Model Reduction Scheme Research Articles

Simulation of 2D Acoustic Wave Propagation with Absorbing Boundary Condition

In this work we design an algorithm to simulate the 2D acoustic wave (P wave) propagation in a medium with different regional velocities. We use the Generalized Conjugate Residual (GCR) algorithm to solve the trapezoidal time integration, where for a given time step of 0.01 seconds, the error is approximately 8% with respect to the reference. We further speed up the simulation by a factor of 900 by implementing a model reduction scheme using Proper Orthogonal Decomposition (POD). Lastly, to suppress the artificial reflections formed when the wave bounces off the fixed boundaries, we created two virtual absorbing layers by tapering the solution of nodes close to the corresponding boundaries after each time iteration. However, we note that the successful implementations of the model reduction and absorbing boundary condition are not compatible with each other and provide explanations for why and how to combine them in future work.

Carbon emission evaluation model and carbon reduction strategies for newly urbanized areas

China is the world's largest energy consumer and carbon emitter. Cities account for 85% of China's total carbon emissions, making them key areas for implementing carbon reduction policies. In recent years, the construction of newly urbanized areas in China has been increasing. Thus, conducting research on carbon emission evaluation and carbon reduction strategies for these areas is important to help cities deal with climate change. This study sorted carbon emission sources and sinks according to the management of the government as well as formulated an inventory and evaluation model of the carbon emissions of newly urbanized areas. Three cases located in cities of different scales were taken as examples and their emissions were calculated under the current planning mode. The key factors of emissions at each stage of urban development were analyzed. Accordingly, corresponding carbon reduction strategies were proposed for the planning and operation stages and the short-term and long-term targets for the implementation of carbon reduction strategies were formulated. The carbon reduction effect of various strategies was quantified by comparing the emissions before and after the implementation of strategies. The results of this study can support low-carbon construction policies and green ecological development of newly urbanized areas and cities.

Identification of MIMO Wiener-type Koopman models for data-driven model reduction using deep learning

We use Koopman theory to develop a data-driven nonlinear model reduction and identification strategy for multiple-input multiple-output (MIMO) input-affine dynamical systems. While the present literature has focused on linear and bilinear Koopman models, we derive and use a Wiener-type Koopman formulation. We discuss that the Wiener structure is particularly suitable for model reduction, and can be naturally derived from Koopman theory. Moreover, the Wiener block-structure unifies the mathematical simplicity of linear dynamical blocks and the accuracy of bilinear dynamics. We present a Koopman deep-learning strategy combining autoencoders and linear dynamics that generates low-order surrogate models of MIMO Wiener type. In three case studies, we apply our framework for identification and reduction of a system with input multiplicity, a chemical reactor and a high-purity distillation column. We compare the prediction performance of the identified Wiener models to linear and bilinear Koopman models. We observe the highest accuracy and strongest model reduction capabilities of low-order Wiener-type Koopman models, making them promising for control.

Real-time Human Response Prediction Using a Non-intrusive Data-driven Model Reduction Scheme

Recent research in non-intrusive data-driven model order reduction (MOR) enabled accurate and efficient approximation of parameterized ordinary differential equations (ODEs). However, previous studies have focused on constant parameters, whereas time-dependent parameters have been neglected. The purpose of this paper is to introduce a novel two-step MOR scheme to tackle this issue. In a first step, classic MOR approaches are applied to calculate a low-dimensional representation of high-dimensional ODE solutions, i.e., to extract the most important features of simulation data. Based on this representation, a long short-term memory (LSTM) is trained to predict the reduced dynamics iteratively in a second step considering the parameters at the respective time step. The potential of this approach is demonstrated on an occupant model within a car driving scenario. The reduced model's response to time-varying accelerations matches the reference data with high accuracy for a limited amount of time. Furthermore, real-time capability is achieved. Accordingly, it is concluded that the presented method is well suited to approximate parameterized ODEs and can handle time-dependent parameters in contrast to common methods.

Reduced order models for Lagrangian hydrodynamics

As a mathematical model of high-speed flow and shock wave propagation in a complex multimaterial setting, Lagrangian hydrodynamics is characterized by moving meshes, advection-dominated solutions, and moving shock fronts with sharp gradients. These challenges hinder the existing projection-based model reduction schemes from being practical. We develop several variations of projection-based reduced order model techniques for Lagrangian hydrodynamics by introducing three different reduced bases for position, velocity, and energy fields. A time-windowing approach is also developed to address the challenge imposed by the advection-dominated solutions. Lagrangian hydrodynamics is formulated as a nonlinear problem, which requires a proper hyper-reduction technique. Therefore, we apply the over-sampling DEIM and SNS approaches to reduce the complexity due to the nonlinear terms. Finally, we also present both a posteriori and a priori error bounds associated with our reduced order model. We compare the performance of the spatial and time-windowing reduced order modeling approaches in terms of accuracy and speed-up with respect to the corresponding full order model for several numerical examples, namely Sedov blast, Gresho vortices, Taylor–Green vortices, and triple-point problems.

Dynamics of a flexible body: a two-field formulation

A two-field formulation of the nonlinear dynamics of an elastic body is presented in which positions/orientations and the resulting velocity field are treated as independent. Combining a nonclassical description of elastic velocity that includes the convection velocity due to elastic deformation with floating reference axes minimizing the relative kinetic energy due to elastic deformation provides a fully uncoupled expression of kinetic energy. A transformation inspired by the classical Legendre transformation concept is introduced to develop the motion equations in canonical form. Finite element discretization is achieved using the same shape function sets for elastic displacements and velocities. Specific attention is brought to the discretization of the gyroscopic forces induced by elastic deformation. A model reduction strategy to construct superelement models suitable for flexible multibody dynamics applications is proposed, which fulfills the essential condition of orthogonality between a rigid body and elastic motions. The problem of expressing kinematic connections at superelement boundaries is briefly addressed. Two academic examples have been developed to illustrate some of the concepts presented.

Microscale Thermal Modelling of Multifunctional Composite Materials Made from Polymer Electrolyte Coated Carbon Fibres Including Homogenization and Model Reduction Strategies

Polymer electrolyte coated carbon fibres embedded in polymeric matrix materials represent a multifunctional material with several application scenarios. Structural batteries, thermal management materials as well as stiffness adaptive composites, made from this material, are exposed to significant joule heat, when electrical energy is transferred via the carbon fibres. This leads to a temperature increase of up to 100 K. The thermal behaviour of this composite material is characterized in this numerical study based on a RVE representation for the first time. Compared to classical fibre reinforced plastics, this material comprises a third material phase, the polymer electrolyte coating, covering each individual fibre. This material has not been evaluated for effective thermal conductivity, specific heat and thermal behaviour on the microscale before. Therefore, boundary conditions, motivated from applications, are applied and joule heating by the carbon fibres is included as heat source by an electro-thermal coupling. The resulting temperature field is discussed towards its effect on the mechanical behaviour of the material. Especially the temperature gradient is pronounced in thickness direction, leading to a temperature drop of 1 °Cmm, which needs to be included in thermal stress analysis in future thermo-mechanically coupled models. Another important emphasis is the identification of suitable homogenization and model reduction strategies in order to reduce the numerical effort spent on the thermal problem. Therefore, traditional analytical homogenization methods as well as a newly proposed “Two-Level Lewis-Nielsen” approach are discussed in comparison to virtually measured effective quantities. This extensive comparison of analytical and numerical methods is original compared to earlier works dealing with PeCCF composites. In addition, the accuracy of the new Two-Level Lewis-Nielsen method is found to fit best compared to classical methods. Finally, a first efficient and accurate 2D representation of the thermal behaviour of the PeCCF composite is shown, which reduces computational cost by up to 97%. This benefit comes with a different Temperature drop prediction in thickness direction of 1.5 °Cmm. In the context of future modelling of multifunctional PeCCF composite materials with multiphysical couplings, this deviation is acceptable with respect to the huge benefit for computational cost.

Time-domain impedance boundary conditions for acoustic reduced order finite element simulations

Transient responses impose additional restrictions concerning model order reduction of acoustic finite element systems. Time-stable model order reduction methods achieve model compaction while guaranteeing frequency domain models transform in a physically meaningful way. Efficiency and stability are of course of little consequence if the model is rendered inaccurate. Krylov subspaces inherently include system input and/or output behavior in the reduction basis making them ideal reduction bases for investigating system behavior outside of steady state. Realistic boundary conditions are demanded and must be preserved in the reduction basis. Frequency dependent impedance boundary conditions help in this regard but complicate both model reduction and time-integration strategies. Multiplications to enforce system damping in the frequency domain become time-domain convolutions. Recursively calculated minimal memory convolution formulations have long proven useful in lowering the associated computational burden. Complex frequency-dependent damping matrices create a challenge for Krylov subspace based model reduction due to the way the reduction basis is constructed. Arnoldi iterations implicitly match the moments of the system transfer function to span a Krylov subspace. This paper demonstrates how to maintain compatibility with such algorithms while including frequency dependent damping. This work proposes combining projection based model order reduction with an efficient time domain-impedance boundary condition formulation. An important benefit of working in the time domain is the ability to directly output binaural audio signals. To this end, discrepancies are discussed in the perceptual context of audibility. A reduction of system degrees of freedom from NDOF=13125 to RDOF=63 and the inclusion of time-domain impedance boundary conditions are shown to enable computational speedups by a factor of 11–36 without introducing audible differences.

Efficient Wildland Fire Simulation via Nonlinear Model Order Reduction

We propose a new hyper-reduction method for a recently introduced nonlinear model reduction framework based on dynamically transformed basis functions and especially well-suited for transport-dominated systems. Furthermore, we discuss applying this new method to a wildland fire model whose dynamics feature traveling combustion waves and local ignition and is thus challenging for classical model reduction schemes based on linear subspaces. The new hyper-reduction framework allows us to construct parameter-dependent reduced-order models (ROMs) with efficient offline/online decomposition. The numerical experiments demonstrate that the ROMs obtained by the novel method outperform those obtained by a classical approach using the proper orthogonal decomposition and the discrete empirical interpolation method in terms of run time and accuracy.

Sparce Subspace Learning and Characteristic Based Split for Modelling Artificial Ground Freezing

Artificial Ground Freezing technique is often used to build underground structures, such as urban tunnels. modeling activities are necessary to optimize the design of the freezing process. The paper introduces a numerical solution of the phenomenon of water freezing for various soil types, using the finite element method combined with the model order reduction technique. In particular, the characteristic based split method is used as a base to build a model reduction scheme by means of Sparse Subspace Learning. The aim of this study is to develop a mathematical model able to carry out numerical results with negligible CPU time, by varying three parameters reproducing the characteristics of the soil, i.e. the porosity, the Darcy number and the Rayleigh number. The obtained results show how the soil parameters influence the freezing process and the dimension of ice wall.

A dual reduction strategy for reduce-order modeling of periodic control system

Model order reduction (MOR) of periodic systems using the Krylov subspace methods received lots of interest in last few decades. In this paper, a structured Krylov subspace based model reduction for linear discrete-time periodic (LDTP) control system has been proposed using the corresponding lifted form. The proposed model reduction strategy first finds a reduced order Lyapunov equation via a projection where the projection subspace is computed by using the periodic Arnoldi method. Then, by applying the balanced truncation model reduction approach, it forms a reduced order model of the original system. In both cases, the periodic structure is ensured in the solutions and systems’ dynamics. This paper also presents numerical results to confirm the efficiency and accuracy of the proposed algorithm.

The development of F-76 and algae-derived HRD-76 diesel fuel surrogates

The surrogate models for HRD-76 and F-76 diesel fuels to capture their combustion properties were developed in the present study. n-C16H34 (n-hexadecane), C15H32-26X (2,6,10-trimethyl dodecane) and C6H5C4H9 (n-butylbenzene) were chosen as surrogate components, and formulated by the FGBS (Function Group Based Surrogate Fuel) methodology. The HRD-76 surrogate consists of 58.0% n-C16H34 and 42.0% C15H32-26X, and the F-76 surrogate consists of 41.0% n-C16H34, 24.0% C15H32-26X and 35.0% C6H5C4H9 by mole fraction, were determined by directly matching four functional groups that obtained from the NMR (Nuclear Magnetic Resonance) spectroscopy analysis. A detailed kinetic mechanism was constructed and then reduced by a kinetic model reduction scheme. After that, the merged surrogate kinetic mechanism was verified against various experiments for each individual surrogate component. Furthermore, the HRD-76 and F-76 surrogate fuel models which contain the determined surrogate formulation and developed surrogate mechanism were validated against experiments. The agreements between the simulation results and the experiments on ignition delay times (IDTs), laminar flame speeds (Su0) and extinction strain rates (Kext) are reasonably good. Based on sensitivity analysis, key reactions of HRD-76 and F-76 surrogate fuels were compared and analyzed. This study can help understand the combustion characteristics of HRD-76 and F-76 fuels in deeply, and lay the foundation for the engine combustion simulation using the CFD.

A model reduction method for fast finite element analysis of continuously symmetric waveguides

This paper presents a model reduction strategy for fast finite element analysis of continuously symmetric elastic waveguides. Accounting for continuous symmetry, the initial three-dimensional problem is first reduced to a two-dimensional quadratic eigenvalue problem based on a semi-analytical finite element method. This problem has to be solved repeatedly at multiple frequencies to determine the modal properties of the waveguide as functions of frequency (e.g. dispersion curves). Although two-dimensional, such a problem becomes costly when the problem size increases. In this paper, a specific reduction basis is proposed to project the matrix operators of the eigenvalue problem. This reduced basis is built from complex eigenvectors computed at the maximum frequency of analysis, enriched by cut-off eigenvectors, and then post-processed through a QR decomposition. Numerical results are presented for three critical test cases, involving high-frequency viscoelastic loss, curve veering phenomena, or leakage loss respectively. The reduction method turns out to be simple to implement, fast, and accurate for the complex spectrum, including attenuated waves. The time needed to build the reduced model, prior to its solution, remains short (roughly equal to a single frequency iteration of the full model). Once reduced, the eigenvalue problem can be solved more than 50 times faster than the full eigenvalue problem.

An efficient optimization framework for tracking multiple quality attributes in supply chains of perishable products

Improved supply chain optimization strategies can play a major role in addressing global food security and safety in years to come. In particular, tighter safety regulations, changing consumer quality requirements and more stringent market competition call upon integrated supply chain decision-making frameworks that explicitly consider product quality control. This effort requires metrics of quality that accurately reflect product physico-chemical properties, as well as consumer purchasing preferences. However, a critical challenge linked to embedding the complex dynamics of the evolution of product quality in time within supply chain models is the large-scale nature of the ensuing optimization problems, which are computationally intractable even for moderate-size, single-item systems. In the present work, we introduce a computationally efficient optimal production and distribution planning framework for perishable products having multiple quality attributes that evolve in time as a function of environmental conditions during shipment and storage. We also propose a model reduction strategy and a decomposition framework that enhance the scalability of our approach. We perform extensive numerical simulations using different network instances to validate our theoretical findings, as well as to demonstrate the advantages of the proposed supply chain management scheme.

A Reformulation of Degree of Disequilibrium Analysis for Automatic Selection of Kinetic Constraints in the Rate-Controlled Constrained-Equilibrium Method

Abstract The rate-controlled constrained equilibrium (RCCE) is a model reduction scheme for chemical kinetics. It describes the evolution of a complex chemical system with acceptable accuracy with a number of rate controlling constraints on the associated constrained-equilibrium states of the system, much lower than the number of species in the underlying detailed kinetic model (DKM). Successful approximation of the constrained-equilibrium states requires accurate identification of the constraints. One promising procedure is the fully automatable Approximate Singular Value Decomposition of the Actual Degrees of Disequilibrium (ASVDADD) method that is capable of identifying the best constraints for a given range of thermodynamic conditions and a required level of approximation. ASVDADD is based on simple algebraic analysis of the results of the underlying DKM simulation and is focused on the behavior of the degrees of disequilibrium (DoD) of the individual chemical reactions. In this paper, we introduce an alternative ASVDADD algorithm. Unlike the original ASVDADD algorithm that require the direct computation of the DKM-derived DoDs and the identification of the set of linearly independent reactions, in the alternative algorithm, the components of the overall degree of disequilibrium vector can be computed directly by casting the DKM as an RCCE simulation considering a set of linearly independent constraints equaling the number of chemical species in size. The effectiveness and robustness of the derived constraints from the alternative procedure is examined in hydrogen/oxygen and methane/oxygen ignition delay simulations and the results are compared with those obtained from DKM.

Data-driven phase-isostable reduction for optimal nonfeedback stabilization of cardiac alternans.

Phase-isostable reduction is an emerging model reduction strategy that can be used to accurately replicate nonlinear behaviors in systems for which standard phase reduction techniques fail. In this work, we derive relationships between the cycle-to-cycle variance of the reduced isostable coordinates for systems subject to both additive white noise and periodic stimulation. Using this information, we propose a data-driven technique for inferring nonlinear terms of the phase-isostable coordinate reduction framework. We apply the proposed model inference strategy to the biologically motivated problem of eliminating cardiac alternans, an arrhythmia that is widely considered to be a precursor to more deadly cardiac arrhythmias. Using this strategy, by simply measuring a series of action potential durations in response to periodic stimulation, we are able to identify energy-optimal, nonfeedback control inputs to stabilize a period-1, alternans-free solution.

Dynamic clustering-based model reduction scheme for damping control of large power systems using series compensators from wide area signals

This paper presents a new scheme of thyristor controlled series compensators (TCSC)-based damping controller for damping inter-area oscillations in bulk power systems based on a wide area measurement system (WAMS). In this regard, the proposed scheme provides an adaptive three-phase step through real-time working mode. In the first step, by proposing dynamic clustering-based technique, wide area signals evaluate the inter-area oscillations of the system which in a real-time procedure identify the corresponding oscillating areas. In the case of reducing the order of power systems through a set of aggregated coherent areas, the proposed scheme implements through the second step which utilizes as a damping controller for damping the inter-area oscillations. In this step, the wide area electromechanical signals use as input data through the proposed TCSC-based damping controller. The corresponding controlling signals consist of inter-area rotor angle (ΔδCOI) and inter-area speed deviation (ΔωCOI) evaluated through center of inertia (COI) frame. In the third step, at each time window (ΔT) through real-time working mode, the proposed scheme examines through a set of fault occurrences which based on online evaluations of the system dynamic responses, the controller parameters adjust adaptively. The proposed controller is an online and non-model-based scheme which properly reduces the system order through a set of coherent areas to provide proper damping performances of inter-area oscillations. The effectiveness of the proposed scheme evaluates through the Iran National Power Grid with the potential of two oscillating areas with proper damping performances for damping the unstable inter-area oscillations.

Comparison of Cycle Reduction and Model Reduction Strategies for the Design Optimization of Hybrid Powertrains on Driving Cycles

Decision-making is a crucial and difficult step in the design process of complex systems such as the hybrid powertrain. Finding an optimal solution requires the system feedback. This can be, depending on the granularity of the models at the component level, highly time-consuming. This is even more true when the system’s performance is determined by its control. In fact, various possibilities can be selected to deliver the required torque to the wheels during a driving cycle. In this work, two different design strategies are proposed to minimize the fuel consumption and the cost of the hybrid powertrain. Both strategies adopt the iterative framework which allows for the separation of the powertrain design problem and its control while leading to system optimality. The first approach is based on model reduction, while the second approach relies on improved cycle reduction techniques. They are then applied to a parallel hybrid vehicle case study, leading to important cost reduction in reasonable delays and are compared using different metrics.

Mixed Positive-Bounded Balanced Truncation

Reducing the order of positive real and bounded real systems has received great interest over the years. Several methods have been published regarding these issues which are able to produce positive real or bounded real reduced models. However, none of them can guarantee that the reduced models are both positive real and bounded real at the same time. In this brief, a novel model reduction scheme is presented which focuses on preserving two possible properties of the system (positive realness and bounded realness) simultaneously. This is done by using one positive real Riccati equation and one bounded real Riccati equation in the reduction procedure. Thereafter, the gramians extracted from these Riccati equations are balanced based on the Balanced Truncation (BT) concept. Finally, a numerical example is provided to investigate the efficacy of the proposed method.

Reducing Stochastic Discrete Models of Biochemical Networks

Biochemical systems have numerous practical applications, in particular to the study of critical intracellular processes. Frequently, biochemical kinetic models depict cellular processes as systems of chemical reactions. Many biological processes in a cell are inherently stochastic, due to the existence of some low molecular amounts. These stochastic fluctuations may have a great effect on the biochemical system’s behaviour. In such cases, stochastic models are necessary to accurately describe the system’s dynamics. Biochemical systems at the cellular level may entail many species or reactions and their mathematical models may be non-linear and with multiple scales in time. In this work, we provide a numerical technique for simplifying stochastic discrete models of well-stirred biochemical systems, which ensures that the main properties of the original system are preserved. The proposed technique employs sensitivity analysis and requires solving an optimization problem. The numerical tests on several models of practical interest show that our model reduction strategy performs very well.