International Workshop on Fluid-Structure Interaction Problems - IMS

International Workshop on Fluid-Structure Interaction Problems
(30 May - 3 Jun 2016)

~ Abstracts ~

Transport between two fluids across their mutual flow interface: the streakline approach
Sanjeeva Balasuriya, University of Adelaide, Australia

Mixing between two different miscible fluids with a mutual interface must be initiated by fluid transporting across this fluid interface, which might be effected by applying an unsteady velocity agitation. "Transport across the interface" would of course mean that the interface is broken in some way, and in this talk I will discuss a dynamical systems approach associated with Lagrangian particle trajectories to make sense of how this can be interpreted. Merely considering the time-evolution of particles "on the interface" does not help, since such a timeline is a material curve across which there is no Lagrangian flux. I will show why streaklines are key to understanding the breaking of the interface, and present a theory for quantifying the time-variation of the streaklines. I will show numerical simulations of how the interface breaks in a cross-channel mixer and a perturbed Kirchhoff's elliptic vortex, and validate the theory in these cases. I will develop a methodology for quantifying the time-varying Lagrangian transport between the two fluids, and obtain a theoretical expression valid under weak velocity agitations. This will be useful in understanding, for example, optimal strategies for maximizing mixing between two fluids in a microfluidic device, or quantifying and controling the mixing of a pollutant blob with its surroundings. I look forward to discussing how these dynamical systems ideas can be combined with the work of the fluid-structure/CFD community, for potential future collaborations.

Modeling and simulation for solid-state dewetting problems
Weizhu Bao, National University of Singapore

In this talk, I will present sharp interface models with anistropic surface energy and a phase field model for simulating solid-state dewetting and the morphological evolution of patterned islands on a substrate. The sharp interface model tracks the moving interface explicitly and it is very easy to be handled in two dimensions via arc-length parametrization. The phase field model is governed by the Cahn-Hilliard equation with isotropic surface tension and variable scalar mobility and it easily deals with the complex boundary conditions and/or complicated geometry arising in the solid-state dewetting problem. Since the phase field model does not explicitly track the moving surface, it naturally captures the topological changes that occur during film/island morphology evolution. Efficient and accurate numerical methods for both sharp interface models and phase field models are proposed. They are applied to study numerically different setups of solid-state dewetting including short and long island films, pinch-off, hole dynamics, semi-infinite film, etc. Our results agree with experimental results very well.
This is joint works with Wei Jiang, David J. Srolovitz, Carl V. Thompson, Yan Wang and Quan Zhao.

Uniform error estimates for finite difference methods applied to fluid motion with interfaces
James Thomas Beale, Duke University, USA

There has been extensive development of numerical methods for fluid flow interacting with moving boundaries or interfaces, using regular finite difference grids which do not conform to the interfaces. When viscosity is significant it is observed, with methods such as the immersed interface method, that the velocity can be accurate to about O(h^2), with grid spacing h, even if the discretization near the interface has truncation error (the error in obeying the equations) as large as O(h). We will describe error estimates which partially explain how such accuracy can be achieved at low Reynolds number. We will first discuss maximum norm estimates for finite difference versions of the Poisson equation and diffusion equation with a gain of regularity as for the exact equations. We will then describe a convergence theorem for one numerical scheme for Navier-Stokes flow with a moving interface, treating the interface location as known; that is, we obtain uniform error estimates in the fluid variables but mostly neglect errors in the interface. As much as possible we use discrete analogues of standard methods for estimating solutions of partial differential equations.

A particle suspension mixture model by smoothed particle hydrodynamics and its application to turbulent sediment flow
Erwan Bertevas, National University of Singapore

We present results from an on-going research effort which aims at applying the Smoothed Particle Hydrodynamics (SPH) method [1] to various cases of complex flows involving particle-suspension / structure interaction. The proposed method relies on the description of particle suspensions via a mixture model [2] which was adapted to the Lagrangian framework of SPH. Particle transport is modeled through the convection-diffusion of the sediment volume fraction. This accounts for particle sedimentation and turbulent diffusion which can be obtained from standard models relating the diffusion coefficient to the turbulent viscosity. The latter is extracted from the solution of standard two-equation turbulence models which are coupled to the momentum equation via the Boussinesq concept of turbulent viscosity. Moreover, the proposed implementation offers the possibility to account for the suspension non-Newtonian rheology through generalized, volume fraction-dependent viscosity models such as the well-known Krieger-Dougherty, Herschel-Bulkley or Papanastasiou models.
The model is then applied to various cases such as Rayleigh-Taylor instability, particle-driven gravity currents and plate/sediment interaction. The proposed method is notably assessed by comparison with available experimental results on particle-driven gravity currents [3], where results for the current front evolution and particle deposition profiles are reported for initial particle concentrations ranging from O(1%) up to O(25%). Considering the wide range of concentrations covered and the uncertainties regarding the suspension rheology and particle impact on turbulence, we obtained satisfactory agreement with experimental results. Another application to this modelling work is related to the perspective of estimating the disturbance created by a harvesting device operating near the seabed for deep-sea applications. With this perspective in mind, a laboratory-scale setup was designed and comprises a horizontally translating inclined plate partially immersed into a layer of clay sediment. The induced sediment dispersion is investigated and the SPH predictions are compared with visual observations obtained from the experiments.

References

[1] J.J Monaghan, Smoothed Particle Hydrodynamics and its diverse applications. Annu. Rev. Fluid Mech. 2012. 44:323-46.
[2] M. Manninen et al. On the Mixture model for Multiphase Flow. VTT publications 288, Finland, 1996.
[3] M.A. Hallworth et al. Abrupt transitions in high-concentration, particle-driven gravity currents. Phys. Fluids 1998 10:1083-1087.

Joint work with Nhan Phan-Thien, Boo Cheong Khoo.

Optimization, control, and sensitivity analysis in fluid-elasticity interactions
Lorena Bociu, North Carolina State University, USA

Free and moving boundary fluid-structure interactions are ubiquitous in nature, with most known examples coming from industrial processes, aero-elasticity, and biomechanics. We consider optimization and control problems subject to interactions between viscous, incompressible fluid and nonlinear elasticity. We discuss existence of optimal controls, sensitivity equations, and optimality conditions. One of the main challenges of applying optimization tools to free and moving boundary interactions is the proper derivation of the adjoint sensitivity information with correct adjoint balancing conditions on the common interface. As the coupled fluid-structure state is the solution of a system of PDEs that are coupled through continuity relations defined on the free interface, sensitivity of the fluid state, which is an Eulerian quantity, with respect to the motion of the solid, which is a Lagrangian quantity, falls into moving shape analysis framework.

Approximation of fluid-structure interaction problems with Lagrange multiplier
Daniele Boffi, Università di Pavia, Italy

The Immersed Boundary Method (IBM) is a consolidated technology for the approximation of problems involving the interaction of fluids and solids. The Finite Element IBM (FE-IBM) provides an efficient analysis for practical problems without the need of approximating the Dirac delta function as in the original IBM. Within this approach, semi-implicit time advancing schemes are stable if a suitable CFL condition is satisfied. A modification of the FE-IBM involves the introduction of a Lagrange multiplier associated to a different approximation of the kinematic condition. We called this formulation DLM-IBM. In the case of solids of codimension one, this new approach can be seen as a natural fictitious domain formulation. It turns out that a semi-implicit time discretization of the coupled system is unconditionally stable with respect to the time step size. The stability of the fully discrete mixed problem is derived in terms of an inf-sup condition which relates the solid and fluid meshes. Our analysis leads to optimal convergence estimates.

FSI analysis of diseased coronary using patient specific data
Mingchao Cai, Morgan State University, USA

Atherosclerotic plaques may rupture without warning and cause acute cardiovascular syndromes such as heart attack and stroke. It is commonly believed that plaque rupture may be linked to critical stress/strain conditions. In this talk, I am going to discuss how to combine fluid-structure interaction models, IVUS(Intravascular Ultrasound) images, patient specific data and blood vessel material properties to do mechanical analysis of diseased human coronary arteries. Some technical details including mesh generation, boundary conditions, mathematical models and numerical algorithms will be addressed.

Fluid-structure interactions of membrane wings
Bharathram Ganapathisubramani, University of Southampton, UK

Natural fliers achieve exceptional aerodynamics by continuous adjustments of their geometry through a mix of dynamic wing compliance and distributed sensing and actuation. This enables them to routinely perform a wide range of manoeuvres including rapid turns, rolls, dives, and climbs with seeming ease. This suggests that the use of active and passive compliant wings will enhance the aeromechanics of a Micro Air Vehicle (MAV).

In this talk, I will present some experimental results that show the superior aerodynamic performance of passive membrane wings compared to traditional rigid wings. Membrane wings show higher lift and higher or comparable efficiency due to flow-induced cambering and unsteady oscillations of the membrane. Simultaneous force, membrane vibration and velocity field measurements reveal that the force fluctuations are coupled to mode shapes in the membrane that are caused by passage of vortices over the top surface of wing. This fluid-structure coupling suggests that it might be possible to tune the material properties on-the-fly to realize "on-demand" aerodynamic performance. Time permitting, I will also present some experimental results using electroactive membrane wings that reveal the possibility of tailoring the aerodynamic performance using integral actuation. This opens the possibility of using these electroactive membrane wings for flight control.

Error analysis of immersed interface method for stationary stokes interface problem
Rui Hu, North Carolina State University, USA

When the Reynolds number be small enough, the stokes equation will become stationary. In the stationary stokes interface problem with proper jump conditions, we do divergence on both sides of the stokes equation such that we can transform the stokes equation into three Poisson equations with jump conditions and boundary conditions. Then use the augment immersed interface method on the Poisson interface problem for pressure. Here we will derive a Neumann boundary condition for pressure such that the divergence-free of velocity will hold. Consider pressure Poisson interface problem. On irregular grid points of the Cartesian grid, the error for the immersed interface method will be O(h). We will show that with O(h) error on irregular points we still can reach O(h^2) for pressure on the whole domain. We will introduce Discrete Neumann-Poincare inequality to help us show this result.

Computational biomechanics of cellular flows: cell adhesion and suspension rheology
Yohsuke Imai, Tohoku University, Japan

We have been developing computational biomechanics of physiological flows from the cell to organ scales. In this workshop, we will present our recent studies on cellular flows, particularly focusing on cell adhesion in microcirculatory blood flow and rheological properties of cell suspension. Cell adhesion in the microcirculation closely relates to various physiological and pathological processes. Examples include the immune system, hemostasis, malaria, and cancer metastasis. Cells adhere to the vascular wall through ligand-receptor bonds. This problem involves not only the biochemical interaction of adhesion proteins, but also the solid and fluid mechanics of cells and their cellular environment. The Monte-Carlo method for ligand-receptor interactions is coupled with the lattice Boltzmann method (LBM) or the boundary element method (BEM) for the fluid mechanics of plasma and cytoplasm and the finite element method (FEM) for the solid mechanics of biological membrane. All the procedures are fully implemented in graphics processing unit (GPU) computing. We have simulated the margination and adhesion of malaria-infected red blood cells, leukocytes and circulating tumor cells (CTCs) under various conditions. We have also applied this method to analyze the rheological property of cell suspensions in the Stokes flow regime. We show the deformation of a spherical cell in oscillating shear flow, and the relationship between the deformation and orientation of cells and the suspension rheology.

Numerical modeling of self-excited vibrations and flapping dynamics
Rajeev Jaiman, National University of Singapore

In this workshop, I would like to share some of our progress and new results for a range of industrial and academic fluid-structure problems. Of particular interest is to solve self-excited vibration and flapping dynamical systems in ocean and wind environments, which constitute an interesting problem for both numerical and theoretical modeling, and can have a profound impact on the performance of coupled dynamical systems in offshore, aerospace, energy harvesting, nuclear reactor and many more. During self-excitation vibration, phase relation between motion and driving force adjusts in such a way that the coupled system experiences sustained energy transfer into the system. Vortex-/wake-induced vibrations and fluttering flags are two common examples of this kind of self-excited and self-limiting phenomena. Oscillating flexible cylinders or flapping of flexible flags in a freestream can form a great variety of vortex wake modes that play a major role on the performance of structural dynamics. We draw some parallels between vortex-induced vibrations and flapping dynamics as a dynamic equilibrium between the instability of the flow and the synchronized response of structures. We also review some of our partitioned iterative and quasi-monolithic schemes for coupled finite element systems. We provide theoretical studies and clarification on the role of added-mass effects in compressible and incompressible flow-structural system. We shed light on the effects of the wake interference, mass ratio, Reynolds number, reduced velocity, and proximity effects with the other solid body and the stationary wall. Finally, we provide a summary of challenging problems that keep intense interest on the topic from the offshore and aerospace industry standpoints.

1. R. K. Jaiman, M. Parmar and P. S. Gurugubelli. Added mass and aeroelastic stability of a flexible plate interacting with mean flow in a confined channel. Journal of Applied Mechanics, 81, 041006, 2013.
2. R. K. Jaiman, S. Sen, and P. Gurugubelli. A fully implicit combined field scheme for freely vibrating square cylinders with sharp and rounded corners. Computers and Fluids, 112:1-18, 2015.
3. P. S. Gurugubelli and R. K. Jaiman. Self-induced flapping dynamics of a flexible inverted foil in a uniform flow. Journal of Fluid Mechanics, 781, pp 657-694, 2015.
4. R. K. Jaiman, N. R. Pillalamarri, and M.Z Guan. A stable second-order partitioned iterative scheme for freely vibrating low-mass bluff bodies in a uniform flow, Comp. Meth. Appl. Mech. Engrg., 301:187-215, 2016.

Solid-state dewetting: equilibrium & dynamics
Wei Jiang, National University of Singapore and Wuhan University, China

Solid-state dewetting (or called agglomeration) of thin films during annealing has been simultaneously observed in various thin film materials, and has attracted ever more attention because of its extensive technological applications and important scientific interest. In this talk, we will talk about two important issues related with theoretical studies on solid-state dewetting problems: one is how to predict the equilibrium shapes of thin island films (i.e., the equilibrium problems); the other is how to derive mathematical models for describing the morphology evolution of thin solid films (i.e., the evolution problems).

Modelling non-Newtonian behaviours of cohesive mixtures using particle-based method
Le-Cao Khoa, National University of Singapore

Many of foods and cosmetic products are cohesive mixtures. Such materials hardly flow if the applied stress is lower than a certain value, but they are liquefied at stresses higher than that value. The marked stress value for the transition is called the yield stress. This non-Newtonian behaviour can be qualitatively explained by the forming and destruction of the microstructure inside the fluid [1]. In continuum mechanics, such materials are modelled by rheological constitutive equations (e.g. Bingham models). These equations are then transformed into discretisation forms and solved numerically by finite volume (FVM), finite element (FEM) or smoothed particle hydrodynamics (SPH) methods. The continuum approaches based on Bingham models are classified as top-down approach which only considered macro behaviour of the systems. This work aims to introduce a bottom-up approach which is based on definition of interaction forces at particle level to obtain the desired macro properties (e.g. yield stress) by using a particle-based simulation technique, known as dissipative particle dynamics (DPD). It is reported that the DPD method satisfies the conservations of mass and momentum and may be applicable to problems of any scale and regarded as a particle-based method for solving continuum problems [2]. In this work, the DPD method is adopted to investigate the dynamics of the cohesive mixtures under viscometric (e.g. shear and Poiseuille) flows. The relation between shear stress and shear rate is studied for those mixtures under different DPD parameters. It is shown numerically that the shear stress ? shear rate relationship is a nonlinear curve and the obtained DPD data can be fitted well to that of Papanastasiou's yield stress model [3]. The present model is also able to yield a fairly good prediction of the normalised velocity profiles in the Poiseuille flows.

References

[1] P. COUSSOT, A.I. LEONOV, J.M. PIAU, Rheology of concentrated dispersed systems in a low molecular weight matrix, J. Nonnewton Fluid Mech, 46, 179-217 (1993).
[2] N. PHAN-THIEN, Understanding Viscoelasticity - An Introduction to Rheology, Edition Number 2, Springer-Verlag Berlin Heidelberg, 2013.
[3] T.C. PAPANASTASIOU, Flows of materials with yield, J Rheology, 31, 385 (1987).

Joint work with Nhan Phan-Thien, Boo Cheong Khoo.

Transport in Biofilms
Isaac Klapper, Temple University, USA

Biofilms are collections of microbes anchored together into sessile communities by self secreted polymers. As such the realities of biofilms as physical materials are important to their function. In particular, function is often constrained by transport of soluble quantities, such as substrates and signals, into or out of the community. However, the combination of transport with reaction can lead to spatial heterogeneity and pattern formation within the biofilm structure, even without direct biological control. Examples include formation of active layers, formation of "external" structure (like mushrooms), and formation of "internal" structure (like lumps). Conversely, such pattern formation can impact biofilm function, particularly through transport but also through mechanics. Examples include formation of microenvironments and impacts on community level transport efficiency. Relatively simple mathematical models of biofilms, coupling growth with transport, will be used to illustrate the importance of physics in biofilm form and function.

Vesicle electrohydrodynamic simulations by coupling immersed boundary and immersed interface method
Ming-Chih Lai, National Chiao Tung University, Taiwan

In this talk, we introduce a coupled immersed boundary (IB) and immersed interface method (IIM) to simulate the electrodeformation and electrohydrodynamics of a vesicle in Navier-Stokes leaky dielectric fluids under a DC electric field. The vesicle membrane is modeled as an inextensible elastic interface with an electric capacitance and an electric conductance. Within the leaky dielectric framework and the piecewise constant electric properties in each fluid, the electric stress can be treated as an interfacial force so that both the membrane electric and mechanical forces can be formulated in a unified immersed boundary method. The electric potential and transmembrane potential are solved simultaneously via an efficient immersed interface method. The fluid variables in Navier-Stokes equations are solved using a projection method on a staggered MAC grid while the electric potential is solved at the cell center. A series of numerical tests have been carefully conducted to illustrate the accuracy and applicability of the present method to simulate vesicle electrohydrodynamics. In particular, we investigate the prolate-oblate-prolate (POP) transition and the effect of electric field and shear flow on vesicle electrohydrodynamics.

An immersed boundary method for fluid-structure interaction
Duc-Vinh Le, Institute of High Performance Computing

We present a moving-least-square immersed boundary method for solving viscous incompressible flow involving both immersed rigid and deformable boundaries on a uniform Cartesian grid. To handle rigid boundaries, noslip condition at the rigid interfaces is enforced using the immersed-boundary direct-forcing method. Noslip boundary condition is taken into account implicitly by reconstructing the velocities at the grid points in the immediate vicinity of the solid boundaries. We propose a reconstruction approach that utilizes moving least squares method to reconstruct the velocity at the forcing points. Moving least squares method allows us to include velocity information from all spatial directions and from more points in both fluid and rigid boundaries in the reconstruction procedure. For deformable boundaries, the moving least squares method is employed to build the interpolation and distribution operators for the implicit immersed boundary method instead of using discrete delta functions. In the original immersed boundary method, the force, torque and power are correctly converted back and forth between Lagrangian and Eulerian grids through discrete delta functions. The discrete delta functions constructed on the uniform grid work well when the deformable boundaries are not very close to the immersed rigid boundaries. However, when the deformable boundaries are within 2 grid points from the rigid boundaries, the discrete delta function may distribute the force into the solid region and this force does not contribute to the motion of the fluid in the direct-forcing method. And when using the discrete delta function for interpolating the velocity at the control points near the rigid boundaries, the velocity of the solid nodes which may not be physical enters the interpolation process and causes errors. To avoid using unphysical values in the velocity interpolation and avoid distributing Lagrangian forces into the rigid regions, we propose using the moving least squares method to construct the interpolation of the velocity and the distribution of the forces instead of using the discrete delta function. The present numerical technique has been validated by several examples including deformations of elastic capsules in shear flow and dynamics of red-blood cell in microfluidic devices.

A least squares augmented IIM for fluid/porous media couplings
Zhilin Li, North Carolina State University, USA

New finite difference methods based on Cartesian meshes and fast Poisson solvers are proposed to solve the coupling of a fluid flow modeled by the incompressible Stokes or Navier-Stokes equations and a porous media modeled by the Darcy's law. Several augmented variables along the interface between the fluid flow and the porous media are introduced so that the coupled equations can be decoupled as several Poisson/Helmholtz equations with jump conditions corresponding to a source and dipole distributions. The augmented variables should be chosen so that the Beavers-Joseph-Saffman (BJS) and other interface conditions are satisfied. Another significant strategy is to enforce the divergence condition at the interface from the fluid side. We have shown that the original and transformed systems are equivalent. In the discretization, the linear system of equations for the augmented variables is a rectangular but it is consistent which leads to a least squares problem for the coupling problem. The entire discretization is second order accurate. Numerical results against known analytical solutions show that both the computed velocity and the pressure are second order accurate in the $L^{\infty}$ norm except maybe for a factor of $\log h$. The proposed new method has also been utilized to simulate different flow/porous media setting with complicated interfaces.

Fluid-structure interaction method for parachute simulation using fabric spring model based on Rayleigh-Ritz analysis
Xiaolin Li, State University of New York, USA

A mesoscale spring model based on Rayleigh-Ritz analysis is used to mimic the fabric surface as an elastic membrane in parachute simulation. Such elastic structure is coupled with fluid solver through the impulse method. We will discuss several challenging problems in this multi-physics system including turbulence modeling, fabric collision, parachutist coupling, and computational parallelization. We will show numerical proof of convergence, verification and validation of numerical components, and programming design for simulations of different air-delivery assemblies.

A thermodynamically consistent phase-field model for incompressible flows and its energy law preserving C⁰ finite element scheme
Ping Lin, University of Dundee, UK

We develop a phase-field model for binary incompressible fluid with thermocapillary effects, which allows for different properties (densities, viscosities and heat conductivities) of each component while maintaining the thermodynamic consistency. A sharp-interface limit analysis is carried out to show that the interfacial conditions of the classical sharp-interface models can be recovered from our phase-field model. We also show how an energy law preserving continuous finite element scheme may be derived and used to compute and validate the model.

A second order stable scheme for fluid structure interaction with Navier slip interface
Jie Liu, National University of Singapore

In this talk, we present a temporally second-order numerical scheme for fluid structure interaction where we enforce Navier slip boundary conditions at the fluid structure interface. We prove the stability of the second order fully discrete scheme. Using the elastic-rigid fish model that we have introduced before, we are able to perform 2D planner fish swimming simulation with Navier slip interface conditions. Numerical results show that with the new interface conditions, a fish can swim much faster than if it uses no slip interface conditions.

The immersed interface method for flow around non-smooth boundaries
Yang Liu, Southern Methodist University, USA

In the immersed interface method, a boundary immersed in a fluid is generated by a singular force in the Navier-Stokes equations, and the singular force enters a numerical scheme as jump conditions across the boundary. In previous work, the method has been developed for smooth boundaries. In this talk, we present how to extend the method for non-smooth boundaries. We use panels to represent a boundary, compute necessary jump conditions explicitly, and compare two different pressure Poisson solvers. We test our extended method by simulating flows past a circular cylinder, a square cylinder or around a flapping plate. Our results show that the method is robust, accurate and efficient.

Simulation of coalescence and breakup of interface with a 3D front-tracking method
Jiacai Lu, University of Notre Dame, USA

Front-tracking method, which use one set of governing equations for the whole computational domain on the fixed, structured grid, and use connected marker points to explicitly track the interface, has been successfully used in direct numerical simulations (DNS) of turbulent channel bubbly flows and many other multiphase flows. With the void fraction increase in multiphase flows, interface topology becomes very complex, and interfaces undergo continuous coalescence and breakup. Modeling such flow is still very primitive, and is identified as a challenge for the front-tracking method. We present an efficient algorithm to change the topology of triangular elements in the interface mesh here. The algorithm is validated by comparing the simulation results with experimental ones of head-on collision of two droplets. Its applications to simulations of flow regime transitions in turbulent channel flows are implemented to study evolutions of interface areas, flow rates, wall shears and many other parameters.

Optimal control in free or moving boundary coupling of navier-stokes and nonlinear elasticity
Kristina Martin, NC State University, USA

We consider optimal control problems involving free or moving boundary fluid-elasticity interaction described by a coupling of Navier-Stokes with the equations of nonlinear elastodynamics in the context of isotropic elasticity. We prove in the free boundary, steady case that a distributed control can minimize the difference between the fluid velocity and a desired velocity. In the dynamic, moving boundary case we prove that a distributed control can minimize turbulence in the fluid flow. We additionally discuss sensitivity analysis, including derivation of the linearized adjoint equations and first order necessary optimality conditions.

Modeling osmotic transmembrane water flow with moving interfacial membranes
Yoichiro Mori, University of Minnesota, USA

Electrolyte and volume balance is fundamental to cell physiology, and its importance is increasingly recognized in cell movement and proliferation. After a quick review of cell volume control, we present a PDE model of cell volume and electrolyte balance in the presence of moving interfacial membranes. A unified treatment of electrodiffusion, osmosis and cell mechanics is made possible by requiring that the model satisfy an appropriate free energy identity. We then discuss our efforts in developing a computational method to solve the resulting equations, based on the immersed boundary method for fluid-structure interaction and cartesian grid embedded boundary method for solute diffusion. This work is a collaboration with Chun Liu, Bob Eisenberg and Lingxing Yao.

A DLM/FD method for simulating particle motion in viscoelastic fluid
Tsorng-Whay Pan, University of Houston, USA

In this talk we present numerical methods for simulating particle motion in viscoelastic fluid. We have combined a fictitious domain/distributed Lagrange multiplier (DLM/FD) method with a factorization approach from Lozinski and Owens [J. Non-Newtonian Fluid Mech. 112 (2003) 161] via an operator splitting technique. The new scheme preserves the positive definiteness of the conformation tensor at the discrete level. The effect of the maximum extension of the immersed polymer coils and that of the elasticity on the particle chaining have been studied.

Numerical study of a monolithic fluid-structure formulation
Olivier Pironneau, University of Paris VI, France

The conservation laws of continuum mechanic are naturally written in an Eulerian frame where the difference between a fluid and a solid is only in the expression of the stress tensors, usually with Newton's hypothesis for the fluids and Helmholtz potentials of energy for hyperelastic solids. There are currently two favoured approaches to Fluid Structured Interactions (FSI) both working with the equations for the solid in the initial domain; one uses an ALE formulation for the fluid and the other matches the fluid-structure interfaces using Lagrange multipliers and the immersed boundary method. By contrast the proposed formulation works in the frame of physically deformed solids and proposes a discretization where the structures have large displacements computed in the deformed domain together with the fluid in the same; in such a monolithic formulation velocities of solids and fluids are computed all at once in a single variational formulation by a semi-implicit in time and the finite element method. Besides the simplicity of the formulation the advantage is a single algorithm for a variety of problems including multi-fluids, free boundaries and FSI. The idea is not new but the progress of mesh generators renders this approach feasible and even reasonably robust. In this lecture the method and its discretization are presented, stability is discussed showing in a loose fashion were are the difficulties and why one is able to show convergence of monolithic algorithms on fixed domains for fluids in compliant shell vessels restricted to small displacements. A numerical section discusses implementation issues and presents a few simple tests.

Simulation of fluid-structure interaction problems arising hemodynamics
Annalisa Quaini, University of Houston, USA

We focus on the interaction of an incompressible fluid and an elastic structure. Two cases are considered: 1. the elastic structure covers part of the fluid boundary and undergoes small displacement and 2. the elastic structure is immersed in the fluid and it features large displacement. For the first case, we propose an Arbitrary Lagrangian-Eulerian (ALE) method based on Lie's operator splitting. The resulting algorithm is unconditionally stable and weakly coupled: it requires the solution of one fluid subproblem and one structure subproblem, both endowed with Robin type boundary conditions, per time step. This algorithm is applied to blood flow in a healthy straight artery and in a diseased artery with implanted stent. Standard ALE methods fail when the structural displacement is large. Thus, for the second case we propose an extended ALE method that avoids remeshing. The extended ALE approach relies on a variational mesh optimization technique, combined with an additional constraint which is imposed to enforce the alignment of the structure with certain edges of the fluid triangulation without changing connectivity. This method is applied to a 2D benchmark problem modeling valves: a thin elastic 1D leaflet, modeled by an inextensible beam equation, is immersed in a 2D incompressible, viscous fluid driven by the time-dependent inlet and outlet data.

An immersed boundary method for simulating vesicle dynamics in three dimensions
Yunchang Seol, National Taiwan University, Taiwan

In this talk, we present the extension of our previous immersed boundary (IB) method for 3D axisymmetric inextensible vesicle in Navier-Stokes flows to general three dimensions. Despite a similar spirit in numerical algorithms to the axisymmetric case, the fully 3D numerical implementation is much more complicated and is far from straightforward. A vesicle membrane surface is known to be incompressible and exhibits bending resistance. As in 3D axisymmetric case, instead of keeping the vesicle locally incompressible, we adopt a modified elastic tension energy to make the vesicle surface patch nearly incompressible so that solving the unknown tension (Lagrange multiplier for the incompressible constraint) can be avoided. Nevertheless, the new elastic force derived from the modified tension energy has exactly the same mathematical form as the original one except the different definitions of tension. The vesicle surface is discretized on a triangular mesh where the elastic tension and bending force are calculated on each vertex (Lagrangian marker in the IB method) of the triangulation. Throughout this presentation, we illustrate the robustness and applicability of our method.

Energy harvesting from flow-induced vibration of piezoelectric membranes
Kourosh Shoele, Johns Hopkins University, USA

We report numerical studies on energy harvesting by self-sustained oscillations (flutter) of flexible piezoelectric membranes. Immersed boundary technique with explicit description of fluid, solid and electric systems as well as their coupling is developed to investigate the electrical energy harvesting efficiency of the piezoelectric structures. Our results show that an inverted configuration of the piezoelectric membrane with the fixed trailing-edge and free leading-edge can
amplify the flag deflection over a wide range of wind speeds where there is lock-on between the flag flutter and the intrinsic wake shedding phenomenon. The state with large, symmetric flutter is identified as being most promising for energy harvesting, occurs when there is a match between the timescales of flutter and the intrinsic time-scale of the piezoelectric circuit. The simulations are used to examine a simple scaling law that could be used to predict the energy harvesting performance of such devices. Three dimensional effects are explored and the relation between the aspect ratio of the flag and the level of electrical power generation by the piezoelectric system is examined. Finally, the numerical predictions are tested with a series of experiments at a tabletop wind tunnel where high-speed videography confirms good agreement between numerical simulations and experimental observations.

Authors:
Kourosh Shoele 1, Santiago Orrego 1,2, Sung Hoon Kang 1,2, Rajat Mittal 1
1 Department of Mechanical Engineering, Johns Hopkins University. Baltimore, MD 21218.
2 Hopkins Extreme Materials Institute, Johns Hopkins University. Baltimore, MD 21218.

On solution reconstruction and evaluation of numerical fluxes in CFD
Chang Shu, National University of Singapore

Finite volume method is one of the most popular numerical approaches in computational fluid dynamics (CFD). It involves conservative flow variables defined at cell centres and numerical fluxes at cell interfaces. Usually, the flow variables at cell centres are defined as unknowns, which can be given from the solution of discrete governing equations. In the solution process, we need to locally reconstruct a solution from flow variables at cell centres to evaluate fluxes at cell interfaces. Currently, there are three major approaches to reconstruct the solution for evaluation of fluxes. They are based on smooth function approximation (general solver), one-dimensional Euler equation (Riemann solver), and continuous Boltzmann equation (gas-kinetic solver). In this talk, a brief review of current 3 flux solvers will be presented first. Then the newly-developed lattice Boltzmann flux solver, circular function-based gas kinetic flux solver and the simple gas kinetic flux solver will be shown in details. These solvers evaluate viscous and inviscid fluxes simultaneously and physically. They combine advantages of conventional Navier-Stokes solvers and lattice Boltzmann/gas kinetic solvers. The fluid flows from incompressible regime to hypersonic regime can be well simulated by these solvers. The proposed new flux solvers will be validated by various numerical examples.

Smoothed particle hydrodynamics simulations for sediment dispersion in ocean
Tran-Duc Thien, National University of Singapore

Polymetallic nodules comprising manganese, nickel, cobalt, copper, etc., are found in mass on ocean bottom. Those nodules have high economic value and collection of those nodules has been a subject for studies on technology for a few decades. Beside technological challenges, possible environmental impacts due to the collection activity are very much concerned. During the operation, sediment in the top-layer of the ocean bottom is disturbed and then could be spread in a large area by ocean currents. Since sediment particles are heavier than ocean water, they tend to re-deposit onto the ocean bottom. Residence time, concentration distribution, and occupied area of the disturbed sediment are expected to be known. In this study, the sediment spreading problem will be studied in the smoothed particle hydrodynamics context. It is the first time an anisotropic dispersion studied with smoothed particle hydrodynamics.
Joint work with Nhan Phan-Thien, Boo Cheong Khoo.

Numerical simulations of complex multiphase flows
Gretar Tryggvason, University of Notre Dame, USA

Direct Numerical Simulations (DNS) of multiphase flows, where every continuum length and time scale are fully resolved, currently allow us to simulate flows of considerable complexity such as the motion of several hundred bubbles, drops and particles in turbulent flows, for sufficiently long time that meaningful statistical quantities can be obtained. Additional physical processes such as heat and mass transfer and phase change have also been included, and we will review a few recent studies. The addition of new physics often results in new length and time scales that are shorter and faster than the dominant flow scales. Similarly, very small features such as thin films, filaments, and drops can also arise during coalescence and breakup of fluid blobs. The geometry of these features is usually simple, since surface tension effects are strong and inertia effects are relatively small and in isolation these features are often well described by analytical or semi-analytical models. Recent efforts to embed analytical and semi-analytical models to capture such features, in combination with direct numerical simulations of the rest of the flow, are described. The ultimate goal of DNS simulations is to help generate closure models for equations for the average flow of industrial scale processes, where DNS are not practical, and we will end the talk by discussing recent efforts to use data mining techniques to extract such closures from DNS data.

An arbitrary lagrangian eulerian discontinuous galerkin approach to fluid-structure interaction and its application to cardiovascular problem
Yifan Wang, University of Houston, USA

The Discontinuous Galerkin (DG) method is ideal to solve convection dominated fluid problems, since it provides a stable and physically correct upwind scheme for the convective term in the Navier-Stokes equations. Additionally, it produces high order (based on hp-adaptivity) and less dissipative solutions in contrast with conventional finite element methods (FEM). To deal with time-dependent (moving) geometries associated with fluid-structure interaction (FSI) problems, we propose to use an Arbitrary Lagrangian Eulerian (ALE) approach implemented within the DG method by using the Internal Penalty (IP) flux to minimize the mis-match in the moving domain mesh. Due to the fact that ALE relies on the continuous mapping between physical and reference domains, DG method is not intuitive for ALE and therefore barely discussed in literature. However, we get around this difficulty by proposing an element-wise mapping and constructing an Internal Penalty (IP) flux such that the continuous ALE mapping is well approximated by the corresponding element-wise DG implementation. There are several advantages of our approach: Firstly, we can utilize the advantages of the DG method to acquire high-order and less dissipative solutions for the fluid problem; Secondly, no extra computation of constructing FEM operation matrix for ALE mapping is required; Thirdly, our DG-ALE approach will prove to be particularly useful when we deal with Abdominal Aortic Aneurysm (AAA) treated with a stent-graft, at which high jumps in the fluid pressure and velocity occur across the stent-graft interface.

We demonstrate our approach by solving fluid structure interaction (FSI) problems motivated by blood flow applications. The FSI problem is solved using either the Dirichlet-Neumann or Robin-Neumann partition scheme with corresponding transmission conditions. The arterial walls are modeled by the linearly elastic membrane model, which captures both radial and longitudinal displacements. DG method based on unstructured triangular elements and 2nd order semi-implicit time splitting scheme is adopted to solve the fluid sub-problem. To update the fluid domain, local DG method with internal penalty is used to calculate the element-wise ALE mapping. We demonstrate that our algorithm captures the correct solution by testing the benchmark FSI problem involving a straight cylinder in 2D and 3D. Then, we show the capability of our algorithm in dealing with discontinuities in the FSI problem by examining the test case of a periodic beam immersed in the fluid, which contains pressure jump across the beam interface. In the end, we present a physiologically relevant FSI problem in a patient-specific geometry of Aortic Abdominal Aneurysm (AAA) and show how our simulations provide some new information about the pressure and displacement distribution in patients with tortuous and stent-graft treated AAA. This is a joint work with S. Canic and A. Quaini.

The accurate and efficient numerical simulation of general fluid-structure interaction
Yongxing Wang, University of Leeds, UK

The accurate and efficient numerical simulation of general fluid-structure interactions (FSI) is a significant computational challenge because of the strong nonlinearities associated with the fluid flow, the structural deformation (in the case of large displacements) and, crucially, with the coupling/interaction between the two. The current gold standard for robust and accurate simulation of such problems is to use a monolithic method that strongly couples the fluid and solid models and discretizes them into a single nonlinear system at each time step (involving fluid velocity and pressure, elastic displacements and Lagrange multipliers to enforce consistency at the interface (which must either be tracked or recovered)). Such schemes are exceedingly computationally expensive however, and require sophisticated numerical schemes to ensure convergence of the nonlinear solver at each time step.

In this talk,we present a new finite element method for simulation of FSI which has the same generality and robustness of monolithic methods (and is therefore able to model a range of solid materials from very soft to very hard, for example) but is semi-explicit and therefore significantly more computationally efficient and easier to implement. Our proposed approach has similarities with classical immersed finite element methods (IFEMs), by approximating a single velocity field in the entire domain (i.e. occupied by fluid and solid) on a single mesh, but differs by treating the corrections due to the solid deformation on the left-hand side of the modified fluid flow equations (i.e. implicitly).

In addition to motivating the derivation of our numerical scheme, we will provide a description of its implementation within an adaptive finite element code and a wide range of computational examples will be presented in order to validate the method across a wide range of flows, solids and interactions.

A sharp Cartesian method for incompressible flows with large density ratios
Lisl Weynans, Bordeaux University, France

A new Cartesian method for bifluid incompressible flows with high density ratios is presented. The Navier-Stokes equations are integrated in time thanks to a fractional step method based on the Chorin scheme and discretized in space on a Cartesian mesh. The bifluid interface is implicitly represented using a level set function. The specificity of the method relies on a sharp second-order numerical scheme for the spatial resolution of the discontinuous elliptic problem for the pressure. The numerical tests show the improvements due to this sharp method compared to classical first order methods.

The immersed interface method for flow around moving solids
Sheng Xu, Southern Methodist University, USA

To simulate flow around moving solids, the immersed interface method treats the objects as the fluid and recovers their effect on the surrounding flow by incorporating jump conditions across the fluid-solid interfaces into numerical schemes. In this talk, I will present some recent enhancement of the method toward robust simulation of flow around numerous rigid solids of complex geometries in 3d. After an overview of the method, I will derive necessary jump conditions for solid surfaces represented by triangular meshes; I will go through the algorithm to implement the method, and I will then discuss how to parallelize the method for a large number of moving solids. Numerical examples will be shown to test the accuracy, efficiency and robustness of the method.

A boundary integral equation formulation based cartesian grid method for the stokes equations
Wenjun Ying, Shanghai Jiao Tong University, China

In this talk, I will present a boundary integral equation (BIE) formulation based Cartesian grid method for the Stokes equations on complex domains in both two and three space dimensions. The method reformulates a boundary value problem or an interface problem of the Stokes equations as a Fredholm boundary integral equation of the second kind. Taking advantage of the well-conditioning property of the BIE, the method solves it by an iterative method with the number of iterations be independent of the system dimension. Evaluation of boundary integrals in the iteration is done by solving an equivalent but much simpler interface problem with the marker-and-cell (MAC) scheme on a staggered Cartesian grid. In this way, the method avoids the need to know any analytical expression of the kernel of the integral operator and the dense matrix-vector multiplication associated with the traditional boundary integral method. On the staggered grid, the discrete equations are treated with a fast Fourier transform (FFT) accelerated Stokes solver. Moreover, the method avoids generation of any unstructured grids for both the volume and the boundary or interface of the domain. It represents the domain boundary or interface by its intersection points with the underlying Cartesian grid. This representation makes numerical interpolation and differentiation as well as relocation of the discretization points much easier. Some numerical results in both two and three space dimensions will be presented.

Parallel multilevel iterative methods for coupled PDEs
Chensong Zhang, Chinese Academy of Sciences, China

Over the last few decades, intensive research has been done on developing efficient parallel iterative solvers for coupled PDEs. One useful mathematical technique that has drawn a lot of attention is multilevel iterative solvers/preconditioners. In this work, we discuss recent development of multilevel iterative solvers for solving practical multi-physics problems, including fluid-structure interaction and petroleum reservoir simulation. In particular, we are going to discuss how to construct effective preconditioners for coupled PDEs. Furthermore, we will talk about how to implement these methods on modern parallel computer architectures.

Keywords(optional): multilevel iterative method, fluid-structure interaction, petroleum reservoir simulation

A parametric finite element method for simulating the solid state dewetting with anisotropic surface energies
Quan Zhao, National University of Singapore

The sharp interface model for solid-state dewetting in two dimensions is a fourth-order (sixth-order for strongly anisotropic case) geometric partial differential equations with moving contact line (points) migration. We propose an efficient parametric finite element method to solve this interfacial problem. Compared to the traditional methods (e.g., marker-particle methods), the proposed PFEM not only has very good accuracy, but also poses very mild restrictions on the numerical stability, and thus it has significant advantages for solving this type of open curve evolution problems with applications in the simulation of solid-state dewetting. Extensive numerical results are reported to demonstrate the accuracy and high efficiency of the proposed PFEM.

A new 3D immersed boundary method with application
Luoding Zhu, Indiana University-Purdue University Indianapolis, USA

In this talk we introduce a new 3D IB method that adopts a classic approach (shell as an assembly of plate elements) to model a thin-walled structure and a special finite element method (the corotational scheme) to solve numerically the partial differential equations governing the motion of the thin-walled structure. The Navier-Stokes equations are solved for numerically by the lattice Boltzmann method. The coupling of the fluid and thin-walled structure is through the penalty approach. As an application, the new IB method is applied to simulation of a 3D viscous flow past a deformable circular thin plate. Our numerical results are in very good agreement with the laboratory experiments.

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