% Step 70
% ------------------------------------
+
+
+@InProceedings{Freund1995,
+ author = {Freund, J. and Stenberg, R.},
+ title = {On weakly imposed boundary conditions for
+ second order problems},
+ booktitle = {Proceedings of the Ninth International Conference on
+ Finite Elements in Fluids},
+ year = 1995,
+ pages = {327--336}}
+
+@article{Angot1999,
+ doi = {10.1007/s002110050401},
+ url = {https://doi.org/10.1007/s002110050401},
+ year = {1999},
+ month = feb,
+ publisher = {Springer Science and Business Media {LLC}},
+ volume = {81},
+ number = {4},
+ pages = {497--520},
+ author = {Philippe Angot and Charles-Henri Bruneau and Pierre Fabrie},
+ title = {A penalization method to take into account obstacles in incompressible viscous flows},
+ journal = {Numerische Mathematik}
+}
+
+
+@article{Glowinski1999,
+ doi = {10.1016/s0301-9322(98)00048-2},
+ url = {https://doi.org/10.1016/s0301-9322(98)00048-2},
+ year = {1999},
+ month = aug,
+ publisher = {Elsevier {BV}},
+ volume = {25},
+ number = {5},
+ pages = {755--794},
+ author = {R. Glowinski and T.-W. Pan and T.I. Hesla and D.D. Joseph},
+ title = {A distributed Lagrange multiplier/fictitious domain method for particulate flows},
+ journal = {International Journal of Multiphase Flow}
+}
+
+@article{Boffi2008,
+ doi = {10.1016/j.cma.2007.09.015},
+ url = {https://doi.org/10.1016/j.cma.2007.09.015},
+ year = {2008},
+ month = apr,
+ publisher = {Elsevier {BV}},
+ volume = {197},
+ number = {25-28},
+ pages = {2210--2231},
+ author = {Daniele Boffi and Lucia Gastaldi and Luca Heltai and Charles S. Peskin},
+ title = {On the hyper-elastic formulation of the immersed boundary method},
+ journal = {Computer Methods in Applied Mechanics and Engineering}
+}
+
+
+@article{Heltai2012,
+ doi = {10.1016/j.cma.2012.04.001},
+ url = {https://doi.org/10.1016/j.cma.2012.04.001},
+ year = {2012},
+ month = jul,
+ publisher = {Elsevier {BV}},
+ volume = {229-232},
+ pages = {110--127},
+ author = {Luca Heltai and Francesco Costanzo},
+ title = {Variational implementation of immersed finite element methods},
+ journal = {Computer Methods in Applied Mechanics and Engineering}
+}
+
@article{Riviere1999,
doi = {10.1023/a:1011591328604},
url = {https://doi.org/10.1023/a:1011591328604},
number = {3/4},
pages = {337--360},
author = {B{\'{e}}atrice Rivi{\`{e}}re and Mary F. Wheeler and Vivette Girault},
- journal = {Computational Geosciences}
+ journal = {Computational Geosciences},
+ title = {Improved energy estimates for interior penalty, constrained and discontinuous {G}alerkin methods for elliptic problems. {P}art {I}}.
}
% ------------------------------------
}
@article{GLHPW2018,
- title={Flexible and Scalable Particle-in-Cell Methods With Adaptive Mesh Refinement for Geodynamic Computations},
- author={Gassm{\"o}ller, Rene and Lokavarapu, Harsha and Heien, Eric and Puckett, Elbridge Gerry and Bangerth, Wolfgang},
- journal={Geochemistry, Geophysics, Geosystems},
- volume={19},
- number={9},
- pages={3596--3604},
- year={2018}
+ doi = {10.1029/2018gc007508},
+ url = {https://doi.org/10.1029/2018gc007508},
+ year = {2018},
+ month = sep,
+ publisher = {American Geophysical Union ({AGU})},
+ volume = {19},
+ number = {9},
+ pages = {3596--3604},
+ author = {Rene Gassm\"{o}ller and Harsha Lokavarapu and Eric Heien and Elbridge Gerry Puckett and Wolfgang Bangerth},
+ title = {Flexible and Scalable Particle-in-Cell Methods With Adaptive Mesh Refinement for Geodynamic Computations},
+ journal = {Geochemistry, Geophysics, Geosystems}
}
@TechReport{Saad1991,
on both the velocity and its gradient, it is an $\mathcal{H}^1$ penalization.
The case of the $\mathcal{L}^2$ penalization is very similar to a Darcy-type
approach. Both $\mathcal{L}^2$ and $\mathcal{H}^1$ penalizations have been
-analyzed extensively (see, for example, Angot 1999).
+analyzed extensively (see, for example, @cite Angot1999).
- The embedded domain $\Gamma$ has an intrinsic dimension `dim` which is smaller
than that of $\Omega$ (`spacedim`), thus its spacedim-dimensional measure is
physically impossible, but one may consider very thin sheets of metal
moving in a fluid as essentially lower-dimensional if the thickness of
the sheet is negligible. In this case, the boundary
-condition is imposed weakly on $\Gamma$ by applying the <a href="https://en.wikipedia.org/wiki/Joachim_Nitsche">Nitsche</a> method (see
-Freund, 1995).
+condition is imposed weakly on $\Gamma$ by applying the
+<a href="https://en.wikipedia.org/wiki/Joachim_Nitsche">Nitsche</a> method (see
+@cite Freund1995).
Both approaches have very similar requirements and result in highly
similar formulations. Thus, we treat them almost in the same way.
+ \beta (\textbf{v},\textbf{g})_{\Gamma}.
@f}
-The integrals over $\Gamma$ are lower-dimensional integrals. It can be shown (see Freund,
-1995) that there exists a positive constant
+The integrals over $\Gamma$ are lower-dimensional integrals. It can be shown (see
+@cite Freund1995) that there exists a positive constant
$C_1$ so that if $\beta > C_1$, the weak imposition of the boundary will
be consistent and stable. The first two additional integrals on $\Gamma$ (the
second line in the equation above) appear naturally after integrating by parts,
infinitesimally small "particles", which carry the required `JxW` information with the
movement of the solid. deal.II has the ability to distribute and
store such a set of particles in large-scale parallel computations in the form of
-the ParticleHandler class (for details on the implementation see Gassmöller et
-al., 2018), and we will make use of this functionality in this tutorial.
+the ParticleHandler class (for details on the implementation see @cite GLHPW2018),
+and we will make use of this functionality in this tutorial.
Thus, the approach taken in this step is as follows:
- Create a parallel::distributed::Triangulation for the domain $\Gamma$;
are advected by the fluid and which return to their original position, thus
demonstrating the time-reversibility of the flow.
-<h3>References</h3>
-<ul>
-<li> Freund, J., Stenberg, R. (1995). "On weakly imposed boundary conditions for
- second order problems". Proceedings of the Ninth International Conference on
- Finite Elements in Fluids. 327-336.
+<h3> More references</h3>
-<li> Angot, Philippe, Charles-Henri Bruneau and Pierre Fabrie. 1999. "A penalization
- method to take into account obstacles in incompressible viscous flows."
- Numerische Mathematik 81.4 : 497-520.
-
-<li> Glowinski, R., T.-W. Pan, T.I. Hesla, and D.D. Joseph. 1999. “A Distributed
- Lagrange Multiplier/fictitious Domain Method for Particulate Flows.”
- International Journal of Multiphase Flow 25 (5). Pergamon: 755–94.
-
-<li> Boffi, D., L. Gastaldi, L. Heltai, and C.S. Peskin. 2008. “On the
- Hyper-Elastic Formulation of the Immersed Boundary Method.” Computer Methods
- in Applied Mechanics and Engineering 197 (25–28).
-
-<li> Heltai, L., and F. Costanzo. 2012. “Variational Implementation of Immersed
- Finite Element Methods.” Computer Methods in Applied Mechanics and Engineering
- 229–232.
-
-<li> Gassmöller, R., H. Lokavarapu, E. Heien, E. G. Puckett, and
- W. Bangerth. 2018. "Flexible and Scalable Particle‐in‐Cell Methods
- With Adaptive Mesh Refinement for Geodynamic Computations."
- Geochemistry, Geophysics, Geosystems 19(9). 3596-3604.
-</ul>
+This tutorial program uses a number of techniques on imposing velocity
+conditions on non-matching interfaces in the interior of the fluid.
+For more background material, you may want to look up the following references:
+@cite Freund1995,
+@cite Angot1999,
+@cite Glowinski1999,
+@cite Boffi2008,
+@cite Heltai2012.