On the order of accuracy of the immersed boundary method: Higher order convergence rates for sufficiently smooth problems

doi:10.1016/J.JCP.2005.02.011

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On the order of accuracy of the immersed boundary method: Higher order convergence rates for sufficiently smooth problems

Journal Article•DOI•

On the order of accuracy of the immersed boundary method: Higher order convergence rates for sufficiently smooth problems

Boyce E. Griffith¹, Charles S. Peskin¹•Institutions (1)

Courant Institute of Mathematical Sciences¹

01 Sep 2005-Journal of Computational Physics (Academic Press Professional, Inc.)-Vol. 208, Iss: 1, pp 75-105

TL;DR: In this article, Lai et al. describe a new formally second order accurate immersed boundary method and demonstrate its performance for a prototypical fluid-structure interaction problem, involving an immersed viscoelastic shell of finite thickness, studied over a broad range of Reynolds numbers.

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About: This article is published in Journal of Computational Physics.The article was published on 2005-09-01. It has received 299 citations till now. The article focuses on the topics: Immersed boundary method & Order of accuracy.

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Journal Article•DOI•

Lattice-Boltzmann Method for Complex Flows

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Cyrus K. Aidun¹, Jonathan R. Clausen•Institutions (1)

Georgia Institute of Technology¹

01 Jan 2010-Annual Review of Fluid Mechanics

TL;DR: This work reviews many significant developments over the past decade of the lattice-Boltzmann method and discusses higherorder boundary conditions and the simulation of microchannel flow with finite Knudsen number.

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Abstract: With its roots in kinetic theory and the cellular automaton concept, the lattice-Boltzmann (LB) equation can be used to obtain continuum flow quantities from simple and local update rules based on particle interactions. The simplicity of formulation and its versatility explain the rapid expansion of the LB method to applications in complex and multiscale flows. We review many significant developments over the past decade with specific examples. Some of the most active developments include the entropic LB method and the application of the LB method to turbulent flow, multiphase flow, and deformable particle and fiber suspensions. Hybrid methods based on the combination of the Eulerian lattice with a Lagrangian grid system for the simulation of moving deformable boundaries show promise for more efficient applications to a broader class of problems. We also discuss higherorder boundary conditions and the simulation of microchannel flow with finite Knudsen number. Additionally, the remarkable scalability of the LB method for parallel processing is shown with examples. Teraflop simulations with the LB method are routine, and there is no doubt that this method will be one of the first candidates for petaflop computational fluid dynamics in the near future.

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1,585 citations

Journal Article•DOI•

Numerical Methods for Fluid-Structure Interaction — A Review

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Gene Hou, Jin Wang, Anita T. Layton

01 Aug 2012-Communications in Computational Physics

TL;DR: This article reviews representative numeri- cal methods based on conforming and non-conforming meshes that are currently avail- able for computing fluid-structure interaction problems, with an emphasis on some of the recent developments in the field.

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Abstract: The interactions between incompressible fluid flows and immersed struc- tures are nonlinear multi-physics phenomena that have applications to a wide range of scientific and engineering disciplines In this article, we review representative numeri- cal methods based onconforming and non-conforming meshes that arecurrentlyavail- able for computing fluid-structure 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 fluid-structure interactions

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555 citations

Journal Article•DOI•

An adaptive, formally second order accurate version of the immersed boundary method

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Boyce E. Griffith¹, Richard D. Hornung², David M. McQueen¹, Charles S. Peskin¹•Institutions (2)

Courant Institute of Mathematical Sciences¹, Lawrence Livermore National Laboratory²

01 Apr 2007-Journal of Computational Physics

TL;DR: The results obtaining by the adaptive method show good qualitative agreement with simulation results obtained by earlier non-adaptive versions of the method, but the flow in the vicinity of the model heart valves indicates that the new methodology provides enhanced boundary layer resolution.

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356 citations

Cites background or methods from "On the order of accuracy of the imm..."

...In the approach of Peskin and McQueen [4–9], the blood is modeled as a viscous incompressible fluid, whereas the muscular heart wall is modeled as a thick viscoelastic structure with time-dependent elastic parameters, and the flexible heart valve leaflets are modeled as thin elastic boundaries....
[...]
...Our algorithm, first introduced in [3], extends to the viscous case the hybrid approximate projection method (‘‘version 5’’) introduced by Almgren et al....
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...The present work extends the uniform grid method of [3] to provide an adaptive discretization of the Eulerian equations of motion (i....
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...The present adaptive algorithm employs the same hierarchical structured grid approach (but a different numerical scheme, see below) as that used by Roma, Peskin, and Berger to discretize the Eulerian equations of motion (i.e., the incompressible Navier–Stokes equations)....
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...where Nnþ 1 2 1⁄4 Nnþ12ðun; uMAC;n; f;Gpn 12Þ is the explicit approximation to 1⁄2ðu rÞu nþ 1 2 described in the appendix to [3] (note that in three spatial dimensions, we make use of full corner transport coupling [39]; see also [27] for a careful presentation of the particular three-dimensional treatment we employ), and...
[...]

Journal Article•DOI•

Immersed boundary methods for simulating fluid-structure interaction

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Fotis Sotiropoulos¹, Xiaolei Yang¹•Institutions (1)

University of Minnesota¹

01 Feb 2014-Progress in Aerospace Sciences

TL;DR: Different IB approaches for imposing boundary conditions, efficient iterative algorithms for solving the incompressible Navier–Stokes equations in the presence of dynamic immersed boundaries, and strong and loose coupling FSI strategies are summarized and juxtapose.

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336 citations

Journal Article•DOI•

Combined multi-direct forcing and immersed boundary method for simulating flows with moving particles

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Zeli Wang¹, Jianren Fan¹, Kun Luo¹•Institutions (1)

Zhejiang University¹

01 Mar 2008-International Journal of Multiphase Flow

TL;DR: In this paper, a combined multi-direct forcing and immersed boundary method was presented to simulate flows laden with finite-size moving particles with full-scale solutions, and the hydrodynamic interactions between moving rigid boundary and fluid were calculated using the multidirect forcing scheme.

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236 citations

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Journal Article•DOI•

Numerical Calculation of Time‐Dependent Viscous Incompressible Flow of Fluid with Free Surface

[...]

Francis H. Harlow, J. Eddie Welch

01 Jan 1965-Physics of Fluids

TL;DR: In this paper, a new technique is described for the numerical investigation of the time-dependent flow of an incompressible fluid, the boundary of which is partially confined and partially free The full Navier-Stokes equations are written in finite-difference form, and the solution is accomplished by finite-time step advancement.

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Abstract: A new technique is described for the numerical investigation of the time‐dependent flow of an incompressible fluid, the boundary of which is partially confined and partially free The full Navier‐Stokes equations are written in finite‐difference form, and the solution is accomplished by finite‐time‐step advancement The primary dependent variables are the pressure and the velocity components Also used is a set of marker particles which move with the fluid The technique is called the marker and cell method Some examples of the application of this method are presented All non‐linear effects are completely included, and the transient aspects can be computed for as much elapsed time as desired

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5,841 citations

"On the order of accuracy of the imm..." refers background in this paper

...By convention, a vector field defined on the Cartesian grid in terms of those vector components that are normal to the faces of the grid cells is called a MAC vector field [44]....
[...]

Journal Article•DOI•

Numerical solution of the Navier-Stokes equations

[...]

Alexandre J. Chorin

01 Jan 1968-Mathematics of Computation

TL;DR: In this paper, a finite-difference method for solving the time-dependent Navier-Stokes equations for an incompressible fluid is introduced, which is equally applicable to problems in two and three space dimensions.

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Abstract: A finite-difference method for solving the time-dependent Navier- Stokes equations for an incompressible fluid is introduced. This method uses the primitive variables, i.e. the velocities and the pressure, and is equally applicable to problems in two and three space dimensions. Test problems are solved, and an ap- plication to a three-dimensional convection problem is presented.

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4,991 citations

Journal Article•DOI•

The immersed boundary method

[...]

Charles S. Peskin¹•Institutions (1)

Courant Institute of Mathematical Sciences¹

01 Jan 2002-Acta Numerica

TL;DR: This paper is concerned with the mathematical structure of the immersed boundary (IB) method, which is intended for the computer simulation of fluid–structure interaction, especially in biological fluid dynamics.

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Abstract: This paper is concerned with the mathematical structure of the immersed boundary (IB) method, which is intended for the computer simulation of fluid–structure interaction, especially in biological fluid dynamics. The IB formulation of such problems, derived here from the principle of least action, involves both Eulerian and Lagrangian variables, linked by the Dirac delta function. Spatial discretization of the IB equations is based on a fixed Cartesian mesh for the Eulerian variables, and a moving curvilinear mesh for the Lagrangian variables. The two types of variables are linked by interaction equations that involve a smoothed approximation to the Dirac delta function. Eulerian/Lagrangian identities govern the transfer of data from one mesh to the other. Temporal discretization is by a second-order Runge–Kutta method. Current and future research directions are pointed out, and applications of the IB method are briefly discussed. Introduction The immersed boundary (IB) method was introduced to study flow patterns around heart valves and has evolved into a generally useful method for problems of fluid–structure interaction. The IB method is both a mathematical formulation and a numerical scheme. The mathematical formulation employs a mixture of Eulerian and Lagrangian variables. These are related by interaction equations in which the Dirac delta function plays a prominent role. In the numerical scheme motivated by the IB formulation, the Eulerian variables are defined on a fixed Cartesian mesh, and the Lagrangian variables are defined on a curvilinear mesh that moves freely through the fixed Cartesian mesh without being constrained to adapt to it in any way at all.

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4,164 citations

"On the order of accuracy of the imm..." refers background in this paper

...This smoothed delta function has a support of four meshwidths in each spatial dimension and satisfies four moment conditions, but it does not satisfy all of the axioms prescribed in [9]....
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...(Even in the more complicated case in which the mass density of the structure differs from that of the fluid, the momentum, velocity, and incompressibility of the coupled system can still be described by the incompressible Navier–Stokes equations; see [9]....
[...]
...One can show [5,9] that the corresponding curvilinear elastic force density functional can be put in the form...
[...]
...In [9], a collection of axioms (including moment conditions and a quadratic condition) are described that lead to the unique definition of a particular smoothed delta function with finite support....
[...]

PETSc Users Manual

[...]

Satish Balay, Shrirang Abhyankar, Mark F. Adams, Jed Brown, Peter R. Brune, Kristopher R. Buschelman, Lisandro Dalcin, Alp Dener, Eijkhout, William Gropp, Dmitry Karpeyev, Dinesh K. Kaushik, Matthew G. Knepley, Dave A. May, L. Curfman McInnes, Richard T. Mills, Todd Munson, Karl Rupp, Patrick Sanan, Barry Smith, Stefano Zampini, Hong Zhang - Show less +18 more

01 Jan 2019

TL;DR: The Portable, Extensible Toolkit for Scientific Computation (PETSc), is a suite of data structures and routines for the scalable (parallel) solution of scientific applications modeled by partial differential equations that supports MPI, and GPUs through CUDA or OpenCL, as well as hybrid MPI-GPU parallelism.

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Abstract: The Portable, Extensible Toolkit for Scientific Computation (PETSc), is a suite of data structures and routines for the scalable (parallel) solution of scientific applications modeled by partial differential equations. It supports MPI, and GPUs through CUDA or OpenCL, as well as hybrid MPI-GPU parallelism. PETSc (sometimes called PETSc/Tao) also contains the Tao optimization software library.

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2,570 citations

Book•

A multigrid tutorial

[...]

William L. Briggs

01 Jan 1987

TL;DR: This paper presents an implementation of Multilevel adaptive methods for Algebraic multigrid (AMG), a version of which has already been described in more detail in the preface.

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Abstract: Preface 1. Model problems 2. Basic iterative methods 3. Elements of multigrid 4. Implementation 5. Some theory 6. Nonlinear problems 7. Selected applications 8. Algebraic multigrid (AMG) 9. Multilevel adaptive methods 10. Finite elements Bibliography Index.

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2,505 citations