Comparisons between reduced order models and full 3D models for fluid-structure interaction problems in haemodynamics

When modelling the cardiovascular system, the effect of the vessel wall on the blood flow has great relevance. Arterial vessels are complex living tissues and three-dimensional specific models have been proposed to represent their behaviour. The numerical simulation of the 3D-3D Fluid-Structure Interaction (FSI) coupled problem has high computational costs in terms of required time and memory storage. Even if many possible solutions have been explored to speed up the resolution of such problem, we are far from having a 3D-3D FSI model that can be solved quickly. In 3D-3D FSI models two of the main sources of complexity are represented by the domain motion and the coupling between the fluid and the structural part. Nevertheless, in many cases, we are interested in the blood flow dynamics in compliant vessels, whereas the displacement of the domain is small and the structure dynamics is less relevant. In these situations, techniques to reduce the complexity of the problem can be used. One consists in using transpiration conditions for the fluid model as surrogate for the wall displacement, thus allowing problem's solution on a fixed domain. Another strategy consists in modelling the arterial wall as a thin membrane under specific assumptions (Figueroa et al., 2006, Nobile and Vergara, 2008) instead of using a more realistic (but more computationally intensive) 3D elastodynamic model. Using this strategy the dynamics of the vessel motion is embedded in the equation for the blood flow. Combining the transpiration conditions with the membrane model assumption, we obtain an attractive formulation, in fact, instead of solving two different models on two moving physical domains, we solve only a Navier-Stokes system in a fixed fluid domain where the structure model is integrated as a generalized Robin condition. In this paper, we present a general formulation in the boundary conditions which is independent of the time discretization scheme choice and on the stress-strain constitutive relation adopted for the vessel wall structure. Our aim is, first, to write a formulation of a reduced order model with zero order transpiration conditions for a generic time discretization scheme, then to compare a 3D-3D PSI model and a reduced FSI one in two realistic patient-specific cases: a femoropopliteal bypass and an aorta. In particular, we are interested in comparing the wall shear stresses, in fact this quantity can be used as a risk factor for some pathologies such as atherosclerosis or thrombogenesis. More in general we want to assess the accuracy and the computational convenience to use simpler formulations based on reduced order models. In particular, we show that, in the case of small displacements, using a 3D-3D PSI linear elastic model or the correspondent reduced order one yields many similar results. (c) 2013 Elsevier B.V. All rights reserved

Numerical Modelling of the Brain Poromechanics by High-Order Discontinuous Galerkin Methods

We introduce and analyze a discontinuous Galerkin method for the numerical modelling of the equations of Multiple-Network Poroelastic Theory (MPET) in the dynamic formulation. The MPET model can comprehensively describe functional changes in the brain considering multiple scales of fluids. Concerning the spatial discretization, we employ a high-order discontinuous Galerkin method on polygonal and polyhedral grids and we derive stability and a priori error estimates. The temporal discretization is based on a coupling between a Newmark $\beta$-method for the momentum equation and a $\theta$-method for the pressure equations. After the presentation of some verification numerical tests, we perform a convergence analysis using an agglomerated mesh of a geometry of a brain slice. Finally we present a simulation in a three dimensional patient-specific brain reconstructed from magnetic resonance images. The model presented in this paper can be regarded as a preliminary attempt to model the perfusion in the brain

Metal artefact reduction in computed tomography images by a fourth-order total variation flow

Permanent metallic implants, such as dental fillings and cardiac devices, generate streaks-like artefacts in computed tomography (CT) images. In this article, we propose a strategy to perform metal artefact reduction (MAR) that relies on the total variation-H-1 inpainting, a variational approach based on a fourth-order total variation (TV) flow. This approach has never been used to perform MAR, although it has been profitably employed in other branches of image processing. A systematic evaluation of the performance is carried out. Comparisons are made with the results obtained using classical linear interpolation and two other partial differential equation-based approaches relying, respectively, on the Fourier's heat equation and on a second order TV flow. Visual inspection of both synthetic and real CT images, as well as computation of similarity indexes, suggests that our strategy for MAR outperforms the others considered here, as it provides best image restoration, highest similarity indexes and for being the only one able to recover hidden structures, a task of primary importance in the medical field

M for Models

In this short notice I illustrate the role of mathematical models and their impact on our daily life. In particular I provide a couple of instances of applications: one to the enhancement of sports performances, the others to the improvement of our knowledge of the human cardio-circulatory system

An online intrinsic stabilization strategy for the reduced basis approximation of parametrized advection-dominated problems - Une stratégie intrinsèque de stabilisation en ligne pour l'approximation bases réduites de problèmes paramètrés avec transport dominant

We propose a new, black-box online stabilization strategy for reduced basis (RB) approximations of parameter-dependent advection–diffusion problems in the advection-dominated case. Our goal is to stabilize the RB problem irrespectively of the stabilization (if any) operated on the high-fidelity (e.g., finite element) approximation, provided a set of stable RB functions have been computed. Inspired by the spectral vanishing viscosity method, our approach relies on the transformation of the basis functions into modal basis, then on the addition of a vanishing viscosity term over the high RB modes, and on a rectification stage – prompted by the spectral filtering technique – to further enhance the accuracy of the RB approximation. Numerical results dealing with an advection-dominated problem parametrized with respect to the diffusion coefficient show the accuracy of the RB solution on the whole parametric range

Isogeometric analysis for second order partial differential equations on surfaces

We consider the numerical solution of second order Partial Differential Equations (PDEs) on lower dimensional manifolds, specifically on surfaces in three dimensional spaces. For the spatial approximation, we consider Isogeometric Analysis which facilitates the encapsulation of the exact geometrical description of the manifold in the analysis when this is represented by B-splines or NURBS. Our analysis addresses linear, nonlinear, time dependent, and eigenvalues problems involving the Laplace-Beltrami operator on surfaces. Moreover, we propose a priori error estimates under h-refinement in the general case of second order PDEs on the lower dimensional manifolds. We highlight the accuracy and efficiency of Isogeometric Analysis with respect to the exactness of the geometrical representations of the surfaces

On the coupling of hyperbolic and parabolic systems: analytical and numerical approach

We deal with the coupling of hyperbolic and parabolic systems in a one-dimentional domain Ω divided into two disjoint subdomains Ω+ and Ω-. Our main concern is to find out the proper interface conditions to be fulfilled at the surface separating the two domains. Next, we will use them in the numerical approximation of the problem. The justification of the interface conditions is based on a singular perturbation analysis, that is, the hyperbolic system is rendered parabolic by adding a small artificial "viscosity". As this goes to zero, the coupled parabolic-parabolic problem degenerates into the original one, yielding some conditions at the interface. These we take as interface conditions for the hyperbolic-parabolic problem. Actually, we discuss two alternative sets of interface conditions according to whether the regularization procedure is variational or nonvariational. We show how these conditions can be used in the frame of a numerical approximation to the given problem. Furthermore, we discuss a method of resolution which alternates the resolution of the hyperbolic problem within Ω- and of the parabolic one within Ω+. The spectral collocation method is proposed, as an example of space discretization (different methods could be used as well); both explicit and implicit time-advancing schemes are considered. The present study is a preliminary step toward the analysis of the coupling between Euler and Navier-Stokes equations for compressible flows. © 1989

Legendre and Chebyshev spectral approximations of Burgers' equation

We give error estimates for both Legendre and Chebyshev spectral approximations of the steady state Burgers' problem {Mathematical expression} To do that we prove some abstract approximation properties of orthogonal projection operators in some weighted Sobolev spaces Hωs(I) for both Legendre and Chebyshev weights. © 1981 Springer-Verlag