Order Reduction Method Research Articles

RETRACTED ARTICLE: Model Order Reduction Method Based on Machine Learning for Parameterized Time-Dependent Partial Differential Equations

RETRACTED ARTICLE: Model Order Reduction Method Based on Machine Learning for Parameterized Time-Dependent Partial Differential Equations

A POD Reduced Order Method Applied to an Instability Problem of Rayleigh–Bénard Convection

The application of a proper orthogonal decomposition (POD) reduced order method to a two-dimensional instability problem of Rayleigh–Bénard convection is described in this work. The partial differential equations that model the problem along with the corresponding eigenvalue problems of the linear stability analysis of stationary solutions are numerically solved for different values of the bifurcation parameter, i.e. the Rayleigh number [Formula: see text]. The POD reduced order method considers spaces derived from different stationary solutions depending on [Formula: see text] as snapshots. The eigenvalue with the largest real part and its corresponding eigenfunction are calculated for the POD solutions. The resulting matrices of the eigenvalue problems are low-dimensional. There is a drastic reduction in the computational cost.

Efficient Long-Term Simulation of the Heat Equation with Application in Geothermal Energy Storage

Long-term evolutions of parabolic partial differential equations, such as the heat equation, are the subject of interest in many applications. There are several numerical solvers marking the state-of-the-art in diverse scientific fields that may be used with benefit for the numerical simulation of such long-term scenarios. We show how to adapt some of the currently most efficient numerical approaches for solving the fundamental problem of long-term linear heat evolution with internal and external boundary conditions as well as source terms. Such long-term simulations are required for the optimal dimensioning of geothermal energy storages and their profitability assessment, for which we provide a comprehensive analytical and numerical model. Implicit methods are usually considered the best choice for resolving long-term simulations of linear parabolic problems; however, in practice the efficiency of such schemes in terms of the combination of computational load and obtained accuracy may be a delicate issue, as it depends very much on the properties of the underlying model. For example, one of the challenges in long-term simulation may arise by the presence of time-dependent boundary conditions, as in our application. In order to provide both a computationally efficient and accurate enough simulation, we give a thorough discussion of the various numerical solvers along with many technical details and own adaptations. By our investigation, we focus on two largely competitive approaches for our application, namely the fast explicit diffusion method originating in image processing and an adaptation of the Krylov subspace model order reduction method. We validate our numerical findings via several experiments using synthetic and real-world data. We show that we can obtain fast and accurate long-term simulations of typical geothermal energy storage facilities. We conjecture that our techniques can be highly useful for tackling long-term heat evolution in many applications.

A hyper-reduced MAC scheme for the parametric Stokes and Navier-Stokes equations

The need for accelerating the repeated solving of certain parametrized systems motivates the development of more efficient reduced order methods. The classical reduced basis method is popular due to an offline-online decomposition and a mathematically rigorous a posterior error estimator which guides a greedy algorithm offline. For nonlinear and nonaffine problems, hyper reduction techniques have been introduced to make this decomposition efficient. However, they may be tricky to implement and often degrade the offline and online computational efficiency.To avoid this degradation, reduced residual reduced over-collocation (R2-ROC) was invented. It integrates the empirical interpolation techniques on the solution snapshots and the well-chosen residuals, the collocation philosophy, and the simplicity of evaluating the hyper-reduced well-chosen residuals. In this paper, we introduce an adaptive enrichment strategy for R2-ROC rendering it capable of handling parametric fluid flow problems. Built on top of an underlying Marker and Cell (MAC) scheme, a novel hyper-reduced MAC scheme is therefore presented and tested on lid-driven cavity and flow past a backward-facing step problems demonstrating its high efficiency, stability and accuracy.

MicroROM: An efficient and accurate reduced order method to solve many-query problems in micro-motility

In the study of micro-swimmers, both artificial and biological ones, many-query problems arise naturally. Even with the use of advanced high performance computing (HPC), it is not possible to solve this kind of problems in an acceptable amount of time. Various approximations of the Stokes equation have been considered in the past to ease such computational efforts but they introduce non-negligible errors that can easily make the solution of the problem inaccurate and unreliable. Reduced order modeling solves this issue by taking advantage of a proper subdivision between a computationally expensive offline phase and a fast and efficient online stage. This work presents the coupling of Boundary Element Method (BEM) and Reduced Basis (RB) Reduced Order Modeling (ROM) in two models of practical interest, obtaining accurate and reliable solutions to different many-query problems. Comparisons of standard reduced order modeling approaches in different simulation settings and a comparison to typical approximations to Stokes equations are also shown. Different couplings between a solver based on a HPC boundary element method for micro-motility problems and reduced order models are presented in detail. The methodology is tested on two different models: a robotic-bacterium-like and an Eukaryotic-like swimmer, and in each case two resolution strategies for the swimming problem, the split and monolithic one, are used as starting points for the ROM. An efficient and accurate reconstruction of the performance of interest is achieved in both cases proving the effectiveness of our strategy.

A refined Bloch mode synthesis framework for fast and accurate analysis of electroelastic metamaterials with piezoelectric resonant shunt damping

Bloch mode synthesis (BMS) is a model order reduction method that was originally formulated to accelerate band-structure calculations of periodic elastic materials. In this paper, the BMS framework is expanded to offer a platform suitable for analysis of electroelastic metamaterials with piezoelectric resonant shunt damping. First, a transformation of coordinates formed by a set of dominant fixed-interface modes and a set of static-constraint modes is used in combination with Bloch periodicity boundary conditions to derive the classical BMS method in the context of electroelastic metamaterials. Then, a set of residual modes that is simply truncated in the classical BMS, is augmented into the reduction process, leading to an enhanced BMS method, denoted RMH-BMS.Currently available model order reduction techniques for electroelastic metamaterials adopt a reduction basis obtained from one of the limiting eigenvalue problems associated with short-circuit (SC) and open-circuit (OC) boundary conditions. In either case, the electrical circuit parameters are ignored during modal reduction. In this paper, it is shown that such approximation leads to inaccurate band-structure predictions for electroelastic metamaterials equipped with periodic array of resonant shunt circuits. To this end, two versions of the proposed BMS methods are evaluated and compared. In the first version, the reduced order model is constructed following the widely used OC basis approach. The second version, on the other hand, exploits a coupled basis approach, which requires knowledge of the electrical circuit parameters prior to the interior modal reduction step. A comprehensive error analysis is conducted to assess the computational performances of the BMS methods, and the results revealed that whether the classical BMS or the RMH-BMS is employed to construct the reduced order model, the coupled basis approach produces significantly more accurate results compared to the OC basis approach. The results presented in this work expose the limitations of implementing the OC basis approach in wave propagation analysis of electroelastic metamaterials with piezoelectric resonant shunt damping and shed light on the associated consequences in predicting the bandgap properties of such metamaterials.

A non-intrusive neural network model order reduction algorithm for parameterized parabolic PDEs

Reduced-order modeling based on projection-driven neural network (PDNN) generally needs sufficient data set while physics-informed machine learning (PINN) and physics-reinforced neural network (PRNN) take the reduced order systems into consideration. However, the physics-informed machine learning technique used in these two methods gives rise to expensive time consumption for complex neural network, higher reduced basis and a large amount of residual points. With understanding of PDNN, PINN and PRNN, a model-based neural network (MBNN) is proposed to cope with nonlinear parabolic partial differential equations (PDEs) without source terms. Compared with PINN, the fully discrete scheme of PDEs is adopted to avoid expensive cost for automatic differentiation technique. Moreover, initial conditions at residual points in parameter space are added to loss function with a proper proportion. Since the reduced order equation is taken into account, the proposed MBNN can predict solutions in a larger time range than the time range to which the snapshot belongs. Numerical results show that MBNN achieves better performance than the projection-driven neural network. Generally, the proposed method for lower order reduction can get a consistent L2 error with POD after the convergent tolerance is reached.

A new reduced order model to represent the creep induced fuel assembly bow in PWR cores

Progressive deformation of nuclear fuel assemblies that occurs during a succession of irradiation cycles within PWR is at the core of numerous interactions. Progresses in the study of this phenomenon resulted from more and more complex models. To date, in the more advanced models, a mechanical model is combined to a hydraulic model of the core to address the fluid–structure interaction. A complete mechanical model of the core is generally built with a simplified finite element representation of each assembly using beam or simpler elements. The simplifying approach is required due to the high complexity of the multi-body multi-physics calculations. In order to improve the representativeness of the core model without penalizing the computational cost of the coupled simulations, a new method of model order reduction is proposed. It is specially adapted to this context and borrows the concepts of NTFA non-linear homogenization. The theoretical foundations and the application to a slender structure are presented. First, the method is validated on a simple case with spatially homogeneous characteristics. Finally, the case of the creep response of a fuel assembly under realistic heterogeneous loads is demonstrated.

A recursive-cluster based reduced order method for numerical prediction of effective properties of heterogeneous viscoelastic materials

A novel recursive-cluster based reduced order method is presented to predict the effective properties of heterogeneous viscoelastic materials. By virtue of a piecewise series expansion based variable separation, a temporally and spatially homogenization process is decoupled into a series of recursive elastic homogenization ones, which require only one-time clustering decomposition with one-time elastic heterogeneous analysis. Such that, the heavy computational burden caused by the direct numerical simulation over the whole domain of viscoelastic heterogeneous RUC can significantly be alleviated. The effective creep compliance or relaxation modulus is presented via a time varying average strain or average stress, and a piecewise adaptive technique is proposed to secure the temporal accuracy when step size varies. Numerical examples are provided to demonstrate the effectiveness and efficiency of proposed approach, and the predicted results agree well with those given by FEM/SBFEM based viscoelastic DNS. • Propose a recursive-cluster based reduced order method to evaluate effective properties of heterogeneous viscoelastic materials. • Decouple a temporally and spatially homogenization process into a series of recursive elastic homogenization ones. • Take only one-time clustering decomposition with one-time elastic heterogeneous analysis. • Execute a piecewise adaptive computing to gain the effective strains of clusters on RUC. • Circumvent the heavy computational burden caused by the high-fidelity viscoelastic DNS on RUC significantly.

Control effects of a high-frequency pulsed discharge on a hypersonic separated flow

The control effects of a high-frequency pulsed discharge on a hypersonic separated flow induced by a 42° compression ramp are studied with high-speed schlieren imaging. A discharge is generated between an electrode pair upstream of the ramp to introduce strong disturbances into the boundary layer. The discharge frequencies used are fd = 10, 20, 30, and 40 kHz, with corresponding single-pulse energies Esp = 22.2, 20.3, 12.0, and 8.7 mJ. The results show that the discharge can reduce the separation area, especially with relatively low frequencies of 10 or 20 kHz. The flow mechanism induced by the discharge is scrutinized using a reduced-order method based on the information from time-resolved schlieren images, including proper orthogonal decomposition and dynamic mode decomposition. The results identify an unsteady mode representing regular energetic eddies within the shear layer of the separation zone. This mode has the same characteristic frequency as the discharge. These eddies enhance the mixing of the shear layer and the main flow stream, accelerating the destruction of the shear layer and reducing the separation zone size. The control effectiveness correlates with the scale of the energetic eddies. At frequencies of 10 or 20 kHz, the discharge is more conducive to attenuating the separation because large-scale eddies, excited by higher single-pulse energy, are enhanced as they propagate downstream in the shear layer. By contrast, the control effectiveness deteriorates when the discharge frequency is 30 or 40 kHz because the induced eddies have smaller scales and decay during propagation.

Assessment and mitigation of DC bias risk in power grids at provincial boundaries

With the wide application of high voltage/ultra-high voltage (HV/UHV) DC transmission technology, the impact of DC grounding electrode location selection on the surrounding power grid has become increasingly prominent, especially the problem of DC bias hazard caused by DC grounding electrodes at provincial grid boundaries needs to be solved urgently. This paper studies the assessment and prevention of trans-regional DC bias risk, and proposes an inversion method of earth resistivity model based on the measured data of neutral current in the provincial boundary area. Firstly, the DC bias risk of provincial boundary power grid is simulated and calculated, and the influence of reasonable selection of earth model on the accuracy of risk assessment results is explained. Based on the classical Fletcher-Reeves conjugate gradient method and the measured neutral current data, a reduced-order inversion method for the earth resistivity parameters of the provincial boundary power grid is proposed. On this basis, a set of DC bias risk control scheme is formulated for the actual project of Gannan power grid. Finally, the feasibility of the scheme is verified by simulation and measured data.

Model order reduction method based on (r)POD-ANNs for parameterized time-dependent partial differential equations

In this paper, we propose a non-intrusive data-driven model order reduction method, also known as (r)POD-ANNs model order reduction method. In this method, (r)POD priori dimension reduction is performed on high-fidelity data set, and then the mapping relationship between parameter space and generalized coordinates of (r)POD is implicitly constructed by using the universal approximation property of neural network. Through pre-training on small batch data sets, updating neural network parameters every several time steps, and combination of L-BFGS optimization algorithm and LM optimization algorithm, time cost of the reduced order model in the off-line calculation stage is reduced. This makes the (r)POD-ANNs model order reduction method suitable for high-fidelity models with larger spatial degrees of freedom and higher complexity. Finally, we verify effectiveness of the proposed method by comparing with the data-driven model reduction method, and then apply it to the Allen-Cahn equations with strong nonlinearity and cylinder flow problem with large spatial degrees of freedom.

A New Passivity Preserving Model Order Reduction Method: Conic Positive Real Balanced Truncation Method

This article is dedicated to model order reduction of linear time-invariant passive systems, i.e., positive real (PR) systems, based on the balanced truncation (BT) concept. The main feature of this article is that we have used the phase angle of the transfer function, <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$G(s)$ </tex-math></inline-formula> , in model order reduction for the first time in the proposed method, which results in more accurate reduced-order models. It is worth noting that this hypothesis has been never made before in any other model order reduction techniques. Furthermore, new gramians are calculated corresponding to their associated new Riccati equations and Lur’e equations, which are in turn achieved by using Kalman–Yakubovic–Popov (KYP) lemma and linear matrix inequalities (LMI). Thereafter, a novel passivity preserving model order reduction algorithm, which takes the phase angle of the transfer function, <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$G(s)$ </tex-math></inline-formula> , into account, is presented. It is demonstrated that the proposed method is a generalization of the positive real balanced truncation (PRBT) method, and it is perfectly capable of providing more accurate approximation error compared to PRBT. Finally, numerical examples are included to figure out the effectiveness of the presented method.

Model Order Reduction for Water Quality Dynamics

AbstractA state‐space representation of water quality (WQ) dynamics describing disinfectant (e.g., chlorine) transport dynamics in drinking water distribution networks has been recently proposed. Such representation is a byproduct of space‐ and time‐discretization of the partial differential equations modeling transport dynamics. This results in a large state‐space dimension even for small networks with tens of nodes. Although such a state‐space model provides a model‐driven approach to predict WQ dynamics, incorporating it into model‐based control algorithms or state estimators for large networks is challenging and at times intractable. To that end, this paper investigates model order reduction (MOR) methods for WQ dynamics with the objective of performing post‐reduction feedback control. The presented investigation focuses on reducing state‐dimension by orders of magnitude, the stability of the MOR methods, and the application of these methods to model predictive control.

Model order reduction for bilinear systems with non-zero initial states – different approaches with error bounds

In this paper, we consider model order reduction for bilinear systems with non-zero initial conditions. We discuss the choices of Gramians for both the homogeneous and the inhomogeneous parts of the system individually and prove how these Gramians characterise the respective dominant subspaces of each of the two subsystems. Proposing different, not necessarily structure preserving, reduced-order methods for each subsystem, we establish several strategies to reduce the dimension of the full system. For all these approaches, error bounds are shown depending on the truncated Hankel singular values of the subsystems. Besides the error analysis, stability is discussed. In particular, a focus is on a new criterion for the homogeneous subsystem guaranteeing the existence of the associated Gramians and an asymptotically stable realisation of the system.

Error bounds for model reduction of feedback-controlled linear stochastic dynamics on Hilbert spaces

We analyze structure-preserving model order reduction methods for Ornstein–Uhlenbeck processes and linear S(P)DEs with multiplicative noise based on balanced truncation. For the first time, we include in this study the analysis of non-zero initial conditions. We moreover allow for feedback-controlled dynamics for solving stochastic optimal control problems with reduced-order models and prove novel error bounds for a class of linear quadratic regulator problems. We provide numerical evidence for the bounds and discuss the application of our approach to enhanced sampling methods from non-equilibrium statistical mechanics.

Riemannian optimization model order reduction method for general linear port-Hamiltonian systems

Abstract This paper presents a Riemannian optimal model order reduction method for general linear stable port-Hamiltonian systems based on the Riemannian trust-region method. We consider the $\mathcal{H}_2$ optimal model order reduction problem of the general linear port-Hamiltonian systems. The problem is formulated as an optimization problem on the product manifold, which is composed of the set of skew symmetric matrices, the manifold of the positive definite matrices, the manifold of the positive semidefinite matrices with fixed rank and the Euclidean space. To solve the optimal problem, the Riemannian geometry of the product manifold is given. Moreover, the Riemannian gradient and the Riemannian Hessian of the objective function are derived. Furthermore, we propose the Riemannian trust-region method for the optimization problem and introduce the truncated conjugate gradient method to solve the trust-region subproblem. Finally, the numerical experiments illustrate the efficiency of the proposed method.

POD-Galerkin model order reduction for parametrized nonlinear time-dependent optimal flow control: an application to shallow water equations

AbstractIn the present paper we propose reduced order methods as a reliable strategy to efficiently solve parametrized optimal control problems governed by shallow waters equations in a solution tracking setting. The physical parametrized model we deal with is nonlinear and time dependent: this leads to very time consuming simulations which can be unbearable, e.g., in a marine environmental monitoring plan application. Our aim is to show how reduced order modelling could help in studying different configurations and phenomena in a fast way. After building the optimality system, we rely on a POD-Galerkin reduction in order to solve the optimal control problem in a low dimensional reduced space. The presented theoretical framework is actually suited to general nonlinear time dependent optimal control problems. The proposed methodology is finally tested with a numerical experiment: the reduced optimal control problem governed by shallow waters equations reproduces the desired velocity and height profiles faster than the standard model, still remaining accurate.

Generation and verification of vertical meteorological data for building energy simulation from a 325-meter Beijing meteorological tower

As the number of high-rise buildings increases, so does the demand for meteorological data at different heights. The difference in meteorological elements in the vertical direction should be considered in heating, ventilation, and air conditioning design and building performance simulations (BPS). This study uses a meteorological tower in Beijing as a model to provide a reduced-order method for data of typical meteorological years (TMY) and outdoor air design condition for high-rise buildings. The vertically-translated TMY results obtained by this method could achieve load simulation results close to the average load level of the observed station meteorological data. The difference between the cumulative heat load obtained from vertically-translated TMY data and the average cumulative heat load obtained from observed station meteorological data was 0.82–6.08%. By verifying the vertically-translated data, we observed that the vertically-translated outdoor air design condition at different altitudes were consistent with the direct statistical results based on observation data. This reduced-order method is helpful for modifying energy consumption simulations and the energy-saving designs of high-rise buildings.

Rank-adaptive structure-preserving model order reduction of Hamiltonian systems

This work proposes an adaptive structure-preserving model order reduction method for finite-dimensional parametrized Hamiltonian systems modeling non-dissipative phenomena. To overcome the slowly decaying Kolmogorov width typical of transport problems, the full model is approximated on local reduced spaces that are adapted in time using dynamical low-rank approximation techniques. The reduced dynamics is prescribed by approximating the symplectic projection of the Hamiltonian vector field in the tangent space to the local reduced space. This ensures that the canonical symplectic structure of the Hamiltonian dynamics is preserved during the reduction. In addition, accurate approximations with low-rank reduced solutions are obtained by allowing the dimension of the reduced space to change during the time evolution. Whenever the quality of the reduced solution, assessed via an error indicator, is not satisfactory, the reduced basis is augmented in the parameter direction that is worst approximated by the current basis. Extensive numerical tests involving wave interactions, nonlinear transport problems, and the Vlasov equation demonstrate the superior stability properties and considerable runtime speedups of the proposed method as compared to global and traditional reduced basis approaches.