Field-to-field coupled fluid structure interaction: A reduced order model study

The standard Fluid-Structure Interaction (fsi ) coupling, that uses as unknowns velocity and pressure for the fluid and displacements for the solid, is compared against two novel types of coupling, the first one a three-field coupling (velocity-pressure-stress/displacement-pressure-stress) introduced by the authors in a recent work, and a two-field coupling (velocity-pressure/displacement-pressure) introduced in this paper, in this way completing our set of Field to Field (f2f ) equations, all stabilized by means of the Variational Multi-Scale (vms ) method using dynamic and orthogonal subscales. The solid two-field fsi coupling formulation is benchmarked statically and dynamically. Proper Orthogonal Decomposition (pod ) is applied to all three fsi formulations to obtain reduced basis and asses their performance in a reduced space. Numerical tests are shown comparing all three formulations. By correctly resolving the Cauchy stress tensor, the three-field fsi coupling proves to provide more accurate results in both Full Order Model (fom ) and Reduced Order Model (rom ) spaces than its counterparts for a similar number of degrees of freedom, making it a reliable formulation. f2f pairing appears to be beneficial, providing more accurate results in all cases shown; mixed pairing with a three-field formulation in the solid appears to produce very precise results as well.     

A reduced order model‐based preconditioner for the efficient solution of transient diffusion equations

This paper presents a novel class of preconditioners for the iterative solution of the sequence of symmetric positive‐definite linear systems arising from the numerical discretization of transient parabolic and self‐adjoint partial differential equations. The preconditioners are obtained by nesting appropriate projections of reduced‐order models into the classical iteration of the preconditioned conjugate gradient (PCG). The main idea is to employ the reduced‐order solver to project the residual associated with the conjugate gradient iterations onto the space spanned by the reduced bases. This approach is particularly appealing for transient systems where the full‐model solution has to be computed at each time step. In these cases, the natural reduced space is the one generated by full‐model solutions at previous time steps. When increasing the size of the projection space, the proposed methodology highly reduces the system conditioning number and the number of PCG iterations at every time step. The cost of the application of the preconditioner linearly increases with the size of the projection basis, and a trade‐off must be found to effectively reduce the PCG computational cost. The quality and efficiency of the proposed approach is finally tested in the solution of groundwater flow models. © 2016 The Authors. International Journal for Numerical Methods in Engineering Published by John Wiley & Sons Ltd.     

Proper orthogonal decomposition‐based model order reduction via radial basis functions for molecular dynamics systems

Model order reduction for molecular dynamics (MD) systems exhibits intrinsic complexities because of the highly nonlinear and nonlocal multi‐atomic interactions in high dimensions. In the present work, we introduce a proper orthogonal decomposition‐based method in conjunction with the radial basis function (RBF) approximation of the nonlinear and nonlocal potential energies and inter‐atomic forces for MD systems. This approach avoids coordinate transformation between the physical and reduced‐order coordinates, and allows the potentials and inter‐atomic forces to be calculated directly in the reduced‐order space. The RBF‐approximated potential energies and inter‐atomic forces in the reduced‐order space are discretized on the basis of the Smolyak sparse grid algorithm to further enhance the effectiveness of the proposed method. The good approximation properties of RBFs in interpolating scattered data make them ideal candidates for the reduced‐order approximation of MD inter‐atomic force calculations. The proposed approach is validated by performing the reduced‐order simulations of DNA molecules under various external loadings. Copyright © 2013 John Wiley & Sons, Ltd.     

Limited‐memory adaptive snapshot selection for proper orthogonal decomposition

Reduced order models are useful for accelerating simulations in many‐query contexts, such as optimization, uncertainty quantification, and sensitivity analysis. However, offline training of reduced order models (ROMs) can have prohibitively expensive memory and floating‐point operation costs in high‐performance computing applications, where memory per core is limited. To overcome this limitation for proper orthogonal decomposition, we propose a novel adaptive selection method for snapshots in time that limits offline training costs by selecting snapshots according an error control mechanism similar to that found in adaptive time‐stepping ordinary differential equation solvers. The error estimator used in this work is related to theory bounding the approximation error in time of proper orthogonal decomposition‐based ROMs, and memory usage is minimized by computing the singular value decomposition using a single‐pass incremental algorithm. Results for a viscous Burgers' test problem demonstrate convergence in the limit as the algorithm error tolerances go to zero; in this limit, the full‐order model is recovered to within discretization error. A parallel version of the resulting method can be used on supercomputers to generate proper orthogonal decomposition‐based ROMs, or as a subroutine within hyperreduction algorithms that require taking snapshots in time, or within greedy algorithms for sampling parameter space. Copyright © 2016 John Wiley & Sons, Ltd.     

Efficient geometrical parametrization for finite-volume-based reduced order methods

In this work, we present an approach for the efficient treatment of parametrized geometries in the context of proper orthogonal decomposition (POD)-Galerkin reduced order methods based on finite-volume full order approximations. On the contrary to what is normally done in the framework of finite-element reduced order methods, different geometries are not mapped to a common reference domain: the method relies on basis functions defined on an average deformed configuration and makes use of the discrete empirical interpolation method to handle together nonaffinity of the parametrization and nonlinearities. In the first numerical example, different mesh motion strategies, based on a Laplacian smoothing technique and on a radial basis function approach, are analyzed and compared on a heat transfer problem. Particular attention is devoted to the role of the nonorthogonal correction. In the second numerical example, the methodology is tested on a geometrically parametrized incompressible Navier-Stokes problem. In this case, the reduced order model is constructed following the same segregated approach used at the full order level.     

A stable Galerkin reduced order model for coupled fluid–structure interaction problems

A stable reduced order model (ROM) of a linear fluid–structure interaction (FSI) problem involving linearized compressible inviscid flow over a flat linear von Kármán plate is developed. Separate stable ROMs for each of the fluid and the structure equations are derived. Both ROMs are built using the ‘continuous’ Galerkin projection approach, in which the continuous governing equations are projected onto the reduced basis modes in a continuous inner product. The mode shapes for the structure ROM are the eigenmodes of the governing (linear) plate equation. The fluid ROM basis is constructed via the proper orthogonal decomposition. For the linearized compressible Euler fluid equations, a symmetry transformation is required to obtain a stable formulation of the Galerkin projection step in the model reduction procedure. Stability of the Galerkin projection of the structure model in the standard L² inner product is shown. The fluid and structure ROMs are coupled through solid wall boundary conditions at the interface (plate) boundary. An a priori energy linear stability analysis of the coupled fluid/structure system is performed. It is shown that, under some physical assumptions about the flow field, the FSI ROM is linearly stable a priori if a stabilization term is added to the fluid pressure loading on the plate. The stability of the coupled ROM is studied in the context of a test problem of inviscid, supersonic flow past a thin, square, elastic rectangular panel that will undergo flutter once the non‐dimensional pressure parameter exceeds a certain threshold. This a posteriori stability analysis reveals that the FSI ROM can be numerically stable even without the addition of the aforementioned stabilization term. Moreover, the ROM constructed for this problem properly predicts the maintenance of stability below the flutter boundary and gives a reasonable prediction for the instability growth rate above the flutter boundary. Copyright © 2013 John Wiley & Sons, Ltd.     

Adaptive reduced order modeling for nonlinear dynamical systems through a new a posteriori error estimator: Application to uncertainty quantification

Multiquery problems such as uncertainty quantification (UQ), optimization of a dynamical system require solving a differential equation at multiple parameter values. Therefore, for large systems, the computational cost becomes prohibitive. This issue can be addressed by using a cheaper reduced order model (ROM) instead. However, the ROM entails error in the solution due to approximation in a lower dimensional subspace. Moreover, the ROM lacks robustness over a wide range of parameter values. To address these issues, first, an upper bound on the norm of the state transition matrix is derived. This bound, along with the residual in the governing equation, are then used to develop an error estimator for general nonlinear dynamical systems. Furthermore, this error estimator is used in conjunction with the modified greedy search algorithm proposed by Hossain and Ghosh (Int J Numer Methods Eng, 2018;116(12-13): 741-758) to adaptively construct a robust proper orthogonal decomposition-based ROM. This adaptive ROM is subsequently deployed for UQ by invoking it in a statistical simulation. Two numerical studies: (i) viscous Burgers' equation and (ii) beam on nonlinear Winkler foundation, showed an improved accuracy of the error estimator compared to the current literature. A significant computational speed-up in UQ is achieved.     

Reduced order model assisted evolutionary algorithms for multi-objective flow design optimization

Evolutionary algorithms (EAs) have been widely used for flow design optimization problems for their well-known robustness and derivative-free property as well as their advantages in dealing with multi-objective optimization problems and providing global optimal solutions. However, EAs usually involve a large number of function evaluations that are sometimes quite time consuming. In this article a reduced order modelling technique that combines proper orthogonal decomposition and radial basis function interpolation is developed to reduce the computational cost. These models provide an efficient way to simulate the whole flow region with varied geometry parameters instead of solving partial differential equations. As a test case, the design optimization of a heat exchanger is considered. Shape variation is conducted through a free form deformation technique, which deforms the computational grid employed by the flow solver. A comparison between the optimization results when using reduced order models and the exact flow solver is presented.     

A combined reduced order-full order methodology for the solution of 3D magneto-mechanical problems with application to magnetic resonance imaging scanners

The design of a new magnetic resonance imaging (MRI) scanner requires multiple numerical simulations of the same magneto-mechanical problem for varying model parameters, such as frequency and electric conductivity, in order to ensure that the vibrations, noise, and heat dissipation are minimized. The high computational cost required for these repeated simulations leads to a bottleneck in the design process due to an increased design time and, thus, a higher cost. To alleviate these issues, the application of reduced order modeling techniques, which are able to find a general solution to high-dimensional parametric problems in a very efficient manner, is considered. Building on the established proper orthogonal decomposition technique available in the literature, the main novelty of this work is an efficient implementation for the solution of 3D magneto-mechanical problems in the context of challenging MRI configurations. This methodology provides a general solution for varying parameters of interest. The accuracy and efficiency of the method are proven by applying it to challenging MRI configurations and comparing with the full-order solution.     

Adaptive training of local reduced bases for unsteady incompressible Navier–Stokes flows

This report presents a numerical study of reduced‐order representations for simulating incompressible Navier–Stokes flows over a range of physical parameters. The reduced‐order representations combine ideas of approximation for nonlinear terms, of local bases, and of least‐squares residual minimization. To construct the local bases, temporal snapshots for different physical configurations are collected automatically until an error indicator is reduced below a user‐specified tolerance. An adaptive time‐integration scheme is also employed to accelerate the generation of snapshots as well as the simulations with the reduced‐order representations. The accuracy and efficiency of the different representations is compared with examples with parameter sweeps. Copyright © 2015 John Wiley & Sons, Ltd.     

Proper orthogonal decomposition‐based reduced basis element thermal modeling of integrated circuits

Various reduced basis element methods are compared for performing transient thermal simulations of integrated circuits. The reduced basis element method is a type of reduced order modeling that takes advantage of repeated geometrical features. It uses a reduced set of basis functions to approximate the solution of a PDE on subdomains (blocks); then these blocks are coupled together to perform a simulation on an entire domain. As the simulations are transient, a proper orthogonal decomposition basis is used, and the proper orthogonal decomposition eigenvalues from each block are used to derive error bounds for the entire simulation. These bounds are used to examine choices of block decompositions for a simplified integrated circuit structure. A decomposition that uses a single block for each transistor device is compared with a decomposition that uses one block for multiple devices. It was found that larger blocks are more computationally efficient; however, the advantage decreases if the devices within a block receive independent signals. Continuous and discontinuous methods of coupling the blocks were also compared. The coupling methods lend themselves to different solution approaches such as static condensation (continuous coupling) and block‐based inversion (discontinuous). Static condensation yielded the best convergence rate, accuracy, and operation count. Copyright © 2017 John Wiley & Sons, Ltd.     

A non‐intrusive model reduction approach for polynomial chaos expansion using proper orthogonal decomposition

In this paper, a non‐intrusive stochastic model reduction scheme is developed for polynomial chaos representation using proper orthogonal decomposition. The main idea is to extract the optimal orthogonal basis via inexpensive calculations on a coarse mesh and then use them for the fine‐scale analysis. To validate the developed reduced‐order model, the method is implemented to: (1) the stochastic steady‐state heat diffusion in a square slab; (2) the incompressible, two‐dimensional laminar boundary‐layer over a flat plate with uncertainties in free‐stream velocity and physical properties; and (3) the highly nonlinear Ackley function with uncertain coefficients. For the heat diffusion problem, the thermal conductivity of the slab is assumed to be a stochastic field with known exponential covariance function and approximated via the Karhunen–Loève expansion. In all three test cases, the input random parameters are assumed to be uniformly distributed, and a polynomial chaos expansion is found using the regression method. The Sobol's quasi‐random sequence is used to generate the sample points. The numerical results of the three test cases show that the non‐intrusive model reduction scheme is able to produce satisfactory results for the statistical quantities of interest. It is found that the developed non‐intrusive model reduction scheme is computationally more efficient than the classical polynomial chaos expansion for uncertainty quantification of stochastic problems. The performance of the developed scheme becomes more apparent for the problems with larger stochastic dimensions and those requiring higher polynomial order for the stochastic discretization. Copyright © 2015 John Wiley & Sons, Ltd.     

Development and application of reduced‐order modeling procedures for subsurface flow simulation

The optimization of subsurface flow processes is important for many applications, including oil field operations and the geological storage of carbon dioxide. These optimizations are very demanding computationally due to the large number of flow simulations that must be performed and the typically large dimension of the simulation models. In this work, reduced‐order modeling (ROM) techniques are applied to reduce the simulation time of complex large‐scale subsurface flow models. The procedures all entail proper orthogonal decomposition (POD), in which a high‐fidelity training simulation is run, solution snapshots are stored, and an eigen‐decomposition (SVD) is performed on the resulting data matrix. Additional recently developed ROM techniques are also implemented, including a snapshot clustering procedure and a missing point estimation technique to eliminate rows from the POD basis matrix. The implementation of the ROM procedures into a general‐purpose research simulator is described. Extensive flow simulations involving water injection into a geologically complex 3D oil reservoir model containing 60 000 grid blocks are presented. The various ROM techniques are assessed in terms of their ability to reproduce high‐fidelity simulation results for different well schedules and also in terms of the computational speedups they provide. The numerical solutions demonstrate that the ROM procedures can accurately reproduce the reference simulations and can provide speedups of up to an order of magnitude when compared with a high‐fidelity model simulated using an optimized solver. Copyright © 2008 John Wiley & Sons, Ltd.     

A sub‐grid scale finite element agglomeration multigrid method with application to the Boltzmann transport equation

This article describes a new element agglomeration multigrid method for solving partial differential equations discretised through a sub‐grid scale finite element formulation. The sub‐grid scale discretisation resolves solution variables through their separate coarse and fine scales, and these are mapped between the multigrid levels using a dual set of transfer operators. The sub‐grid scale multigrid method forms coarse linear systems, possessing the same sub‐grid scale structure as the original discretisation, that can be resolved without them being stored in memory. This is necessary for the application of this article in resolving the Boltzmann transport equation as the linear systems become extremely large. The novelty of this article is therefore a matrix‐free multigrid scheme that is integrated within its own sub‐grid scale discretisation using dual transfer operators and applied to the Boltzmann transport equation. The numerical examples presented are designed to show the method's preconditioning capabilities for a Krylov space‐based solver. The problems range in difficulty, geometry and discretisation type, and comparisons made with established methods show this new approach to perform consistently well. Smoothing operators are also analysed and this includes using the generalized minimal residual method. Here, it is shown that an adaptation to the preconditioned Krylov space is necessary for it to work efficiently. Copyright © 2012 John Wiley & Sons, Ltd.     

Multivariate predictions of local reduced‐order‐model errors and dimensions

This paper introduces multivariate input‐output models to predict the errors and bases dimensions of local parametric Proper Orthogonal Decomposition reduced‐order models. We refer to these mappings as the multivariate predictions of local reduced‐order model characteristics (MP‐LROM) models. We use Gaussian processes and artificial neural networks to construct approximations of these multivariate mappings. Numerical results with a viscous Burgers model illustrate the performance and potential of the machine learning‐based regression MP‐LROM models to approximate the characteristics of parametric local reduced‐order models. The predicted reduced‐order models errors are compared against the multifidelity correction and reduced‐order model error surrogates methods predictions, whereas the predicted reduced‐order dimensions are tested against the standard method based on the spectrum of snapshots matrix. Since the MP‐LROM models incorporate more features and elements to construct the probabilistic mappings, they achieve more accurate results. However, for high‐dimensional parametric spaces, the MP‐LROM models might suffer from the curse of dimensionality. Scalability challenges of MP‐LROM models and the feasible ways of addressing them are also discussed in this study.     

Certification of projection‐based reduced order modelling in computational homogenisation by the constitutive relation error

In this paper, we propose upper and lower error bounding techniques for reduced order modelling applied to the computational homogenisation of random composites. The upper bound relies on the construction of a reduced model for the stress field. Upon ensuring that the reduced stress satisfies the equilibrium in the finite element sense, the desired bounding property is obtained. The lower bound is obtained by defining a hierarchical enriched reduced model for the displacement. We show that the sharpness of both error estimates can be seamlessly controlled by adapting the parameters of the corresponding reduced order model. Copyright © 2013 John Wiley & Sons, Ltd.     

A local multiple proper generalized decomposition based on the partition of unity

It is well known that model order reduction techniques that project the solution of the problem at hand onto a low-dimensional subspace present difficulties when this solution lies on a nonlinear manifold. To overcome these difficulties (notably, an undesirable increase in the number of required modes in the solution), several solutions have been suggested. Among them, we can cite the use of nonlinear dimensionality reduction techniques or, alternatively, the employ of linear local reduced order approaches. These last approaches usually present the difficulty of ensuring continuity between these local models. Here, a new method is presented, which ensures this continuity by resorting to the paradigm of the partition of unity while employing proper generalized decompositions at each local patch.     

On the stability and convergence of a Galerkin reduced order model (ROM) of compressible flow with solid wall and far‐field boundary treatment

A reduced order model (ROM) based on the proper orthogonal decomposition (POD)/Galerkin projection method is proposed as an alternative discretization of the linearized compressible Euler equations. It is shown that the numerical stability of the ROM is intimately tied to the choice of inner product used to define the Galerkin projection. For the linearized compressible Euler equations, a symmetry transformation motivates the construction of a weighted L² inner product that guarantees certain stability bounds satisfied by the ROM. Sufficient conditions for well‐posedness and stability of the present Galerkin projection method applied to a general linear hyperbolic initial boundary value problem (IBVP) are stated and proven. Well‐posed and stable far‐field and solid wall boundary conditions are formulated for the linearized compressible Euler ROM using these more general results. A convergence analysis employing a stable penalty‐like formulation of the boundary conditions reveals that the ROM solution converges to the exact solution with refinement of both the numerical solution used to generate the ROM and of the POD basis. An a priori error estimate for the computed ROM solution is derived, and examined using a numerical test case. Published in 2010 by John Wiley & Sons, Ltd.     

Enhanced goal‐oriented error assessment and computational strategies in adaptive reduced basis solver for stochastic problems

This work focuses on providing accurate low‐cost approximations of stochastic finite elements simulations in the framework of linear elasticity. In a previous work, an adaptive strategy was introduced as an improved Monte‐Carlo method for multi‐dimensional large stochastic problems. We provide here a complete analysis of the method including a new enhanced goal‐oriented error estimator and estimates of CPU (computational processing unit) cost gain. Technical insights of these two topics are presented in details, and numerical examples show the interest of these new developments. Copyright © 2016 John Wiley & Sons, Ltd.     

Fast Bayesian approach for parameter estimation

This paper presents two techniques, i.e. the proper orthogonal decomposition (POD) and the stochastic collocation method (SCM), for constructing surrogate models to accelerate the Bayesian inference approach for parameter estimation problems associated with partial differential equations. POD is a model reduction technique that derives reduced‐order models using an optimal problem‐adapted basis to effect significant reduction of the problem size and hence computational cost. SCM is an uncertainty propagation technique that approximates the parameterized solution and reduces further forward solves to function evaluations. The utility of the techniques is assessed on the non‐linear inverse problem of probabilistically calibrating scalar Robin coefficients from boundary measurements arising in the quenching process and non‐destructive evaluation. A hierarchical Bayesian model that handles flexibly the regularization parameter and the noise level is employed, and the posterior state space is explored by the Markov chain Monte Carlo. The numerical results indicate that significant computational gains can be realized without sacrificing the accuracy. Copyright © 2008 John Wiley & Sons, Ltd.