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/** \mainpage Introduction
\section whatis What is FCLIB ?
FCLIB is
<ul>
<li> A open source collection of Frictional Contact (FC) problems stored in a specific <a href="http://www.hdfgroup.org/HDF5/"> HDF5 format </a></li>
<li> A open source light implementation of Input/Output functions in C Language to read and write problems </li>
</ul>
<br>
\section goals Goals of the project (2011-2017)
The goal of this work is to set up a collection of 2D and 3D Frictional Contact (FC) problems in order to
<ul>
<li> set up a list of benchmarks </li>
<li> provide a standard framework for testing available and new algorithms for solving discrete frictional contact problems </li>
<li> share common formulations of problems in order to exchange data</li>
</ul>
<br>
\section References
<p>
<a href="https://hal.inria.fr/hal-00945820v2/document" >
FCLIB: a collection of discrete 3D Frictional Contact problems.</a>
<br> Vincent Acary, Maurice Brémond, Tomasz Koziara, Franck Pérignon.
[Technical Report] RT-0444, INRIA. 2014, pp.34.<br></br>
<br></br>
<a href="https://hal.inria.fr/inria-00423530" > Numerical Methods for Nonsmooth Dynamical Systems: Applications in Mechanics and Electronics </a><br></br>
Vincent Acary, Bernard Brogliato<br></br>
Springer Verlag, 35, pp.526, 2008, Lecture Notes in Applied and Computational Mechanics, 978-3-540-75391-9 <br></br>
<br/>
</p>
\section howtodownload How to download ?
How to download sources files of the API?
<ul>
<li> <a href="https://github.com/FrictionalContactLibrary/fclib">latest version on github</a></li>
<li> <a href="https://github.com/FrictionalContactLibrary/fclib/releases"> releases on github </a></li>
</ul>
<br>
<p></p>
How to download the collection of problems ?
<ul>
<li> The problems are stored on a github repo with the help of git-lfs <a href="https://github.com/FrictionalContactLibrary/fclib-library"> fclib-library </a> </li>
<li> The release of the library of problems are tagged and can be found <a href="https://github.com/FrictionalContactLibrary/fclib-library/releases"> there </a> </li>
<br/>
<br>
<p></p>
\section Wahtis What is a Frictional contact problem ?
A Frictional contact problem is algebraic problem obtained after possible time and space discretizations of problems of mechanics of solid involving contact and Coulomb's friction. The mathematical structure of the problem is a second-order cone complementarity problem. For more details, you could have a look to the <a href="doc/FCLib.pdf"> fclib specifications </a>
\subsection Localfclib The local Frictional Contact problem with equality constraints
Given
<ul>
<li> a positive semi--definite matrix \f${W} \in {\mathrm{I\!R}}^{m \times m}\f$</li>
<li> a matrix \f${V} \in {\mathrm{I\!R}}^{m \times p}\f$</li>
<li> a matrix \f${R} \in {\mathrm{I\!R}}^{p \times p}\f$</li>
<li> a vector \f$q \in {\mathrm{I\!R}}^{m}\f$,</li>
<li> a vector \f$s \in {\mathrm{I\!R}}^{p}\f$,</li>
<li> a vector of coefficients of friction \f$\mu \in {\mathrm{I\!R}}^{n_c}\f$
</ul>
the Mixed 3DFC problem is to find three vectors \f$u\in{\mathrm{I\!R}}^m\f$, \f$r\in {\mathrm{I\!R}}^m\f$ and \f$\lambda \in {\mathrm{I\!R}}^p\f$ denoted by \f$\mathrm{M3DFC}(R,V,W,q,s,\mu)\f$ such that
\f{eqnarray*}\label{eq:lcp1}
\begin{cases}
V^T {r} + R \lambda + s = 0 \\ \\
\hat u = W {r} + V\lambda + q +\left[
\left[\begin{array}{c}
\mu^\alpha \|u^\alpha_T\|\\
0 \\
0
\end{array}\right]^T, \alpha = 1 \ldots n_c
\right]^T \\ \\
K^\star_{\mu} \ni {\hat u} \perp r \in K_{\mu}
\end{cases}
\f}
where the Coulomb friction cone for a contact \f$\alpha\f$ is defined by
\f{eqnarray*}{
\label{eq:CCC}
K
_{\mu^\alpha}^{\alpha} = \{r^\alpha, \|r^\alpha_T \| \leq \mu^\alpha |r^\alpha_N| \}
\f}
and the set \f$K^{\alpha,\star}_{\mu^\alpha}\f$ is its dual.
\subsection globalfclib The Global Frictional Contact problem with equality constraints
We are also dealing with global FC problem defined by
Given
<ul>
<li> a positive definite matrix \f${M} \in {\mathrm{I\!R}}^{n \times n}\f$</li>
<li> a vector \f$ {f} \in {\mathrm{I\!R}}^n\f$,</li>
<li> a matrix \f${H} \in {\mathrm{I\!R}}^{n \times m}\f$</li>
<li> a matrix \f${G} \in {\mathrm{I\!R}}^{n \times p}\f$</li>
<li> a vector \f$w \in {\mathrm{I\!R}}^{m}\f$,</li>
<li> a vector \f$b \in {\mathrm{I\!R}}^{p}\f$,</li>
<li> a vector of coefficients of friction \f$\mu \in {\mathrm{I\!R}}^{n_c}\f$
</ul>
the Global Mixed 3DFC problem is to find four vectors \f$ {v} \in {\mathrm{I\!R}}^n\f$, \f$u\in{\mathrm{I\!R}}^m\f$, \f$r\in {\mathrm{I\!R}}^m\f$ and \f$\lambda \in {\mathrm{I\!R}}^p\f$ denoted by \f$\mathrm{GM3DFC}(M,H,G,w,b,\mu)\f$ such that
\f{eqnarray*}{
\begin{cases}
M v = {H} {r} + G\lambda + {f} \\ \\
G^T v +b =0 \\ \\
\hat u = H^T v + w +\left[
\left[\begin{array}{c}
\mu \|u^\alpha_T\|\\
0 \\
0
\end{array}\right]^T, \alpha = 1 \ldots n_c
\right]^T \\ \\
K^\star_{\mu} \ni {\hat u} \perp r \in K_{\mu}
\end{cases}
\f}
\subsection without Problems without equality constraints
If the original problems do not contain inequality constraints, or if they are reduced, the problems do no have the variables \f$\lambda\f$ as unknowns and can be simplified. However, the storage in HDF5 file remains the same.
\subsection merit Merit functions.
The API provides also some Merit functions which measures if one set of vectors satisfies the previous problems.
*/
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