339 research outputs found
Results and questions on a nonlinear approximation approach for solving high-dimensional partial differential equations
We investigate mathematically a nonlinear approximation type approach
recently introduced in [A. Ammar et al., J. Non-Newtonian Fluid Mech., 2006] to
solve high dimensional partial differential equations. We show the link between
the approach and the greedy algorithms of approximation theory studied e.g. in
[R.A. DeVore and V.N. Temlyakov, Adv. Comput. Math., 1996]. On the prototypical
case of the Poisson equation, we show that a variational version of the
approach, based on minimization of energies, converges. On the other hand, we
show various theoretical and numerical difficulties arising with the non
variational version of the approach, consisting of simply solving the first
order optimality equations of the problem. Several unsolved issues are
indicated in order to motivate further research
A well-posed optimal spectral element approximation for the Stokes problem
A method is proposed for the spectral element simulation of incompressible flow. This method constitutes in a well-posed optimal approximation of the steady Stokes problem with no spurious modes in the pressure. The resulting method is analyzed, and numerical results are presented for a model problem
Large-eddy simulation of the flow in a lid-driven cubical cavity
Large-eddy simulations of the turbulent flow in a lid-driven cubical cavity
have been carried out at a Reynolds number of 12000 using spectral element
methods. Two distinct subgrid-scales models, namely a dynamic Smagorinsky model
and a dynamic mixed model, have been both implemented and used to perform
long-lasting simulations required by the relevant time scales of the flow. All
filtering levels make use of explicit filters applied in the physical space (on
an element-by-element approach) and spectral (modal) spaces. The two
subgrid-scales models are validated and compared to available experimental and
numerical reference results, showing very good agreement. Specific features of
lid-driven cavity flow in the turbulent regime, such as inhomogeneity of
turbulence, turbulence production near the downstream corner eddy, small-scales
localization and helical properties are investigated and discussed in the
large-eddy simulation framework. Time histories of quantities such as the total
energy, total turbulent kinetic energy or helicity exhibit different evolutions
but only after a relatively long transient period. However, the average values
remain extremely close
Reduced-basis output bound methods for parabolic problems
In this paper, we extend reduced-basis output bound methods developed earlier for elliptic problems, to problems described by ‘parameterized parabolic’ partial differential equations. The essential new ingredient and the novelty of this paper consist in the presence of time in the formulation and solution of the problem. First, without assuming a time discretization, a reduced-basis procedure is presented to ‘efficiently’ compute accurate approximations to the solution of the parabolic problem and ‘relevant’ outputs of interest. In addition, we develop an error estimation procedure to ‘a posteriori validate’ the accuracy of our output predictions. Second, using the discontinuous Galerkin method for the temporal discretization, the reduced-basis method and the output bound procedure are analysed for the semi-discrete case. In both cases the reduced-basis is constructed by taking ‘snapshots’ of the solution both in time and in the parameters: in that sense the method is close to Proper Orthogonal Decomposition (POD)
Following red blood cells in a pulmonary capillary
The red blood cells or erythrocytes are biconcave shaped cells and consist
mostly in a membrane delimiting a cytosol with a high concentration in
hemoglobin. This membrane is highly deformable and allows the cells to go
through narrow passages like the capillaries which diameters can be much
smaller than red blood cells one. They carry oxygen thanks to hemoglobin, a
complex molecule that have very high affinity for oxygen. The capacity of
erythrocytes to load and unload oxygen is thus a determinant factor in their
efficacy. In this paper, we will focus on the pulmonary capillary where red
blood cells capture oxygen. We propose a camera method in order to numerically
study the behavior of the red blood cell along a whole capillary. Our goal is
to understand how erythrocytes geometrical changes along the capillary can
affect its capacity to capture oxygen. The first part of this document presents
the model chosen for the red blood cells along with the numerical method used
to determine and follow their shapes along the capillary. The membrane of the
red blood cell is complex and has been modelled by an hyper-elastic approach
coming from Mills et al (2004). This camera method is then validated and
confronted with a standard ALE method. Some geometrical properties of the red
blood cells observed in our simulations are then studied and discussed. The
second part of this paper deals with the modeling of oxygen and hemoglobin
chemistry in the geometries obtained in the first part. We have implemented a
full complex hemoglobin behavior with allosteric states inspired from
Czerlinski et al (1999).Comment: 17 page
Flow pattern transition accompanied with sudden growth of flow resistance in two-dimensional curvilinear viscoelastic flows
We find three types of steady solutions and remarkable flow pattern
transitions between them in a two-dimensional wavy-walled channel for low to
moderate Reynolds (Re) and Weissenberg (Wi) numbers using direct numerical
simulations with spectral element method. The solutions are called
"convective", "transition", and "elastic" in ascending order of Wi. In the
convective region in the Re-Wi parameter space, the convective effect and the
pressure gradient balance on average. As Wi increases, the elastic effect
becomes suddenly comparable and the first transition sets in. Through the
transition, a separation vortex disappears and a jet flow induced close to the
wall by the viscoelasticity moves into the bulk; The viscous drag significantly
drops and the elastic wall friction rises sharply. This transition is caused by
an elastic force in the streamwise direction due to the competition of the
convective and elastic effects. In the transition region, the convective and
elastic effects balance. When the elastic effect dominates the convective
effect, the second transition occurs but it is relatively moderate. The second
one seems to be governed by so-called Weissenberg effect. These transitions are
not sensitive to driving forces. By the scaling analysis, it is shown that the
stress component is proportional to the Reynolds number on the boundary of the
first transition in the Re-Wi space. This scaling coincides well with the
numerical result.Comment: 33pages, 23figures, submitted to Physical Review
Heat flow and calculus on metric measure spaces with Ricci curvature bounded below - the compact case
We provide a quick overview of various calculus tools and of the main results
concerning the heat flow on compact metric measure spaces, with applications to
spaces with lower Ricci curvature bounds.
Topics include the Hopf-Lax semigroup and the Hamilton-Jacobi equation in
metric spaces, a new approach to differentiation and to the theory of Sobolev
spaces over metric measure spaces, the equivalence of the L^2-gradient flow of
a suitably defined "Dirichlet energy" and the Wasserstein gradient flow of the
relative entropy functional, a metric version of Brenier's Theorem, and a new
(stronger) definition of Ricci curvature bound from below for metric measure
spaces. This new notion is stable w.r.t. measured Gromov-Hausdorff convergence
and it is strictly connected with the linearity of the heat flow.Comment: To the memory of Enrico Magenes, whose exemplar life, research and
teaching shaped generations of mathematician
Reduced-basis Output Bound Methods for Parametrised Partial Differential Equations
An efficient and reliable method for the prediction of outputs of interest of partial differential equations with affine parameter dependence is presented. To achieve efficiency we employ the reduced-basis method: a weighted residual Galerkin-type method, where the solution is projected onto low-dimensional spaces with certain problem-specific approximation properties. Reliability is obtained by a posteriori error estimation methods - relaxations of the standard error-residual equation that provide inexpensive but sharp and rigorous bounds for the error in outputs of interest. Special affine parameter dependence of the differential operator is exploited to develop a two-stage off-line/on-line blackbox computational procedure. In the on-line stage, for every new parameter value, we calculate the output of interest and an associated error bound. The computational complexity of the on-line stage of the procedure scales only with the dimension of the reduced-basis space and the parametric complexity of the partial differential operator; 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. The theory and corroborating numerical results are presented for: symmetric coercive problems (e.g. problems in conduction heat transfer), parabolic problems (e.g. unsteady heat transfer), noncoercive problems (e.g. the reduced-wave, or Helmholtz, equation), the Stokes problem (e.g flow of highly viscous fluids), and certain nonlinear equations (e.g. eigenvalue problems)
A generalized empirical interpolation method : application of reduced basis techniques to data assimilation
In an effort to extend the classical lagrangian interpolation tools, new interpolating methods that use general interpolating functions are explored. The method analyzed in this paper, called Generalized Empirical Interpolation Method (GEIM), belongs to this class of new techniques. It generalizes the plain Empirical Interpolation Method by replacing the evaluation at interpolating points by application of a class of interpolating linear functions. The paper is divided into two parts: first, the most basic properties of GEIM (such as the well-posedness of the generalized interpolation problem that is derived) will be analyzed. On a second part, a numerical example will illustrate how GEIM, if considered from a reduced basis point of view, can be used for the real-time reconstruction of experiments by coupling data assimilation with numerical simulations in a domain decomposition framework
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