Application Of Model Order Reduction Research Articles

A stochastic averaging mathematical framework for design and optimization of nonlinear energy harvesters with several electrical DOFs

Energy harvesters for mechanical vibrations are electro-mechanical systems designed to capture ambient dispersed kinetic energy, and to convert it into usable electrical power. The random nature of mechanical vibrations, combined with the intrinsic non-linearity of the harvester, implies that long, time domain Monte-Carlo simulations are required to assess the device performance, making the analysis burdensome when a large parameter space must be explored. Therefore a simplified, albeit approximate, semi-analytical analysis technique is of paramount importance. In this work we present a methodology for the analysis and design of nonlinear piezoelectric energy harvesters for random mechanical vibrations. The methodology is based on the combined application of model order reduction, to project the dynamics onto a lower dimensional space, and of stochastic averaging, to calculate the stationary probability density function of the reduced variables. The probability distribution is used to calculate expectations of the most relevant quantities, like output voltage, harvested power and power efficiency. Based on our previous works, we consider an energy harvester with a matching network, interposed between the harvester and the load, that reduces the impedance mismatch between the two stages. The methodology is applied to the optimization of the matching network, allowing to maximize the global harvested power and the conversion efficiency. We show that the proposed methodology gives accurate predictions of the harvester’s performance, and that it can be used to significantly simplify the analysis, design and optimization of the device.

STOCHASTIC GALERKIN METHOD AND PORT-HAMILTONIAN FORM FOR LINEAR FIRST-ORDER ORDINARY DIFFERENTIAL EQUATIONS

We consider linear first-order systems of ordinary differential equations (ODEs) in port-Hamiltonian (pH) form. Physical parameters are remodeled as random variables to conduct an uncertainty quantification. A stochastic Galerkin projection yields a larger deterministic system of ODEs, which does not exhibit a pH form in general. We apply transformations of the original systems such that the stochastic Galerkin projection becomes structure-preserving. Furthermore, we investigate meaning and properties of the Hamiltonian function belonging to the stochastic Galerkin system. A large number of random variables implies a high-dimensional stochastic Galerkin system, which suggests itself to apply model order reduction (MOR) generating a low-dimensional system of ODEs. We discuss structure preservation in projection-based MOR, where the smaller systems of ODEs feature pH form again. Results of numerical computations are presented using two test examples.

Adaptive POD-DEIM correction for Turing pattern approximation in reaction–diffusion PDE systems

Abstract We investigate a suitable application of Model Order Reduction (MOR) techniques for the numerical approximation of Turing patterns, that are stationary solutions of reaction–diffusion PDE (RD-PDE) systems. We show that solutions of surrogate models built by classical Proper Orthogonal Decomposition (POD) exhibit an unstable error behaviour over the dimension of the reduced space. To overcome this drawback, first of all, we propose a POD-DEIM technique with a correction term that includes missing information in the reduced models. To improve the computational efficiency, we propose an adaptive version of this algorithm in time that accounts for the peculiar dynamics of the RD-PDE in presence of Turing instability. We show the effectiveness of the proposed methods in terms of accuracy and computational cost for a selection of RD systems, i.e., FitzHugh–Nagumo, Schnakenberg and the morphochemical DIB models, with increasing degree of nonlinearity and more structured patterns.

Applying Model Order Reduction Algorithm for Control Design of the Digital Filter

Applying Model Order Reduction Algorithm for Control Design of the Digital Filter

Application of model order reduction with Cauer ladder networks to industrial inductors

PurposeThe Cauer ladder network (CLN) model order reduction (MOR) method is applied to an industrial inductor. This paper aims to to anaylse the influence of different meshes on the CLN method and their parameters.Design/methodology/approachThe industrial inductor is simulated with the CLN method for different meshes. Meshes considering skin effect are compared with equidistant meshes. The inductor is also simulated with the eddy current finite element method (ECFEM) for frequencies 1 kHz to 1 MHz. The solution of the CLN method is compared with the ECFEM solutions for the current density in the conductor and the total impedance.FindingsThe increase of resistance resulting from the skin effect can be modelled with the CLN method, using a uniform mesh with elements much larger than the skin depth. Meshes taking account of the skin depth are only needed if the electromagnetic fields have to be reconstructed. Additionally, the convergence of the impedance is used to define a stopping criterion without the need for a benchmark solution.Originality/valueThe work shows that the CLN method can generate a network, which is capable of mimicking the increase of resistance usually accompanied by the skin effect without using a mesh that takes the skin depth into account. In addition, the proposed stopping criterion makes it possible to use the CLN method as an a priori MOR technique.

Reduced Flow Model for Plastics Melt Inside an Extrusion Die

AbstractIn this work, we study the application of Model Order Reduction (MOR) to the flow of plastics melt in an extrusion die. The process is described with a non‐linear model consisting of the Stokes equations in combination with a shear‐thinning material model. The reduction is performed through a Proper Orthogonal Decomposition (POD) approach with subsequent projection. We perform an a‐posteriori error analysis for a quantity of interest, based on the full‐ and reduced‐order solutions. Finally, we explore the computational speedup to prove the aptitude of the reduced model in the context of many‐query scenarios.

Proper orthogonal decomposition for reduced order dynamic modeling of vapor compression systems

A computationally efficient but accurate dynamic modeling approach for vapor compression systems is important for many applications. Nonlinear model order reduction techniques which generate reduced order models based on high fidelity vapor compression cycle (VCC) models are attractive for the purposes. In this paper, a number of technical challenges of applying model order reduction methods to VCCs are described and corresponding solution approaches are presented. It starts with a reformulation of a standard finite volume heat exchanger model for matching the baseline model reduction structure. Reduced order models for evaporator and condenser are constructed from numerical snapshots of the high fidelity models using Proper Orthogonal Decomposition (POD). Methodologies for system stability and numerical efficiency of POD reduced order models are described. The reduced order heat exchanger models are then coupled with quasi-static models of other components to form a reduced order cycle model. Transient simulations were conducted over a wide range of operating conditions and results were compared with the full order model as well as measurements. The validation results indicate that the reduced order model can execute much faster than a high-fidelity finite volume model with negligible prediction errors.

Real-time prediction of the deformation of microcapsules using Proper Orthogonal Decomposition

Microcapsules are liquid droplets enclosed by a thin elastic membrane. Being suspended in an external fluid, they undergo large deformations when flowing. Their deformation can be solved numerically, but the resolution of the fluid–structure interactions (FSI) requires extremely long computation times. This is a major constraint for instance when identifying the membrane mechanical properties from experimentations of microcapsules flowing in a microfluidic channel of similar size (Hu et al. PFE 2013). We propose to apply Model Order Reduction (MOR) to predict in real time the steady-state capsule deformed shape, needed to determine the membrane elasticity. A database of the capsule deformed shapes was obtained numerically by solving the three-dimensional (3D) FSI through a finite element-boundary integral method coupling, and by varying systematically the two non-dimensionalized parameters of the problem: the capillary number, ratio of the viscous to the elastic forces, and the capsule-to-tube size ratio. Among the MOR techniques, we chose to apply Proper Orthogonal Decomposition (POD) onto the database, which provides a vector basis of principal components, defining a multi-dimensional vector space. The advantage is that, when all the capsule shapes of the database are mapped into this new vector space, they form a manifold (smooth hypersurface) that represents all the admissible solutions of the problem. We show that POD with a manifold walking technique can be successfully applied to 3D microcapsule data sets, whether they are Lagrangian (e.g. known position vector fields) or Eulerian (i.e. when data is acquired experimentally or numerically using methods like level sets). In both cases, the problem dimensionality is reduced, the predicted capsule shapes are obtained within computation times of milliseconds, and they accurately fit the full FSI simulations. This paves the way to real-time computations for capsules in flow, while retaining all the physical ingredients of the FSI problem.

Modelling and analysis of vibration on power electronic module structure and application of model order reduction

Modelling and analysis of vibration of an IGBT power electronic module (PEM) structure were undertaken. PEM structure considered in this study was without molding compound and wirebonds. The most critical resonant frequency was identified by modal analysis. At the critical frequency of 1345 Hz, for the vertical displacement of the base excitation, subsequent stress distribution on the PEM structure was analysed. Concurrent vibration and thermo-mechanical fatigue loads on the reliability of PEM structure solder interconnects were also estimated by widely used linear damage superposition approach. It was concluded that the at critical resonant frequency the vibration induced damage is more severe than the thermo-mechanical fatigue loading. In addition, a quarter car model (QCM) was used to mimic the dynamic interaction between the rough road surface and an electric vehicle (EV) in order to analyse the road surface roughness induced excitation on the PEM structure in the engine compartment. Stress and strain distribution on the PEM structure due to road surface roughness were analysed. Furthermore, three Krylov subspace based model order reduction (MOR) techniques were applied to the resulting dynamic system in vibration analysis. Due to the limits on computing resources, a submodel was utilized for MOR analysis. Within the three MOR techniques, Passive Reduced order Interconnect Macromodeling Algorithm (PRIMA) MOR technique performs better than the other techniques. Computational time ratio between reduced system iteration and the full system iteration is 1: 53.

Reduced basis method for managed pressure drilling based on a model with local nonlinearities

AbstractTo circumvent restrictions of conventional drilling methods, such as slow control actions and inability to drill depleted reservoirs, a drilling method called managed pressure drilling (MPD) has been developed. In MPD, single‐phase flow processes can be modeled as a feedback interconnection of a high‐order linear system and a low‐order nonlinear system. These nonlinearities appear locally both inside and at the boundaries of the computational domain. To obtain a fast simulation platform for real‐time purposes (eg, online model‐based controller implementation), model order reduction is required for MPD. However, the local nonlinearities render applying model order reduction techniques challenging. In this study, a new approach is proposed to deal with such nonlinearities within the reduced basis (RB) context and it is successfully tested on a model for MPD. Contrary to the classical RB technique, the proposed approach not only does not generate nonphysical spikes at the locations of these local nonlinearities but also yields high speedup factors. The obtained reduced‐order model can be used for efficient online simulation and controller design for drilling systems with MPD.

A HJB-POD approach for the control of nonlinear PDEs on a tree structure

The Dynamic Programming approach allows to compute a feedback control for nonlinear problems, but suffers from the curse of dimensionality. The computation of the control relies on the resolution of a nonlinear PDE, the Hamilton-Jacobi-Bellman equation, with the same dimension of the original problem. Recently, a new numerical method to compute the value function on a tree structure has been introduced. The method allows to work without a structured grid and avoids any interpolation.Here, we aim at testing the algorithm for nonlinear two dimensional PDEs. We apply model order reduction to decrease the computational complexity since the tree structure algorithm requires to solve many PDEs. Furthermore, we prove an error estimate which guarantees the convergence of the proposed method. Finally, we show efficiency of the method through numerical tests.

Application of model order reduction to a finite element model of cryogenic turning

AbstractMetastable austenitic steels offer the opportunity of a surface layer hardening integrated in the machining process. The hardening effect is achieved by a deformation induced austenite‐martensite phase transformation, for which high mechanical loads and low temperatures are necessary, typically below room temperature. These conditions can be accomplished during cryogenic turning, allowing a phase transformation in the surface layer of the workpiece. To study the austenite‐martensite transformation behavior, information about the temperatures in the contact zone between tool and workpiece during machining is necessary, which can hardly be measured. Therefore, FE simulations of the process are utilized to calculate the transient process temperatures. In this paper the idea of projection‐based model order reduction and its implementation into the finite element program FEAP are described, allowing a solution speed‐up while maintaining accuracy of the full‐order model.

Sensitivity analysis of random linear dynamical systems using quadratic outputs

In uncertainty quantification, a stochastic modelling is often applied, where parameters are substituted by random variables. We investigate linear dynamical systems of ordinary differential equations with a quantity of interest as output. Our objective is to analyse the sensitivity of the output with respect to the random variables. A variance-based approach generates partial variances and sensitivity indices. We expand the random output using the generalised polynomial chaos. The stochastic Galerkin method yields a larger system of ordinary differential equations. The partial variances represent quadratic outputs of this system. We examine system norms of the stochastic Galerkin formulation to obtain sensitivity measures. Furthermore, we apply model order reduction by balanced truncation, which allows for an efficient computation of the system norms with guaranteed error bounds. Numerical results are shown for a test example.

Approximate Green's Function Coupled Cluster Method Employing Effective Dimension Reduction.

The Green's function coupled cluster (GFCC) method, originally proposed in the early 1990s, is a powerful many-body tool for computing and analyzing the electronic structure of molecular and periodic systems, especially when electrons of the system are strongly correlated. However, in order for the GFCC to become a method that may be routinely used in the electronic structure calculations, robust numerical techniques and approximations must be employed to reduce its extremely high computational overhead. In our recent studies, it has been demonstrated that the GFCC equations can be solved directly in the frequency domain using iterative linear solvers, which can be easily distributed in a massively parallel environment. In the present work, we demonstrate a successful application of model-order-reduction (MOR) techniques in the GFCC framework. Briefly speaking, for a frequency regime of interest that requires high-resolution descriptions of spectral function, instead of solving the GFCC linear equation of full dimension for every single frequency point of interest, an efficiently solvable linear system model of a reduced dimension may be built upon projecting the original GFCC linear system onto a subspace. From this reduced order model is obtained a reasonable approximation to the full dimensional GFCC linear equations in both interpolative and extrapolative spectral regions. Here, we show that the subspace can be properly constructed in an iterative manner from the auxiliary vectors of the GFCC linear equations at some selected frequencies within the spectral region of interest. During the iterations, the quality of the subspace, as well as the linear system model, can be systematically improved. The method is tested in this work in terms of the efficiency and accuracy of computing spectral functions for some typical molecular systems such as carbon monoxide, 1,3-butadiene, benzene, and adenine. To reach the same level of accuracy as that of the original GFCC method, the application of MOR in the GFCC method is able to significantly lower the original computational cost for the aforementioned molecules in designated frequency regimes. As a byproduct, the reduced order model obtained by this method is found to provide a high-quality initial guess, which improves the convergence rate for the existing iterative linear solver.

Load Snapshot Decomposition to Consider Heat Radiation in Thermal Model Order Reduction

Heat exchange by radiation in thermal field problems which corresponds to the fourth power of the temperature leads to nonlinear system equations. Hence, simulation times increase drastically, especially when parameter studies or sensitivity analyses are performed. Model order reduction is used to overcome these difficulties by projecting the model into a subspace of much smaller dimension. Heat radiation is treated as a part of the load vector resulting in constant system matrices. However, thermal loads like convection and radiation are distributed over the whole surface. That is why a high percentage of the nodes within the finite element model serves as inputs and outputs of the system. Thus, the Krylov subspace method cannot be applied directly as the reduced dimension depends on the number of inputs and would be too large.Therefore, input reduction becomes necessary and loads changing synchronously are clustered into one input. Furthermore, in the approach presented here load vector snapshots are taken from a training simulation of a reference configuration. The input matrix determination bases on the singular value decomposition of the snapshot matrix. Using only the singular vectors corresponding to the highest singular values reduces the number of inputs by several orders of magnitude allowing the application of model order reduction. Parameter variations can be performed yielding good results within a broad parameter range. The computational effort for the derivation of the snapshots is significantly decreased in comparison to conventional methods.

A computationally efficient implementation of a full and reduced-order electrochemistry-based model for Li-ion batteries

Lithium-ion batteries are commonly employed in various applications owing to high energy density and long service life. Lithium-ion battery models are used for analysing batteries and enabling power control in applications. The Doyle-Fuller-Newman (DFN) model is a popular electrochemistry-based lithium-ion battery model which represents solid-state and electrolyte diffusion dynamics and accurately predicts the current/voltage response using a set of nonlinear partial differential equations. However, implementation of the full DFN model requires significant computation time. This paper proposes a computationally efficient implementation of the full DFN battery model, which is convenient for real-time applications. The proposed implementation is based on applying model order reduction to a spatial and temporal discretisation of the governing model equations. For model order reduction, we apply proper orthogonal decomposition and discrete empirical interpolation method, which leads to a set of reduced order nonlinear algebraic equations. These equations are solved using a particular numerical scheme, based on a damped Newton’s method. In a simulation study, the computational efficiency of the proposed implementation is shown and the resulting accuracy is presented.

Interface and model reduction for efficient explicit simulations - a case study with nonlinear vehicle crash models

ABSTRACTThis work presents an approach to save simulation time in explicit crash simulations of vehicles by applying model order reduction (MOR). The model of a kart frame is split into linear and nonlinear parts. The linear part is reduced with linear MOR techniques in MatMorembs. Optimal substructuring methods are used to calculate suitable reduced models which are exported to LS-DYNA. Afterwards, the model consisting of a linear reduced part and a nonlinear part is simulated in the online step. The plastic deformation of the kart frame as well as the accelerations of the driver calculated with various reduction and parameter settings are compared with the accelerations measured when the original, unreduced nonlinear model is simulated. For optimal simulation results, the large interface between the models needs to be approximated by suitable interface-reduction approaches. The novel contribution is the application of advanced interface-reduction techniques in nonlinear explicit crash simulations. Craig–Bampton and Gramian matrix-based-like techniques with global, local and no interface reduction are compared to find optimally reduced substructures in terms of approximation quality of the assembled system and computational effort. For the kart frame, the applicability of the method is proven by gaining a small simulation speed up which cannot be achieved with the standard reduction methods provided by LS-DYNA.

An hp-proper orthogonal decomposition–moving least squares approach for molecular dynamics simulation

Abstract The application of model order reduction for molecular dynamic systems exhibits intrinsic complexities due to the highly nonlinear and nonlocal properties of such systems. The most costly computational part of such work is the calculation of potential energy and inter-atomic force fields. We introduce the so-called hp -proper orthogonal decomposition–moving least squares ( hp -POD-MLS) method not only to tackle this force computation, but also to improve the accuracy of reduced displacement approximations in an adaptive and controllable manner. The approach combines the proper orthogonal decomposition (POD) method with the moving least squares (MLS) one to interpolate the inter-atomic force field at an arbitrary time instance; this approximated force field is then fed into the reduced molecular dynamic (MD) governing equation to compute the reduced order displacement field. In fact, this POD–MLS approach is already able to handle efficiently the force computation in the online computational stage (i.e., the online force computational cost is independent of the dimension of the full MD space); however its corresponding approximated displacement error will be very large if the considered total time interval is long. The proposed hp -POD-MLS algorithm will divide adaptively and automatically the global temporal interval into many smaller local subintervals ( h -refinement), and then build the sets of local reduced basis functions over each of these subintervals ( p -refinement). The goal is to reduce the corresponding displacement errors to a desired tolerance over these subintervals. The effectiveness of the algorithm is demonstrated by performing reduced-order simulations of several molecular dynamic systems.

Model order reduction with oblique projections for large scale wave propagation

There are many applications in which it is necessary to solve large scale parametrized wave propagation problems repeatedly. This is still quite a challenging task, even with the largest available computer clusters. In this paper we will discuss the application of Model Order Reduction (MOR) to problems in seismic petroleum exploration, with the aim of diminishing the necessary computing time by a significant factor. We consider POD and some variants. POD is a Model Order Reduction technique that uses snapshots of a few simulations in order to quickly compute related problems with similar accuracy. The method of lines via a Petrov–Galerkin approximation that uses the snapshots as basis functions is the considered approach. The order reduction comes from projecting the wave equation discretized in space to the subspace spanned by the snapshots. This has been shown earlier to work well in two dimensions. The challenge in three dimensions comes from the size of the spatial meshes required and the fact that the method usually requires a number of snapshots that do not fit in fast memory, even for current high end multicore machines. Parallelization is not an option since it is already used for other aspects of this massive problem.

030 Block SOAR法による大規模モデルの縮退と有限要素モデルへの適用

030 Block SOAR法による大規模モデルの縮退と有限要素モデルへの適用