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Journal ArticleDOI

A Reduced Basis Model with Parametric Coupling for Fluid-Structure Interaction Problems

01 Apr 2012-SIAM Journal on Scientific Computing (Society for Industrial and Applied Mathematics)-Vol. 34, Iss: 2, pp 1187-1213
TL;DR: A reduced order model with reliable a posteriori error bounds is obtained for steady fluid-structure interaction problems and rapid convergence of the reduced solution of the parametrically coupled problem as the number of geometric parameters is increased.
Abstract: We present a new model reduction technique for steady fluid-structure interaction problems. When the fluid domain deformation is suitably parametrized, the coupling conditions between the fluid and the structure can be formulated in the low-dimensional space of geometric parameters. Moreover, we apply the reduced basis method to reduce the cost of repeated fluid solutions necessary to achieve convergence of fluid-structure iterations. In this way a reduced order model with reliable a posteriori error bounds is obtained. The proposed method is validated with an example of steady Stokes flow in an axisymmetric channel, where the structure is described by a simple one-dimensional generalized string model. We demonstrate rapid convergence of the reduced solution of the parametrically coupled problem as the number of geometric parameters is increased.

Summary (1 min read)

Schematic of the control points and resulting free-form parametric deformation

  • This allows the user to keep the number of FFD parameters to a desired low level (in their case roughly 5-10 parameters).
  • Parametric coupling of fluid and structure.
  • 1. Formulation of the coupled problem in the parameter space.
  • This requires showing that the nearest point projection is continuous in the strong H 2 -norm topology.

5.2. Empirical interpolation method for nonaffine problems.

  • Terms, and similarly for the other forms.
  • In practice the EIM has been quite useful for solving nonaffinely parametrized PDEs with the reduced basis method [20, 36, 47] .
  • For the free-form deformation detailed in Sect. 3.2 in fact the forms B and F are affine due to the fact that the map T FFD is polynomial.
  • For generic nonpolynomial shape parametrizations the situation remains more challenging.

6.3. Convergence and accuracy of the coupling algorithm.

  • By introducing a parametric free-form deformation of the flow geometry the fluid equations can be written as parametric partial differential equations on a fixed domain.
  • The authors then applied the reduced basis method to the fluid equations to obtain an efficient reduced model with certified error bounds.
  • The geometric deformation parameters were also used to couple the fluid domain to a 1-d wall equation, where the parameters acted as the coupling variables.
  • The authors demonstrated that for a modest number of free-form deformation parameters an approximate coupling between fluid and structure can be achieved.
  • Numerical simulations were based on the rbMIT toolkit [23] developed by the group of Anthony Patera as well as the MLife fluid mechanics solvers originally authored by Fausto Saleri.

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This is a repository copy of A reduced basis model with parametric coupling for
fluid-structure interation problems.
White Rose Research Online URL for this paper:
http://eprints.whiterose.ac.uk/81804/
Version: Submitted Version
Article:
Lassila, T., Quarteroni, A. and Rozza, G. (2012) A reduced basis model with parametric
coupling for fluid-structure interation problems. SIAM Journal on Scientific Computing , 34
(2). A1187 - A1213. ISSN 1064-8275
https://doi.org/10.1137/110819950
eprints@whiterose.ac.uk
https://eprints.whiterose.ac.uk/
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A REDUCED BASIS MODEL WITH PARAMETRIC COUPLING
FOR FLUID-STRUCTURE INTERACTION PROBLEMS
TONI LASSILA
, ALFIO QUARTERONI
§
, AND GIANLUIGI ROZZA
Abstract. We present a new model reduction technique for steady fluid-structure interaction
problems. When the fluid domain deformation is suitably parametrized, the coupling conditions
between the fluid and structure can be formulated in the low-dimensional space of geometric p aram -
eters. Moreover, we apply the reduced basis method to reduce the cost of repeated fluid solutions
necessary to achieve convergence of fluid-structure iterations. In this way a reduced order m odel
with reliable a posteriori error bounds is obtained. The proposed method is validated with an exam-
ple of steady Stokes flow in an axisymmetric channel, where the structure is described by a simple
1-d generalized string model. We demonstrate rapid convergence of the reduced solution of the
parametrically coupled problem as the number of geometric parameters is increased.
AMS subject classifications. 65N30, 74F10, 76D07
Key words. fluid-structure interaction, mo del reduction, incompressible Stokes equations, re-
duced basis method, free-form deformation
1. Introduction. The numerical simulation of Fluid-Structure Interaction (FSI)
problems is an important topic in wide areas of engineering and medical research.
Concerning the latter, of great importance is the modelling of blood flow in the large
arteries of the human cardiovascular system, where pulsatile flows combined with a
high degree of deformability of the arterial walls togeth er cause large displacement
effects that cannot be neglected when attempting to accurately model the flow dy-
namics of the system. High fidelity computational fluid dynamics and structural
mechanics solvers b ased on, for example, th e Finite Element Method (FEM) need to
be combined in a framework that is challenging both from a mathematical as well as
implementation viewpoint. For an ove rv i ew of cardiovascular modelling techniques we
refer to [42, 44] and the bo ok [14]. The comple xi ty and nonlinearity of FSI problems
has u ntil rece ntly lim i te d the scope of physically meaningful simulations to just small
and isolated sections of arteries. When at t em pt i n g to consider the entire cardiovas-
cular sy st e m as a complex network of different time and spatial scales, Model Order
Reduction (MOR) techniques can accurately and reliably re du ce the nonlinear FSI
models to computationally more cost-efficient ones.
In the geom e tr i c multiscale ap pr oach to MOR [13] the flow network is decomposed
to smaller parts that are joined t oget h er using physical coupling conditions, and each
part of which is model l ed at a level necessary to capture the relevant local dynamics
of the system. The target for our proposed reduced model is those parts of the car-
diovascular network, where a full fideli ty 3-d Navier-Stokes solution is not nece ss ary,
but whe re fluid-stru ct u re interaction effects are still important. The reduced model
should fulfill two conditions: (i) it should have certified a posteriori error bou nd s that
can be tuned to the user’s requirements, and (ii) it should have sufficiently low online
computational memory require ments to fit on one parallel node of a supercomputer.
Mode lling and Scientific Computing (CMCS),
´
Ecole Polytechnique ed´erale de Lausanne, Lau-
sanne, Switzerland (toni.lassila@epfl.ch, gi anlu igi.ro zza@ epfl. ch, alfio.quarteroni@epfl.ch). Support
provided by ERC-Mathcard Project (ERC-2008-AdG 227058).
Department of Mathematics and Systems Analysis, Aalto University, Helsinki, Finland. Sup-
ported by the Emil Aaltonen Foun datio n.
§
Mode lling and Scientific Computing (MOX), Politecnico di Milano, Milan, Italy.
1

2 T. LASSILA, A. QUARTERONI AND G. ROZZA
An important aspect of any large-displacement FSI problem is finding the config-
uration of the interface between fluid and structure. The pro c ess is typically iterative:
a trial configu rat i on of the geometry is used to solve the fluid and structure subprob-
lems, the coupling conditions are test ed , and if th ey are not satisfied within a desired
degree of accuracy then the trial configuration is updated and the step is repeated. A
traditional approach to FSI is that the discrete mesh is updated on each iteration by
moving the boundary nodes and adjusting the interior mesh points to ensure mesh
quality. This approach lead s to a large number of coupling variables (the total number
of mesh points on the fr ee boundary). An external parametrization of the geometr y
can be used to drive down the number of coupling variables. When considering sim-
ple flow geometries the shape of the deformable wall can be direct ly parametrized
e.g. with splines. For realistic geometries it might be necessary to parametrize the
geometry in a way that is relatively indepen de nt of its description .
There are many shape parametrization methods to choose from. Comparisons of
different shape parametrization techniques from a fluid dynam ic s viewpoint can b e
found in [52], and from a model reduction viewpoi nt in [33]. We propose to descri be
the deformat ion s of the fluid channel with Fre e -Form Deformations (FFDs) [53]. They
are a technique for smooth parametric deformations of arbitr ar y shapes embedded in
the grid of control points. FFDs can be used to give a flexible and global parametric
deformation of a fixed reference domain that is comp l et e ly independent of the shape
and its computational mesh. Model reduction for FFD-based shape p aram et ri z at ion s
has been previously considered for the shape design of airfoils in potential [27] and
thermal flows [50]. In cardiovascular applications, FFDs have been used to track the
motion of the left ventricle (see [34] for a review), and to solve an optimal shape
design problem of an aorto-coronaric bypass anastomoses [32].
After parametrizing the geometry with a FFDs we need to address the coupling
between fluid and structure. We use the deformation parameters of the FFD as
coupling variables. A fixed point coupling algorithm can be written in the parameter
space rather than the displacement space. Again an iterat i ve procedure is needed to
ensure the coupling conditions are satisfied to a desired tolerance. Thus a pot e ntially
large number of parametric PDE solutions for the fluid equations need t o be performed
in different parametric configurations.
To reduce the memory requirements and the online computat i onal cost of solving
the fluid system, we apply the Reduced Basis (RB) method (originally proposed and
analyzed in [1, 11, 37 , 41]). It is a rel iabl e MOR method for paramet r ic PDEs. An
overview can be found in [49] and a more detailed exposition in [38]. The attractive-
ness of these methods is based on their ability to give certified a posteriori bounds
on the error of the field solutions and their outputs when compared to the underlying
FEM solution. We use the reduced basis method to reduce the computational cost of
the steady Stokes equations in different configurations of the geometry.
The s t ru ct u r e is t he following: in Sect. 2 we introduce the steady FSI problem of
incompressible Stokes e q uat i ons cou pl e d to an elli p t ic 1-d gen er ali z ed st ri n g equ ati on .
This is a benchmark probl em for which the existence of solutions has been demon-
strated in [18, 19] and whose numerical solution has been previously considered e.g.
in [29, 35, 54]. In Sect. 3 we discuss the geometric parametrization and introduce the
free-form deformations. In Sect. 4 we couple the fluid and structure in the space of
parametric deformations. In Sect. 5 the reduced basis method for the fluid equations
is detailed, and we discuss a posteriori error bounds of the solutions. In Sect. 6 we
present numerical results valid ati n g our approach. Sect. 7 contains some conclusions.

REDUCED BASIS WITH PARAMETRIC COUPLING FOR FSI 3
2. Th e steady fluid-stru ctu r e interaction model. We use the following
standard notations: R
d
, d = 1, 2, 3, is a bounded open set, H
k
(Ω) is the
Sobolev space of functions with weak derivatives up t o order k square-integrable on
X, H
k1/2
(Ω) i s the space of functions that are traces of H
k
(Ω) on the boundary
Ω, H
k
0
(Ω) is the subspace of functions whose trace vanishes on Ω; C
k,α
(Ω) is the
space of functions with derivatives up t o order k being older-continuous with expo-
nent 0 < α 1 (if α = 1 these are the Lipschitz-continuous functions); L
2
(Ω) is the
space of square-integrable functions, and L
(Ω) is the space of essentially bounded
functions on Ω.
o
Γ
w
Γ
in
x
2
x
1
φ
η(x
1
)
R(x
1
)
Γ
out
Fig. 2. 1. Axisymmetric flow geometry for the fluid-structure intera c tio n model problem
2.1. Fluid model: the steady incompressible Stokes equations. We as-
sume the flow geometry represented in Fig. 2.1 that is axisymmetric with cylindrical
coordinates (x, φ) = (x
1
, x
2
, φ)
o
× [0, 2π). The lengthwise cross-section of the
domain
o
:= (0, L) × (0, R) depend s on the unknown radius R(x
1
) of the channel,
which satisfies R(x
1
) := R
0
+ η(x
1
) > 0, where η H
2
0
(0, L) is a function describing
the smooth displacement of the outer wall from its reference configurati on ( a cyli n -
drical tube of radius R
0
> 0). We assume also axis ym met r i c forces, f = f (x) and
g = g(x
2
). Owing to the axial symmetry we can consider the steady Stokes equa-
tions for incompressible fluid flow i n the two-dimensional d omai n
o
(η) with m i xe d
boundary conditions on its boundary Γ(η) = Γ
in
Γ
out
Γ
w
(η), that is
· σ + f = 0 in
o
(η)
· u = 0 in
o
(η)
u = 0 on Γ
w
, u = g on Γ
in
, σ · n = 0 on Γ
out
, (2.1)
where u is the fluid velocity field, and σ is the symmetric Cauchy stress tensor.
The data are assumed to have the following regularity: f [L
2
(Ω
o
)]
2
and g
H
1/2
(Γ), where the space [H
1/2
(Γ)]
2
= γ
Γ
([H
1
(Ω
o
)]
2
) is defined as usual wit h the
continuous trace operator γ
Γ
on Γ. We denote by
b
g [H
1
0
(Ω
o
)]
2
any continuous
extension of the Dirichlet data to the fluid domain. Assuming a Newtonian fluid,
the stress-s t rain relationship is given by σ = pI + ν (u + u
t
) , where ν denotes
the dynamic viscosity and p is t he pressure field. After choosin g the velocity space
V := [H
1
Γ
d
(Ω
o
(η))]
2
of functions that vanish on Γ
d
= Γ
in
Γ
w
and the pressure space
Q := L
2
(Ω
o
(η)), a mixed weak formulation of the eq u at ion s is to find u V and

4 T. LASSILA, A. QUARTERONI AND G. ROZZA
p Q s.t.
Z
o
[νu : v p · v] d =
Z
o
f · v d
Z
o
ν
b
g · v d for all v V
Z
o
q · u d =
Z
o
q ·
b
g d for all q Q
.
(2.2)
The treatment of the inhomogeneous Dirichlet condition is done by lifting this is the
standard way when ai mi n g at reduced basis approximations in parameter-dependent
domains
1
. For notational b r ev i ty we define the bilinear forms
A(u, v) := ν
Z
o
u : v d , B(q, v) :=
Z
o
q · v d (2.3)
and the linear f orm
F(v) :=
Z
o
f · v d. (2.4)
Then (2.2) can be compactly written as
(
A(u, v) + B(p, v) = F(v) A(
b
g, v) for all v V
B(q, u) = −B(q,
b
g) for all q Q
. (2.5)
With our assumptions on the displacement function η the domain
o
is of class C
0,1
and the Stokes equations have a unique solution (u, p) V × Q [16].
2.2. Structural model: the 1-d generalized string equation. Next we give
the equ at i ons for the structural displacement function η. These equations are in the
Lagrangian form on the undeformed configuration of the wall, which we identify as
the interval (0, L) in our simplifi ed 1-d case. We assume the displacements are small
and always in the normal direction of Γ
w
, the tangential displacement being equal
to zero. The equilibrium equation for the structural displacement is chosen as the
second order equation with a fourth orde r perturbation (with ε > 0 small)
ε
4
η
x
4
1
kGh
2
η
x
2
1
+
Eh
1 ν
2
P
η
R
0
(x
1
)
2
= τ
Γ
w
, x
1
(0, L) (2.6)
where h is the wall thickness, k is the Timoshenko shear cor re ct i on factor, G the
shear modulus, E the Young modulus, ν
P
the Poisson ratio, R
0
the radius of the
reference configuration, and τ
Γ
w
denotes the applied traction. This is a simplified
1-d equation for the structure that is often used in haemodynamic fluid-struct u r e
interaction problems as a “first approximation” [44]. We h ave added a fourth order
term in order to have added regularity for the displacement. The weak form of (2.6)
is to find the structural displacement in the normal dire ct i on η D s.t .
τ
Γ
w
(φ) = ε
R
L
0
2
η
x
2
1
2
φ
x
2
1
dx
1
+kGh
R
L
0
η
x
1
φ
x
1
dx
1
+
Eh
1ν
2
P
R
L
0
η φ
R
0
(x
1
)
2
dx
1
:= C(η, φ) (2.7)
for all φ in the space D := H
2
0
(0, L) of kinematically admissible displacements.
1
Other approaches, such as Lagrange multipliers or Nitsche’s method, that might seem more
attractive from a mathematical viewpoint may run into pro blems when dealing with the reduced
basis method.

Citations
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Book
01 Sep 2015
TL;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

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TL;DR: The reduced basis methods (built upon a high-fidelity ‘truth’ finite element approximation) for a rapid and reliable approximation of parametrized partial differential equations are reviewed, and their potential impact on applications of industrial interest is commented on.
Abstract: Reduction strategies, such as model order reduction (MOR) or reduced basis (RB) methods, in scientific computing may become crucial in applications of increasing complexity. In this paper we review the reduced basis methods (built upon a high-fidelity ‘truth’ finite element approximation) for a rapid and reliable approximation of parametrized partial differential equations, and comment on their potential impact on applications of industrial interest. The essential ingredients of RB methodology are: a Galerkin projection onto a low-dimensional space of basis functions properly selected, an affine parametric dependence enabling to perform a competitive Offline-Online splitting in the computational procedure, and a rigorous a posteriori error estimation used for both the basis selection and the certification of the solution. The combination of these three factors yields substantial computational savings which are at the basis of an efficient model order reduction, ideally suited for real-time simulation and many-query contexts (for example, optimization, control or parameter identification). After a brief excursus on the methodology, we focus on linear elliptic and parabolic problems, discussing some extensions to more general classes of problems and several perspectives of the ongoing research. We present some results from applications dealing with heat and mass transfer, conduction-convection phenomena, and thermal treatments.

277 citations

Book ChapterDOI
01 Jan 2014
TL;DR: In this paper, a review of model reduction techniques for fluid dynamics systems is presented, with a focus on steady and unsteady viscous flows modelled by the incompressible Navier-Stokes equations.
Abstract: 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.

213 citations

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TL;DR: This review article will address the two principal components of the cardiovascular system: arterial circulation and heart function, and systematically describe all aspects of the problem, ranging from data imaging acquisition to the development of reduced-order models that are of paramount importance when solving problems with high complexity, which would otherwise be out of reach.
Abstract: Mathematical and numerical modelling of the cardiovascular system is a research topic that has attracted remarkable interest from the mathematical community because of its intrinsic mathematical difficulty and the increasing impact of cardiovascular diseases worldwide. In this review article we will address the two principal components of the cardiovascular system: arterial circulation and heart function. We will systematically describe all aspects of the problem, ranging from data imaging acquisition, stating the basic physical principles, analysing the associated mathematical models that comprise PDE and ODE systems, proposing sound and efficient numerical methods for their approximation, and simulating both benchmark problems and clinically inspired problems. Mathematical modelling itself imposes tremendous challenges, due to the amazing complexity of the cardiocirculatory system, the multiscale nature of the physiological processes involved, and the need to devise computational methods that are stable, reliable and efficient. Critical issues involve filtering the data, identifying the parameters of mathematical models, devising optimal treatments and accounting for uncertainties. For this reason, we will devote the last part of the paper to control and inverse problems, including parameter estimation, uncertainty quantification and the development of reduced-order models that are of paramount importance when solving problems with high complexity, which would otherwise be out of reach.

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TL;DR: A domain parametrization technique is applied to reduce both the geometrical and computational complexities of the forward problem and replace the finite element solution of the incompressible Navier-Stokes equations by a computationally less-expensive reduced-basis approximation.
Abstract: The solution of inverse problems in cardiovascular mathematics is computationally expensive. In this paper, we apply a domain parametrization technique to reduce both the geometrical and computational complexities of the forward problem and replace the finite element solution of the incompressible Navier-Stokes equations by a computationally less-expensive reduced-basis approximation. This greatly reduces the cost of simulating the forward problem. We then consider the solution of inverse problems both in the deterministic sense, by solving a least-squares problem, and in the statistical sense, by using a Bayesian framework for quantifying uncertainty. Two inverse problems arising in hemodynamics modeling are considered: (i) a simplified fluid-structure interaction model problem in a portion of a stenosed artery for quantifying the risk of atherosclerosis by identifying the material parameters of the arterial wall on the basis of pressure measurements; (ii) a simplified femoral bypass graft model for robust shape design under uncertain residual flow in the main arterial branch identified from pressure measurements.

116 citations

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Mixed Approximation and Application to Finite Elements of Degree l.- 2.3. The Technique of Mesh-Dependent Norms.- 3. Further Topics on the "Stream Function-Vorticity-Pressure" Scheme.- 3.1. Refinement of the Error Analysis.- 3.2. Super Convergence Using Quadrilateral Finite Elements of Degree l.- 4. A "Stream Function-Gradient of Velocity Tensor" Method in Two Dimensions.- 4.1. The Hellan-Herrmann-Johnson Formulation.- 4.2. Approximation with Triangular Finite Elements of Degree l.- 4.3. Additional Results for the Hellan-Herrmann-Johnson Scheme.- 4.4. Discontinuous Approximation of the Pressure.- 5. A "Vector Potential-Vorticity" Scheme in Three Dimensions.- 5.1. A Mixed Formulation of the Three-Dimensional Stokes Problem.- 5.2. Mixed Approximation in H(curl ?).- 5.3. A Family of Conforming Finite Elements in H(curl ?).- 5.4. Error Analysis for Finite Elements of Degree l.- 5.5. Discontinuous Approximation of the Pressure.- IV. Theory and Approximation of the Navier-Stokes Problem.- 1. A Class of Nonlinear Problems.- s Problem for the Laplace Operator.- 1.5. Example 3: Dirichlet's Problem for the Biharmonic Operator.- 2. Function Spaces for the Stokes Problem.- 2.1. Preliminary Results.- 2.2. Some Properties of Spaces Related to the Divergence Operator.- 2.3. Some Properties of Spaces Related to the Curl Operator.- 3. A Decomposition of Vector Fields.- 3.1. Decomposition of Two-Dimensional Vector Fields.- 3.2. Application to the Regularity of Functions of H(div ?) ? H(curl ?).- 3.3. Decomposition of Three-Dimensional Vector Fields.- 3.4. The Imbedding of H(div ?) ? H0 (curl ?) into H1(?)3.- 3.5. The Imbedding of H0(div ?) ? H (curl ?) into H1(?)3.- 4. Analysis of an Abstract Variational Problem.- 4.1. A General Result.- 4.2. A Saddle-Point Approach.- 4.3. Approximation by Regularization or Penalty.- 4.4. Iterative Methods of Gradient Type.- 5. The Stokes Equations.- 5.1. The Dirichlet Problem in the Velocity-Pressure Formulation.- 5.2. The Stream Function Formulation of the Dirichlet Problem in Two Dimensions.- 5.3. The Three-Dimensional Case.- Appendix A. Results of Standard Finite Element Approximation.- A.l. Triangular Finite Elements.- A.2. Quadrilateral Finite Elements.- A.3. Interpolation of Discontinuous Functions.- II. Numerical Solution of the Stokes Problem in the Primitive Variables.- 1. General Approximation.- 1.1. An Abstract Approximation Result.- 1.2. Decoupling the Computation of uh and ?h.- 1.3. Application to the Homogeneous Stokes Problem.- 1.4. Checking the inf-sup Condition.- 2. Simplicial Finite Element Methods Using Discontinuous Pressures.- 2.1. A First Order Approximation on Triangular Elements.- 2.2. Higher-Order Approximation on Triangular Elements.- 2.3. The Three-Dimensional case: First and Higher-Order Schemes.- 3. Quadrilateral Finite Element Methods Using Discontinuous Pressures.- 3.1. A quadrilateral Finite Element of Order One.- 3.2. Higher-Order Quadrilateral Elements.- 3.3. An Example of Checkerboard Instability: the Q1 - P0 Element.- 3.4. Error Estimates for the Q1 - P0 Element.- 4. Continuous Approximation of the Pressure.- 4.1. A First Order Method: the "Mini" Finite Element.- 4.2. The "Hood-Taylor" Finite Element Method.- 4.3. The "Glowinski-Pironneau" Finite Element Method.- 4.4. Implementation of the Glowinski-Pironneau Scheme.- III. Incompressible Mixed Finite Element Methods for Solving the Stokes Problem.- 1. Mixed Approximation of an Abstract Problem.- 1.1. A Mixed Variational Problem.- 1.2. Abstract Mixed Approximation.- 2. The "Stream Function-Vorticity-Pressure" Method for the Stokes Problem in Two Dimensions.- 2.1. A Mixed Formulation.- 2.2. Mixed Approximation and Application to Finite Elements of Degree l.- 2.3. The Technique of Mesh-Dependent Norms.- 3. Further Topics on the "Stream Function-Vorticity-Pressure" Scheme.- 3.1. Refinement of the Error Analysis.- 3.2. Super Convergence Using Quadrilateral Finite Elements of Degree l.- 4. A "Stream Function-Gradient of Velocity Tensor" Method in Two Dimensions.- 4.1. The Hellan-Herrmann-Johnson Formulation.- 4.2. Approximation with Triangular Finite Elements of Degree l.- 4.3. Additional Results for the Hellan-Herrmann-Johnson Scheme.- 4.4. Discontinuous Approximation of the Pressure.- 5. A "Vector Potential-Vorticity" Scheme in Three Dimensions.- 5.1. A Mixed Formulation of the Three-Dimensional Stokes Problem.- 5.2. Mixed Approximation in H(curl ?).- 5.3. A Family of Conforming Finite Elements in H(curl ?).- 5.4. Error Analysis for Finite Elements of Degree l.- 5.5. Discontinuous Approximation of the Pressure.- IV. Theory and Approximation of the Navier-Stokes Problem.- 1. A Class of Nonlinear Problems.- 2. Theory of the Steady-State Navier-Stokes Equations.- 2.1. The Dirichlet Problem in the Velocity-Pressure Formulation.- 2.2. The Stream Function Formulation of the Homogeneous Problem..- 3. Approximation of Branches of Nonsingular Solutions.- 3.1. An Abstract Framework.- 3.2. Approximation of Branches of Nonsingular Solutions.- 3.3. Application to a Class of Nonlinear Problems.- 3.4. Non-Differentiable Approximation of Branches of Nonsingular Solutions.- 4. Numerical Analysis of Centered Finite Element Schemes.- 4.1. Formulation in Primitive Variables: Methods Using Discontinuous Pressures.- 4.2. Formulation in Primitive Variables: the Case of Continuous Pressures.- 4.3. Mixed Incompressible Methods: the "Stream Function-Vorticity" Formulation.- 4.4. Remarks on the "Stream Function-Gradient of Velocity Tensor" Scheme.- 5. Numerical Analysis of Upwind Schemes.- 5.1. Upwinding in the Stream Function-Vorticity Scheme.- 5.2. Error Analysis of the Upwind Scheme.- 5.3. Approximating the Pressure with the Upwind Scheme.- 6. Numerical Algorithms.- 2.11. General Methods of Descent and Application to Gradient Methods.- 2.12. Least-Squares and Gradient Methods to Solve the Navier-Stokes Equations.- 2.13. Newton's Method and the Continuation Method.- References.- Index of Mathematical Symbols.

5,572 citations


"A Reduced Basis Model with Parametr..." refers background or methods in this paper

  • ...discretized problem, and thus a sufficient condition for stability is that the finite element velocity and pressure spaces Vh and Qh should be chosen such that they satisfy the discrete Ladyzhenskaya-Babuška-Brezzi (LBB) condition [16]...

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  • ...With our assumptions on the displacement function η the domain Ωo is of class C 0,1 and the Stokes equations have a unique solution (u, p) ∈ V ×Q [16]....

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Journal ArticleDOI
TL;DR: In this paper, a trust region approach for minimizing nonlinear functions subject to simple bounds is proposed, where the trust region is defined by minimizing a quadratic function subject only to an ellipsoidal constraint and the iterates generated by these methods are always strictly feasible.
Abstract: We propose a new trust region approach for minimizing nonlinear functions subject to simple bounds. By choosing an appropriate quadratic model and scaling matrix at each iteration, we show that it is not necessary to solve a quadratic programming subproblem, with linear inequalities, to obtain an improved step using the trust region idea. Instead, a solution to a trust region subproblem is defined by minimizing a quadratic function subject only to an ellipsoidal constraint. The iterates generated by these methods are always strictly feasible. Our proposed methods reduce to a standard trust region approach for the unconstrained problem when there are no upper or lower bounds on the variables. Global and quadratic convergence of the methods is established; preliminary numerical experiments are reported.

3,026 citations

Journal ArticleDOI
31 Aug 1986
TL;DR: A technique is presented for deforming solid geometric models in a free-form manner based on trivariate Bernstein polynomials, and provides the designer with an intuitive appreciation for its effects.
Abstract: A technique is presented for deforming solid geometric models in a free-form manner. The technique can be used with any solid modeling system, such as CSG or B-rep. It can deform surface primitives of any type or degree: planes, quadrics, parametric surface patches, or implicitly defined surfaces, for example. The deformation can be applied either globally or locally. Local deformations can be imposed with any desired degree of derivative continuity. It is also possible to deform a solid model in such a way that its volume is preserved.The scheme is based on trivariate Bernstein polynomials, and provides the designer with an intuitive appreciation for its effects.

2,896 citations


"A Reduced Basis Model with Parametr..." refers methods in this paper

  • ...We propose to describe the deformations of the fluid channel with Free-Form Deformations (FFDs) [53]....

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Book
01 Aug 1994
TL;DR: In this article, the authors provide a thorough illustration of numerical methods, carry out their stability and convergence analysis, derive error bounds, and discuss the algorithmic aspects relative to their implementation.
Abstract: This is the softcover reprint of the very popular hardcover edition. This book deals with the numerical approximation of partial differential equations. Its scope is to provide a thorough illustration of numerical methods, carry out their stability and convergence analysis, derive error bounds, and discuss the algorithmic aspects relative to their implementation. A sound balancing of theoretical analysis, description of algorithms and discussion of applications is one of its main features. Many kinds of problems are addressed. A comprehensive theory of Galerkin method and its variants, as well as that of collocation methods, are developed for the spatial discretization. These theories are then specified to two numerical subspace realizations of remarkable interest: the finite element method and the spectral method.

2,383 citations

Journal ArticleDOI
TL;DR: The rapidly expanding body of work on the development and application of deformable models to problems of fundamental importance in medical image analysis, including segmentation, shape representation, matching and motion tracking is reviewed.

2,222 citations


"A Reduced Basis Model with Parametr..." refers methods in this paper

  • ...In cardiovascular applications, FFDs have been used to track the motion of the left ventricle (see [34] for a review), and to solve an optimal shape design problem of an aorto-coronaric bypass anastomoses [32]....

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Frequently Asked Questions (1)
Q1. What are the contributions in "A reduced basis model with parametric coupling for fluid-structure interaction problems" ?

The authors present a new model reduction technique for steady fluid-structure interaction problems. The authors demonstrate rapid convergence of the reduced solution of the parametrically coupled problem as the number of geometric parameters is increased.