Reduced-order modeling of time-dependent PDEs with multiple parameters in the boundary data

Gunzburger, Max; Peterson, Janet; Shadid, John N.

doi:10.1016/j.cma.2006.08.004

Cited by 143 publications

(144 citation statements)

References 39 publications

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“…These two methods can also be applied to parameter-dependent problems with multiple parameters in the boundary data. For a comparison of the two approaches we refer to [54], where both methods were found to produce similar results.…”

Section: Proper Orthogonal Decompositionmentioning

confidence: 99%

“…the parametric studies in [33] for rotating transitional flow, in [54] for modeling the airflow in a large public building, and in [67] for the analysis of turbulent plane channel flow). In fact, if the system (1) depends in addition on a vector µ ∈ P ⊂ R P of P parameters of interest, we can follow the same procedure, except that the snapshots are now sampled also in the parameter space.…”

Section: Proper Orthogonal Decompositionmentioning

confidence: 99%

See 1 more Smart Citation

Model Order Reduction in Fluid Dynamics: Challenges and Perspectives

Lassila

Manzoni

Quarteroni

et al. 2014

Reduced Order Methods for Modeling and Computational Reduction

161

179

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This chapter reviews techniques of model reduction of fluid dynamics systems. Fluid systems are known to be difficult to reduce efficiently due to several reasons. First of all, they exhibit strong nonlinearities -which are mainly related either to nonlinear convection terms and/or some geometric variability -that often cannot be treated by simple linearization. Additional difficulties arise when attempting model reduction of unsteady flows, especially when long-term transient behavior needs to be accurately predicted using reduced order models and more complex features, such as turbulence or multiphysics phenomena, have to be taken into consideration. We first discuss some general principles that apply to many parametric model order reduction problems, then we apply them on steady and unsteady viscous flows modelled by the incompressible Navier-Stokes equations. We address questions of inf-sup stability, certification through error estimation, computational issues and -in the unsteady case -long-time stability of the reduced model. Moreover, we provide an extensive list of literature references.

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Section: Proper Orthogonal Decompositionmentioning

confidence: 99%

Section: Proper Orthogonal Decompositionmentioning

confidence: 99%

Model Order Reduction in Fluid Dynamics: Challenges and Perspectives

Lassila

Manzoni

Quarteroni

et al. 2014

Reduced Order Methods for Modeling and Computational Reduction

161

179

View full text Add to dashboard Cite

show abstract

“…In order to write the affine decomposition with respect to the parameter σ 1 (t), a suitable choice to embed condition (3) into the variational formulation has to be made. In the literature two possible approaches are proposed: a strong imposition, using a lifting function or suitable Lagrange multipliers [17], and a weak imposition adding suitable penalty variational terms [2,26]. Due to the fact that the Dirichlet data can be written in the form (4), a single time independent lifting function can be constructed and properly weighted by a scalar in order to represent the lifting at each temporal instant.…”

Section: Boundary Conditionmentioning

confidence: 99%

“…Other applications of reduced basis techniques applied to fluid problems can be found (e.g., in [9][10][11][12][13][14][15][16][17][18] and in the recent volume [6]). …”

Section: Introductionmentioning

confidence: 99%

Reduced Numerical Approximation of Reduced Fluid-Structure Interaction Problems With Applications in Hemodynamics

Colciago

Deparis

2018

Front. Appl. Math. Stat.

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This paper deals with fast simulations of the hemodynamics in large arteries by considering a reduced model of the associated fluid-structure interaction problem, which in turn allows an additional reduction in terms of the numerical discretisation. The resulting method is both accurate and computationally cheap. This goal is achieved by means of two levels of reduction: first, we describe the model equations with a reduced mathematical formulation which allows to write the fluid-structure interaction problem as a Navier-Stokes system with non-standard boundary conditions; second, we employ numerical reduction techniques to further and drastically lower the computational costs. The non standard boundary condition is of a generalized Robin type, with a boundary mass and boundary stiffness terms accounting for the arterial wall compliance. The numerical reduction is obtained coupling two well-known techniques: the proper orthogonal decomposition and the reduced basis method, in particular the greedy algorithm. We start by reducing the numerical dimension of the problem at hand with a proper orthogonal decomposition and we measure the system energy with specific norms; this allows to take into account the different orders of magnitude of the state variables, the velocity and the pressure. Then, we introduce a strategy based on a greedy procedure which aims at enriching the reduced discretization space with low offline computational costs. As application, we consider a realistic hemodynamics problem with a perturbation in the boundary conditions and we show the good performances of the reduction techniques presented in the paper. The results obtained with the numerical reduction algorithm are compared with the one obtained by a standard finite element method.The gains obtained in term of CPU time are of three orders of magnitude.

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“…The other approach consists in solving the non-reduced model in the whole time interval, whose solution allows defining the reduced approximation basis that then could be used for solving "similar" models, as the ones that involve for example slight variations in some material parameters or in the boundary conditions. Some recent advances in such approaches can be found in [36], [31], [39], [40], [41] [9], [13], [25] and the references therein.…”

Section: The Proper Orthogonal Decomposition Revisitedmentioning

confidence: 99%

Recirculating Flows Involving Short Fiber Suspensions: Numerical Difficulties and Efficient Advanced Micro-Macro Solvers

Prulière

Ammar

Kissi

et al. 2008

Arch Computat Methods Eng

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is an open access repository that collects the work of Arts et Métiers ParisTech researchers and makes it freely available over the web where possible. SummaryNumerical modelling of non-Newtonian flows usually involves the coupling between equations of motion characterized by an elliptic character, and the fluid constitutive equation, which defines an advection problem linked to the fluid history. There are different numerical techniques to treat the hyperbolic advection equations. In non-recirculating flows, Eulerian discretizations can give a convergent solution within a short computing time. However, the existence of steady recirculating flow areas induces additional difficulties. Actually, in these flows neither boundary conditions nor initial conditions are known. In this paper we compares different advanced strategies (some of them recently proposed and extended here for addressing complex flows) when they are applied to the solution of the kinetic theory description of a short fiber suspension fluid flows.

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Reduced-order modeling of time-dependent PDEs with multiple parameters in the boundary data

Cited by 143 publications

References 39 publications

Model Order Reduction in Fluid Dynamics: Challenges and Perspectives

Model Order Reduction in Fluid Dynamics: Challenges and Perspectives

Reduced Numerical Approximation of Reduced Fluid-Structure Interaction Problems With Applications in Hemodynamics

Recirculating Flows Involving Short Fiber Suspensions: Numerical Difficulties and Efficient Advanced Micro-Macro Solvers

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