Bifurcations in valveless pumping techniques from a coupled fluid-structure-electrophysiology model in heart development

We explore an embryonic heart model that couples electrophysiology and muscle-force generation to flow induced using a $2D$ fluid-structure interaction framework based on the immersed boundary method. The propagation of action potentials are coupled to muscular contraction and hence the overall pumping dynamics. In comparison to previous models, the electro-dynamical model does not use prescribed motion to initiate the pumping motion, but rather the pumping dynamics are fully coupled to an underlying electrophysiology model, governed by the FitzHugh-Nagumo equations. Perturbing the diffusion parameter in the FitzHugh-Nagumo model leads to a bifurcation in dynamics of action potential propagation. This bifurcation is able to capture a spectrum of different pumping regimes, with dynamic suction pumping and peristaltic-like pumping at the extremes. We find that more bulk flow is produced within the realm of peristaltic-like pumping.Comment: 11 pages, 13 figures. arXiv admin note: text overlap with arXiv:1610.0342

Computing the force distribution on the surface of complex, deforming geometries using vortex methods and Brinkman penalization

The distribution of forces on the surface of complex, deforming geometries is an invaluable output of flow simulations. One particular example of such geometries involves self-propelled swimmers. Surface forces can provide significant information about the flow field sensed by the swimmers, and are difficult to obtain experimentally. At the same time, simulations of flow around complex, deforming shapes can be computationally prohibitive when body-fitted grids are used. Alternatively, such simulations may employ penalization techniques. Penalization methods rely on simple Cartesian grids to discretize the governing equations, which are enhanced by a penalty term to account for the boundary conditions. They have been shown to provide a robust estimation of mean quantities, such as drag and propulsion velocity, but the computation of surface force distribution remains a challenge. We present a method for determining flow- induced forces on the surface of both rigid and deforming bodies, in simulations using re-meshed vortex methods and Brinkman penalization. The pressure field is recovered from the velocity by solving a Poisson's equation using the Green's function approach, augmented with a fast multipole expansion and a tree- code algorithm. The viscous forces are determined by evaluating the strain-rate tensor on the surface of deforming bodies, and on a 'lifted' surface in simulations involving rigid objects. We present results for benchmark flows demonstrating that we can obtain an accurate distribution of flow-induced surface-forces. The capabilities of our method are demonstrated using simulations of self-propelled swimmers, where we obtain the pressure and shear distribution on their deforming surfaces

Simulations of propelling and energy harvesting articulated bodies via vortex particle-mesh methods

The emergence and understanding of new design paradigms that exploit flow induced mechanical instabilities for propulsion or energy harvesting demands robust and accurate flow structure interaction numerical models. In this context, we develop a novel two dimensional algorithm that combines a Vortex Particle-Mesh (VPM) method and a Multi-Body System (MBS) solver for the simulation of passive and actuated structures in fluids. The hydrodynamic forces and torques are recovered through an innovative approach which crucially complements and extends the projection and penalization approach of Coquerelle et al. and Gazzola et al. The resulting method avoids time consuming computation of the stresses at the wall to recover the force distribution on the surface of complex deforming shapes. This feature distinguishes the proposed approach from other VPM formulations. The methodology was verified against a number of benchmark results ranging from the sedimentation of a 2D cylinder to a passive three segmented structure in the wake of a cylinder. We then showcase the capabilities of this method through the study of an energy harvesting structure where the stocking process is modeled by the use of damping elements

Shallow Water and Navier-Stokes SPH-like numerical modelling of rapidly varying free-surface flows

In coastal engineering, Lagrangian meshless numerical methods have reached a good popularity and they have been applied with success to describe wave breaking, impact of wave on structures and other rapid phenomena. This is due to the fact that they have a number of advantages in comparison with classical Eulerian schemes: no explicit treatment of the free surface and no computational grid mean that sophisticated meshing is not needed for complex geometries and therefore a number of problems that were considered largely intractable using classical Eulerian numerical methods such as finite volume or finite elements can now be simulated. As a relatively new method in Computational Fluid Dynamics, this kind of methods may be considered immature and many fundamental aspects and key characteristics remain to be fully investigated. The solid boundary condition is such an example: imposing closed boundary conditions in meshless methods in general, and in Smoothed Particle Hydrodynamic (SPH) in particular, is still an open problem. In the first chapter of this thesis an approximate Virtual Boundary Particle Method (VBP) for solid boundary conditions in two-dimensional (2-D) SPH models is presented; this is a development of the original VBP method recently proposed by Ferrari et al. (A new 3-D parallel SPH scheme for free-surface flows, Computers \& Fluids, 38(6), 1203-1217, 2009). The aim is to maintain the zeroth moment of the kernel function as closely as possible to unity, (a property referred to as zero-consistency), for particles close to solid boundaries. The main advantage of the MVBP in comparison with other methods such as Mirrored Particles is that curved boundaries or boundaries with angles can be easily reproduced. Some authors applied the Smoothed Particle Hydrodynamics (SPH) method to integrate the Shallow Water Equations (SWE) obtaining promising results for simple test cases where no open boundaries are present and the analytical formulation of source terms are applied: with SPH the wet-dry fronts do not need any special treatment, the equations are solved just where the fluid is present and this can potentially speed up the calculations if there are large dry areas in the domain. A 2D Shallow Water code based on the SPH interpolation is developed in the chapters 2 – 4 of this work, with the aim of further improving the capability of these numerical schemes of simulating real flooding events. The SPH-SWEs code is developed following the variational formulation, thanks to this approach the numerical scheme is robust and both the total mass and the momentum are conserved. Some major improvement has been introduced in the SPH-SWEs model in order to make the simulation of real floodings feasible. The Modified Virtual Boundary Particles (MVBP) is used to describe the closed boundaries, the bottom and the friction source term is described by a set of bottom particle. This discretization is effective not just for simple test case but also in for real bathymetries. Moreover, a particle splitting procedure has been inserted: it has the purpose to avoid the lack of resolution due to the variable kernel size being inversely proportional to water depth. This splitting procedure conserves mass and momentum by varying the smoothing length, velocity and acceleration of each refined particle. This improves predictions but does not necessarily provide good shock capturing. This is improved by treating particle interactions as a Riemann problem with MUSCL reconstruction providing stability. The last limitation that inhibits the use of the SPH-SWEs for real flooding simulation is the absence of any method to impose open boundary conditions. These are introduced in chapter 4 by adopting a simplified version of the Characteristic boundary method. Both supercritical and subcritical inflow and outflow boundary conditions can be simulated. Thanks to all the improvements described above, the simulation of two real events by a SPH-SWEs is presented in chapter 4, for the first time. The first case is the Okushiri tsunami occurred in Japan in 1993, whereas the second one is a flooding flood inundation at Thamesmead (UK). In Chapter 5 the simulation of rapidly varying flows is analysed removing the hypothesis of Shallow Water flows: a meshless Lagrangian numerical model called Finite Pointset Method (FPM) for the integration of Navier-Stokes equations in presence of free-surface flow is presented. The Finite Pointset Method (FPM) is a Lagrangian meshless method for numerical integration of pure incompressible Navier-Stokes equations, applied to date just to internal flows. It belongs to SPH like family because each particle carries a vector of field quantities such as pressure, density, velocity etc. and information and physical quantities are approximated using particles in a circular neighbourhood. FPM holds also some remarkable advantages in comparison with classical SPH methods: it is based on a moving least squares approach, where particles are just interpolation points without any associated mass and this means that any order of accuracy can be reached regardless to the particle’s position. In FPM the fluid is described as purely incompressible and the Navier-Stokes equation are solved numerically by means of the projection method therefore no spurious oscillations in the pressure field are present. Moreover in FPM boundary conditions can be analytically enforced using boundary particles and fluid particles can be added and removed in order to preserve the stability of the solution. This fact represents another fundamental advantage in comparison with classical SPH. Originally the FPM has been confined to single or two phase flow, but in chapter 5 it has extended also to free-surface flows by introducing a novel algorithm for free surface detection. In addition to that, a novel formulation of the Projection Method, called Incremental Pressure Projection Method, has been applied in order to preserve the hydrostatic condition

Enhanced SPH modeling of free-surface flows with large deformations

The subject of the present thesis is the development of a numerical solver to study the violent interaction of marine flows with rigid structures. Among the many numerical models available, the Smoothed Particle Hydrodynamics (SPH) has been chosen as it proved appropriate in dealing with violent free-surface flows. Due to its Lagrangian and meshless character it can naturally handle breaking waves and fragmentation that generally are not easily treated by standard methods. On the other hand, some consolidated features of mesh-based methods, such as the solid boundary treatment, still remain unsolved issues in the SPH context. In the present work a great part of the research activity has been devoted to tackle some of the bottlenecks of the method. Firstly, an enhanced SPH model, called delta-SPH, has been proposed. In this model, a proper numerical diffusive term has been added in the continuity equation in order to remove the spurious numerical noise in the pressure field which typically affects the weakly-compressible SPH models. Then, particular attention has been paid to the development of suitable techniques for the enforcement of the boundary conditions. As for the free-surface, a specific algorithm has been designed to detect free-surface particles and to define a related level-set function with two main targets: to allow the imposition of peculiar conditions on the free-surface and to analyse and visualize more easily the simulation outcome (especially in 3D cases). Concerning the solid boundary treatment, much effort has been spent to devise new techniques for handling generic body geometries with an adequate accuracy in both 2D and 3D problems. Two different techniques have been described: in the first one the standard ghost fluid method has been extended in order to treat complex solid geometries. Both free-slip and no-slip boundary conditions have been implemented, the latter being a quite complex matter in the SPH context. The proposed boundary treatment proved to be robust and accurate in evaluating local and global loads, though it is not easy to extend to generic 3D surfaces. The second technique has been adopted for these cases. Such a technique has been developed in the context of Riemann-SPH methods and in the present work is reformulated in the context of the standard SPH scheme. The method proved to be robust in treating complex 3D solid surfaces though less accurate than the former. Finally, an algorithm to correctly initialize the SPH simulation in the case of generic geometries has been described. It forces a resettlement of the fluid particles to achieve a regular and uniform spacing even in complex configurations. This pre-processing procedure avoids the generation of spurious currents due to local defects in the particle distribution at the beginning of the simulation. The delta-SPH model has been validated against several problems concerning fluid-structure interactions. Firstly, the capability of the solver in dealing with water impacts has been tested by simulating a jet impinging on a flat plate and a dam-break flow against a vertical wall. In this cases, the accuracy in the prediction of local loads and of the pressure field have been the main focus. Then, the viscous flow around a cylinder, in both steady and unsteady conditions, has been simulated comparing the results with reference solutions. Finally, the generation and propagation of 2D gravity waves has been simulated. Several regimes of propagation have been tested and the results compared against a potential flow solver. The developed numerical solver has been applied to several cases of free-surface flows striking rigid structures and to the problem of the generation and evolution of ship generated waves. In the former case, the robustness of the solver has been challenged by simulating 2D and 3D water impacts against complex solid surfaces. The numerical outcome have been compared with analytical solutions, experimental data and other numerical results and the limits of the model have been discussed. As for the ship generated waves, the problem has been firstly studied within the 2D+t approximation, focusing on the occurrence and features of the breaking bow waves. Then, a dedicated 3D SPH parallel solver has been developed to tackle the simulation of the entire ship in constant forward motion. This simulation is quite demanding in terms of complexities of the boundary geometry and computational resources required. The wave pattern obtained has been compared against experimental data and results from other numerical methods, showing in both the cases a fair and promising agreement

Problèmes d'interactions entre une structure déformable et un fluide visqueux et incompressible

Dans cette thèse, nous étudions un système fluide-solide qui modélise les interactions entre une struc- ture déformable, et un fluide visqueux et incompressible qui l'entoure. Il couple les équations de Navier- Stokes incompressibles (pour l'état du fluide) avec les lois de Newton (pour la dynamique du solide). L'existence de solutions fortes est étudiée dans les deux premiers chapitres, pour des déformations du solide limitées ou non en régularité. Puis nous prouvons la stabilisation à zéro de ce système couplé, pour des perturbations extérieures petites, par des déformations du solide soumises à des contraintes physiques qui lui garantissent en particulier d'être autopropulsé. Ensuite nous décrivons des moyens pratiques de générer de telles déformations. Enfin nous développons une méthode numérique pour un problème de Stokes avec conditions de Dirichlet non homogènes. Elle nous permet d'obtenir une bonne approximation de la trace normale du tenseur des contraintes de Cauchy, pour des frontières qui ne dépendent pas du maillage. Cette méthode combine une approche de type domaines fictifs basée sur les idées de Xfem, et une méthode de Lagrangien augmenté. Du point de vue des interactions fluide-structure, l'intérêt de cette méthode réside dans l'importance du rôle joué par les forces du fluide à l'interface fluide-solide.In this thesis, we study a fluid-solid system which is a model for the interactions between a deformable structure, and a viscous incompressible fluid surrounding it. It couples the incompressible Navier-Stokes equations (for the fluid flow) with the Newton's laws (for the solid's dynamics). The existence of strong solutions is studied in the first two chapters, for solid's deformations which are limited or not in regularity. Then we prove the stabilization to zero of this coupled system, for small external perturbations, by solid's deformations submitted to physical constraints which guarantee its self-propel led nature. After that we describe practical means of generating such deformations. Finally we develop a numerical method for a Stokes problem with nonhomogeneous Dirichlet conditions. It enables us to get a good approximation of the normal trace of the Cauchy stress tensor, for boundaries which does not depend on the mesh. This method combines a fictitious domain type approach based on the ideas of Xfem, and an augmented Lagrangian method. In a fluid-structure interaction perspective, the interest of this method lies in the importance of the role played by the fluid's forces at the fluid-solid interface

The Fluid Dynamics of Heart Development: The effect of morphology on flow at several stages

Proper cardiogenesis requires a delicate balance between genetic and environmental (epigenetic) signals, and mechanical forces. While many cellular biologists and geneticists have extensively studied heart morphogenesis using various experimental techniques, only a few scientists have begun using mathematical modeling as a tool for studying cardiogenic events. Hemodynamic processes, such as vortex formation, are important in the generation of shear at the endothelial surface layer and strains at the epithelial layer, which aid in proper morphology and functionality. The purpose of this thesis is to study the underlying fluid dynamics in various stages on heart development, in particular, the morphogenic stages when the heart is a linear heart tube as well as during the onset of ventricular trabeculation. Previous mathematical models of the linear heart tube stage have focused on mechanisms of valveless pumping, whether dynamic suction pumping (impedance pumping) or peristalsis; however, they all have neglected hematocrit. The impact of blood cells was examined by fluid-structure interaction simulations, via the immersed boundary method. Moreover, electrophysiology models were incorporated into an immersed boundary framework, and bifurcations within the morphospace were studied that give rise to a spectrum of pumping regimes, with peristaltic-like waves of contraction and impedance pumping at the extremes. Lastly, effects of resonant pumping, damping, and boundary inertial effects (added mass) were studied for dynamic suction pumping. The other stage of heart development considered here is during the onset of ventricular trabeculation. This occurs after the heart has undergone the cardiac looping stage and now is a multi-chambered pumping system with primitive endocardial cushions, which act as precursors to valve leaflets. Trabeculation introduces complex morphology onto the inner lining of the endocardium in the ventricle. This transition of a smooth endocardium to one with complex geometry, may have significant effect on the intracardial fluid dynamics and stress distribution within emrbyonic hearts. Previous studies have not included these geometric perturbations along the ventricular endocardium. The role of trabeculae on intracardial (and intertrabecular) flows was studied using two different mathematical models implemented within an immersed boundary framework. It is shown that the trabecular geometry and number density have a significant effect on such flows. Furthermore this thesis also focused attention to the creation of software for scientists and engineers to perform fluid-structure interaction simulations at an accelerated rate, in user-friendly environments for beginner programmers, e.g., MATLAB or Python 3.5. The software, IB2d, performs fully coupled fluid-structure interaction problems using Charles Peskin's immersed boundary method. IB2d is capable of running a vast range of biomechanics models and contains multiple options for constructing material properties of the fiber structure, advection-diffusion of a chemical gradient, muscle mechanics models, Boussinesq approximations, and artificial forcing to drive boundaries with a preferred motion. The software currently contains over 50 examples, ranging from rubber-bands oscillating to flow past a cylinder to a simple aneurysm model to falling spheres in a chemical gradient to jellyfish locomotion to a heart tube pumping coupled with electrophysiology, muscle, and calcium dynamics modelsDoctor of Philosoph