Immersed boundary method: Difference between revisions

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{{More footnotes|date=April 2017}}

In [[computational fluid dynamics]], the '''immersed boundary method''' originally referred to an approach developed by [[Charles S. Peskin|Charles Peskin]] in 1972 to simulate fluid-structure (fiber) interactions.<ref>{{Cite journal|last=Peskin|first=Charles S|date=1972-10-01|title=Flow patterns around heart valves: A numerical method|journal=Journal of Computational Physics|volume=10|issue=2|pages=252–271|doi=10.1016/0021-9991(72)90065-4|bibcode=1972JCoPh..10..252P |issn=0021-9991}}</ref> Treating the coupling of the structure deformations and the fluid flow poses a number of challenging problems for [[Computer simulation|numerical simulations]] (the elastic boundary changes the flow of the fluid and the fluid moves the elastic boundary simultaneously). In the immersed boundary method the fluid is represented onin an [[Lagrangian and Eulerian coordinates|Eulerian coordinate]] system and the structure is represented on ain [[Lagrangian and Eulerian coordinates|Lagrangian coordinatecoordinates]]. For [[Newtonian fluids]] governed by the incompressible [[Navier–Stokes equations]], the fluid equations are

:<math>

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</math>

and inif casethe offlow is incompressible fluids (assuming constant density), we have the further condition that

:<math>

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</math>

The immersed structures are typically represented as a collection of one-dimensional fibers, denoted by <math> \Gamma </math>. Each fiber can be viewed as a parametric curve <math> X(s,t) </math> where <math> s </math> is the parameterLagrangian coordinate along the fiber and <math>t </math> is time. PhysicsThe physics of the fiber is represented via thea fiber force distribution function <math> F(s,t) </math>. Spring forces, bending resistance or any other type of behavior can be built into this term. The force exerted by the structure on the fluid is then interpolated as a source term in the momentum equation using

:<math>

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Variants of this basic approach have been applied to simulate a wide variety of mechanical systems involving elastic structures which interact with fluid flows.

Since the original development of this method by Peskin, a variety of approaches have been developed. These include stochastic formulations for microscopic systems, viscoelastic soft materials, complex fluids, such as the Stochastic Immersed Boundary Methods of Atzberger, Kramer, and Peskin,<ref>

Since the original development of this method by Peskin, a variety of approaches have been developed to simulate flow over complicated immersed bodies on grids that do not conform to the surface of the body. These include methods such as the immersed interface method, the Cartesian grid method, the ghost fluid method and the cut-cell method. Mittal and Iaccarino<ref>{{harvnb|Mittal|Iaccarino|2005}}.</ref> refer to all these (and other related) methods as Immersed Boundary Methods and provide various categorizations of these methods. From the point of view of implementation, they categorize immersed boundary methods into ''continuous forcing'' and ''discrete forcing'' methods. In the former, a force term is added to the continuous Navier-Stokes equations before discretization, whereas in the latter, the forcing is applied (explicitly or implicitly) to the discretized equations. Under this taxonomy, Peskin's original method is a ''continuous forcing'' method whereas Cartesian grid, cut-cell and the ghost-fluid methods are ''discrete forcing'' methods.

{{Cite journal

| last = Atzberger

| first = Paul J.

| title = Stochastic Eulerian Lagrangian Methods for Fluid Structure Interactions with Thermal Fluctuations

| journal = Journal of Computational Physics

| volume = 230

| issue = 8

| pages = 2821–2837

| year = 2011

| doi = 10.1016/j.jcp.2010.12.028

| arxiv = 1009.5648

| bibcode = 2011JCoPh.230.2821A

| s2cid = 6067032

}}

</ref><ref>

{{Citation

| last1=Atzberger

| first1=Paul

| title=Incorporating Shear into Stochastic Eulerian Lagrangian Methods for Rheological Studies of Complex Fluids and Soft Materials

| journal=Physica D

| volume=265

| pages=57–70

| year=2013

| doi=10.1016/j.physd.2013.09.002

| arxiv = 2212.10651

}}

</ref> methods for simulating flows over complicated immersed solid bodies on grids that do not conform to the surface of the body Mittal and Iaccarino,<ref>{{harvnb|Mittal|Iaccarino|2005}}.</ref> and other approaches that incorporate mass and rotational degrees of freedom Olson, Lim, Cortez.<ref>{{Cite journal |last1=Olson |first1=S.

|last2=Lim |first2=S. |last3=Cortez |first3=R.

|title=Modeling the dynamics of an elastic rod with intrinsic curvature and twist using a regularized Stokes formulation |journal=Journal of Computational Physics |date=2013 |volume=238 |pages=169–187

|doi=10.1016/j.jcp.2012.12.026}}</ref> Methods for complicated body shapes include the immersed interface method, the Cartesian grid method, the ghost fluid method and the cut-cell methods categorizing immersed boundary methods into ''continuous forcing'' and ''discrete forcing'' methods. Methods have been developed for simulations of viscoelastic fluids, curved fluid interfaces, microscopic biophysical systems (proteins in lipid bilayer membranes, swimmers), and engineered devices, such as the Stochastic Immersed Boundary Methods of Atzberger, Kramer, and Peskin,<ref>

{{Cite journal

| last = Atzberger

| first = Paul J.

| title = Stochastic Eulerian Lagrangian Methods for Fluid Structure Interactions with Thermal Fluctuations

| journal = Journal of Computational Physics

| volume = 230

| issue = 8

| pages = 2821–2837

| year = 2011

| doi = 10.1016/j.jcp.2010.12.028

| arxiv = 1009.5648

| bibcode = 2011JCoPh.230.2821A

| s2cid = 6067032

}}

</ref><ref>{{cite journal |last1=Rower |first1=David A. |last2=Padidar |first2=Misha |last3=Atzberger |first3=Paul J. |title=Surface fluctuating hydrodynamics methods for the drift-diffusion dynamics of particles and microstructures within curved fluid interfaces |journal=Journal of Computational Physics |date=April 2022 |volume=455 |pages=110994 |doi=10.1016/j.jcp.2022.110994 |arxiv=1906.01146}}</ref>

Stochastic Eulerian Lagrangian Methods of Atzberger,<ref>{{Cite journal

| last = Atzberger

| first = Paul J.

| title = Stochastic Eulerian Lagrangian Methods for Fluid Structure Interactions with Thermal Fluctuations

| journal = Journal of Computational Physics

| volume = 230

| issue = 8

| pages = 2821–2837

| year = 2011

| doi = 10.1016/j.jcp.2010.12.028

| arxiv = 1009.5648

| bibcode = 2011JCoPh.230.2821A

| s2cid = 6067032

}}

</ref><ref>

{{Citation

| last1=Atzberger

| first1=Paul

| title=Incorporating Shear into Stochastic Eulerian Lagrangian Methods for Rheological Studies of Complex Fluids and Soft Materials

| journal=Physica D

| volume=265

| pages=57–70

| year=2013

| doi=10.1016/j.physd.2013.09.002

| arxiv = 2212.10651

}}

</ref><ref>{{cite journal |last1=Atzberger |first1=Paul |title=Hydrodynamic Coupling of Particle Inclusions Embedded in Curved Lipid Bilayer Membranes |journal=Soft Matter |date=2016 |volume=12 |issue=32 |pages=6685–6707 |doi=10.1039/C6SM00194G |pmid=27373277 | arxiv=1601.06461}}

</ref> Massed Immersed Boundary Methods of Mori,<ref>

{{Cite journal

| last1 = Mori

| first1 = Yoichiro

| last2 = Peskin

| first2 = Charles S.

| title = Implicit Second-Order Immersed Boundary Methods with Boundary Mass

| journal = Computer Methods in Applied Mechanics and Engineering

| volume = 197

| issue = 25–28

| pages = 2049–2067

| year = 2008

| doi = 10.1016/j.cma.2007.05.028

| bibcode = 2008CMAME.197.2049M

}}

</ref> and Rotational Immersed Boundary Methods of Olson, Lim, Cortez.<ref>{{Cite journal |last1=Olson |first1=S.

|last2=Lim |first2=S. |last3=Cortez |first3=R.

|title=Modeling the dynamics of an elastic rod with intrinsic curvature and twist using a regularized Stokes formulation |journal=Journal of Computational Physics |date=2013 |volume=238 |pages=169–187

|doi=10.1016/j.jcp.2012.12.026}}</ref>

In general, for immersed boundary methods and related variants, there is an active research community that is still developing new techniques and related software implementations and incorporating related techniques into simulation packages and CAD engineering software. For more details see below.

== See also ==

*[[Stochastic Eulerian Lagrangian method]]s

*[[Stokesian dynamics]]

*[[Volume of fluid method]]

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* [https://www.mentor.com/products/mechanical/floefd/ FloEFD: Commercial CFD IBM code ]

* [[Advanced Simulation Library]]

* [http://mango-selm.org/ Mango-Selm : Immersed Boundary Methods and SELM Simulations, 3D Package, (Python interface, LAMMPS MD Integration), P. Atzberger, UCSB]

* [http://wwwsoftware.mathatzberger.ucsb.eduorg/~atzberg/SIB_Codes/index.html Stochastic Immersed Boundary Methods in 3D, P. Atzberger, UCSB]

* [http://www.math.utah.edu/IBIS/ Immersed Boundary Method for Uniform Meshes in 2D, A. Fogelson, Utah]

* [https://github.com/IBAMR/IBAMR IBAMR : Immersed Boundary Method for Adaptive Meshes in 3D, B. Griffith, NYU.]

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* [https://openfoamwiki.net/index.php/Extend-bazaar/Toolkits/ImmersedBoundary CFD IBM code based on OpenFoam]

* [https://github.com/ChenguangZhang/sdfibm sdfibm: Another CFD IBM code based on OpenFoam]

* [https://www.simscale.com/docs/analysis-types/immersed-boundary-analysis/ SimScale: Immersed Boundary Method for fluid mechanics and conjugate heat transfer simulation in the cloud]

==Notes==

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| s2cid = 17977915

}}

*{{Citation

| last1=Atzberger

| first1=Paul

| title=Incorporating Shear into Stochastic Eulerian Lagrangian Methods for Rheological Studies of Complex Fluids and Soft Materials

| journal=Physica D

| volume=265

| pages=57–70

| year=2013

| doi=10.1016/j.physd.2013.09.002

| arxiv = 2212.10651

}}

*{{Citation

| last1 = Jindal

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| first4 = K.

| chapter = The Immersed Boundary CFD Approach for Complex Aerodynamics Flow Predictions

| series = SAE Technical Paper

| issue = 2007–01–0109

| year = 2007

| doi = 10.4271/2007-01-0109

}}.

* {{Cite journal |last1=Atzberger |first1=Paul |title=Hydrodynamic Coupling of Particle Inclusions Embedded in Curved Lipid Bilayer Membranes |journal=Soft Matter |date=2016 |volume=12 |issue=32 |pages=6685–6707 |doi=10.1039/C6SM00194G |pmid=27373277 | arxiv=1601.06461}}.

*{{Cite journal

| last1 = Kim

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| bibcode = 2001JCoPh.171..132K

}}

*{{Cite journal |last1=Rower |first1=David A. |last2=Padidar |first2=Misha |last3=Atzberger |first3=Paul J. |title=Surface fluctuating hydrodynamics methods for the drift-diffusion dynamics of particles and microstructures within curved fluid interfaces |journal=Journal of Computational Physics |date=April 2022 |volume=455 |pages=110994 |doi=10.1016/j.jcp.2022.110994 |arxiv=1906.01146}}

*{{Cite journal

| last1 = Mittal

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}}

*{{Cite journal

| last1 = MoriaMori

| first1 = Yoichiro

| last2 = Peskin