Accelerating oil reservoir simulations using POD on the fly

A reduced order model (ROM) is presented for the long‐term calculation of subsurface oil/water flows. As in several previous ROMs in the field, the Newton iterations in the full model (FM) equations, which are implicit in time, are projected onto a set of modes obtained by applying proper orthogonal decomposition (POD) to a set of snapshots computed by the FM itself. The novelty of the present ROM is that the POD modes are (i) first calculated from snapshots computed by the FM in a short initial stage, and then (ii) updated on the fly along the simulation itself, using new sets of snapshots computed by the FM in even shorter additional runs. Thus, the POD modes adapt themselves to the local dynamics along the simulation, instead of being completely calculated at the outset, which requires a computationally expensive preprocess. This strategy is robust and computationally efficient, which is tested in 10‐ and 30‐year simulations for a realistic reservoir model taken from the SAIGUP project. Copyright © 2016 John Wiley & Sons, Ltd.     

Global and local POD models for the prediction of compressible flows with DG methods

Proper orthogonal decomposition (POD) allows to compress information by identifying the most energetic modes obtained from a database of snapshots. In this work, POD is used to predict the behavior of compressible flows by means of global and local approaches, which exploit some features of a discontinuous Galerkin spatial discretization. The presented global approach requires the definition of high‐order and low‐order POD bases, which are built from a database of high‐fidelity simulations. Predictions are obtained by performing a cheap low‐order simulation whose solution is projected on the low‐order basis. The projection coefficients are then used for the reconstruction with the high‐order basis. However, the nonlinear behavior related to the advection term of the governing equations makes the use of global POD bases quite problematic. For this reason, a second approach is presented in which an empirical POD basis is defined in each element of the mesh. This local approach is more intrusive with respect to the global approach but it is able to capture better the nonlinearities related to advection. The two approaches are tested and compared on the inviscid compressible flow around a gas‐turbine cascade and on the compressible turbulent flow around a wind turbine airfoil.     

基于特征正交分解的三维壁板非线性颤振分析

基于von Karman 大变形理论及活塞理论建立超音速流中壁板的气动弹性方程。采用特征正交分解法 (POD)结合向伽辽金法(Galerkin)的映射这样一种半解析法建立降阶模型(ROM)求解三维壁板的非线性气动弹性问题,并与传统的Galerkin法对比。发现并证明了POD数值模态与伽辽金法简谐基函数之间转换矩阵的正交性,从而简化了POD降阶模型的建立过程。通过数值算例考察了POD法的准确性、收敛性及高效性。结果表明POD降阶模型能够以更少的模态,更高的计算效率达到与Galerkin法同样的精度。以长宽比4为例,POD法以2个模态,3s的时间计算了壁板的振动响应;而Galerkin法需要16个模态,900s的时间。     

Numerical simulation of flapping wings using a panel method and a high‐order Navier–Stokes solver

The design of efficient flapping wings for human engineered micro aerial vehicles (MAVs) has long been an elusive goal, in part because of the large size of the design space. One strategy for overcoming this difficulty is to use a multifidelity simulation strategy that appropriately balances computation time and accuracy. We compare two models with different geometric and physical fidelity. The low‐fidelity model is an inviscid doublet lattice method with infinitely thin lifting surfaces. The high‐fidelity model is a high‐order accurate discontinuous Galerkin Navier–Stokes solver, which uses an accurate representation of the flapping wing geometry. To compare the performance of the two methods, we consider a model flapping wing with an elliptical planform and an analytically prescribed spanwise wing twist, at size scales relevant to MAVs. Our results show that in many cases, including those with mild separation, low‐fidelity simulations can accurately predict integrated forces, provide insight into the flow structure, indicate regions of likely separation, and shed light on design–relevant quantities. But for problems with significant levels of separation, higher‐fidelity methods are required to capture the details of the flow field. Inevitably high‐fidelity simulations are needed to establish the limits of validity of the lower fidelity simulations.Copyright © 2012 John Wiley & Sons, Ltd.     

Error modeling for surrogates of dynamical systems using machine learning

A machine learning–based framework for modeling the error introduced by surrogate models of parameterized dynamical systems is proposed. The framework entails the use of high‐dimensional regression techniques (eg, random forests, and LASSO) to map a large set of inexpensively computed “error indicators” (ie, features) produced by the surrogate model at a given time instance to a prediction of the surrogate‐model error in a quantity of interest (QoI). This eliminates the need for the user to hand‐select a small number of informative features. The methodology requires a training set of parameter instances at which the time‐dependent surrogate‐model error is computed by simulating both the high‐fidelity and surrogate models. Using these training data, the method first determines regression‐model locality (via classification or clustering) and subsequently constructs a “local” regression model to predict the time‐instantaneous error within each identified region of feature space. We consider 2 uses for the resulting error model: (1) as a correction to the surrogate‐model QoI prediction at each time instance and (2) as a way to statistically model arbitrary functions of the time‐dependent surrogate‐model error (eg, time‐integrated errors). We apply the proposed framework to model errors in reduced‐order models of nonlinear oil‐water subsurface flow simulations, with time‐varying well‐control (bottom‐hole pressure) parameters. The reduced‐order models used in this work entail application of trajectory piecewise linearization in conjunction with proper orthogonal decomposition. When the first use of the method is considered, numerical experiments demonstrate consistent improvement in accuracy in the time‐instantaneous QoI prediction relative to the original surrogate model, across a large number of test cases. When the second use is considered, results show that the proposed method provides accurate statistical predictions of the time‐ and well‐averaged errors.     

Development of a reduced-order model for large-scale Eulerian–Lagrangian simulations

Multiphase flows with solid particles are commonly encountered in various industries. The CFD–DEM method is extensively used to simulate their dynamical behavior. However, the application of the CFD–DEM method to simulate industrial-scale powder processes unavoidably leads to huge computational costs. With the aim of overcoming this issue, we propose a nonintrusive reduced-order model for Eulerian–Lagrangian simulations (ROM-EL) to efficiently reproduce gas–solid flow in fluidized beds. In the model, a Lanczos based proper orthogonal decomposition (LPOD) is newly employed to efficiently generate a set of POD bases. After the numerical snapshots are projected onto the reduced space spanned by the POD bases, a series of multidimensional functions of POD coefficients are constructed using a surrogate interpolation method. To demonstrate the effectiveness of this model, validation studies are performed based on the simulations of a fluidized bed. The macroscopic properties, such as the particle distribution, bed height, pressure drop, and distribution of bubble size, are shown to agree well in the CFD–DEM model and ROM-EL. Further, our proposed ROM-EL reduces the computational cost by several orders of magnitude compared with the CFD–DEM simulation. Accordingly, the ROM-EL could significantly contribute to the progress of modeling and simulation for industrial granular flows.     

Highly efficient optimization mesh movement method based on proper orthogonal decomposition

In this paper, we implement the method of proper orthogonal decomposition (POD) to generate a reduced order model (ROM) of an optimization‐based mesh movement scheme. In this study it is shown that POD can be used effectively to generate an ROM, that accurately reproduces the full order mesh movement algorithm, with a decrease in computational time of over 99%. We further introduce a novel training procedure whereby the POD models are generated in a fully automated fashion. The technology is applicable to any mesh movement method and enables potential reductions of up to four orders of magnitude in mesh movement related costs. The proposed model can be implemented without having to pre‐train the POD model, to any fluid–structure interaction code with an existing mesh movement scheme. Copyright © 2010 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 multiscale mass scaling approach for explicit time integration using proper orthogonal decomposition

One of the main computational issues with explicit dynamics simulations is the significant reduction of the critical time step as the spatial resolution of the finite element mesh increases. In this work, a selective mass scaling approach is presented that can significantly reduce the computational cost in explicit dynamic simulations, while maintaining accuracy. The proposed method is based on a multiscale decomposition approach that separates the dynamics of the system into low (coarse scales) and high frequencies (fine scales). Here, the critical time step is increased by selectively applying mass scaling on the fine scale component only. In problems where the response is dominated by the coarse (low frequency) scales, significant increases in the stable time step can be realized. In this work, we use the proper orthogonal decomposition (POD) method to build the coarse scale space. The main idea behind POD is to obtain an optimal low‐dimensional orthogonal basis for representing an ensemble of high‐dimensional data. In our proposed method, the POD space is generated with snapshots of the solution obtained from early times of the full‐scale simulation. The example problems addressed in this work show significant improvements in computational time, without heavily compromising the accuracy of the results. Copyright © 2013 John Wiley & Sons, Ltd.     

New formulation of Cholesky decomposition and applications in stochastic simulation

Monte Carlo simulation plays a significant role in the mechanical and structural analysis due to its versatility and accuracy. Classical spectral representation method is based on the direct decomposition of the power spectral density (PSD) or evolutionary power spectral density (EPSD) matrix through Cholesky decomposition. This direct decomposition of complex matrix usually results in large computational time and storage memory.In this study, a new formulation of the Cholesky decomposition for the EPSD/PSD matrix and corresponding simulation scheme are presented. The key idea to this approach is to separate the phase from the complex EPSD/PSD matrix. The derived real modulus matrix evidently expedites decomposition compared to the direct Cholesky decomposition of the complex EPSD/PSD matrix. In the proposed simulation scheme, the separated phase can be easily assembled. The modulus of EPSD/PSD matrix could be further decomposed into the modulus of coherence matrix (or lagged coherence matrix), which describes the basic coherence structure of stochastic process. The lagged coherence matrix is independence of time and thus remarkably improves the Cholesky decomposition efficiency.The application of the proposed schemes to Gaussian stochastic simulations is presented. Firstly, the previous closed-form wind speed simulation algorithm for equally-spaced locations is extended to a more general situation. Secondly, the proposed approach facilitates the application of interpolation technique in stochastic simulation. The application of interpolation techniques in the wind field simulation is studied as an example.     

An efficient parallel computation of unsteady incompressible viscous flow with elastic moving and compliant boundaries on unstructured grids

This paper presents the development and validation of a parallel unstructured‐grid fluid–structure interaction (FSI) solver for the simulation of unsteady incompressible viscous flow with long elastic moving and compliant boundaries. The Navier–Stokes solver on unstructured moving grid using the arbitrary Lagrangian Eulerian formulation is based on the artificial compressibility approach and a high‐order characteristics‐based finite‐volume scheme. Both unsteady flow and FSI are calculated with a matrix‐free implicit dual time‐stepping scheme. A membrane model has been formulated to study fluid flow in a channel with an elastic membrane wall and their interactions. This model can be employed to calculate arbitrary wall movement and variable tension along the membrane, together with a dynamic mesh method for large deformation of the flow field. The parallelization of the fluid–structure solver is achieved using the single program multiple data programming paradigm and message passing interface for communication of data. The parallel solver is used to simulate fluid flow in a two‐dimensional channel with and without moving membrane for validation and performance evaluation purposes. The speedups and parallel efficiencies obtained by this method are excellent, using up to 16 processors on a SGI Origin 2000 parallel computer. A maximum speedup of 23.14 could be achieved on 16 processors taking advantage of an improved handling of the membrane solver. The parallel results obtained are compared with those using serial code and they are found to be identical. Copyright © 2005 John Wiley & Sons, Ltd.     

Nonhierarchical multi‐model fusion using spatial random processes

New model fusion techniques based on spatial‐random‐process modeling are developed in this work for combining multi‐fidelity data from simulations and experiments. Existing works in multi‐fidelity modeling generally assume a hierarchical structure in which the levels of fidelity of the simulation models can be clearly ranked. In contrast, we consider the nonhierarchical situation in which one wishes to incorporate multiple models whose levels of fidelity are unknown or cannot be differentiated (e.g., if the fidelity of the models changes over the input domain). We propose three new nonhierarchical multi‐model fusion approaches with different assumptions or structures regarding the relationships between the simulation models and physical observations. One approach models the true response as a weighted sum of the multiple simulation models and a single discrepancy function. The other two approaches model the true response as the sum of one simulation model and a corresponding discrepancy function, and differ in their assumptions regarding the statistical behavior of the discrepancy functions, such as independence with the true response or a common spatial correlation function. The proposed approaches are compared via numerical examples and a real engineering application. Furthermore, the effectiveness and relative merits of the different approaches are discussed. Copyright © 2015 John Wiley & Sons, Ltd.     

Normalized explicit approximate inverse preconditioning for solving 3D boundary value problems on uniprocessor and distributed systems

Normalized explicit approximate inverse matrix techniques, based on normalized approximate factorization procedures, for solving sparse linear systems resulting from the finite difference discretization of partial differential equations in three space variables are introduced. Normalized explicit preconditioned conjugate gradient schemes in conjunction with normalized approximate inverse matrix techniques are presented for solving sparse linear systems. The convergence analysis with theoretical estimates on the rate of convergence and computational complexity of the normalized explicit preconditioned conjugate gradient method are also derived. A Parallel Normalized Explicit Preconditioned Conjugate Gradient method for distributed memory systems, using message passing interface (MPI) communication library, is also given along with theoretical estimates on speedups, efficiency and computational complexity. Application of the proposed method on a three‐dimensional boundary value problem is discussed and numerical results are given for uniprocessor and multicomputer systems. Copyright © 2005 John Wiley & Sons, Ltd.     

A hybrid stabilization approach for reduced-order models of compressible flows with shock-vortex interaction

Model order reduction approaches, such as proper orthogonal decomposition (POD)-Galerkin projection, provide a systematic manner to construct Reduced-Order Models (ROM) from pregenerated high-fidelity datasets. The current study focuses on the stabilization of ROMs built from high-fidelity simulation data of a supersonic flow passing a circular cylinder, which features strong interactions between shockwaves and vortices. As shown in previous literatures and the current study, an implicit subspace correction (ISC) method is efficient in the stabilization of similar problems, but its accuracy is not consistent when applied on different ROMs; on the other hand, an eigenvalue reassignment (ER) method delivers superb accuracy when the mode number is small, but becomes too expensive and less robust as the number increases. A Hybrid method is proposed here to balance the computational cost while improving the overall robustness/accuracy in ROM stabilization. The Hybrid method first handles the majority of the modes using the ISC method and then applies the ER method to fine tune a smaller number of modes under a constraint for accuracy. Furthermore, when the typical L2 inner product is changed to a symmetry inner product in both POD computation and Galerkin projection, the performance of the stabilized ROMs is substantially improved for all methods.     

Application of proper orthogonal decomposition to the discrete Euler equations

The response of a fluid moving above a panel to localized oscillation of the panel is predicted using reduced‐order modelling (ROM) with the proper orthogonal decomposition technique. The flow is assumed to be inviscid and is modelled with the Euler equations. These non‐linear equations are discretized with a total‐variation diminishing algorithm and are projected onto an energy‐optimal subspace defined by an energy‐threshold criterion applied to a modal representation of time series data. Results are obtained for a bump oscillating in a Mach 1.2 flow. ROM is found to reduce the degrees of freedom necessary to simulate the flowfield by three orders of magnitude while preserving solution accuracy. Other observed benefits of ROM include increased allowable time step and robustness to variation of oscillation amplitude. Published in 2002 by John Wiley & Sons, Ltd.     

A trust-region framework for constrained optimization using reduced order modeling

Reduced order models (ROM), such as proper orthogonal decomposition (POD), lead to powerful techniques to address computational challenges in PDE-constrained optimization. However, when incorporated within optimization strategies, ROMs are sufficiently accurate only in a restricted zone around the point in decision variable space where they are constructed. Consequently, the ROM needs to be updated in a systematic manner over the course of the optimization. As an enabling strategy, trust-region methods provide an excellent adaptive framework for ROM-based optimization. They not only restrict the optimization step within the ROM’s validity, but also synchronize ROM updates with information obtained during the course of optimization, thus providing a robust and globally convergent framework. This study extends the trust-region framework to constrained optimization problems, using two approaches. We first develop an exact penalty-based trust-region algorithm with correction schemes to ensure global convergence with ROM-based approximate models. Next, we develop a novel filter trust-region algorithm which utilizes refinement of the ROM, and a feasibility restoration phase. Both algorithms are applied to the optimization of a two-bed four-step isothermal pressure swing adsorption (PSA) system, for CO₂ capture. Although both algorithms converge to the same local optimum, the filter approach takes fewer trust-region iterations and requires less computational effort.     

Reduced‐order modeling of parameterized PDEs using time–space‐parameter principal component analysis

This paper presents a methodology for constructing low‐order surrogate models of finite element/finite volume discrete solutions of parameterized steady‐state partial differential equations. The construction of proper orthogonal decomposition modes in both physical space and parameter space allows us to represent high‐dimensional discrete solutions using only a few coefficients. An incremental greedy approach is developed for efficiently tackling problems with high‐dimensional parameter spaces. For numerical experiments and validation, several non‐linear steady‐state convection–diffusion–reaction problems are considered: first in one spatial dimension with two parameters, and then in two spatial dimensions with two and five parameters. In the two‐dimensional spatial case with two parameters, it is shown that a 7 × 7 coefficient matrix is sufficient to accurately reproduce the expected solution, while in the five parameters problem, a 13 × 6 coefficient matrix is shown to reproduce the solution with sufficient accuracy. The proposed methodology is expected to find applications to parameter variation studies, uncertainty analysis, inverse problems and optimal design. Copyright © 2009 John Wiley & Sons, Ltd.     

Wavelet BEM for large‐scale Stokes flows based on the direct integral formulation

This paper describes a new wavelet boundary element method (WBEM) for large‐scale simulations of three‐dimensional Stokes problems. It is based on a Galerkin formulation and uses only one set of wavelet basis. A method for the efficient discretization and compression of the double‐layer integral operator of Stokes equation is proposed. In addition, a compression strategy for further reducing the setting‐up time for the sparse system matrix is also developed. With these new developments, the method has demonstrated a high matrix compression rate for problems with complicated geometries. Applications of the method are illustrated through several examples concerning the modeling of damping forces acting on MEMS resonators including a cantilever resonator oscillating in an unbounded air and a perforated plate resonator oscillating next to a fixed substrate. The numerical results clearly illustrate the efficiency and accuracy of the developed WBEM in these large‐scale Stokes flow simulations. Copyright © 2011 John Wiley & Sons, Ltd.     

Non-intrusive parallelization of multibody system dynamic simulations

This paper evaluates two non-intrusive parallelization techniques for multibody system dynamics: parallel sparse linear equation solvers and OpenMP. Both techniques can be applied to existing simulation software with minimal changes in the code structure; this is a major advantage over Message Passing Interface, the standard parallelization method in multibody dynamics. Both techniques have been applied to parallelize a starting sequential implementation of a global index-3 augmented Lagrangian formulation combined with the trapezoidal rule as numerical integrator, in order to solve the forward dynamics of a variable-loop four-bar mechanism. Numerical experiments have been performed to measure the efficiency as a function of problem size and matrix filling. Results show that the best parallel solver (Pardiso) performs better than the best sequential solver (CHOLMOD) for multibody problems of large and medium sizes leading to matrix fillings above 10. OpenMP also proved to be advantageous even for problems of small sizes. Both techniques delivered speedups above 70% of the maximum theoretical values for a wide range of multibody problems.     

基于双POD模型的空间相关三维随机风场数值模拟

POD法提供了一种高效、准确的风荷载模拟方法,通过对风场的功率谱密度矩阵进行Schur分解,得到一系列的特征值和特征向量,选取主要的几阶特征模态进行计算就可以得到比较精确的结果。该文讨论了具有空间相关性三维随机风场的数学模型,利用双POD模型和蒙特卡罗模拟法,详细描述了空间相关三维风场的数值模拟方法。通过大跨越输电塔三维风场的数值模拟研究表明,模拟的顺风向、横风向、竖直向风速的功率谱密度函数与理论值较为符合,并且具有较好的随机性。证实了该文提出的方法是一种高效、准确的结构三维风场模拟方法,并可应用于大跨空间结构、高层建筑及大跨度桥梁等结构之中。