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177
A Boundary Condition Capturing Method for Poisson's Equation on Irregular Domains
, 1999
"... Interfaces have a variety of boundary conditions (or jump conditions) that need to be enforced. In [3], the Ghost Fluid Method (GFM) was developed to capture the boundary conditions at a contact discontinuity in the inviscid Euler equations. This method was extended to treat more general discontinui ..."
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Cited by 105 (20 self)
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Interfaces have a variety of boundary conditions (or jump conditions) that need to be enforced. In [3], the Ghost Fluid Method (GFM) was developed to capture the boundary conditions at a contact discontinuity in the inviscid Euler equations. This method was extended to treat more general discontinuities such as shocks, detonations, and deflagrations in [2] and compressible viscous flows in [4]. In this paper, a similar boundary condition capturing approach is used to develop a new numerical method for the variable coefficient Poisson equation in the presence of interfaces where both the variable coefficients and the solution itself may be discontinuous. This new method is robust and easy to implement even in three spatial dimensions. Furthermore, the coefficient matrix of the associated linear system is the standard symmetric matrix for the variable coefficient Poisson equation in the absence of interfaces allowing for straightforward application of standard "black box" solvers.
Gerris: A TreeBased Adaptive Solver For The Incompressible Euler Equations In Complex Geometries
 J. Comp. Phys
, 2003
"... An adaptive mesh projection method for the timedependent incompressible Euler equations is presented. The domain is spatially discretised using quad/octrees and a multilevel Poisson solver is used to obtain the pressure. Complex solid boundaries are represented using a volumeoffluid approach. Sec ..."
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Cited by 100 (16 self)
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An adaptive mesh projection method for the timedependent incompressible Euler equations is presented. The domain is spatially discretised using quad/octrees and a multilevel Poisson solver is used to obtain the pressure. Complex solid boundaries are represented using a volumeoffluid approach. Secondorder convergence in space and time is demonstrated on regular, statically and dynamically refined grids. The quad/octree discretisation proves to be very flexible and allows accurate and efficient tracking of flow features. The source code of the method implementation is freely available.
A fast variational framework for accurate solidfluid coupling
 ACM Trans. Graph
, 2007
"... Figure 1: Left: A solid stirring smoke runs at interactive rates, two orders of magnitude faster than previously. Middle: Fully coupled rigid bodies of widely varying density, with flow visualized by marker particles. Right: Interactive manipulation of immersed rigid bodies. Physical simulation has ..."
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Cited by 75 (4 self)
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Figure 1: Left: A solid stirring smoke runs at interactive rates, two orders of magnitude faster than previously. Middle: Fully coupled rigid bodies of widely varying density, with flow visualized by marker particles. Right: Interactive manipulation of immersed rigid bodies. Physical simulation has emerged as a compelling animation technique, yet current approaches to coupling simulations of fluids and solids with irregular boundary geometry are inefficient or cannot handle some relevant scenarios robustly. We propose a new variational approach which allows robust and accurate solution on relatively coarse Cartesian grids, allowing possibly orders of magnitude faster simulation. By rephrasing the classical pressure projection step as a kinetic energy minimization, broadly similar to modern approaches to rigid body contact, we permit a robust coupling between fluid and arbitrary solid simulations that always gives a wellposed symmetric positive semidefinite linear system. We provide several examples of efficient fluidsolid interaction and rigid body coupling with subgrid cell flow. In addition, we extend the framework with a new boundary condition for freesurface flow, allowing fluid to separate naturally from solids.
A secondorderaccurate symmetric discretization of the Poisson equation on irregular domain
 J. Comput. Phys
"... In this paper, we consider the variable coefficient Poisson equation with Dirichlet boundary conditions on an irregular domain and show that one can obtain second order accuracy with a rather simple discretization. Moreover, since our discretization matrix is symmetric, it can be inverted rather qui ..."
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Cited by 75 (17 self)
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In this paper, we consider the variable coefficient Poisson equation with Dirichlet boundary conditions on an irregular domain and show that one can obtain second order accuracy with a rather simple discretization. Moreover, since our discretization matrix is symmetric, it can be inverted rather quickly as opposed to the more complicated nonsymmetric discretization matrices found in other second order accurate discretizations of this problem. Multidimensional computational results are presented to demonstrate the second order accuracy of this numerical method. In addition, we use our approach to formulate a second order accurate symmetric implicit time discretization of the heat equation on irregular domains. Then, we briefly consider Stefan problems.
A sharp interface Cartesian grid method for simulating flows with complex moving boundaries
 J. Comput. Phys
, 2001
"... A Cartesian grid method for computing flows with complex immersed, moving boundaries is presented. The flow is computed on a fixed Cartesian mesh and the solid boundaries are allowed to move freely through the mesh. A mixed Eulerian– Lagrangian framework is employed, which allows us to treat the imm ..."
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Cited by 66 (4 self)
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A Cartesian grid method for computing flows with complex immersed, moving boundaries is presented. The flow is computed on a fixed Cartesian mesh and the solid boundaries are allowed to move freely through the mesh. A mixed Eulerian– Lagrangian framework is employed, which allows us to treat the immersed moving boundary as a sharp interface. The incompressible Navier–Stokes equations are discretized using a secondorderaccurate finitevolume technique, and a secondorderaccurate fractionalstep scheme is employed for time advancement. The fractionalstep method and associated boundary conditions are formulated in a manner that properly accounts for the boundary motion. A unique problem with sharp interface methods is the temporal discretization of what are termed “freshly cleared ” cells, i.e., cells that are inside the solid at one time step and emerge into the fluid at the next time step. A simple and consistent remedy for this problem is also presented. The solution of the pressure Poisson equation is usually the most timeconsuming step in a fractional step scheme and this is even more so for moving boundary problems where the flow domain changes constantly. A multigrid method is presented and is shown to
A Fourth Order Accurate Discretization for the Laplace and Heat Equations on Arbitrary Domains, with Applications to the Stefan Problem
 J. Comput. Phys
, 2004
"... In this paper, we first describe a fourth order accurate finite di#erence discretization for both the Laplace equation and the heat equation with Dirichlet boundary conditions on irregular domains. In the case of the heat equation we use an implicit time discretization to avoid the stringent tim ..."
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Cited by 52 (5 self)
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In this paper, we first describe a fourth order accurate finite di#erence discretization for both the Laplace equation and the heat equation with Dirichlet boundary conditions on irregular domains. In the case of the heat equation we use an implicit time discretization to avoid the stringent time step restrictions associated with explicit schemes. We then turn our focus to the Stefan problem and construct a third order accurate method that also includes an implicit time discretization.
A cellcentered adaptive projection method for the incompressible NavierStokes equations
"... We present an algorithm to compute adaptive solutions for incompressible flows using blockstructured local refinement in both space and time. This method uses a projection formulation based on a cellcentered approximate projection, which allows the use of a single set of cellcentered solvers. Bec ..."
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Cited by 51 (14 self)
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We present an algorithm to compute adaptive solutions for incompressible flows using blockstructured local refinement in both space and time. This method uses a projection formulation based on a cellcentered approximate projection, which allows the use of a single set of cellcentered solvers. Because of refinement in time, additional steps are taken to accurately discretize the advection and projection operators at grid refinement boundaries using composite operators which span the coarse and refined grids. This ensures that the method is approximately freestream preserving and satisfies an appropriate form of the divergence constraint. c ○ 2000 Academic Press Key Words: adaptive mesh refinement; incompressible flow; projection methods. 1.
High order matched interface and boundary method for elliptic equations with discontinuous coefficients and singular sources
, 2006
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An adaptive, formally second order accurate version of the immersed boundary method
, 2006
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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