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99
The immersed boundary method: a projection approach.
 J. Comput. Phys.,
, 2007
"... Abstract A new formulation of the immersed boundary method with a structure algebraically identical to the traditional fractional step method is presented for incompressible flow over bodies with prescribed surface motion. Like previous methods, a boundary force is applied at the immersed surface t ..."
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Cited by 59 (12 self)
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Abstract A new formulation of the immersed boundary method with a structure algebraically identical to the traditional fractional step method is presented for incompressible flow over bodies with prescribed surface motion. Like previous methods, a boundary force is applied at the immersed surface to satisfy the noslip constraint. This extra constraint can be added to the incompressible NavierStokes equations by introducing regularization and interpolation operators. The current method gives prominence to the role of the boundary force acting as a Lagrange multiplier to satisfy the noslip condition. This role is analogous to the effect of pressure on the momentum equation to satisfy the divergencefree constraint. The current immersed boundary method removes slip and nondivergencefree components of the velocity field through a projection. The boundary force is determined implicitly without any constitutive relations allowing the present formulation to use larger CFL numbers compared to some past methods. Symmetry and positivedefiniteness of the system are preserved such that the conjugate gradient method can be used to solve for the flow field. Examples show that the current formulation achieves secondorder temporal accuracy and better than firstorder spatial accuracy in L 2 norms for oneand twodimensional test problems. Results from twodimensional simulations of flows over stationary and moving cylinders are in good agreement with those from previous experimental and numerical studies.
An adaptive, formally second order accurate version of the immersed boundary method
, 2006
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An immersed interface method for viscous incompressible flows involving rigid and flexible boundaries
 J. Comp. Phys
, 2006
"... We present an immersed interface method for the incompressible NavierStokes equations capable of handling rigid immersed boundaries. The immersed boundary is represented by a set of Lagrangian control points. In order to guarantee that the noslip condition on the boundary is satisfied, singular fo ..."
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Cited by 37 (3 self)
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We present an immersed interface method for the incompressible NavierStokes equations capable of handling rigid immersed boundaries. The immersed boundary is represented by a set of Lagrangian control points. In order to guarantee that the noslip condition on the boundary is satisfied, singular forces are applied on the fluid. The forces are related to the jumps in pressure and the jumps in the derivatives of both pressure and velocity, and are interpolated using cubic splines. The strength of the singular forces is determined by solving a small system of equations iteratively at each time step. The NavierStokes equations are discretized on a staggered Cartesian grid by a second order accurate projection method for pressure and velocity. Keywords: Immersed interface method, NavierStokes equations, Cartesian grid method, finite difference, fast Poisson solvers, irregular domains.
Moving Overlapping Grids with Adaptive Mesh Refinement for HighSpeed Reactive and Nonreactive Flow
 Journal of Computational Physics
, 1979
"... We consider the solution of the reactive and nonreactive Euler equations on twodimensional domains that evolve in time. The domains are discretized using moving overlapping grids. In a typical grid construction, boundarytted grids are used to represent moving boundaries, and these grids overlap w ..."
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Cited by 23 (6 self)
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We consider the solution of the reactive and nonreactive Euler equations on twodimensional domains that evolve in time. The domains are discretized using moving overlapping grids. In a typical grid construction, boundarytted grids are used to represent moving boundaries, and these grids overlap with stationary background Cartesian grids. Blockstructured adaptive mesh renement (AMR) is used to resolve nescale features in the
ow such as shocks and detonations. Renement grids are added to baselevel grids according to an estimate of the error, and these re nement grids move with their corresponding baselevel grids. The numerical approximation of the governing equations takes place in the parameter space of each component grid which is dened by a mapping from ( xed) parameter space to (moving) physical space. The mapped equations are solved numerically using a secondorder extension of Godunov's method. The sti source term in the reactive case is handled using a RungeKutta errorcontrol scheme. We consider cases when the boundaries move according to a prescribed function of time and when
Logically Rectangular Grids and Finite Volume Methods for PDEs in Circular and Spherical Domains
 In preparation; http://www.amath. washington.edu/~rjl/pubs/circles
, 2005
"... Abstract. We describe a class of logically rectangular quadrilateral and hexahedral grids for solving PDEs in circular and spherical domains, including grid mappings for the circle, the surface of the sphere and the threedimensional ball. The grids are logically rectangular and the computational do ..."
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Cited by 22 (6 self)
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Abstract. We describe a class of logically rectangular quadrilateral and hexahedral grids for solving PDEs in circular and spherical domains, including grid mappings for the circle, the surface of the sphere and the threedimensional ball. The grids are logically rectangular and the computational domain is a single Cartesian grid. Compared to alternative approaches based on a multiblock data structure or unstructured triangulations, this approach simplifies the implementation of numerical methods and the use of adaptive refinement. A more general domain with a smooth boundary can be gridded by composing one of the mappings from this paper with another smooth mapping from the circle or sphere to the desired domain. Although these grids are highly nonorthogonal, we show that the highresolution wavepropagation algorithm implemented in clawpack can be effectively used to approximate hyperbolic problems on these grids. Since the ratio between the largest and smallest grid is below 2 for most of our grid mappings, explicit finite volume methods such as the wave propagation algorithm do not suffer from the center or pole singularities that arise with polar or latitudelongitude grids. Numerical test calculations illustrate the potential use of these grids for a variety of applications including Euler equations, shallow water equations, and acoustics in a heterogeneous medium. Pattern formation from a reactiondiffusion equation on the sphere is also considered. All examples are implemented in the clawpack software package and full source code is available on the web, along with matlab routines for the various mappings.
A spectral boundary integral method for flowing blood cells
 J. Comp. Phys., 229:3726
"... a b s t r a c t A spectral boundary integral method for simulating large numbers of blood cells flowing in complex geometries is developed and demonstrated. The blood cells are modeled as finitedeformation elastic membranes containing a higher viscosity fluid than the surrounding plasma, but the so ..."
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Cited by 22 (1 self)
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a b s t r a c t A spectral boundary integral method for simulating large numbers of blood cells flowing in complex geometries is developed and demonstrated. The blood cells are modeled as finitedeformation elastic membranes containing a higher viscosity fluid than the surrounding plasma, but the solver itself is independent of the particular constitutive model employed for the cell membranes. The surface integrals developed for solving the viscous flow, and thereby the motion of the massless membrane, are evaluated using an OðN log NÞ particlemesh Ewald (PME) approach. The cell shapes, which can become highly distorted under physiologic conditions, are discretized with spherical harmonics. The resolution of these global basis functions is, of course, excellent, but more importantly they facilitate an approximate dealiasing procedure that stabilizes the simulations without adding any numerical dissipation or further restricting the permissible numerical time step. Complex geometry noslip boundaries are included using a constraint method that is coupled into an implicit system that is solved as part of the time advancement routine. The implementation is verified against solutions for axisymmetric flows reported in the literature, and its accuracy is demonstrated by comparison against exact solutions for relaxing surface deformations. It is also used to simulate flow of blood cells at 30% volume fraction in tubes between 4.9 and 16.9 lm in diameter. For these, it is shown to reproduce the wellknown nonmonotonic dependence of the effective viscosity on the tube diameter.
Penalty immersed boundary method for an elastic boundary with mass
 Physics of Fluids
"... The immersed boundary (IB) method has been widely applied to problems involving a moving elastic boundary that is immersed in fluid and interacting with it. But most of the previous applications of the IB method have involved a massless elastic boundary and used efficient Fourier transform methods f ..."
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Cited by 22 (3 self)
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The immersed boundary (IB) method has been widely applied to problems involving a moving elastic boundary that is immersed in fluid and interacting with it. But most of the previous applications of the IB method have involved a massless elastic boundary and used efficient Fourier transform methods for the numerical solutions. Extending the method to cover the case of a massive boundary has required spreading the boundary mass out onto the fluid grid and then solving the NavierStokes equations with a variable mass density. The variable mass density of this previous approach makes Fourier transform methods inapplicable, and requires a multigrid solver. Here we propose a new and simple way to give mass to the elastic boundary and show that the new method can be applied to many problems for which the boundary mass is important. The new method does not spread mass to the fluid grid, retains the use of Fourier transform methodology, and is easy to implement in the context of an existing IB method code for the massless case.
Numerical Methods for FluidStructure Interaction  A Review
, 2012
"... The interactions between incompressible fluid flows and immersed structures are nonlinear multiphysics phenomena that have applications to a wide range of scientific and engineering disciplines. In this article, we review representative numerical methods based on conforming and nonconforming me ..."
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Cited by 21 (0 self)
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The interactions between incompressible fluid flows and immersed structures are nonlinear multiphysics phenomena that have applications to a wide range of scientific and engineering disciplines. In this article, we review representative numerical methods based on conforming and nonconforming meshes that are currently available for computing fluidstructure interaction problems, with an emphasis on some of the recent developments in the field. A goal is to categorize the selected methods and assess their accuracy and efficiency. We discuss challenges faced by researchers in this field, and we emphasize the importance of interdisciplinary effort for advancing the study in fluidstructure interactions.