Proper Orthogonal Decomposition Basis Research Articles

A numerical comparison of simplified Galerkin and machine learning reduced order models for vaginal deformations

High-fidelity computer simulations of childbirth remain prohibitively expensive and time consuming, making them impractical for guiding decision-making during obstetric emergencies. Cheap computer simulations that preserve the accuracy of high-fidelity models can be developed using surrogate modeling. Two common approaches to surrogate modeling are physics-based reduced order modeling (ROM) and machine learning (ML), with the latter gaining popularity as the scientific computing community seeks to leverage advances from other, mostly non-physics-based, computational strategies. Although ROM and ML have been compared for various problems, to our knowledge, such a comparison for simulations of vaginal deformations is currently missing. This study provides a baseline numerical comparison between methods from these two fundamentally different approaches. Since there are many methods falling into each modeling approach, to provide a fair and natural comparison, we select a basic model from each category, with each allowing (i) a straightforward implementation in commercial software packages, and (ii) use by practitioners with limited experience in the field. As a benchmark for the numerical comparison of the ROM and ML approaches, we use the finite element (FE) modeling of the ex vivo deformations of rat vaginal tissue subjected to inflation testing to study the effect of a pre-imposed tear. From the ROM strategies, we consider a simplified Galerkin ROM (G-ROM) that is based on the linearization of the underlying nonlinear equations. From the ML strategies, we select a feed-forward neural network to create mappings from constitutive model parameters and luminal pressure values to either the FE displacement history (in which case we denote the resulting model ML) or the proper orthogonal decomposition (POD) coefficients of the displacement history (in which case we denote the resulting model POD-ML). The numerical investigation of G-ROM, ML, and POD-ML takes place in the reconstructive regime. The numerical results show that the G-ROM outperforms the ML model in terms of offline central processing unit (CPU) time for model training, online CPU time required to generate approximations, and relative error with respect to the FE models. The G-ROM achieves superior error performance to the best ML model with 11 POD basis functions. With higher-dimensional POD bases, the G-ROM achieves a relative error 3 orders of magnitude lower than that of the best ML model with an online CPU time still on the same order of magnitude as the best ML model. The POD-ML model improves on the speed performance of the ML, having online CPU times comparable to those of the G-ROM given the same size of POD bases. However, the POD-ML model does not improve on the error performance of the ML and is still outperformed by the G-ROM for POD bases of size greater than 11. This baseline numerical investigation serves as a starting point for future computer simulations that consider state-of-the-art G-ROM and ML strategies, and the in vivo geometry, boundary conditions, and material properties of the human vagina, as well as their changes during labor.

Deep learning methods for blood flow reconstruction in a vessel with contrast enhanced x-ray computed tomography.

The reconstruction of blood velocity in a vessel from contrast enhanced x-ray computed tomography projections is a complex inverse problem. It can be formulated as reconstruction problem with a partial differential equation constraint. A solution can be estimated with the a variational adjoint method and proper orthogonal decomposition (POD) basis. In this work, we investigate new inversion approaches based on PODs coupled with deep learning methods. The effectiveness of the reconstruction methods is shown with simulated realistic stationary blood flows in a vessel. The methods outperform the reduced adjoint method and show large speed-up at the online stage.

An adaptive model order reduction technique for parameter-dependent modular structures

AbstractThis work is concerned with an adaptive reduced order model of modular structures assembled from parameter-dependent substructures. The substructures are reduced by proper orthogonal decomposition (POD) and connected by means of a tied contact formulation. We present a method to adapt the matrices of the substructures to parameter changes. We employ interpolation on Grassmann manifolds for the parametric adaption of the projection matrices. For the adaptation of the stiffness matrices, we use the direct empirical interpolation method (DEIM). Manifold interpolation of the reduced stiffness matrices, cannot be applied here since it would require semi-positive definiteness, which is here not fulfilled because of necessary rigid body motion modes. The novelty of this work is the application of these interpolation methods to the special problem class of POD-based tied contact model order reduction. Furthermore, we show a methodology to compute significant snapshots on the substructure level to compute a POD basis that can be used in different global structures.

A rapid method to predict biaxial fatigue life of automotive wheels using proper orthogonal decomposition and radial basis function algorithm

This paper presents a rapid method for predicting the biaxial fatigue life of automotive wheels using a combination of proper orthogonal decomposition and radial basis function algorithm. Currently, numerical simulations of biaxial fatigue tests are being developed to evaluate wheel performance. However, these simulations are computationally expensive due to the need to simulate multiple discrete loading cases within a given biaxial spectrum. To address this issue, we propose a novel approach that utilizes proper orthogonal decomposition and radial basis function algorithm to improve computational efficiency. By leveraging high-fidelity simulation results from a small number of loading cases, a reduced order model is constructed to accurately predict the tire-rim interface force fields required for wheel strength calculations. The reduced order model significantly reduces the computational time by 65.4% for simulating all loading cases, while maintaining a maximum predicted error of less than 2% compared to the high-fidelity model. Subsequently, the predicted interface forces are mapped onto the rim surface for strength calculation, and the wheel fatigue life is determined using the Brown-Miller multiaxial damage criterion. Comparative analysis with experimental results demonstrates the desirable accuracy of our method in simulating the stress-strain history, crack initiation position, and minimum fatigue life of the wheel. Overall, the proposed method offers a powerful tool for the rapid fatigue analysis of spectrum-loaded wheels, providing an efficient and accurate means of predicting biaxial fatigue life.

A simulation-based software to support the real-time operational parameters selection of tunnel boring machines

With the fact that the main operational parameters of the construction process in mechanized tunneling are currently selected based on monitoring data and engineering experience without exploiting the advantages of computer methods, the focus of this work is to develop a simulation-based real-time assistant system to support the selection of operational parameters. The choice of an appropriate set of these parameters (i.e., the face support pressure, the grouting pressure, and the advance speed) during the operation of tunnel boring machines (TBM) is determined by evaluating different tunneling-induced soil-structure interactions such as the surface settlement, the associated risks on existing structures and the tunnel lining behavior. To evaluate soil-structure behavior, an advanced process-oriented numerical simulation model based on the finite cell method is utilized. To enable the real-time prediction capability of the simulation model for a practical application during the advancement of TBMs, surrogate models based on the Proper Orthogonal Decomposition and Radial Basis Functions (POD-RBF) are adopted. The proposed approach is demonstrated through several synthetic numerical examples inspired by the data of real tunnel projects. The developed methods are integrated into a user-friendly application called SMART to serve as a support platform for tunnel engineers at construction sites. Corresponding to each user adjustment of the input parameters, i.e., each TBM driving scenario, approximately two million outputs of soil-structure interactions are quickly predicted and visualized in seconds, which can provide the site engineers with a rough estimation of the impacts of the chosen scenario on structural responses of the tunnel and above ground structures.

A new method to interpolate POD reduced bases–Application to the parametric model order reduction of a gas bearings supported rotor

SummaryThe proper orthogonal decomposition (POD) is successfully employed in a variety of projection‐based methods for parametric model order reduction (pMOR) of large dynamical systems. It extracts the most energetic modes describing the dynamics of a system from time snapshots of the solution. In practice, these snapshots are computed at user‐defined parameters of the system with a high fidelity model. Then either all the snapshots are concatenated to extract a global POD basis, or a different POD basis is extracted for each parameter value. In the latter case, for unseen target parameters, the POD bases need to be adapted with a dedicated technique. An established method addressing this task is the interpolation on the tangent space to the Grassmann manifold (ITSGM). It interpolates the linear subspaces spanned by the known POD bases, discarding by construction the knowledge on the modes ordering by energy levels. In this paper, we formalize an ordered reduced basis interpolation (ORBI) technique which preserves such ordering. This approach improves the adaptation accuracy of the resulting POD basis as more information is taken into account in the construction of the interpolation operator. Numerical investigations are conducted on the parametric reduced order model of a dynamical system representing a small‐scale gas‐bearings supported rotor. Results show that the proposed method is more accurate than the ITSGM method at a similar computation cost. Trained at only three points of the parameters space, the developed hyper reduced order model (h‐ROM) performs to faster simulations with satisfactory accuracy, even far from the training points.

A data-driven model reduction method for parabolic inverse source problems and its convergence analysis

In this paper, we propose a data-driven model reduction method to solve parabolic inverse source problems with uncertain data efficiently. Our method consists of offline and online stages. In the offline stage, we explore the low-dimensional structures in the solution space of parabolic partial differential equations (PDEs) in the forward problems with a given class of source functions and construct a small number of proper orthogonal decomposition (POD) basis functions to achieve significant dimension reduction. Equipped with the POD basis functions, we can solve the forward problems extremely fast in the online stage. Thus, we develop a fast algorithm to solve the optimization problem in parabolic inverse source problems, which is referred to as the POD method. Moreover, we design an a posteriori algorithm to find the optimal regularization parameter in the optimization problem using the proposed POD method without knowing the noise level. Under a weak regularity assumption on the solution of the parabolic PDEs, we prove the convergence of the POD method in solving the forward parabolic PDEs. In addition, we obtain the error estimate of the POD method for parabolic inverse source problems. Finally, we present numerical examples to demonstrate the accuracy and efficiency of the proposed method. Numerical results show that the POD method provides considerable computational savings over the finite element method while maintaining the same accuracy.

A reduced-order immersed interface method based on POD basis for parabolic interface problem

In this paper, we propose a reduced-order immersed interface method (RO-IIM) to cope with the parabolic interface problem. Second-order central difference scheme in spatial direction is constructed. In time direction, we proposed a reduced-order method with fewer degree of freedoms. Our work contains the initial study of the combination of the immersed interface method and proper orthogonal decomposition (POD) technique. Numerical examples show that the proposed RO-IIM is feasible and efficiency for solving the parabolic interface problems.

Proper orthogonal decomposition based seismic source wavefield reconstruction for finite element reverse time migration

The large storage requirement is a critical issue in cross-correlation imaging-condition based reverse time migration (RTM), because it requires the operation of the source and receiver wavefields at the same time. The boundary value method (BVM), based on the finite difference method (FDM), can be used to reconstruct the source wavefield in the reverse time propagation in the same way as the receiver wavefield, which can reduce the storage burden of the RTM data. Considering that the FDM cannot well handle models with discontinuous material properties and rough interfaces, we develop a source wavefield reconstruction strategy based on the finite element method (FEM), using proper orthogonal decomposition (POD) to enhance computational efficiency. In this method, we divide the whole time period into several segments, and construct the POD basis functions to get a reduced order model (ROM) for the source wavefield reconstruction in each segment. We show the corresponding quantitative analysis of the storage requirement of the POD-FEM. Numerical tests on the homogeneous model show the effectiveness of the proposed method, while the layered model and part of the Marmousi model tests indicate that the POD-FEM can keep an excellent balance between computational efficiency and memory usage compared with the full-stored method (FSM) and the BVM, and can be effectively applied in imaging.

A POD/Galerkin model from eigenfunctions of non-converged Newton iterations in a convection problem

A new reduced-order model is applied to study bifurcations in a Rayleigh–Bénard convection problem, using as bifurcation parameter the Rayleigh number. The problem is modeled in a rectangular domain with the incompressible Navier–Stokes equations and the heat equation. The reduced-order model is based on proper orthogonal decomposition and Galerkin projection. The snapshots are eigenfunctions from a linear stability analysis of transient Newton iterations produced by a run of a high-fidelity solver at a single value of the bifurcation parameter. The symmetry properties of the equations have been used to enrich the proper orthogonal decomposition bases. The reduced-order method allows computing the whole bifurcation diagram in an interval of values of the Rayleigh number. Four types of solutions and three bifurcation points are computed with small errors. The results have been validated with the high-fidelity solver. The computational time is reduced with this methodology.

Energy preservation in POD based reduced order models for linearly vibrating systems

The cost of computing the response of a dynamical system grows significantly with the fidelity of the model. Development of a reduced order model (ROM) can alleviate this cost greatly. A proper orthogonal decomposition (POD) based ROM is created by projecting the full-scale model in a lower dimensional subspace obtained from a singular value decomposition of response snapshots. It can be created either in the configuration space or state space. The POD bases have been successful in capturing the maximum displacement and velocity. However, their accuracy in preserving the energy of the vibrating system has not been explored. In this work, this aspect is explored and a new mathematical formulation is proposed that improves the accuracy of the ROM in preserving the energy. Accordingly a new method is proposed where an optimal set of POD bases is found by using preservation of time averaged total mechanical energy – the sum of kinetic and potential energies – in the state space. This energy is of primary concern for undamped or lightly damped systems. The energy being one fundamental quantity in vibration, this new formulation gives POD bases a strong theoretical underpinning. Through detailed numerical studies it is observed that for a given size the proposed method is more accurate than the existing ROM in state space. The improved accuracy is more pronounced for undamped or lightly damped systems. The proposed ROM is accurate even for systems with clustered natural frequencies.

A non-intrusive reduced order model for the characterisation of the spatial power distribution in large thermal reactors

Large thermal reactors endowed with heavy reflectors are very sensitive to localised perturbations. To adequately characterise the spatial effects of these localised disturbances on the reactor power and, thus, to study the spatial stability of the core in a computationally efficient manner, without modifying the legacy codes that may be inefficient for this kind of parametric safety analysis, a non-intrusive surrogate model to approximate the power distributions is proposed in this paper. The meta-model is composed of two steps. First, the group constants, originally produced with the Monte Carlo method for the full-core diffusion calculations, are approximated via a polynomial chaos expansion regression model, with an acceptable accuracy. Then, a model combining Proper Orthogonal Decomposition and Radial Basis Functions is trained to replace the expensive full-core diffusion calculations. Due to the spatial dependence of the input perturbations, some ad hoc strategies are proposed and successfully applied in the reduced order model training phase. Finally, the bootstrap technique is employed to assess the quality of the meta-model approximations and to provide an estimation of the modelling error distribution.

A non-intrusive model order reduction approach for parameterized time-domain Maxwell's equations

<p style='text-indent:20px;'>We present a non-intrusive model order reduction (NIMOR) approach with an offline-online decoupling for the solution of parameterized time-domain Maxwell's equations. During the offline stage, the training parameters are chosen by using Smolyak sparse grid method with an approximation level <inline-formula><tex-math id="M1">\begin{document}$ L $\end{document}</tex-math></inline-formula> (<inline-formula><tex-math id="M2">\begin{document}$ L\geq1 $\end{document}</tex-math></inline-formula>) over a target parameterized space. For each selected parameter, the snapshot vectors are first produced by a high order discontinuous Galerkin time-domain (DGTD) solver formulated on an unstructured simplicial mesh. In order to minimize the overall computational cost in the offline stage and to improve the accuracy of the NIMOR method, a radial basis function (RBF) interpolation method is then used to construct more snapshot vectors at the sparse grid with approximation level <inline-formula><tex-math id="M3">\begin{document}$ L+1 $\end{document}</tex-math></inline-formula>, which includes the sparse grids from approximation level <inline-formula><tex-math id="M4">\begin{document}$ L $\end{document}</tex-math></inline-formula>. A nested proper orthogonal decomposition (POD) method is employed to extract time- and parameter-independent POD basis functions. By using the singular value decomposition (SVD) method, the principal components of the reduced coefficient matrices of the high-fidelity solutions onto the reduced-order subspace spaned by the POD basis functions are extracted. Moreover, a Gaussian process regression (GPR) method is proposed to approximate the dominating time- and parameter-modes of the reduced coefficient matrices. During the online stage, the reduced-order solutions for new time and parameter values can be rapidly recovered via outputs from the regression models without using the DGTD method. Numerical experiments for the scattering of plane wave by a 2-D dielectric cylinder and a multi-layer heterogeneous medium nicely illustrate the performance of the NIMOR method.</p>

Regression Trees on Grassmann Manifold for Adapting Reduced-Order Models

Low-dimensional and computationally less-expensive reduced-order models (ROMs) have been widely used to capture the dominant behaviors of high-4dimensional systems. An ROM can be obtained, using the well-known proper orthogonal decomposition (POD), by projecting the full-order model to a subspace spanned by modal basis modes that are learned from experimental, simulated, or observational data, i.e., training data. However, the optimal basis can change with the parameter settings. When an ROM, constructed using the POD basis obtained from training data, is applied to new parameter settings, the model often lacks robustness against the change of parameters in design, control, and other real-time operation problems. This paper proposes to use regression trees on Grassmann manifold to learn the mapping between parameters and POD bases that span the low-dimensional subspaces onto which full-order models are projected. Motivated by the observation that a subspace spanned by a POD basis can be viewed as a point in the Grassmann manifold, we propose to grow a tree by repeatedly splitting the tree node to maximize the Riemannian distance between the two subspaces spanned by the predicted POD bases on the left and right daughter nodes. Five numerical examples are presented to comprehensively demonstrate the performance of the proposed method, and compare the proposed tree-based method to the existing interpolation method for POD basis and the use of global POD basis. The results show that the proposed tree-based method is capable of establishing the mapping between parameters and POD bases, and thus adapt ROMs for new parameters.

A Preserving Precision Mixed Finite Element Dimensionality Reduction Method for Unsaturated Flow Problem

The unsaturated flow problem is of important applied background and its mixed finite element (MFE) method can be used to simultaneously calculate both water content and flux in soil, which is the most ideal calculation method. Nonetheless, it includes many unknowns. Thereby, herein we will employ the proper orthogonal decomposition (POD) to lower the dimension of unknown solution coefficient vectors in the MFE method for the unsaturated flow problem. Thus, we first examine the MFE method for the unsaturated flow problem and the existence and convergence of the classical MFE solutions. We then take advantage of the initial L MFE solution coefficient vectors to generate a set of POD basis vectors and utilize the most POD basis vectors to create the preserving precision MFE reduced-dimension (PPMFERD) format. Under the circumstances, the PPMFERD format has the same basis functions as the classical MFE format so that it can maintain the same accuracy as the classical MFE format, but it only includes a few unknowns, so it greatly lightens the calculating load, retards the accumulation of computing errors, saves CPU runtime, and improves the accuracy of the real-time calculation in the computational process. Next, we employ the analysis of matrices to demonstrate the existence and convergence of the PPMFERD solutions such that the theoretical analysis becomes very simple and elegant. Finally, we take advantage of some numerical simulations to check on the correctness of the PPMFERD method. It shows that the PPMFERD method is effective and feasible for simulating both water content and flux in unsaturated flow soil.

Single- and multi-objective optimization of an aircraft hot-air anti-icing system based on Reduced Order Method

Geometric optimization of piccolo tube provides a cost-effective approach to shorten the design circle of hot-air anti-icing system and maximize the utilization efficiency of bleed air. An optimization methodology for aircraft hot-air anti-icing systems based on Reduced Order Method (ROM) is developed. ROM based on Proper Orthogonal Decomposition (POD) and Radial Basis Function neural network (RBF) is constructed to evaluate objective and constraint functions of single- and multi-objective optimizations. POD is adopted for data compression and characteristics extraction for the anti-icing performance snapshot matrix. Then, a lower-dimensional approximation for the snapshot matrix is derived from the projection subspace consisting of a set of basis modes. RBF neural network is introduced to construct the mapping between the design variables of piccolo tube geometric configuration samples and the linear combination coefficients of basis modes. The piccolo tube is parameterized by five design variables and is optimized by the developed methodology. The predicted results indicate that the constructed ROM can provide the distributions of surface temperature and runback water with high accuracy and computational efficiency. The optimal results indicate that the developed optimization methodology is efficient and reliable for handling single- and multi-objective design problems of aircraft piccolo tube hot-air anti-icing systems.

Augmented reduced order models for turbulence

The authors introduce an augmented-basis method (ABM) to stabilize reduced-order models (ROMs) of turbulent incompressible flows. The method begins with standard basis functions derived from proper orthogonal decomposition (POD) of snapshot sets taken from a full-order model. These are then augmented with divergence-free projections of a subset of the nonlinear interaction terms that constitute a significant fraction of the time-derivative of the solution. The augmenting bases, which are rich in localized high wavenumber content, are better able to dissipate turbulent kinetic energy than the standard POD bases. Several examples illustrate that the ABM significantly out-performs L2-, H1- and Leray-stabilized POD ROM approaches. The ABM yields accuracy that is comparable to constraint-based stabilization approaches yet is suitable for parametric model-order reduction in which one uses the ROM to evaluate quantities of interests at parameter values that differ from those used to generate the full-order model snapshots. Several numerical experiments point to the importance of localized high wavenumber content in the generation of stable, accurate, and efficient ROMs for turbulent flows.

Remanence deviations in permanent magnet synchronous machines evaluated using a model order reduction approach

AbstractIn this article remanence deviations of a permanent magnet synchronous machine are studied with the proper orthogonal decomposition (POD). The design of experiments enables to create designs with a reduced sample size compared to full factorial designs without decreasing the stochastic significance of the remanence distributions. Next, a POD basis is created with the belonging design, that is distinctly smaller than the reference. From the POD basis the projection matrix is built to compute the flux density solutions. The machine's output is then analyzed regarding accuracy originating from numerical errors. Different parameters are evaluated and compared to the reference to appraise the accuracy of the flux density solutions, which includes the POD basis size, rotor angle step size and dimension of the reduced system. Results show, that the most influence has the dimension of the reduced system. Depending on the dimension and other parameters about one quarter of computational effort can be conserved without a significant loss in accuracy.

Simulation of the interaction of light with 3‐D metallic nanostructures using a proper orthogonal decomposition‐Galerkin reduced‐order discontinuous Galerkin time‐domain method

AbstractIn this artice, we report on a reduced‐order model (ROM) based on the proper orthogonal decomposition (POD) technique for the system of 3‐D time‐domain Maxwell's equations coupled to a Drude dispersion model, which is employed to describe the interaction of light with nanometer scale metallic structures. By using the singular value decomposition (SVD) method, the POD basis vectors are extracted offline from the snapshots produced by a high order discontinuous Galerkin time‐domain (DGTD) solver. With a Galerkin projection and a second order leap‐frog (LF2) time discretization, a discrete ROM is constructed. The stability condition of the ROM is then analyzed. In particular, when the boundary is a perfect electric conductor condition, the global energy of the ROM is bounded, which is consistent with the characteristics of global energy in the DGTD method. It is shown that the ROM based on Galerkin projection can maintain the stability characteristics of the original high dimensional model. Numerical experiments are presented to verify the accuracy, demonstrate the capabilities of the POD‐based ROM and assess its efficiency for 3‐D nanophotonic problems.

Analysis of an unsteady quasi-capillary channel flow with time-resolved PIV and RBF-based super-resolution

We investigate the interface dynamics in an unsteady quasi-capillary channel flow. The configuration consists of a liquid column that moves along a vertical 2D channel, open to the atmosphere and driven by a controlled pressure head. Both advancing and receding contact lines were analyzed to test the validity of classic models for dynamic wetting and to study the flow field near the interface. The operating conditions are characterized by a large acceleration, thus dominated by inertia. The shape of the moving meniscus was retrieved using Laser-Induced Fluorescence-based image processing, while the flow field near was analyzed via Time-Resolved Particle Image Velocimetry (TR-PIV). The TR-PIV measurements were enhanced in the post-processing, using a combination of Proper Orthogonal Decomposition and Radial Basis Functions to achieve super-resolution of the velocity field. Large counter-rotating vortices were observed, and their evolution was monitored in terms of the maximum intensity of the Q-field. The results show that classic contact angle models based on interface velocity cannot describe the evolution of the contact angle at a macroscopic scale. Moreover, the impact of the interface dynamics on the flow field is considerable and extends to several capillary lengths below the interface.