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.     

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.     

POD–ISAT: An efficient POD‐based surrogate approach with adaptive tabulation and fidelity regions for parametrized steady‐state PDE discrete solutions

A combination of proper orthogonal decomposition (POD) analysis and in situ adaptive tabulation (ISAT) is proposed for the representation of parameter‐dependent solutions of coupled partial differential equation problems. POD is used for the low‐order representation of the spatial fields and ISAT for the local representation of the solution in the design parameter space. The accuracy of the method is easily controlled by free threshold parameters that can be adjusted according to user needs. The method is tested on a coupled fluid‐thermal problem: the design of a simplified aircraft air control system. It is successfully compared with the standard POD; although the POD is inaccurate in certain areas of the design parameters space, the POD–ISAT method achieves accuracy thanks to trust regions based on residuals of the fluid‐thermal problem. The presented POD–ISAT approach provides flexibility, robustness and tunable accuracy to represent solutions of parametrized partial differential equations.Copyright © 2013 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.     

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.     

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.     

A multiscale large time increment/FAS algorithm with time‐space model reduction for frictional contact problems

A multiscale strategy using model reduction for frictional contact computation is presented. This new approach aims to improve computation time of finite element simulations involving frictional contact between linear and elastic bodies. This strategy is based on a combination between the LATIN (LArge Time INcrement) method and the FAS multigrid solver. The LATIN method is an iterative solver operating on the whole time‐space domain. Applying an a posteriori analysis on solutions of different frictional contact problems shows a great potential as far as reducibility for frictional contact problems is concerned. Time‐space vectors forming the so‐called reduced basis depict particular scales of the problem. It becomes easy to make analogies with multigrid method to take full advantage of multiscale information. Copyright © 2013 John Wiley & Sons, Ltd.     

Taguchi-based design of experiments in training POD-RBF surrogate model for inverse material modelling using nanoindentation

This paper investigates the use of a surrogate model based on Proper Orthogonal Decomposition (POD) and Radial Basis Functions (RBF) for calibrating the nanoindentation-based loading of an elastic–plastic material. Using Taguchi design of experiments and Analysis of Variance (ANOVA), the total number of finite element-based training points is reduced for input parameters that exhibited lower significance. It is found that ANOVA-based sensitivity information can be used to reduce the number of training points without significantly affecting model accuracy. It is also observed that RBFs capable of conforming nonlinearly perform better when the spatial distance between training points increases. Furthermore, for some RBFs the performance is further tuned by choice of the shape parameters. Finally, it is demonstrated that the surrogate model’s performance remains stable under the effect of random noise. Thereby, this study provides a general framework for solving a nanoindentation-based material modelling inverse problem using the POD–RBF technique.     

Local/global model order reduction strategy for the simulation of quasi‐brittle fracture

This paper proposes a novel technique to reduce the computational burden associated with the simulation of localized failure. The proposed methodology affords the simulation of damage initiation and propagation while concentrating the computational effort where it is most needed, that is, in the localization zones. To do so, a local/global technique is devised where the global (slave) problem (far from the zones undergoing severe damage and cracking) is solved for in a reduced space computed by the classical proper orthogonal decomposition while the local (master) degrees of freedom (associated with the part of the structure where most of the damage is taking place) are fully resolved. Both domains are coupled through a local/global technique. This method circumvents the difficulties associated with model order reduction for the simulation of highly nonlinear mechanical failure and offers an alternative or complementary approach to the development of multiscale fracture simulators. Copyright © 2011 John Wiley & Sons, Ltd.     

Accelerating iterative solution methods using reduced‐order models as solution predictors

We propose the use of reduced‐order models to accelerate the solution of systems of equations using iterative solvers in time stepping schemes for large‐scale numerical simulation. The acceleration is achieved by determining an improved initial guess for the iterative process based on information in the solution vectors from previous time steps. The algorithm basically consists of two projection steps: (1) projecting the governing equations onto a subspace spanned by a low number of global empirical basis functions extracted from previous time step solutions, and (2) solving the governing equations in this reduced space and projecting the solution back on the original, high dimensional one. We applied the algorithm to numerical models for simulation of two‐phase flow through heterogeneous porous media. In particular we considered implicit‐pressure explicit‐saturation (IMPES) schemes and investigated the scope to accelerate the iterative solution of the pressure equation, which is by far the most time‐consuming part of any IMPES scheme. We achieved a substantial reduction in the number of iterations and an associated acceleration of the solution. Our largest test problem involved 93 500 variables, in which case we obtained a maximum reduction in computing time of 67%. The method is particularly attractive for problems with time‐varying parameters or source terms. Copyright © 2006 John Wiley & Sons, Ltd.