Model Reduction Methodology Research Articles

A Fast Reduced Order Method for Assessment of Wind Farm Layouts

Due to wake losses, the power production of wind farms depend on the positioning of turbines in relation to each other, i.e. the wind farm layout. Computational Fluid Dynamics (CFD) can be used for solving the complex flow in a wind farm since wake effects are inherently simulated. However, it is too slow for interactively assessing the power production for variations of wind farm layouts. Therefore, there is a need for an accurate and fast model based on CFD for computing the flow field and the wake losses in a wind farm for a range of wind speeds and wind directions. In this paper, we adapt model reduction methodology to wind farm flow. The governing equations are formulated in a reduced finite dimensional space spanned by a set of orthogonal modes created from CFD simulations. The accuracy of the reduced model is assessed on a set of test cases. In the reduced model, new solutions for different turbine layouts and wind speeds are produced in a matter of seconds, as opposed to hours for full CFD simulations.

Preserving Lagrangian Structure in Nonlinear Model Reduction with Application to Structural Dynamics

This work proposes a model-reduction methodology that preserves Lagrangian structure and achieves computational efficiency in the presence of high-order nonlinearities and arbitrary parameter dependence. As such, the resulting reduced-order model retains key properties such as energy conservation and symplectic time-evolution maps. We focus on parameterized simple mechanical systems subjected to Rayleigh damping and external forces, and consider an application to nonlinear structural dynamics. To preserve structure, the method first approximates the system's “Lagrangian ingredients''---the Riemannian metric, the potential-energy function, the dissipation function, and the external force---and subsequently derives reduced-order equations of motion by applying the (forced) Euler--Lagrange equation with these quantities. From the algebraic perspective, key contributions include two efficient techniques for approximating parameterized reduced matrices while preserving symmetry and positive definiteness: matrix gappy proper orthogonal decomposition and reduced-basis sparsification. Results for a parameterized truss-structure problem demonstrate the practical importance of preserving Lagrangian structure and illustrate the proposed method's merits: it reduces computation time while maintaining high accuracy and stability, in contrast to existing nonlinear model-reduction techniques that do not preserve structure.

Model reduction: When singular perturbation analysis simplifies to partial equilibrium approximation

A geometrical interpretation is provided for the validity of the model reduction methodology based on Partial Equilibrium Approximation, by comparing its algorithm to that of the methodology based on Singular Perturbation Analysis. The cases where the former methodology fails to provide a valid reduced model, while the latter one succeeds, are explored. It is shown that the failure of the Partial Equilibrium Approximation is due to the inability of (i) the stoichiometric vectors of the reactions considered in partial equilibrium to provide a leading order approximation of the fast directions in phase space or/and (ii) the equilibration of the related forward and backward rates to provide a leading order approximation of the equilibrations that develop under the action of the fast time scales. These issues are illustrated geometrically, by analyzing a number of simple examples.

A case study in multi‐scale model reduction: The effect of cell density on catalytic converter performance

One of the challenges of full‐scale computer simulation of a catalytic reactor is to consider the different scales involved in the problem in a practical fashion. In a monolith catalytic converter, these scales range from the molecular scale for the reactions, through the pore scale, washcoat scale, channel scale, and finally the full converter scale. This paper describes the implementation of a model reduction methodology using look‐up tables to perform a consistent comparison of six different catalytic converters used for the catalytic combustion of methane. A detailed mechanistic model for methane combustion is used. Diffusion in the non‐uniform washcoat is considered. The converters have different cell densities and wall thicknesses. Steady state and transient light‐off simulations are performed. Efficient computational speed is achieved by successive model reduction, which allows the preservation of detailed small‐scale information. The results obtained show that there is a non‐intuitive relationship between the various operating parameters, which can only be deduced from a comprehensive model.

Restrictions in Model Reduction for Polymer Chain Models in Dissipative Particle Dynamics

We model high molecular weight homopolymers in semidilute concentration via Dissipative Particle Dynamics (DPD). We show that in model reduction methodologies for polymers it is not enough to preserve system properties (i.e., density ρ, pressure p, temperature T, radial distribution function g(r)) but preserving also the characteristic shape and length scale of the polymer chain model is necessary. In this work we apply a DPD-model-reduction methodology for linear polymers recently proposed; and demonstrate why the applicability of this methodology is limited upto certain maximum polymer length, and not suitable for solvent coarse graining.

Model reductions with special CSP data

The model reduction methodology of computational singular perturbation (CSP) is enhanced for chemical kinetics problems. The leading order converged CSP-refined fast basis vectors are obtained analytically by exploiting the special format of the given reaction-specific chemical kinetics data. The leading order net reaction rates of the fast reactions (after they are exhausted) are also obtained analytically. Taking advantage of these leading order analytical results, a new model reduction strategy is advocated. The implementation of this new strategy is very straightforward, and extraction of general insights on the reaction system is also easier.

Reduction of chemical reaction networks through delay distributions

Accurate modelling and simulation of dynamic cellular events require two main ingredients: an adequate description of key chemical reactions and simulation of such chemical events in reasonable time spans. Quite logically, posing the right model is a crucial step for any endeavour in Computational Biology. However, more often than not, it is the associated computational costs which actually limit our capabilities of representing complex cellular behaviour. In this paper, we propose a methodology aimed at representing chains of chemical reactions by much simpler, reduced models. The abridgement is achieved by generation of model-specific delay distribution functions, consecutively fed to a delay stochastic simulation algorithm. We show how such delay distributions can be analytically described whenever the system is solely composed of consecutive first-order reactions, with or without additional "backward" bypass reactions, yielding an exact reduction. For models including other types of monomolecular reactions (constitutive synthesis, degradation, or "forward" bypass reactions), we discuss why one must adopt a numerical approach for its accurate stochastic representation, and propose two alternatives for this. In these cases, the accuracy depends on the respective numerical sample size. Our model reduction methodology yields significantly lower computational costs while retaining accuracy. Quite naturally, computational costs increase alongside network size and separation of time scales. Thus, we expect our model reduction methodologies to significantly decrease computational costs in these instances. We anticipate the use of delays in model reduction will greatly alleviate some of the current restrictions in simulating large sets of chemical reactions, largely applicable in pharmaceutical and biological research.

Robust Gust Alleviation and Stabilization of Very Flexible Aircraft

Robust linear control, combined with model-reduction methodologies, is investigated for gust rejection on large, very flexible aircraft using trailing-edge control surfaces. Controllers are designed on linearizations of the nonlinear aeroelastic equations, and the closed-loop response of both the linearized and nonlinear system to discrete gust distributions is compared to the open-loop dynamics. Results show that an controller performs well on a relatively large linearized system, with 9% load alleviation in root bending moments for the critical gust length. When applied to the nonlinear model of the same vehicle, the robust controller gives a good performance in response to short gusts, including the critical length, with even better load reductions than the linear case. However, this performance gap decreases as the gust length is increased. It is also shown how standard model-reduction techniques can provide metrics for the selection of a minimum size of the aeroelastic system. Finally, it is shown that both the full and reduced controllers are capable of stabilizing a large-amplitude phugoid mode on the nonlinear model of the vehicle.

A Short Review on Model Order Reduction Based on Proper Generalized Decomposition

This paper revisits a new model reduction methodology based on the use of separated representations, the so called Proper Generalized Decomposition—PGD. Space and time separated representations generalize Proper Orthogonal Decompositions—POD—avoiding any a priori knowledge on the solution in contrast to the vast majority of POD based model reduction technologies as well as reduced bases approaches. Moreover, PGD allows to treat efficiently models defined in degenerated domains as well as the multidimensional models arising from multidimensional physics (quantum chemistry, kinetic theory descriptions,…) or from the standard ones when some sources of variability are introduced in the model as extra-coordinates.

Predicting stochastic systems by noise sampling, and application to the El Niño-Southern Oscillation

Interannual and interdecadal prediction are major challenges of climate dynamics. In this article we develop a prediction method for climate processes that exhibit low-frequency variability (LFV). The method constructs a nonlinear stochastic model from past observations and estimates a path of the "weather" noise that drives this model over previous finite-time windows. The method has two steps: (i) select noise samples--or "snippets"--from the past noise, which have forced the system during short-time intervals that resemble the LFV phase just preceding the currently observed state; and (ii) use these snippets to drive the system from the current state into the future. The method is placed in the framework of pathwise linear-response theory and is then applied to an El Niño-Southern Oscillation (ENSO) model derived by the empirical model reduction (EMR) methodology; this nonlinear model has 40 coupled, slow, and fast variables. The domain of validity of this forecasting procedure depends on the nature of the system's pathwise response; it is shown numerically that the ENSO model's response is linear on interannual time scales. As a result, the method's skill at a 6- to 16-month lead is highly competitive when compared with currently used dynamic and statistic prediction methods for the Niño-3 index and the global sea surface temperature field.

Efficient simulations of detailed combustion fields via the lattice Boltzmann method

PurposeThe paper aims to be a first step toward the efficient, yet accurate, solution of detailed combustion fields using the lattice Boltzmann (LB) method, where applications are still limited due to both the stiffness of the governing equations and the large amount of fields to solve.Design/methodology/approachThe suggested methodology for model reduction is developed in the setting of slow invariant manifold construction, including details of the while. The simplest LB equation is used in order to work out the procedure of coupling of the reduced model with the flow solver.FindingsThe proposed method is validated with the 2D simulation of a premixed laminar flame in the hydrogen‐air mixture, where a remarkable computational speedup and memory saving are demonstrated.Research limitations/implicationsBecause of the chosen detailed LB model, the flow field may be described with unsatisfactory accuracy: this motivates further investigation in this direction in the near future.Practical implicationsA new framework of simulation of reactive flows is available, based on a coupling between accurate reduced reaction mechanism and the LB representation of the flow phenomena. Hence, the paper includes implications on how to perform accurate reactive flow simulations at a fraction of the cost required in the detailed model.Originality/valueThis paper meets an increasing need to have efficient and accurate numerical tools for modelling complex phenomena, such as pollutant formation during combustion.

Off-line model reduction for on-line linear MPC of nonlinear large-scale distributed systems

Model predictive control (MPC) is an efficient method for the controller design of a large number of processes. However, linear MPC is often inappropriate for controlling nonlinear large-scale systems, while non-linear MPC can be computationally costly. The resulting optimization-based procedure can lead to local minima due to the, non-convexities that non-linear systems can exhibit. To overcome the excessive computational cost of MPC application for large-scale nonlinear systems, model reduction methodology in conjunction with efficient system linearizations have been exploited to enable the efficient application of linear MPC for nonlinear distributed parameter systems (DPS). An off-line model reduction technique, the proper orthogonal decomposition (POD) method, combined with a finite element Galerkin projection is first used to extract accurate non-linear low-order models from the large-scale ones. Trajectory Piecewise-Linear (TPWL) methodologies are subsequently developed to construct a piecewise linear representation of the reduced nonlinear model, both in a static and in a dynamic fashion. Linear MPC, based on quadratic programming, can then be efficiently performed on the resulting low-order, piece-wise affine system. Our combined methodology is readily applicable in combination with advanced MPC methodologies such as multi-parametric MPC (MP-MPC) ( Pistikopoulos, 2009). The stabilisation of the oscillatory behaviour of a tubular reactor with recycle is used as an illustrative example to demonstrate our methodology.

Empirical Model Reduction of Geometrical Constrained Gossamer Structures

The purpose of this research paper is to construct a low-order model of a geometrical constrained gossamer structure, especially for a solar sail. Currently, JAXA is working on a spin type solar power sail spacecraft. Computational analysis of the sail membrane's large mathematical model is difficult and time consuming. Therefore, this issue has a negative impact on design aspects. As a result, the model reduction technique is required. In other words, with a feasible, shortened, computational analysis period, a trade-off would be possible when designing and determining operation procedures. However, there is no model reduction methodology about solar sail dynamics. In this research paper, firstly, the full-order model of the spin type solar sail craft is modeled by the geometrically nonlinear FEM based on the energy momentum method (EMM). As a result, the numerical time integration is unconditionally stable. Secondly, the low-order model is constructed by the empirical model reduction method. And finally and complementarily to the former and latter sentence, the geometrical constraint is modeled by the penalty method based on the EMM. In this research paper, we will show that the low-order model can approximate the full-order model with sufficient accuracy.

Energy-Based Model Reduction Methodology for Automated Modeling

Abstract In recent years, algorithms have been developed to help automate the production of dynamic system models. Part of this effort has been the development of algorithms that use modeling metrics for generating minimum complexity models with realization preserving structure and parameters. Existing algorithms, add or remove ideal compliant elements from a model, and consequently do not equally emphasize the contribution of the other fundamental physical phenomena, i.e., ideal inertial or resistive elements, to the overall system behavior. Furthermore, these algorithms have only been developed for linear or linearized models, leaving the automated production of models of nonlinear systems unresolved. Other model reduction techniques suffer from similar limitations due to linearity or the requirement that the reduced models be realization preserving. This paper presents a new modeling metric, activity, which is based on energy. This metric is used to order the importance of all energy elements in a system model. The ranking of the energy elements provides the relative importance of the model parameters and this information is used as a basis to reduce the size of the model and as a type of parameter sensitivity information for system design. The metric is implemented in an automated modeling algorithm called model order reduction algorithm (MORA) that can automatically generate a hierarchical series of reduced models that are realization preserving based on choosing the energy threshold below which energy elements are not included in the model. Finally, MORA is applied to a nonlinear quarter car model to illustrate that energy elements with low activity can be eliminated from the model resulting in a reduced order model, with physically meaningful parameters, which also accurately predicts the behavior of the full model. The activity metric appears to be a valuable metric for automating the reduction of nonlinear system models—providing in the process models that provide better insight and may be more numerically efficient.

Non‐linear model reduction for uncertainty quantification in large‐scale inverse problems

AbstractWe present a model reduction approach to the solution of large‐scale statistical inverse problems in a Bayesian inference setting. A key to the model reduction is an efficient representation of the non‐linear terms in the reduced model. To achieve this, we present a formulation that employs masked projection of the discrete equations; that is, we compute an approximation of the non‐linear term using a select subset of interpolation points. Further, through this formulation we show similarities among the existing techniques of gappy proper orthogonal decomposition, missing point estimation, and empirical interpolation via coefficient‐function approximation. The resulting model reduction methodology is applied to a highly non‐linear combustion problem governed by an advection–diffusion‐reaction partial differential equation (PDE). Our reduced model is used as a surrogate for a finite element discretization of the non‐linear PDE within the Markov chain Monte Carlo sampling employed by the Bayesian inference approach. In two spatial dimensions, we show that this approach yields accurate results while reducing the computational cost by several orders of magnitude. For the full three‐dimensional problem, a forward solve using a reduced model that has high fidelity over the input parameter space is more than two million times faster than the full‐order finite element model, making tractable the solution of the statistical inverse problem that would otherwise require many years of CPU time. Copyright © 2009 John Wiley & Sons, Ltd.

The effect of multiple sources of uncertainty on the convex hull of material properties of polycrystals

Quantification and propagation of uncertainty in process conditions and initial microstructure on the final product properties in a deformation process are presented. The stochastic deformation problem is modeled using a sparse grid collocation approach that allows the utilization of a deterministic simulator to build interpolants of the main solution variables in the stochastic support space. The ability of the method in estimating the statistics of the macro-scale microstructure-sensitive properties and constructing the convex hull of these properties is shown through examples featuring randomness in initial texture and process parameters. A data-driven model reduction methodology together with a maximum entropy approach are used for representing randomness in initial texture in Rodrigues space. Comparisons are made with the results obtained from the Monte-Carlo method.

An invariant manifold approach for CSTR model reduction in the presence of multi-step biochemical reaction schemes. Application to anaerobic digestion

A systematic methodology for CSTR model reduction has been developed for multi-step biochemical reaction schemes. The proposed method neglects the dynamics of the fast steps by projecting the overall system dynamics on the slow-motion invariant manifold of the system. In particular, using reaction invariants in the description of the overall system dynamics, the slow-motion invariant manifold is calculated by solving the pertinent invariance equation via series-solution or singular perturbation techniques. The proposed method is an alternative to the quasi-steady state approximation which does not rely on a priori physical information. The proposed approach is applied to two model reduction problems arising in anaerobic digestion. The results provide a rigorous answer on how to properly eliminate the fast dynamics of the acidogenesis step.

Exponential observers for distributed tubular (bio)reactors

AbstractThe dissipative nature of spatially distributed process systems is exploited to develop efficient exponential state observers based on a low‐dimensional dynamic representation of the original set of partial differential equations. The suggested approach combines standard observer design techniques for reactors, where the reaction rates are unknown with efficient model reduction methodologies based on projection of the original concentration and temperature fields on low‐dimensional subspaces capturing the slow dynamics of the process. The global exponential stability of the resulting observer is derived combining classical Lyapunov analysis with a transformation that allows us to obtain a diffusion system from a diffusion‐convection system. In addition, aspects related to the location of sensors and their influence on the ability to reconstruct the necessary fields to feed the observer will also be considered. © 2008 American Institute of Chemical Engineers AIChE J, 2008

A grey-box modeling approach for the reduction of nonlinear systems

A novel model reduction methodology is proposed to approximate large-scale nonlinear dynamical systems. The methodology amounts to finding computationally efficient substitute models for the nonlinear subsystems. Model reduction is pursued by viewing the system as a grey-box (or hybrid) model with a mechanistic (white-box) component and an empirical (black-box) component. Before identifying the substitute model, the mechanistic subsystem is reduced by projection using proper orthogonal decomposition. Subsequently, the empirical component is identified by parameter estimation to substitute the nonlinear subsystem. As a consequence, a reduced model with less nonlinear complexity is obtained. An example involving a distributed parameter system is provided.

A GREY-BOX MODELING APPROACH FOR THE REDUCTION OF NONLINEAR SYSTEMS

A novel model reduction methodology is proposed to approximate large-scale nonlinear dynamical systems. The methodology amounts to finding computationally efficient substitute models for an uncertain nonlinear system. Model uncertainty is incorporated by viewing the system as a grey-box or hybrid model with a mechanistic (first-principle) component and an empirical (black-box) component. The mechanistic part is approximated using proper orthogonal decomposition. Subsequently, the empirical part is identified by parameter estimation using the reduced order mechanistic part. As a consequence, the parameter estimation is computationally more efficient. An example with a distributed parameter system is provided.