IGS: an IsoGeometric approach for Smoothing on surfaces

We propose an Isogeometric approach for smoothing on surfaces, namely estimating a function starting from noisy and discrete measurements. More precisely, we aim at estimating functions lying on a surface represented by NURBS, which are geometrical representations commonly used in industrial applications. The estimation is based on the minimization of a penalized least-square functional. The latter is equivalent to solve a 4th-order Partial Differential Equation (PDE). In this context, we use Isogeometric Analysis (IGA) for the numerical approximation of such surface PDE, leading to an IsoGeometric Smoothing (IGS) method for fitting data spatially distributed on a surface. Indeed, IGA facilitates encapsulating the exact geometrical representation of the surface in the analysis and also allows the use of at least globally $C^1-$continuous NURBS basis functions for which the 4th-order PDE can be solved using the standard Galerkin method. We show the performance of the proposed IGS method by means of numerical simulations and we apply it to the estimation of the pressure coefficient, and associated aerodynamic force on a winglet of the SOAR space shuttle

Effect of fibre orientation and bulk modulus on the electromechanical modelling of human ventricles

This work concerns the mathematical and numerical modeling of the heart. The aim is to enhance the understanding of the cardiac function in both physiological and pathological conditions. Along this road, a challenge arises from the multi-scale and multi-physics nature of the mathematical problem at hand. In this paper, we propose an electromechanical model that, in bi-ventricle geometries, combines the monodomain equation, the Bueno-Orovio minimal ionic model, and the Holzapfel-Ogden strain energy function for the passive myocardial tissue modelling together with the active strain approach combined with a model for the transmurally heterogeneous thickening of the myocardium. Since the distribution of the electric signal is dependent on the fibres orientation of the ventricles, we use a Laplace-Dirichlet Rule-Based algorithm to determine the myocardial fibres and sheets configuration in the whole bi-ventricle. In this paper, we study the influence of different fibre directions and incompressibility constraint and penalization on the compressibility of the material (bulk modulus) on the pressure-volume relation simulating a full heart beat. The coupled electromechanical problem is addressed by means of a fully segregated scheme. The numerical discretization is based on the Finite Element Method for the spatial discretization and on Backward Differentiation Formulas for the time discretization. The arising non-linear algebraic system coming from application of the implicit scheme is solved through the Newton method. Numerical simulations are carried out in a patient-specific biventricle geometry to highlight the most relevant results of both electrophysiology and mechanics and to compare them with physiological data and measurements. We show how various fibre configurations and bulk modulus modify relevant clinical quantities such as stroke volume, ejection fraction and ventricle contractility

Mathematical analysis and numerical approximation of a general linearized poro-hyperelastic model

Abstract We describe the behavior of a deformable porous material by means of a poro-hyperelastic model that has been previously proposed in Chapelle and Moireau (2014) under general assumptions for mass and momentum balance and isothermal conditions for a two-component mixture of fluid and solid phases. In particular, we address here a linearized version of the model, based on the assumption of small displacements. We consider the mathematical analysis and the numerical approximation of the problem. More precisely, we carry out firstly the well-posedness analysis of the model. Then, we propose a numerical discretization scheme based on finite differences in time and finite elements for the spatial approximation; stability and numerical error estimates are proved. Particular attention is dedicated to the study of the saddle-point structure of the problem, that turns out to be interesting because velocities of the fluid phase and of the solid phase are combined into a single quasi-incompressibility constraint. Our analysis provides guidelines to select the componentwise polynomial degree of approximation of fluid velocity, solid displacement and pressure, to obtain a stable and robust discretization based on Taylor–Hood type finite element spaces. Interestingly, we show how this choice depends on the porosity of the mixture, i.e. the volume fraction of the fluid phase

Modeling cardiac muscle fibers in ventricular and atrial electrophysiology simulations

Since myocardial fibers drive the electric signal propagation throughout the myocardium, accurately modeling their arrangement is essential for simulating heart electrophysiology (EP). Rule-Based-Methods (RBMs) represent a commonly used strategy to include cardiac fibers in computational models. A particular class of such methods is known as Laplace-Dirichlet-Rule-Based-Methods (LDRBMs) since they rely on the solution of Laplace problems. In this work we provide a unified framework, based on LDRBMs, for generating full heart muscle fibers. First, we review existing ventricular LDRBMs providing a communal mathematical description and introducing also some modeling improvements with respect to the existing literature. We then carry out a systematic comparison of LDRBMs based on meaningful biomarkers produced by numerical EP simulations. Next we propose, for the first time, a LDRBM to be used for generating atrial fibers. The new method, tested both on idealized and realistic atrial models, can be applied to any arbitrary geometries. Finally, we present numerical results obtained in a realistic whole heart where fibers are included for all the four chambers using the discussed LDRBMs

Isogeometric approximation of cardiac electrophysiology models on surfaces: An accuracy study with application to the human left atrium

We consider Isogeometric Analysis in the framework of the Galerkin method for the spatial approximation of cardiac electrophysiology models defined on NURBS surfaces; specifically, we perform a numerical comparison between basis functions of degree p ≥ 1 and globally C k -continuous, with k = 0 or p − 1, to find the most accurate approximation of a propagating front with the minimal number of degrees of freedom. We show that B-spline basis functions of degree p ≥ 1, which are C p−1 -continuous capture accurately the front velocity of the transmembrane potential even with moderately refined meshes; similarly, we show that, for accurate tracking of curved fronts, high-order continuous B-spline basis functions should be used. Finally, we apply Isogeometric Analysis to an idealized human left atrial geometry described by NURBS with physiologically sound fiber directions and anisotropic conductivity tensor to demonstrate that the numerical scheme retains its favorable approximation properties also in a more realistic setting

Reduced basis method and error estimation for parametrized optimal control problems with control constraints

We propose a Reduced Basis method for the solution of parametrized optimal control problems with control constraints for which we extend the method proposed in Dedè, L. (SIAM J. Sci. Comput. 32:997, 2010) for the unconstrained problem. The case of a linear-quadratic optimal control problem is considered with the primal equation represented by a linear parabolic partial differential equation. The standard offline-online decomposition of the Reduced Basis method is employed with the Finite Element approximation as the "truth" one for the offline step. An error estimate is derived and an heuristic indicator is proposed to evaluate the Reduced Basis error on the optimal control problem at the online step; also, the indicator is used at the offline step in a Greedy algorithm to build the Reduced Basis space. We solve numerical tests in the two-dimensional case with applications to heat conduction and environmental optimal control problems. © 2011 Springer Science+Business Media, LLC

Semi-implicit BDF time discretization of the Navier-Stokes equations with VMS-LES modeling in a High Performance Computing framework

In this paper, we propose a semi-implicit approach for the time discretization of the Navier-Stokes equations with Variational Multiscale-Large Eddy Simulation turbulence modeling (VMS-LES). For the spatial approximation of the problem, we use the Finite Element method, while we employ the Backward Differentiation Formulas (BDF) for the time discretization. We treat the nonlinear terms arising in the variational formulation of the problem with a semi-implicit approach leading to a linear system associated to the fully discrete problem which needs to be assembled and solved only once at each discrete time instance. We solve this linear system by means of the GMRES method by employing a multigrid (ML) right preconditioner for the parallel setting. We validate the proposed fully discrete scheme towards the benchmark problem of the flow past a squared cylinder at high Reynolds number and we show the computational efficiency and scalability results of the solver in a High Performance Computing framework