HiMod Reduction of Advection–Diffusion–Reaction Problems with General Boundary Conditions

Aletti, Matteo; Perotto, Simona; Veneziani, Alessandro

doi:10.1007/s10915-017-0614-5

Cited by 27 publications

(49 citation statements)

References 34 publications

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“…Conversely, for small values of h , the error is dominated by the modal component. The convergence rate is linear with respect to the reciprocal of the number of modes, which is consistent with the results in Aletti et al on slabs.HiMod is sensibly competitive with respect to the standard FEM in terms of accuracy and efficiency. For a given number of DOFs, the HiMod error is consistently about one order of magnitude lower than the FEM error (Figure , right).…”

Section: Scalar Advection‐diffusion‐reaction Problemsmentioning

confidence: 99%

“… In the “top‐down” approach, we construct directly the bivariate basis function in

false(\overset{r}{true}, \overset{ϑ}{true} false)

by solving an SLE problem associated with an appropriate self‐adjoint differential operator defined on the unit‐disk, completed with the homogeneous boundary conditions of the original problem to solve. We mimic in this way the “educated approach” introduced in Aletti et al for ADR problems in slabs, where tensor product splitting allows to simplify the 2D computation to 1D SLE problems. In the “bottom‐up” approach, we factorize the construction of the basis by working along

\overset{r}{true}

and

\overset{ϑ}{true}

, separately, and assembling the bivariate basis afterwards. Boundary conditions are then enforced at a second stage.…”

Section: Himod In Cylindrical Domainsmentioning

confidence: 99%

“…Under the assumptions of the spectral theorem, the eigenfunctions associated with the operator

scriptL

are orthogonal with respect to the

L_{w}^{2}

‐weighted scalar product and form a complete set in the same space. Since functions

{\overset{ϕ}{true}}_{k}

automatically include the lateral boundary conditions, the set

false{{\overset{ϕ}{true}}_{k} false}

has been called an “educated” basis …”

Section: Himod In Cylindrical Domainsmentioning

confidence: 99%

“…HiMod is intended to provide a unified numerical tool able to promptly incorporate the transverse components of the 3D solution into a “psychologically 1D” solver. The HiMod reduction has been successfully used to solve advection‐diffusion‐reaction (ADR) problems in 2‐dimensional (2D) domains and parallelepipeds …”

Section: Introductionmentioning

confidence: 99%

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Hierarchical model reduction for incompressible fluids in pipes

Guzzetti

Perotto

Veneziani

2018

Numerical Meth Engineering

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Summary Hierarchical model reduction is intended to solve efficiently partial differential equations in domains with a geometrically dominant direction. In many engineering applications, these problems are often reduced to 1‐dimensional differential systems. This guarantees computational efficiency yet dumps local accuracy as nonaxial dynamics are dropped. Hierarchical model reduction recovers the secondary components of the dynamics of interest with a combination of different discretization techniques, following up a natural separation of variables. The dominant direction is generally solved by the finite element method or isogeometric analysis to guarantee flexibility, while the transverse components are solved by spectral methods, to guarantee a small number of degrees of freedom. By judiciously selecting the number of transverse modes, the method has been proven to improve significantly the accuracy of purely 1‐dimensional solvers, with great computational efficiency. A Cartesian framework has been used so far both in slab domains and cylindrical pipes (including arteries) mapped to Cartesian reference domains. In this paper, we investigate the alternative use of a polar coordinates system for the transverse dynamics in circular or elliptical pipes. This seems a natural choice for applications like computational hemodynamics. In spite of this, the selection of a basis function set for the transverse dynamics is troublesome. As pointed out in the literature—even for simple elliptical problems—there is no “best” basis available. In this paper, we perform an extensive investigation of hierarchical model reduction in polar coordinates to discuss different possible choices for the transverse basis, pointing out pros and cons of the polar coordinate system.

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Section: Scalar Advection‐diffusion‐reaction Problemsmentioning

confidence: 99%

“… In the “top‐down” approach, we construct directly the bivariate basis function in

false(\overset{r}{true}, \overset{ϑ}{true} false)

\overset{r}{true}

and

\overset{ϑ}{true}

, separately, and assembling the bivariate basis afterwards. Boundary conditions are then enforced at a second stage.…”

Section: Himod In Cylindrical Domainsmentioning

confidence: 99%

“…Under the assumptions of the spectral theorem, the eigenfunctions associated with the operator

scriptL

are orthogonal with respect to the

L_{w}^{2}

‐weighted scalar product and form a complete set in the same space. Since functions

{\overset{ϕ}{true}}_{k}

automatically include the lateral boundary conditions, the set

false{{\overset{ϕ}{true}}_{k} false}

has been called an “educated” basis …”

Section: Himod In Cylindrical Domainsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Hierarchical model reduction for incompressible fluids in pipes

Guzzetti

Perotto

Veneziani

2018

Numerical Meth Engineering

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“…For more details about the choice of the modal basis, we refer to [2,14,20], while V h 1D may be a classical finite element space [4,17,18,20] or an isogeometric function space [21].…”

Section: The Hi-mod Settingmentioning

confidence: 99%

Hi-POD Solution of Parametrized Fluid Dynamics Problems: Preliminary Results

Baroli

Cova

Perotto

et al. 2017

Model Reduction of Parametrized Systems

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Transversally enriched pipe element method (TEPEM): An effective numerical approach for blood flow modeling

Alvarez

Blanco

Bulant

et al. 2016

Numer Methods Biomed Eng

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In this work, we present a novel approach tailored to approximate the Navier-Stokes equations to simulate fluid flow in three-dimensional tubular domains of arbitrary cross-sectional shape. The proposed methodology is aimed at filling the gap between (cheap) one-dimensional and (expensive) three-dimensional models, featuring descriptive capabilities comparable with the full and accurate 3D description of the problem at a low computational cost. In addition, this methodology can easily be tuned or even adapted to address local features demanding more accuracy. The numerical strategy employs finite (pipe-type) elements that take advantage of the pipe structure of the spatial domain under analysis. While low order approximation is used for the longitudinal description of the physical fields, transverse approximation is enriched using high order polynomials. Although our application of interest is computational hemodynamics and its relevance to pathological dynamics like atherosclerosis, the approach is quite general and can be applied in any internal fluid dynamics problem in pipe-like domains. Numerical examples covering academic cases as well as patient-specific coronary arterial geometries demonstrate the potentialities of the developed methodology and its performance when compared against traditional finite element methods. Copyright © 2016 John Wiley & Sons, Ltd.

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HiMod Reduction of Advection–Diffusion–Reaction Problems with General Boundary Conditions

Abstract: We extend the hierarchical model reduction procedure previously introduced in

Cited by 27 publications

References 34 publications

Hierarchical model reduction for incompressible fluids in pipes

Hierarchical model reduction for incompressible fluids in pipes

Hi-POD Solution of Parametrized Fluid Dynamics Problems: Preliminary Results

Transversally enriched pipe element method (TEPEM): An effective numerical approach for blood flow modeling

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