An Immersed Interface Method for Discrete Surfaces

Fluid-structure systems occur in a range of scientific and engineering applications. The immersed boundary(IB) method is a widely recognized and effective modeling paradigm for simulating fluid-structure interaction(FSI) in such systems, but a difficulty of the IB formulation is that the pressure and viscous stress are generally discontinuous at the interface. The conventional IB method regularizes these discontinuities, which typically yields low-order accuracy at these interfaces. The immersed interface method(IIM) is an IB-like approach to FSI that sharply imposes stress jump conditions, enabling higher-order accuracy, but prior applications of the IIM have been largely restricted to methods that rely on smooth representations of the interface geometry. This paper introduces an IIM that uses only a C0 representation of the interface,such as those provided by standard nodal Lagrangian FE methods. Verification examples for models with prescribed motion demonstrate that the method sharply resolves stress discontinuities along the IB while avoiding the need for analytic information of the interface geometry. We demonstrate that only the lowest-order jump conditions for the pressure and velocity gradient are required to realize global 2nd-order accuracy. Specifically,we show 2nd-order global convergence rate along with nearly 2nd-order local convergence in the Eulerian velocity, and between 1st-and 2nd-order global convergence rates along with 1st-order local convergence for the Eulerian pressure. We also show 2nd-order local convergence in the interfacial displacement and velocity along with 1st-order local convergence in the fluid traction. As a demonstration of the method's ability to tackle complex geometries,this approach is also used to simulate flow in an anatomical model of the inferior vena cava.Comment: - Added a non-axisymmetric example (flow within eccentric rotating cylinder in Sec. 4.3) - Added a more in-depth analysis and comparison with a body-fitted approach for the application in Sec. 4.

A moving mesh method for non-isothermal multiphase flows

In this thesis, a numerical method is developed for simulating non-isothermal multiphase flows, which are important in many technical applications such as crystal growth and welding. The method is based on the arbitrary Lagrangian Eulerian method of Li (2013). The interface is represented explicitly by mesh lines, and is tracked by an adaptive moving unstructured mesh. The $P2-P1d$ finite element method (FEM) is used for discretisation and the incompressible Navier-Stokes equations are solved by the uzawa method. Firstly, a thorough study is presented on the method’s capability in numerically representing the force balance condition on the interface. An inaccurate representation of this condition induces the non-physical spurious currents, which degrade the simulation accuracy especially when the viscous damping is weak (small Ohnesorge number, $Oh$). For the example of a circular/spherical droplet, the interfacial tension and the associated pressure jump are exactly balanced numerically and thus the static Laplace solution exists in our method. The stability of this solution is examined numerically. The amplitude of the dimensionless spurious currents is found to be around $10^{−15}$ for $Oh \geq 10^{−3}$. Another benchmark test is the axisymmetric oscillation of a freesurface droplet/bubble. The simulation results are in good agreement with the analytical solution for $Oh = 10^{−3}$. This is by far the first successful simulation of droplet/bubble oscillation with such weak viscous damping and it demonstrates the ability of our method in simulating flows with strong capillary forces. Secondly, a numerical treatment of interface topology changes is incorporated into our method for studying problems with interface breakup. Thanks to the adaptive mesh generator, the thin region between the interface boundary and another boundary consists of one layer of elements. The interface topology change is performed once the minimum distance between the two boundaries falls below a pre-set scale $l_{breakup}$ . The numerical implementation is verified through two different examples: dripping faucet and droplet coalescence. Remarkably good agreement has been obtained with the experimental results. The simulation of the low Oh dripping problem shows both the accuracy and robustness of our method. The simulation of droplet coalescence demonstrates the great advantage of our method in solving problems with a large disparity in length scales. Finally, an FEM solver for temperature is developed and the non-isothermal effects are included in our method for the purpose of simulating non-isothermal multiphase flows. The modified method is validated to be accurate through three benchmark examples: natural convection in a cavity, thermocapillary convection of two layers, and droplet migration subject to a temperature gradient. Our method is then applied to investigate the liquid bridge breakup with thermocapillary effect. The non-isothermal liquid bridge breakup in the viscous and inertial regimes are studied. It has been found that the inertial regime breakup exhibits different pinchoff shapes as the Capillary number increases, and that the viscous regime breakup is accelerated by the thermocapillary motion.Cambridge Trust and the China Scholarship Counci

Parallel Processing of Eulerian-Lagrangian, Cell-based Adaptive Method for moving Boundary Problems.

In this study, issues and techniques related to the parallel processing of the Eulerian-Lagrangian method for multi-scale moving boundary computation are investigated. The scope of the study consists of the Eulerian approach for field equations, explicit interface-tracking, Lagrangian interface modification and reconstruction algorithms, and a cell-based unstructured adaptive mesh refinement (AMR) in a distributed-memory computation framework. We decomposed the Eulerian domain spatially along with AMR to balance the computational load of solving field equations, which is a primary cost of the entire solver. The Lagrangian domain is partitioned based on marker vicinities with respect to the Eulerian partitions to minimize inter-processor communication. Overall, the performance of an Eulerian task peaks at 10,000-20,000 cells per processor, and it is the upper bound of the performance of the Eulerian- Lagrangian method. Moreover, the load imbalance of the Lagrangian task is not as influential as the communication overhead of the Eulerian-Lagrangian tasks on the overall performance. To assess the parallel processing capabilities, a high Weber number drop collision is simulated. The high convective to viscous length scale ratios result in disparate length scale distributions; together with the moving and topologically irregular interfaces, the computational tasks require temporally and spatially resolved treatment adaptively. The techniques presented enable us to perform original studies to meet such computational requirements. Coalescence, stretch, and break-up of satellite droplets due xvii to the interfacial instability are observed in current study, and the history of interface evolution is in good agreement with the experimental data. The competing mechanisms of the primary and secondary droplet break up, along with the gas-liquid interfacial dynamics are systematically investigated. This study shows that Rayleigh-Taylor instability on the edge of an extruding sheet can be profound at the initial stage of collision, and Rayleigh-Plateau instability dominates the longitudinal disturbance on the fringe of the liquid sheet at a long time, which eventually results in primary breakups.PHDAerospace EngineeringUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/99988/1/ckkuan_1.pd

An ALE method for simuations of elastic surfaces in flow

Die Dynamik von elastischen Membranen, Kapseln und Schalen hat sich zu einem aktiven Forschungsgebiet in der simulationsgestützten Physik und Biologie entwickelt. Die dünne Oberfläche dieser elastischen Materialien ermöglicht es, sie effizient als Hyperfläche zu approximieren. Solche Oberflächen reagieren auf Dehnungen in Oberflächenrichtung und Verformungen in Normalenrichtung mit einer elastischen Kraft. Zusätzlich können Oberflächenspannungskräfte auftreten. In dieser Arbeit präsentieren wir eine neuartige Arbitrary Lagrangian-Eulerian (ALE) Methode um solche in (Navier-Stokes) Fluiden eingebetteten elastischen Schalen zu simulieren. Dadurch, dass das Gitter an die elastische Oberfläche angepasst ist, kombiniert die vorgeschlagene Methode hohe Genauigkeit mit Effizienz in der Berechnung der Lösungen. Folglich kann man die Simulationen mit einer verhältnismäßig geringen Gitterauflösung durchführen. Der Fokus dieser Arbeit liegt bei achsensymmetrischen Formen und Strömungen, wie sie bei vielen biophysikalischen Anwendungen zu finden sind. Neben einer allgemeinen dreidimensionalen Beschreibung formulieren wir achsensymmetrische Kräfte auf der Oberfläche, für welche wir eine Diskretisierung mit der Finite Differenzen Methode vorschlagen, welche an eine Finite-Elemente Methode für die umgebenden Fluide gekoppelt ist. Weiterhin entwickeln wir eine Strategie zur impliziten Kopplung der Kräfte, um Zeitschrittrestriktionen zu reduzieren. In verschiedenen numerischen Tests werden wir zeigen, dass akkurate Ergebnisse schon in einer Größenordnung von Minuten auf einer Single-Core CPU erreicht werden können. Die Methode wurde in drei aktuellen Anwendungen verwendet, wobei mindestens zwei davon nach unserer Kenntnis im Moment mit keiner anderen numerischen Methode simuliert werden können: Zunächst präsentieren wir Simulationen von biologischen Zellen, die im Zuge eines RT-DC (Real-Time Deformability Cytometry) Experiments durch einen schmalen mikrofluidischen Kanal advektiert und dabei verformt werden. Danach zeigen wir die Ergebnisse erster Simulationen der uniaxialen Kompression biologischer Zellen zwischen zwei parallelen Platten im Zuge eines AFM Experiments. Schließlich präsentieren wir Resultate erster Simulationen von neuartigen mikroschwimmenden Schalen, welche lediglich durch äußere Einflüsse (wie z.B. Ultraschall), zum Schwimmen angeregt werden können.The dynamics of membranes, shells, and capsules in fluid flow has become an active research area in computational physics and computational biology. The small thickness of these elastic materials enables their efficient approximation as a hypersurface, which exhibits an elastic response to in-plane stretching and out-of-plane bending, possibly accompanied by a surface tension force. In this work, we present a novel arbitrary Lagrangian-Eulerian (ALE) method to simulate such elastic surfaces immersed in Navier-Stokes fluids. The method combines high accuracy with computational efficiency, since the grid is matched to the elastic surface and can therefore be resolved with relatively few grid points. The focus of this work is on axisymmetric shapes and flow conditions, which are present in a wide range of biophysical problems. Next to a general three-dimensional description, we formulate axisymmetric elastic surface forces and propose a discretization with surface finite-differences coupled to evolving finite elements. We further develop an implicit coupling strategy to reduce time step restrictions. Several numerical test cases show that accurate results can be achieved at computational times on the order of minutes on a single core CPU. Three state-of-the-art applications are demonstrated, where to our knowledge at least two of them cannot be simulated with any other numerical method so far. First, simulations of biological cells being advected through a microfluidic channel and therefore being deformed during an RT-DC (Real-Time Deformability Cytometry) experiment are presented. Then, the uniaxial compression of the cortex of a biological cell during an AFM experiment is investigated. Finally, we present the results of first simulations of the observed shape oscillations of novel microswimming shells which can be locomoted by exterior influences (e.g. ultrasound waves) only

Developments on Computer Simulation of Injection Moulding - Modelling With Boundary Element and Finite Element Methods

Several mathematical models which are based mainly on the boundary integral equations are developed for computer simulation of injection moulding. The models are then implemented for the viscous flows in the filling stage, and the temperature field during the cooling stage of the process. Starting with the modelling of nonisothermal laminar flow in ducts, the dependence of viscosity on the pressure, temperature and shear rate is taken into account, and the velocity and temperature solutions to fully developed flow are developed. The solutions are used to obtain the approximate axial solutions and possibility of choking is discussed. The solutions are further extended to the cases of a slightly tapered circular pipe, and of cross sections of any shape the latter of which uses a boundary element model

Runge-Kutta discontinuous Galerkin method for the approximation of Baer and Nunziato type multiphase models

International audienceA high-order numerical method is developed for the computation of compressible multiphase flows. The model we use is based on the Baer \& Nunziato type systems [M.R. Baer, J.W. Nunziato, A two-phase mixture theory for the deflagration-to-detonation transition (ddt) in reactive granular materials, Int. J. Multiphase Flows, 12 (12) (1986), pp. 861-889]. Among all the other available models in the literature, these systems present the advantage to be able to simulate either interface or mixture problems. Nevertheless, they still raise some issues, mainly based on their non-conservative feature. The numerical method we propose is a discontinuous Galerkin type. In this work, the interior side integrals are computed thanks to [R. Abgrall, R. Saurel, Discrete equations for physical and numerical compressible multiphase mixtures, J. Comput. Phys., 186 (361-396) (2003)]. Robustness and high order of accuracy of the method are proved on classical interface problems, but also on suitably derived analytical solutions

The 1999 Center for Simulation of Dynamic Response in Materials Annual Technical Report

Introduction: This annual report describes research accomplishments for FY 99 of the Center for Simulation of Dynamic Response of Materials. The Center is constructing a virtual shock physics facility in which the full three dimensional response of a variety of target materials can be computed for a wide range of compressive, ten- sional, and shear loadings, including those produced by detonation of energetic materials. The goals are to facilitate computation of a variety of experiments in which strong shock and detonation waves are made to impinge on targets consisting of various combinations of materials, compute the subsequent dy- namic response of the target materials, and validate these computations against experimental data

Balanced-force two-phase flow modelling on unstructured and adaptive meshes

Two-phase flows occur regularly in nature and industrial processes and their understanding is of significant interest in engineering research and development. Various numerical methods to predict two-phase phase flows have been developed as a result of extensive research efforts in past decades, however, most methods are limited to Cartesian meshes. A fully-coupled implicit numerical framework for two-phase flows on unstructured meshes is presented, solving the momentum equations and a specifically constructed continuity constraint in a single equation system. The continuity constraint, derived using a momentum interpolation method, satisfies continuity, provides a strong pressure-velocity coupling and ensures a discrete balance between pressure gradient and body forces. The numerical framework is not limited to specific density ratios or a particular interface topology and includes several novelties. A further step towards a more accurate prediction of two-phase flows on unstructured meshes is taken by proposing a new method to evaluate the interface curvature. The curvature estimates obtained with this new method are shown to be as good as or better than methods reported in literature, which are mostly limited to Cartesian meshes, and the accuracy on structured and unstructured meshes is shown to be comparable. Furthermore, lasting contributions are made towards the understanding of convolution methods for two-phase flow modelling and the underlying mechanisms of parasitic currents are studied in detailed. The mesh resolution is of particular importance for two-phase flows due to the inherent first-order accuracy of the interface position using interface capturing methods. A mesh adaption algorithm for tetrahedral meshes with application to two-phase flows and its implementation are presented. The algorithm is applied to study mesh resolution requirements at interfaces and force-balancing for surface-tension-dominated two-phase flows on adaptive meshes.Open Acces