the generalized multiscale finite element method (GMsFEM) and mode decomposition methods to construct a robust global–local approach for model reduction of flows in high-contrast porous media. This is achieved by implementing Proper Orthogonal Decomposition (POD) and Dynamic Mode Decomposition (DMD) techniques on a coarse grid computed using GMsFEM. The resulting reduced-order approach enables a significant reduction in the flow problem size while accurately capturing the behavior of fully-resolved solutions. We consider a variety of high-contrast coefficients and present the corresponding numerical results to illustrate the effectiveness of the proposed technique. This paper is a continuation of our work presented in Ghommem et al. (2013) [1] where we examine the applicability of POD and DMD to derive simplified and reliable representations of flows in high-contrast porous media on fully resolved models. In the current paper, we discuss how these global model reduction approaches can be combined with local techniques to speed-up the simulations. The speed-up is due to inexpensive, while sufficiently accurate, computations of global snapshots.

POD Galerkin Schemes for Nonlinear Elliptic-Parabolic Systems

In this paper the authors study a nonlinear elliptic-parabolic system, which is motivated by mathematical models for lithium ion batteries. For the reliable and fast numerical solution of the system a reduced-order approach based on proper orthogonal decomposition (POD) is applied. The strategy is justified by a priori error analysis for the error between the solution to the coupled system and its POD approximation. The nonlinear coupling is realized by variants of the empirical interpolation introduced by Barrault et al. [C. R. Acad. Sci. Paris Ser. I, 339 (2004), pp. 667--672] and Chaturantabut and Sorensen [Application of POD and DEIM on a Dimension Reduction of Nonlinear Miscible Viscous Fingering in Porous Media, Technical Report, RICE University, 2009]. Numerical examples illustrate the efficiency of the proposed reduced-order modeling.

Fast Adaptive 3-D Nonstationary Electrical Impedance Tomography Based on Reduced-Order Modeling

Computational cost of image reconstruction in electrical impedance tomography (EIT) is generally very high. Time consumption of data processing can be prohibitive particularly in systems intended for continuous monitoring of time-varying targets in various applications. Recently, two promising approximate computational approaches have been proposed to reduce the computational cost of image reconstruction. These approaches are based on reduced-order approximation of the associated computational models. In this paper, we utilize these techniques to reduce the computational cost of 3-D nonstationary EIT imaging when high image reconstruction rate is required due to rapid changes or instabilities in the target of interest. The feasibility of the proposed reduced-order approach is evaluated in simulation and experimental studies. The results show that computational cost in nonstationary image reconstruction can be decreased significantly with reduced-order modeling, and in addition, with an appropriate reduced-order representation of the system state, the effects on the accuracy are very small.

Unified control of n-link underactuated manipulator with single passive joint: A reduced order approach

This paper addresses the motion control problem of an n-link planar underactuated manipulator with any one of its joints being passive. A reduced order approach is proposed to achieve a swing-up control and a balancing control. The presented approach involves two stages. In Stage 1, (n−2) control laws are first designed to force both the angles and angular velocities of (n−2) actuated links to converge to zero, which guarantees that the n-link manipulator is reduced to like a 2-link one. Then the control law of the rest actuated link is temporarily set to constant to simplify the design of the control laws. In Stage 2, the states of the reduced system are guaranteed by maintaining the (n−2) control laws designed in Stage 1. Meanw'hile, a swing-up control law and a balancing control law are designed for the rest actuated link. Integrated with the (n−2) nonlinear control laws designed in Stage 1, the n-link underactuated manipulator is swung up quickly and balanced at the straight-up unstable equilibrium position effectively. Simulation results on a 4-link underactuated manipulator with different passive joint positions demonstrate the effectiveness of the proposed control approach.

Precision Tracking of a Rotating Shaft With Magnetic Bearings by Nonlinear Decoupled Disturbance Observers

Static offset and synchronous vibration are obstacles to precision tracking of a rotating shaft supported by active magnetic bearings (AMBs). A constant and harmonic disturbance observer is an effective approach to estimate and suppress these undesired effects. A disturbance observer is particularly well-suited for tracking applications since it can directly compensate for AMB force nonlinearity. In this paper, we propose a nonlinear reduced-order disturbance observer which is incorporated into a flatness-based tracking control. Since a reduced-order approach requires knowledge of velocity, we design an inner-loop velocity observer which can achieve convergence at a faster rate than the disturbance observer. This hierarchical observer simplifies the control because disturbance estimates are decoupled for each degree of freedom (DOF) and disturbance compensation is modularized. Experimental results on a commercially available 5-DOF system demonstrate accurate position tracking over the bearing air gap for a wide range of shaft velocities.

Robust modal vibration suppression of a flexible rotor

This study deals with active robust modal vibration control of rotor systems supported by magnetic bearings. The inherent divergent rigid body modes are suppressed by using a dual-level control approach. Finite element method is applied to formulate the rotor model. The Timoshenko beam theory, including the effects of shearing deformations and rotary inertia, is considered in this work. Because practical control systems are often limited by its sensing, hardware, and computation speed capabilities, the reduced order approach is often used for a control system design. This study applies the independent modal space control (IMSC) approach to extract the accurate lower modes from the complex rotor systems with the gyroscopic effect considered. In practice, it is extremely difficult to model the complete dynamic characteristics of a rotor system. The model may contain unmodelled dynamics and parameter changes, which can be viewed as uncertainties of a system. As opposed to the conventional control approach, which requires fixed and accurate system parameters, this study considers robust control approach to design a controller capable of tolerating external disturbance and model uncertainties. It is demonstrated that the proposed approach is effective for vibration suppression when the system is subjected to impulsive or step loading, speed variation, and sudden loss of disk mass.

A reduced-order strategy for 4D-Var data assimilation

This paper presents a reduced-order approach for four-dimensional variational data assimilation, based on a prior EOF analysis of a model trajectory. This method implies two main advantages: a natural model-based definition of a multivariate background error covariance matrix B r , and an important decrease of the computational burden of the method, due to the drastic reduction of the dimension of the control space. An illustration of the feasibility and the effectiveness of this method are given in the academic framework of twin experiments for a model of the equatorial Pacific Ocean. It is shown that the multivariate aspect of B r brings additional information which substantially improves the identification procedure. Moreover the computational cost can be decreased by one order of magnitude with regard to the full-space 4D-Var method.

On an adaptive filter for altimetric data assimilation and its application to a primitive equation model, MICOM

In a reduced-order adaptive filtering approach, we demonstrate a possibility to overcome two major difficulties in estimating oceanic circulation: a very high dimension of the system state and uncertainties in specification of the model error statistics. This approach is based essentially on the assumption that a particular parametrized gain matrix has been selected and the tuning parameters are adjusted by minimizing the mean prediction error. In the present paper we apply a reduced-order adaptive filter for solving the problem of assimilating altimetric sea surface height into a primitive equation model: the Miami isopycnic coordinate ocean model (MICOM). A gain structure is described which is proven to be very efficient in the twin experiments. The assimilation algorithm to be employed in the identical twin experiment is a reduced-order filter whose reduced state consists of the layer thickness. The velocity update is calculated from the geostrophic hypothesis. The gain structure for the non-adaptive filter is obtained on the basis of three principal hypotheses: (H1) analysis error for the system output is cancelled in the case of noise-free observations (as is done naturally in a standard Kalman filter for noise-free observation); (H2) conservation of linear potential vorticity; (H3) no correction for the velocity at the bottom layer. The initial values of the parameters in the gain will be selected in such a way that the filter behaves exactly as the Cooper’Haines filter (CHF) at the first data update step. It is shown that the adaptive filter, which relaxes one or several of the above hypotheses, is capable of producing the better estimates for the ocean state (layer thickness and velocity) compared to that produced by the CHF in all layers, surface or subsurface. Numerical experiments demonstrate the excellent capacity of the adaptive filter to extract useful information from surface observations for inferring the oceanic circulation in the MICOM.

Control of laser surface hardening by a reduced-order approach using proper orthogonal decomposition

Laser surface hardening of steel is formulated in terms of an optimal control problem, where the state equations are a semilinear heat equation and an ordinary differential equation, which describe the evolution of the high temperature phase. The optimal control problem is analyzed and first-order necessary optimality conditions are derived. An error estimate for POD (proper orthogonal decomposition) Galerkin methods for the state system is proved. Finally, a strategy to obtain suboptimal controls using POD is applied to solve some numerical examples.