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A fast algorithm for simulating vesicle flows in three dimensions
, 2010
"... Vesicles are locallyinextensible fluid membranes that can sustain bending. In this paper, we extend “A numerical method for simulating the dynamics of 3D axisymmetric vesicles suspended in viscous flows”, Veerapaneni et al. Journal of Computational Physics, 228(19), 2009 to general nonaxisymmetric ..."
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Vesicles are locallyinextensible fluid membranes that can sustain bending. In this paper, we extend “A numerical method for simulating the dynamics of 3D axisymmetric vesicles suspended in viscous flows”, Veerapaneni et al. Journal of Computational Physics, 228(19), 2009 to general nonaxisymmetric vesicle flows in three dimensions. Although the main components of the algorithm are similar in spirit to the axisymmetric case (spectral approximation in space, semiimplicit timestepping scheme), important new elements need to be introduced for a full 3D method. In particular, spatial quantities are discretized using spherical harmonics, and quadrature rules for singular surface integrals need to be adapted to this case; an algorithm for surface reparameterization is neeed to ensure sufficient of the timestepping scheme, and spectral filtering is introduced to maintain reasonable accuracy while minimizing computational costs. To characterize the stability of the scheme and to construct preconditioners for the iterative linear system solvers used in the semiimplicit timestepping scheme, we perform a spectral analysis of the evolution operator on the unit sphere. By introducing these algorithmic components, we obtain a timestepping scheme that, in our numerical experiments, is unconditionally stable. We present results to analyze the cost and convergence rates of the overall scheme. To illustrate the applicability of the new method, we consider a few vesicleflow interaction problems: a single vesicle in relaxation, sedimentation, shear flows, and manyvesicle flows. 1
Electricallogic simulation
 In International Conference on ComputerAided Design
, 1984
"... the dynamics of inextensible vesicles by the penalty immersed boundary method ..."
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the dynamics of inextensible vesicles by the penalty immersed boundary method
Analytical and Numerical Solutions for Shapes of Quiescent 2D
, 2008
"... We describe an analytic method for the computation of equilibrium shapes for twodimensional vesicles characterized by a Helfrich elastic energy. We derive boundary value problems and solve them analytically in terms of elliptic functions and elliptic integrals. We derive solutions by prescribing le ..."
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We describe an analytic method for the computation of equilibrium shapes for twodimensional vesicles characterized by a Helfrich elastic energy. We derive boundary value problems and solve them analytically in terms of elliptic functions and elliptic integrals. We derive solutions by prescribing length and area, or displacements and angle boundary conditions. The solutions are compared to solutions obtained by a boundary integral equationbased numerical scheme. Our method enables the identification of different configurations of deformable vesicles and accurate calculation of their shape, bending moments, tension, and the pressure jump across the vesicle membrane. Furthermore, we perform numerical experiments that indicate that all these configurations are stable minima. 1
An unstructured solver for simulations of deformable particles in flows at arbitrary Reynolds numbers
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A fractional step immersed boundary method for stokes flow with an inextensible interface enclosing a solid particle
 SIAM J. Sci. Comput
"... Abstract. In this paper, we develop a fractional step method based on the immersed boundary (IB) formulation for Stokes flow with an inextensible (incompressible) interface enclosing a solid particle. In addition to solving for the fluid variables such as the velocity and pressure, the present probl ..."
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Abstract. In this paper, we develop a fractional step method based on the immersed boundary (IB) formulation for Stokes flow with an inextensible (incompressible) interface enclosing a solid particle. In addition to solving for the fluid variables such as the velocity and pressure, the present problem involves finding an extra unknown elastic tension such that the surface divergence of the velocity is zero along the interface, and an extra unknown particle surface force such that the velocity satisfies the noslip boundary condition along the particle surface. While the interface moves with local fluid velocity, the enclosed particle hereby undergoes a rigid body motion, and the system is closed by the forcefree and torquefree conditions along the particle surface. The equations are then discretized by standard centered difference schemes on a staggered grid, and the interactions between the interface and particle with the fluid are discretized using a discrete delta function as in the IB method. The resultant linear system of equations is symmetric and can be solved by fractional steps so that only fast Poisson solvers are involved. The present method can be extended to Navier– Stokes flow with moderate Reynolds number by treating the nonlinear advection terms explicitly for the time integration. The convergent tests for a Stokes solver with or without an inextensible interface are performed and confirm the desired accuracy. The tanktreading to tumbling motion for an inextensible interface enclosing a solid particle with different filling fractions under a simple shear flow has been studied extensively, and the results here are in good agreement with those obtained in literature.
A Velocity Decomposition Approach for Moving Interfaces in Viscous Fluids
"... We present a secondorder accurate method for computing the coupled motion of a viscous fluid and an elastic material interface with zero thickness. The fluid flow is described by the NavierStokes equations, with a singular force due to the stretching of the moving interface. We decompose the veloc ..."
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We present a secondorder accurate method for computing the coupled motion of a viscous fluid and an elastic material interface with zero thickness. The fluid flow is described by the NavierStokes equations, with a singular force due to the stretching of the moving interface. We decompose the velocity into a “Stokes ” part and a “regular ” part. The first part is determined by the Stokes equations and the singular interfacial force. The Stokes solution is obtained using the immersed interface method, which gives secondorder accurate values by incorporating known jumps for the solution and its derivatives into a finite difference method. The regular part of the velocity is given by the NavierStokes equations with a body force resulting from the Stokes part. The regular velocity is obtained using a timestepping method that combines the semiLagrangian method with the backward difference formula. Because the body force is continuous, jump conditions are not necessary. For problems with stiff boundary forces, the decomposition approach can be combined with fractional timestepping, using a smaller time step to advance the interface quickly by Stokes flow, with the velocity computed using boundary integrals. The small time steps maintain numerical stability, while the overall solution is updated on a larger time step to reduce computational cost.
A PARTIALLY IMPLICIT HYBRID METHOD FOR COMPUTING INTERFACE MOTION IN STOKES FLOW
"... We present a partially implicit hybrid method for simulating the motion of a stiff interface immersed in Stokes flow, in free space or in a rectangular domain with boundary conditions. The implicit time integration is based on the smallscale decomposition approach and does not require the iterativ ..."
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We present a partially implicit hybrid method for simulating the motion of a stiff interface immersed in Stokes flow, in free space or in a rectangular domain with boundary conditions. The implicit time integration is based on the smallscale decomposition approach and does not require the iterative solution of a system of nonlinear equations. Firstorder and secondorder versions of the timestepping method are derived systematically, and numerical results indicate that both methods are substantially more stable than explicit methods. At each time level, the Stokes equations are solved using a hybrid approach combining nearly singular integrals on a band of mesh points near the interface and a meshbased solver. The solutions are secondorder accurate in space and preserve the jump discontinuities across the interface. Finally, the hybrid method can be used as an alternative to adaptive mesh refinement to resolve boundary layers that are frequently present around a stiff immersed interface.
A semiimplicit gradient augmented level set method
 SIAM Journal on Scientific Computing
, 2013
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An Immersed Finite Element Method with Integral Equation Correction
"... We propose a robust immersed finite element method in which an integral equation formulation is used to enforce essential boundary conditions. The solution of a boundary value problem is expressed as the superposition of a finite element solution and an integral equation solution. For computing the ..."
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We propose a robust immersed finite element method in which an integral equation formulation is used to enforce essential boundary conditions. The solution of a boundary value problem is expressed as the superposition of a finite element solution and an integral equation solution. For computing the finite element solution, the physical domain is embedded into a slightly larger Cartesian (boxshaped) domain and is discretized using a blockstructured mesh. The defect in the essential boundary conditions, which occurs along the physical domain boundaries, is subsequently corrected with an integral equation method. In order to facilitate the mapping between the finite element and integral equation solutions, the physical domain boundary is represented with a signed distance function on the blockstructured mesh. As a result, only a boundary mesh of the physical domain is necessary and no domain mesh needs to be generated, except for the nonboundaryconforming blockstructured mesh. The overall approach is first presented for the Poisson equation and then generalised to incompressible viscous flow equations. As an example for fluidstructure coupling, the settling of a heavy rigid particle in a closed tank is considered.
Electrohydrodynamics of threedimensional vesicles: A numerical approach
 SIAM Journal On Scientific Computing
"... A threedimensional numerical model of vesicle electrohydrodynamics in the presence of DC electric fields is presented. The vesicle membrane is modeled as a thin capacitive interface through the use of a semiimplicit, gradientaugmented level set Jet scheme. The enclosed volume and surface area are ..."
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A threedimensional numerical model of vesicle electrohydrodynamics in the presence of DC electric fields is presented. The vesicle membrane is modeled as a thin capacitive interface through the use of a semiimplicit, gradientaugmented level set Jet scheme. The enclosed volume and surface area are conserved both locally and globally by a new NavierStokes projection method. The electric field calculations explicitly take into account the capacitive interface by an implicit Immersed Interface Method formulation, which calculates the electric potential field and the transmembrane potential simultaneously. The results match well with previously published experimental, analytic and twodimensional computational works.