The Reduced Basis Method for Incompressible Viscous Flow Calculations
01 Jul 1989-Siam Journal on Scientific and Statistical Computing (Society for Industrial and Applied Mathematics)-Vol. 10, Iss: 4, pp 777-786
TL;DR: In this article, the reduced basis method is used in conjunction with a standard continuation technique to approximate the solution curve for the nonlinear equations resulting from discretizing the Navier-Stokes equations by finite element methods.
Abstract: The reduced basis method is a type of reduction method that can be used to solve large systems of nonlinear equations involving a parameter. In this work, the method is used in conjunction with a standard continuation technique to approximate the solution curve for the nonlinear equations resulting from discretizing the Navier–Stokes equations by finite–element methods. This paper demonstrates that the reduced basis method can be implemented to approximate efficiently solutions to incompressible viscous flows. Choices of basis vectors, issues concerning the implementation of the method, and numerical calculations are discussed. Two fluid flow calculations are considered, the driven cavity problem and flow over a forward facing step.
Citations
More filters
TL;DR: A dimension reduction method called discrete empirical interpolation is proposed and shown to dramatically reduce the computational complexity of the popular proper orthogonal decomposition (POD) method for constructing reduced-order models for time dependent and/or parametrized nonlinear partial differential equations (PDEs).
Abstract: A dimension reduction method called discrete empirical interpolation is proposed and shown to dramatically reduce the computational complexity of the popular proper orthogonal decomposition (POD) method for constructing reduced-order models for time dependent and/or parametrized nonlinear partial differential equations (PDEs). In the presence of a general nonlinearity, the standard POD-Galerkin technique reduces dimension in the sense that far fewer variables are present, but the complexity of evaluating the nonlinear term remains that of the original problem. The original empirical interpolation method (EIM) is a modification of POD that reduces the complexity of evaluating the nonlinear term of the reduced model to a cost proportional to the number of reduced variables obtained by POD. We propose a discrete empirical interpolation method (DEIM), a variant that is suitable for reducing the dimension of systems of ordinary differential equations (ODEs) of a certain type. As presented here, it is applicable to ODEs arising from finite difference discretization of time dependent PDEs and/or parametrically dependent steady state problems. However, the approach extends to arbitrary systems of nonlinear ODEs with minor modification. Our contribution is a greatly simplified description of the EIM in a finite-dimensional setting that possesses an error bound on the quality of approximation. An application of DEIM to a finite difference discretization of the one-dimensional FitzHugh-Nagumo equations is shown to reduce the dimension from 1024 to order 5 variables with negligible error over a long-time integration that fully captures nonlinear limit cycle behavior. We also demonstrate applicability in higher spatial dimensions with similar state space dimension reduction and accuracy results.
1,695 citations
TL;DR: (hierarchical, Lagrange) reduced basis approximation and a posteriori error estimation for linear functional outputs of affinely parametrized elliptic coercive partial differential equations are considered.
Abstract: In this paper we consider (hierarchical, La-grange)reduced basis approximation anda posteriori error estimation for linear functional outputs of affinely parametrized elliptic coercive partial differential equa-tions. The essential ingredients are (primal-dual)Galer-kin projection onto a low-dimensional space associated with a smooth “parametric manifold” - dimension re-duction; efficient and effective greedy sampling meth-ods for identification of optimal and numerically stable approximations - rapid convergence;a posteriori er-ror estimation procedures - rigorous and sharp bounds for the linear-functional outputs of interest; and Offine-Online computational decomposition strategies - min-imummarginal cost for high performance in the real-time/embedded (e.g., parameter-estimation, control)and many-query (e.g., design optimization, multi-model/ scale)contexts. We present illustrative results for heat conduction and convection-diffusion,inviscid flow, and linear elasticity; outputs include transport rates, added mass,and stress intensity factors.
1,090 citations
Cites background from "The Reduced Basis Method for Incomp..."
...Much work focuses on the stationary incompressible (quadratically nonlinear) Navier-Stokes equations [29, 50, 57] of incompressible fluid flow: suitable stable approximations are considered in [57, 67, 114, 123, 137, 139]; rigorous a posteriori error estimation—within the general Brezzi-Rappaz-Raviart (“BRR”) a posteriori framework [30, 34]—is considered in [45, 97, 151, 152]....
[...]
...In the context of affine parameter dependence, in which the operator is expressible as the sum of Q products of parameter-dependent functions and parameter-independent operators, the Offline-Online idea is quite self-apparent and indeed has been re-invented often [15, 66, 70, 114]; however, application of the concept to a posteriori error estimation—note the Online complexity of both the output and the output error bound calculation must be independent of N—is more involved and more recent [64, 121, 122]....
[...]
...2The special issues associated with saddle problems [28, 29], in particular the Stokes equations of incompressible flow, are addressed for divergence-free spaces in [57, 67, 114] and non-divergence-free spaces in [135, 139]....
[...]
Book•
01 Sep 2015TL;DR: In this article, the authors provide a thorough introduction to the mathematical and algorithmic aspects of certified reduced basis methods for parametrized partial differential equations, including model construction, error estimation and computational efficiency.
Abstract: This book provides a thorough introduction to the mathematical and algorithmic aspects of certified reduced basis methods for parametrized partial differential equations. Central aspects ranging from model construction, error estimation and computational efficiency to empirical interpolation methods are discussed in detail for coercive problems. More advanced aspects associated with time-dependent problems, non-compliant and non-coercive problems and applications with geometric variation are also discussed as examples.
831 citations
TL;DR: The method is ideally suited for the repeated and rapid evaluations required in the context of parameter estimation, design, optimization, and real-time control.
Abstract: We present a technique for the rapid and reliable prediction of linear-functional outputs of elliptic (and parabolic) partial differential equations with affine parameter dependence. The essential components are (i) (provably) rapidly convergent global reduced basis approximations, Galerkin projection onto a space W(sub N) spanned by solutions of the governing partial differential equation at N selected points in parameter space; (ii) a posteriori error estimation, relaxations of the error-residual equation that provide inexpensive yet sharp and rigorous bounds for the error in the outputs of interest; and (iii) off-line/on-line computational procedures, methods which decouple the generation and projection stages of the approximation process. The operation count for the on-line stage, in which, given a new parameter value, we calculate the output of interest and associated error bound, depends only on N (typically very small) and the parametric complexity of the problem; the method is thus ideally suited for the repeated and rapid evaluations required in the context of parameter estimation, design, optimization, and real-time control.
588 citations
TL;DR: In this paper, the authors extended the reduced-basis approximations developed earlier for linear elliptic and parabolic partial differential equations with affine parameter dependence to problems in volving.
Abstract: In this paper, we extend the reduced-basis approximations developed earlier for linear elliptic and parabolic partial differential equations with affine parameter dependence to problems in- volving (a) nonaffine dependence on the parameter, and (b) nonlinear dependence on the field variable. The method replaces the nonaffine and nonlinear terms with a coefficient function approximation which then permits an efficient offline-online computational decomposition. We first review the coefficient function approximation procedure: the essential ingredients are (i) a good collateral reduced-basis approximation space, and (ii) a stable and inexpensive interpolation procedure. We then apply this approach to linear nonaffine and nonlinear elliptic and parabolic equations; in each instance, we discuss the reduced-basis approximation and the associated offline-online computational procedures. Numeri- cal results are presented to assess our approach.
532 citations
Cites methods from "The Reduced Basis Method for Incomp..."
...The reduced-basis method was first introduced in the late 1970s for the nonlinear analysis of structures [1,25] and subsequently abstracted and analyzed [5,11,28,33] and extended [16,18,26] to a much larger class of parametrized partial differential equations....
[...]
References
More filters
TL;DR: The vorticity-stream function formulation of the two-dimensional incompressible NavierStokes equations is used to study the effectiveness of the coupled strongly implicit multigrid (CSI-MG) method in the determination of high-Re fine-mesh flow solutions.
4,018 citations
TL;DR: In this paper, a reduced basis technique and a computational algorithm are presented for predicting the nonlinear static response of structures, where a total Lagrangian formulation is used and the structure is discretized by using displacement finite element models.
Abstract: A reduced basis technique and a computational' algorithm are presented for predicting the nonlinear static response of structures. A total Lagrangian formulation is used and the structure is discretized by using displacement finite element models. The nodal displacement vector is expressed as a linear combination of a small number of basis vectors and a Rayleigh-Ritz technique is used to approximate the finite element equations by a reduced system of nonlinear equations. The Rayleigh-Ritz approximation functions (basis vectors) are chosen to be those commonly used in the static perturbation technique namely, a nonlinear solution and a number of its path derivatives. A procedure is outlined for automatically selecting the load (or displacement) step size and monitoring the solution accuracy. The high accuracy and effectiveness of the proposed approach is demonstrated by means of numerical examples.
414 citations
276 citations
TL;DR: The design of a package of continuation procedures called PITCON to handle the following tasks is described: follow numerically any a priori specified curve on an equilibrium manifold determine the exact location of target points where a given variable has a specified value.
Abstract: The design of a package of continuation procedures called PITCON to handle the following tasks is described' (1) follow numerically any a priori specified curve on an equilibrium manifold; (2) on such a curve determine the exact location of target points where a given variable has a specified value; and (3) on such a curve identify and compute exactly the (simple) limit points where stability may be lost. The process is based on the local parametenzatmn wh]ch uses an estimate of the curvature to control the chome of parameter varmble. Categorms and Subject Descriptors. G 1.5 [Numerical Analysis]Roots of Nonlinear Equations-tteratwe methods, systems of equattons General Terms" Algorithms, Design Addltmnal
212 citations
TL;DR: In this paper, a theoretical foundation for the reduced basis technique and mathematical reasons for its effectiveness are provided. But they do not consider the problem of estimating the exact solution curve for a large class of nonlinear problems.
Abstract: This paper provides a theoretical foundation for the reduced basis technique and gives mathematical reasons for its effectiveness. Some rather general results about the existence of solution curves for a large class of nonlinear problems are developed. These results are then used to derive estimates between an exact solution curve and its reduced basis approximation.
Die vorliegende Arbeit bietet eine theoretische Begrundung fur das Verfahren der Reduzierten Basis und gibt mathematische Argumente fur deren Effektivitat au. Es werden einige recht allgemeine Ergebnisse uber die Existenz von Losungskurven fur eine umfangreiche Klasse nichtlinearer Probleme entwickelt. Diese Ergebnisse werden dann dazu verwendet, Fehlerabschatzungen zwischen einer exakten Losungskurve und deren Approximation nach dem Verfahren der Reduzierten Basis abzuleiten.
167 citations