Parametric Model Order Reduction Research Articles

Design optimization of a miniaturized thermoelectric generator via parametric model order reduction

Implantable medical devices play a major role for regenerative therapies. A novel, miniaturized thermoelectric generator can be utilized to harvest the thermal energy from temperature gradients inside human body. Thus it functions as an additional power supply for electrically active implants. In this work, we present an optimization strategy for thermoelectric generator based on mathematical methods of parametric model order reduction. Those methods enable an automatized reduction of the finite element model of thermoelectric generator and surrounding human tissue. Parameterized reduced order models are compact, but highly accurate and are well suited for efficient parametric studies.

Predicting near optimal interpolation points for parametric model order reduction using regression models

AbstractThe design of vibro‐acoustic systems, such as vehicle interiors, regarding optimal vibrational or sound radiation properties requires the solution of many numerical models under varying parameters, as often material or geometric uncertainties have to be considered. Vibro‐acoustic systems are typically large and numerically expensive to solve, so it is desirable to use an efficient parametrized surrogate model for optimization tasks. High quality reduced models of non‐parametric systems can be computed by projection, given a set of optimal expansion points. However, finding a set of optimal expansion points can be computationally expensive and each set is valid for a specific parameter realization only. The method presented in this contribution learns the map between a model's parameter realizations and the corresponding sets of expansion points using data‐driven methods. Queried with an unknown set of parameters, the learned model returns a set of expansion points which are used to compute the corresponding reduced model efficiently. Numerical experiments on two vibro‐acoustic models of different complexity are performed and three data driven regression methods are evaluated: multivariate polynomial regression, k‐nearest neighbors, and support vector regression. Especially k‐nearest neighbors regression yields accurate results for different types of physical models while being computationally inexpensive to fit.

Reduced-order modeling of advection-dominated systems with recurrent neural networks and convolutional autoencoders

A common strategy for the dimensionality reduction of nonlinear partial differential equations (PDEs) relies on the use of the proper orthogonal decomposition (POD) to identify a reduced subspace and the Galerkin projection for evolving dynamics in this reduced space. However, advection-dominated PDEs are represented poorly by this methodology since the process of truncation discards important interactions between higher-order modes during time evolution. In this study, we demonstrate that encoding using convolutional autoencoders (CAEs) followed by a reduced-space time evolution by recurrent neural networks overcomes this limitation effectively. We demonstrate that a truncated system of only two latent space dimensions can reproduce a sharp advecting shock profile for the viscous Burgers equation with very low viscosities, and a six-dimensional latent space can recreate the evolution of the inviscid shallow water equations. Additionally, the proposed framework is extended to a parametric reduced-order model by directly embedding parametric information into the latent space to detect trends in system evolution. Our results show that these advection-dominated systems are more amenable to low-dimensional encoding and time evolution by a CAE and recurrent neural network combination than the POD-Galerkin technique.

A local basis approximation approach for nonlinear parametric model order reduction

The efficient condition assessment of engineered systems requires the coupling of high fidelity models with data extracted from the state of the system ‘as-is’. In enabling this task, this paper implements a parametric Model Order Reduction (pMOR) scheme for nonlinear structural dynamics, and the particular case of material nonlinearity. A physics-based parametric representation is developed, incorporating dependencies on system properties and/or excitation characteristics. The pMOR formulation relies on use of a Proper Orthogonal Decomposition applied to a series of snapshots of the nonlinear dynamic response. A new approach to manifold interpolation is proposed, with interpolation taking place on the reduced coefficient matrix mapping local bases to a global one. We demonstrate the performance of this approach firstly on the simple example of a shear-frame structure, and secondly on the more complex 3D numerical case study of an wind turbine tower under a ground motion excitation. Parametric dependence pertains to structural properties, as well as the temporal and spectral characteristics of the applied excitation. The developed parametric Reduced Order Model (pROM) can be exploited for a number of tasks including monitoring and diagnostics, control of vibrating structures, and residual life estimation of critical components.

Parametric model order reduction based on parallel tensor compression

In this paper, we for the first time explore the model order reduction (MOR) of parametric systems based on the tensor techniques and a parallel tensor compression algorithm. For the parametric system characterising multidimensional parameter space and nonlinear parametric dependence, we first approximate the system matrices by tensor functions of the parameters, whose first-order coefficients are third-order tensors. In order to effectively reduce the computational cost and the storage burden, we propose a parallel tensor compression algorithm based on Tensor-SVD to deal with the tensors in the tensor functions. Then, we obtain the low-rank approximation in Kruskal form of third-order tensors. After that, by computing the first several expansion coefficients of the state variable with the selected parameter vectors, the projection matrix is constructed to obtain the reduced parametric system. Theoretical analysis shows that the reduced parametric system can match the first several expansion coefficients of the output variable of the original system at the selected parameter vectors. Moreover, the stability of the proposed MOR method is discussed. Finally, the efficiency of the proposed method is illustrated by two numerical examples.

An Experimental and Computational Investigation of a Pulsed Air-Jet Excitation System on a Rotating Bladed Disk

Abstract Nonsynchronous vibrations are a difficult problem to address for turbomachines due to the complex nature of the forcing. Such vibrations can be caused by vortex shedding, flow instabilities, stall cells, or flutter. Testing a design with such excitations can be difficult in practice due to the required forcing. This work demonstrates an experimental excitation method using pulsed air jet excitation to create nonsynchronous vibrations in engine hardware rotating at nominal design speeds. Experimental runs were conducted to excite a number of engine orders (EOs). Blade tip timing was used to measure the blade response without interfering with the blade dynamics. The bladed disk was held at a constant rotational speed while the air jets were pulsed at a sweeping frequency to simulate rotating forcing. Computational models of the physical system were constructed using parametric reduced order models that incorporate the effects of rotational speed and small mistuning. The computational model was used in simulations that mimic the experiment; the forcing was swept across the blades while being pulsed. This results in a system response that cannot be captured using traditional harmonic analyses. The computational and experimental datasets were compared through mistuning values, amplitudes, and the nodal diameter (ND) content in the system response.

State and parameter estimation for model-based retinal laser treatment

Abstract We present an approach for state and parameter estimation in retinal laser treatment by a novel setup where both measurement and heating is performed by a single laser. In this medical application, the temperature that is induced by the laser in the patient’s eye is critical for a successful and safe treatment. To this end, we pursue a model-based approach using a model given by a heat diffusion equation on a cylindrical domain, where the source term is given by the absorbed laser power. The model is parametric in the sense that it involves an absorption coefficient, which depends on the treatment spot and plays a central role in the input-output behavior of the system. After discretization, we apply a particularly suited parametric model order reduction to ensure real-time tractability while retaining parameter dependence. We augment known state estimation techniques, i.e., extended Kalman filtering and moving horizon estimation, with parameter estimation to estimate the absorption coefficient and the current state of the system. Eventually, we show first results for simulated and experimental data from porcine eyes. We find that, regarding convergence speed, the moving horizon estimation slightly outperforms the extended Kalman filter on measurement data in terms of parameter and state estimation, however, on simulated data the results are very similar.

Towards model order reduction for fluid-thermal analysis

The authors develop basic components of a parametric model-order reduction (pMOR) procedure targeting high Rayleigh-number buoyancy-driven flows. The pMOR is based on Galerkin formulation of the governing Boussinesq equations using eigenfunction bases derived from proper orthogonal decomposition. The advantages of pMOR over parametric interpolation are demonstrated for quantities of interest (QOIs) with linear and nonlinear dependencies on the solution in low Rayleigh-number cases. Constraint-base and Leray-type regularizations are shown to offer significant advantages over the standard reduced-order Galerkin formulation in a variety of examples including 3D Rayleigh–Bénard flow at Rayleigh number 107 and turbulent flow in a half-pipe at Reynolds number 5300.

Non intrusive method for parametric model order reduction using a bi-calibrated interpolation on the Grassmann manifold

Approximating solutions of non-linear parametrized physical problems by interpolation presents a major challenge in terms of accuracy. In fact, pointwise interpolation of such solutions is rarely efficient and generally leads to incorrect predictions. To overcome this issue, instead of interpolating solutions directly by a straightforward approach, reduced order models (ROMs) can be efficiently used. To this end, the ITSGM (Interpolation On a Tangent Space of the Grassmann Manifold) is an efficient technique used to interpolate parameterized POD (Proper Orthogonal Decomposition) bases. The temporal dynamics is afterwards determined by the Galerkin projection of the predicted basis onto the high fidelity model. However, such interpolated ROMs based on ITSGM/Galerkin are intrusive, given the fact that their construction requires access to the equations of the underlying high fidelity model. In the present paper a non-intrusive approach (Galerkin free) for the construction of reduced order models is proposed. This method, referred to as Bi-CITSGM (Bi-Calibrated ITSGM) consists of two major steps. First, the untrained spatial and temporal bases are predicted by the ITSGM method and the POD eigenvalues by spline cubic. Then, two orthogonal matrices, determined as analytical solutions of two optimization problems, are introduced in order to calibrate the interpolated bases with their corresponding eigenvalues. Results on the flow problem past a circular cylinder where the parameter of interpolation is the Reynolds number, show that for new untrained Reynolds number values, the developed approach produces sufficiently accurate solutions in a real-computational time.

A framework for parametric reduction in large-scale nonlinear dynamical systems

This manuscript presents a general parametric model order reduction (pMOR) framework for nonlinear dynamical systems. At first, a family of local nonlinear reduced order models (ROMs) is generated for different parametric space vectors using the nonlinear moment matching (NLMM) scheme along-with the discrete empirical interpolation method (DEIM). Then, the nonlinear reduced order model for any new parameter is obtained by interpolating the neighbouring reduced order models after projecting them onto a universal subspace. The advantage of such a scheme is that the parametric dependency is maintained in the reduced nonlinear models. Finally, we substantiate our observations by a suite of numerical tests.

Model order reduction of thermo-mechanical models with parametric convective boundary conditions: focus on machine tools

Thermo-mechanical finite element (FE) models predict the thermal behavior of machine tools and the associated mechanical deviations. However, one disadvantage is their high computational expense, linked to the evaluation of the large systems of differential equations. Therefore, projection-based model order reduction (MOR) methods are required in order to create efficient surrogate models. This paper presents a parametric MOR method for weakly coupled thermo-mechanical FE models of machine tools and other similar mechatronic systems. This work proposes a reduction method, Krylov Modal Subspace (KMS), and a theoretical bound of the reduction error. The developed method addresses the parametric dependency of the convective boundary conditions using the concept of system bilinearization. The reduced-order model reproduces the thermal response of the original FE model in the frequency range of interest for any value of the parameters describing the convective boundary conditions. Additionally, this paper investigates the coupling between the reduced-order thermal system and the mechanical response. A numerical example shows that the reduced-order model captures the response of the original system in the frequency range of interest.

Randomized Algorithms for Non-Intrusive Parametric Reduced Order Modeling

This paper demonstrates the development of purely data-driven, nonintrusive parametric reduced-order models for the emulation of high-dimensional field outputs using randomized linear algebra techniques. Typically, low-dimensional representations are built using the proper orthogonal decomposition combined with interpolation/regression in the latent space via supervised learning. However, even moderately large simulations can lead to data sets on which the cost of computing the proper orthogonal decomposition becomes intractable due to storage and computational complexity of the numerical procedure. In an attempt to reduce the offline cost, the proposed method demonstrates the application of randomized singular value decomposition and sketching-based randomized singular value decomposition to compute the proper orthogonal decomposition basis. The predictive capability of reduced-order models resulting from regular singular value decomposition and randomized/sketching-based algorithms are compared with each other to ensure that the decrease in computational cost does not result in a loss in accuracy. Demonstrations on canonical and practical fluid flow problems show that the reduced-order models constructed using randomized methods are competitive in their predictive accuracy with reduced-order models that employ the conventional deterministic method. Through this new method, it is expected that truly large-scale parametric reduced-order models can be constructed under a significantly limited computational resource budget.

Non-autoregressive time-series methods for stable parametric reduced-order models

Advection-dominated dynamical systems, characterized by partial differential equations, are found in applications ranging from weather forecasting to engineering design where accuracy and robustness are crucial. There has been significant interest in the use of techniques borrowed from machine learning to reduce the computational expense and/or improve the accuracy of predictions for these systems. These rely on the identification of a basis that reduces the dimensionality of the problem and the subsequent use of time series and sequential learning methods to forecast the evolution of the reduced state. Often, however, machine-learned predictions after reduced-basis projection are plagued by issues of stability stemming from incomplete capture of multiscale processes as well as due to error growth for long forecast durations. To address these issues, we have developed a non-autoregressive time series approach for predicting linear reduced-basis time histories of forward models. In particular, we demonstrate that non-autoregressive counterparts of sequential learning methods such as long short-term memory (LSTM) considerably improve the stability of machine-learned reduced-order models. We evaluate our approach on the inviscid shallow water equations and show that a non-autoregressive variant of the standard LSTM approach that is bidirectional in the principal component directions obtains the best accuracy for recreating the nonlinear dynamics of partial observations. Moreover—and critical for many applications of these surrogates—inference times are reduced by three orders of magnitude using our approach, compared with both the equation-based Galerkin projection method and the standard LSTM approach.

DC-Centric Parameterized Reduced-Order Model via Moment-Based Interpolation Projection (MIP) Algorithm

This article presents a new approach to construct parameterized reduced-order models for nonlinear circuits. The reduced model is obtained such that it matches the variations in the DC operating point of the original full circuit in response to variations in several of its key design parameters. The new approach leverages, through the use of circuit moments with respect to the design parameters, the discrete empirical interpolation approach developed in a model reduction in other domains and enables its efficient application to the problem of DC operating point in nonlinear circuits. Utilizing the idea of rooted trees, the proposed approach constructs orthogonal bases that are used in projecting the full equations of the large original nonlinear circuit onto a reduced system of nonlinear equations in a space with a much smaller dimension. The variations in the DC operating point of the full circuit are then obtained by solving the reduced system of equations, yielding significant computational savings. Numerical examples are presented to demonstrate the efficiency and accuracy of the reduced model in predicting the change in the DC operating point of the original circuit.

Rigorous and effective a-posteriori error bounds for nonlinear problems\u2014application to RB methods

Quantifying the error that is induced by numerical approximation techniques is an important task in many fields of applied mathematics. Two characteristic properties of error bounds that are desirable are reliability and efficiency. In this article, we present an error estimation procedure for general nonlinear problems and, in particular, for parameter-dependent problems. With the presented auxiliary linear problem (ALP)-based error bounds and corresponding theoretical results, we can prove large improvements in the accuracy of the error predictions compared with existing error bounds. The application of the procedure in parametric model order reduction setting provides a particularly interesting setup, which is why we focus on the application in the reduced basis framework. Several numerical examples illustrate the performance and accuracy of the proposed method.

Development of a parametric model order reduction method for laminated composite structures

This paper presents a parametric model order reduction (PMOR) method for laminated composite structures. The approach is applied to a lattice structure made of carbon fibre reinforced polymer (CFRP) beams. The global reduced order basis (ROB) used to apply the reduction is calculated with two different methods: the Modal-SVD (singular value decomposition) and the proper orthogonal decomposition (POD). Then, the reduced model is obtained by a Galerkin projection. The parametrization is carried out for the angle between the layers of the structure and for the thickness of the structure. The results show how the use of the reduced basis (RB) method to create reduced parametric models of composite structures is an efficient approach that allows to decrease the computational cost of the simulation.

Using parametric model order reduction for inverse analysis of large nonlinear cardiac simulations.

Predictive high-fidelity finite element simulations of human cardiac mechanics commonly require a large number of structural degrees of freedom. Additionally, these models are often coupled with lumped-parameter models of hemodynamics. High computational demands, however, slow down model calibration and therefore limit the use of cardiac simulations in clinical practice. As cardiac models rely on several patient-specific parameters, just one solution corresponding to one specific parameter set does not at all meet clinical demands. Moreover, while solving the nonlinear problem, 90% of the computation time is spent solving linear systems of equations. We propose to reduce the structural dimension of a monolithically coupled structure-Windkessel system by projection onto a lower-dimensional subspace. We obtain a good approximation of the displacement field as well as of key scalar cardiac outputs even with very few reduced degrees of freedom, while achieving considerable speedups. For subspace generation, we use proper orthogonal decomposition of displacement snapshots. Following a brief comparison of subspace interpolation methods, we demonstrate how projection-based model order reduction can be easily integrated into a gradient-based optimization. We demonstrate the performance of our method in a real-world multivariate inverse analysis scenario. Using the presented projection-based model order reduction approach can significantly speed up model personalization and could be used for many-query tasks in a clinical setting.

Towards efficient design optimization of a miniaturized thermoelectric generator for electrically active implants via model order reduction and submodeling technique.

Thermoelectric generators (TEG) convert the thermal energy into electrical energy and are under investigation as a power supply for medical implants. To improve the performance of TEG, the design optimization process through finite element model simulation is preferred by biomedical engineers. This paper aims to provide an efficient method of speeding up the design optimization process of TEG. A three-dimensional realistic human torso model incorporating the TEG is investigated, where the internal heat transfer in human tissue is characterized by Pennes bioheat equation. In addition, convection, radiation, and evaporation effects at the skin surface are applied to identify the heat transfer effects between the human body and the environment. To speed up finite element simulation of the large-scale human torso model, projection-based model order reduction (MOR) is applied for generation of a compact but highly accurate model. Parametric MOR (pMOR) further enables generating a parameter-independent compact model. For an efficient design optimization of TEG, this compact human torso model is applied within a thermal submodeling approach. Its temperature distribution results are back-projected and used as boundary conditions for the TEG submodel. The achieved speed-up in simulation time, demonstrated in this work, clearly indicates that the design optimization process of TEG is more efficient with the combination of MOR and submodeling techniques.

Feedback control of parametrized PDEs via model order reduction and dynamic programming principle

In this paper, we investigate infinite horizon optimal control problems for parametrized partial differential equations. We are interested in feedback control via dynamic programming equations which is well-known to suffer from the curse of dimensionality. Thus, we apply parametric model order reduction techniques to construct low-dimensional subspaces with suitable information on the control problem, where the dynamic programming equations can be approximated. To guarantee a low number of basis functions, we combine recent basis generation methods and parameter partitioning techniques. Furthermore, we present a novel technique to construct non-uniform grids in the reduced domain, which is based on statistical information. Finally, we discuss numerical examples to illustrate the effectiveness of the proposed methods for PDEs in two space dimensions.

Component-level proper orthogonal decomposition for flexible multibody systems

Nonlinear finite element methods such as isogeometric analysis and absolute nodal coordinate formulation can be applied to modeling the flexible multibody systems (FMBS) subject to both large rotation and deformation. However, the computational expense is prohibitive for complex systems, especially for a parameterized FMBS. Proper orthogonal decomposition (POD), nonmodal model order reduction (MOR) technique, has been widely applied to nonlinear problems in order to reduce the computational expense of simulating the dynamic responses of FMBSs. This paper presents a component-level POD method, which applies the singular value decomposition process to snapshots of each component of an FMBS and utilizes constraint equations to satisfy the connecting conditions among the components. Furthermore, the paper provides a systematic study of the parameterized MOR (PMOR) for an FMBS, including the reduction of the constraint equations, energy conserving mesh sampling and weighting (ECSW), and the greedy-POD method. The ECSW method aims to further reducing the computational cost of matrix multiplications for evaluating the reduced stiffness matrix. Greedy-POD is applied to get a robust set of reduced order bases with parametric changes in simulation. Finally, three numerical examples validate the feasibility of the component-level POD as well as the whole PMOR process for an FMBS, thereby demonstrating that the component-level POD performs better numerical simulations in accuracy computational costs than the classical POD.