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83
An immersed interface method for incompressible navierstokes equations
 SIAM J. Sci. Comput
, 2003
"... Abstract. The method developed in this paper is motivated by Peskin’s immersed boundary (IB) method, and allows one to model the motion of flexible membranes or other structures immersed in viscous incompressible fluid using a fluid solver on a fixed Cartesian grid. The IB method uses a set of discr ..."
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Cited by 99 (3 self)
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Abstract. The method developed in this paper is motivated by Peskin’s immersed boundary (IB) method, and allows one to model the motion of flexible membranes or other structures immersed in viscous incompressible fluid using a fluid solver on a fixed Cartesian grid. The IB method uses a set of discrete delta functions to spread the entire singular force exerted by the immersed boundary to the nearby fluid grid points. Our method instead incorporates part of this force into jump conditions for the pressure, avoiding discrete dipole terms that adversely affect the accuracy near the immersed boundary. This has been implemented for the twodimensional incompressible Navier–Stokes equations using a highresolution finitevolume method for the advective terms and a projection method to enforce incompressibility. In the projection step, the correct jump in pressure is imposed in the course of solving the Poisson problem. This gives sharp resolution of the pressure across the interface and also gives better volume conservation than the traditional IB method. Comparisons between this method and the IB method are presented for several test problems. Numerical studies of the convergence and order of accuracy are included.
A fast solver for the Stokes equations with distributed forces in complex geometries
 J. Comput. Phys
"... We present a new method for the solution of the Stokes equations. The main features of our method are: (1) it can be applied to arbitrary geometries in a blackbox fashion; (2) it is second order accurate; and (3) it has optimal algorithmic complexity. Our approach, to which we refer as the Embedded ..."
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Cited by 41 (10 self)
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We present a new method for the solution of the Stokes equations. The main features of our method are: (1) it can be applied to arbitrary geometries in a blackbox fashion; (2) it is second order accurate; and (3) it has optimal algorithmic complexity. Our approach, to which we refer as the Embedded Boundary Integral method, is based on Anita Mayo’s work for the Poisson’s equation: “The Fast Solution of Poisson’s and the Biharmonic Equations on Irregular Regions”, SIAM Journal on Numerical Analysis, 21 (1984), pp. 285–299. We embed the domain in a rectangular domain, for which fast solvers are available, and we impose the boundary conditions as interface (jump) conditions on the velocities and tractions. We use an indirect boundary integral formulation for the homogeneous Stokes equations to compute the jumps. The resulting equations are discretized by Nyström’s method. The rectangular domain problem is discretized by finite elements for a velocitypressure formulation with equal order interpolation bilinear elements (£¥ ¤£¥ ¤). Stabilization is used to circumvent the ¦¨§�©������� � condition for the pressure space. For the integral equations, fast matrix vector multiplications are achieved via an ���¨���� � algorithm based on a block representation of the discrete integral operator, combined with (kernel independent) singular value decomposition to sparsify lowrank blocks. The regular grid solver is a Krylov method (Conjugate Residuals) combined with an optimal twolevel Schwartzpreconditioner. For the integral equation we use GMRES. We have tested our algorithm on several numerical examples and we have observed optimal convergence rates. Key Words: Stokes equations, fast solvers, integral equations, doublelayer potential, fast multipole methods, embedded domain methods, immersed interface methods, fictitious
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.
A Cartesian grid method for solving the twodimensional streamfunctionvorticity equations in irregular regions
 J. Comput. Phys
"... We describe a method for solving the twodimensional Navier–Stokes equations in irregular physical domains. Our method is based on an underlying uniform Cartesian grid and secondorder finitedifference/finitevolume discretizations of the streamfunctionvorticity equations. Geometry representing st ..."
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Cited by 32 (2 self)
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We describe a method for solving the twodimensional Navier–Stokes equations in irregular physical domains. Our method is based on an underlying uniform Cartesian grid and secondorder finitedifference/finitevolume discretizations of the streamfunctionvorticity equations. Geometry representing stationary solid obstacles in the flow domain is embedded in the Cartesian grid and special discretizations near the embedded boundary ensure the accuracy of the solution in the cut cells. Along the embedded boundary, we determine a distribution of vorticity sources needed to impose the noslip flow conditions. This distribution appears as a righthandside term in the discretized fluid equations, and so we can use fast solvers to solve the linear systems that arise. To handle the advective terms, we use the highresolution algorithms in CLAWPACK. We show that our Stokes solver is secondorder accurate for steady state solutions and that our full Navier–Stokes solver is between first and secondorder accurate and reproduces results from wellstudied benchmark problems in viscous fluid flow. Finally, we demonstrate the robustness of our code on flow in
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.
Numerical treatment of twodimensional interfaces for acoustic and elastic waves
 J. Comput. Phys
"... We present a numerical method to take into account 2D arbitraryshaped interfaces in classical finitedifference schemes, on a uniform Cartesian grid. This work extends the “Explicit Simplified Interface Method ” (ESIM), previously proposed in 1D (2001, J. Comput. Phys. 168, pp. 227248). The physi ..."
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Cited by 19 (10 self)
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We present a numerical method to take into account 2D arbitraryshaped interfaces in classical finitedifference schemes, on a uniform Cartesian grid. This work extends the “Explicit Simplified Interface Method ” (ESIM), previously proposed in 1D (2001, J. Comput. Phys. 168, pp. 227248). The physical problem under study concerns the linear hyperbolic systems of acoustics and elastodynamics, with stationary interfaces. Our method maintains, near the interfaces, properties of the schemes in homogeneous medium, such as the order of accuracy and the stability limit. Moreover, it enforces the numerical solution to satisfy the exact interface conditions. Lastly, it provides subcell geometrical features of the interface inside the meshing. The ESIM can be coupled automatically with a wide class of numerical schemes (LaxWendroff, fluxlimiter schemes,...) for a negligible additional computational cost. Throughout the paper, we focus on the challenging case of an interface between a fluid and an elastic solid. In numerical experiments, we provide comparisons between numerical solutions and original analytic solutions, showing the efficiency of the method.
A levelset method for interfacial flows with surfactant
, 2006
"... ... drop deformations and more complex drop–drop interactions compared to the analogous cases for clean drops. The effects of surfactant are found to be most significant in flows with multiple drops. To our knowledge, this is the first time that the levelset method has been used to simulate fluid i ..."
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Cited by 17 (1 self)
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... drop deformations and more complex drop–drop interactions compared to the analogous cases for clean drops. The effects of surfactant are found to be most significant in flows with multiple drops. To our knowledge, this is the first time that the levelset method has been used to simulate fluid interfaces with surfactant.
A new highorder immersed interface method for solving elliptic equations with imbedded interface of discontinuity
 J. Comput. Phys
, 2007
"... Abstract This paper presents a new highorder immersed interface method for elliptic equations with imbedded interface of discontinuity. Compared with the original secondorder immersed interface method of [R.J. LeVeque, Z. Li. The immersed interface method for elliptic equations with discontinuous ..."
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Cited by 13 (1 self)
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Abstract This paper presents a new highorder immersed interface method for elliptic equations with imbedded interface of discontinuity. Compared with the original secondorder immersed interface method of [R.J. LeVeque, Z. Li. The immersed interface method for elliptic equations with discontinuous coefficients and singular sources. SIAM J. Numer. Anal. 31 (1994) 100125], the new method achieves arbitrarily highorder accuracy for derivatives at an irregular grid point by imposing only two physical jump conditions together with a wider set of grid stencils. The new interface difference formulas are expressed in a general explicit form so that they can be applied to different multidimensional problems without any modification. The new interface algorithms of up to O(h 4 ) accuracy have been derived and tested on several one and twodimensional elliptic equations with imbedded interface. Compared to the standard secondorder immersed interface method, the test results show that the new fourthorder immersed interface method leads to a significant improvement in accuracy of the numerical solutions. The proposed method has potential advantages in the application to twophase flow because of its highorder accuracy and simplicity in applications.