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<DIV ALIGN=center><IMG BORDER=0 ALIGN=MIDDLE ALT="" SRC="img82.gif"></DIV>
<P>
Let us consider the decomposition of <IMG BORDER=0 ALIGN=BOTTOM ALT="" SRC="img71.gif"> and <IMG BORDER=0 ALIGN=BOTTOM ALT="" SRC="img83.gif">, such that
<P><A NAME=eq9>&#160;</A><IMG BORDER=0 ALIGN=BOTTOM ALT="" SRC="img84.gif"><P>
where <IMG BORDER=0 ALIGN=MIDDLE ALT="" SRC="img85.gif"> is a <IMG BORDER=0 ALIGN=MIDDLE ALT="" SRC="img86.gif"> matrix.
<P>
With this decomposition, and by using the properties of <IMG BORDER=0 ALIGN=MIDDLE ALT="" SRC="img30.gif"> and <IMG BORDER=0 ALIGN=MIDDLE ALT="" SRC="img53.gif"> in subspace <IMG BORDER=0 ALIGN=MIDDLE ALT="" SRC="img21.gif">
into account, equation (<A HREF="node15.html#eq8">1.10</A>) is equivalent to:
<P>
<P><A NAME=eq10>&#160;</A><IMG BORDER=0 ALIGN=BOTTOM ALT="" SRC="img87.gif"><P>
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Thus, in order to calculate <IMG BORDER=0 ALIGN=MIDDLE ALT="" SRC="img30.gif">, it is necessary to solve (<A HREF="node15.html#eq10">1.12</A>), or equivalently:
<P>
<P><A NAME=eq11>&#160;</A><IMG BORDER=0 ALIGN=BOTTOM ALT="" SRC="img88.gif"><P>
<P>
where  <IMG BORDER=0 ALIGN=MIDDLE ALT="" SRC="img89.gif"> is a <IMG BORDER=0 ALIGN=MIDDLE ALT="" SRC="img90.gif">-dimensional array, 
whose components are the first <IMG BORDER=0 ALIGN=MIDDLE ALT="" SRC="img91.gif"> components of <IMG BORDER=0 ALIGN=MIDDLE ALT="" SRC="img30.gif">.
<P>
<P>
<P>
<b> Remarks:</b>  
<OL><LI> From a mathematical point of view, the relation (<A HREF="node15.html#eq10">1.12</A>) implies (<A HREF="node15.html#eq11">1.13</A>), therefore, to 
obtain <b>u</b>, it is possible to solve (<A HREF="node15.html#eq10">1.12</A>) or (<A HREF="node15.html#eq11">1.13</A>). However, from a practical (computer)
 point of view, it is preferable to solve (<A HREF="node15.html#eq10">1.12</A>) rather than  (<A HREF="node15.html#eq11">1.13</A>), as  (<A HREF="node15.html#eq11">1.13</A>) 
requires a second  
enumeration of the degrees of freedom, or otherwise the imposed boundary conditions will not be taken into account.
<LI> The direct computation of <IMG BORDER=0 ALIGN=MIDDLE ALT="" SRC="img92.gif"> (equation (<A HREF="node15.html#eq10">1.12</A>)), by assembling  the elements
<IMG BORDER=0 ALIGN=MIDDLE ALT="" SRC="img93.gif">,  necessitates a distinction between the elements <IMG BORDER=0 ALIGN=MIDDLE ALT="" SRC="img93.gif"> for 
<IMG BORDER=0 ALIGN=BOTTOM ALT="" SRC="img94.gif">  and <IMG BORDER=0 ALIGN=MIDDLE ALT="" SRC="img95.gif">, so that the condition <IMG BORDER=0 ALIGN=MIDDLE ALT="" SRC="img96.gif"> is taken into
account.  <BR>
<P>
It is therefore simpler to first calculate <IMG BORDER=0 ALIGN=MIDDLE ALT="" SRC="img97.gif"> and then compute <IMG BORDER=0 ALIGN=MIDDLE ALT="" SRC="img92.gif"> . This operation is 
called &quot;imposing the boundary conditions&quot;.
</OL>
<P>
<P>
<P>
The above analysis enables us to extract the different mathematical operators, so that the problem can be
decomposed into the following steps: 
<OL><LI> mesh of the domain,  <A NAME=item1>&#160;</A>
<LI> choice of finite elements,  <A NAME=item2>&#160;</A>
<LI> physical data and calculation of the element matrices and arrays,
<LI> solution of the linear problem, and
<LI> display of the results. 
</OL>
<P>
<b> Remark:</b> Step <A HREF="node15.html#item1">1</A>) and <A HREF="node15.html#item2">2</A>) result from the chosen variational formulation.
<P>
<P>
<P>
Each step can be decomposed into sub-steps. The decomposition of each step, as well as the choice of modules
which compute the mathematical operators, are briefly discussed in the remainder of this section.  <BR>
<P>
<P>
<P>
<H3><A NAME=SECTION04112400000000000000> MODULEF terminology</A></H3>
<P>
<P>
<P>
Let us first define the following terminology:
<DL COMPACT><DT>Variational unknown:
<DD> unknown of the continuous problem;
<DT>Degree of freedom (d.o.f.):
<DD> unknown of a discrete system (here a linear system);
<DT>Mnemonic:
<DD>  associated with a d.o.f. in order to define it clearly, for example: <b> VN</b> is the value at a node,  
<b> DX</b> is the derivative with respect to <b>x</b>, etc. The complete list is given in array <b> MAI8</b> of the 
data structure (D.S.) <b> MAIL</b> (consult [<A HREF="node65.html#guide_2"><A NAME=tex2html35 HREF="../Guide2/welcome.html">MODULEF User Guide - 2</A></A>] for a description of data structures);  
<DT>Node:
<DD> support for the degrees of freedom;
<DT>Point:
<DD> defines the geometry of the element;  
<DT>Vertex:
<DD> usual geometric sense;  
<DT>Sub-domain number:
<DD> number associated with each material, permitting the distinction of the 
sub-domains with different characteristics or different treatments of the same material;
<DT>Reference number:
<DD> similar notion to that of the sub-domains but this time applied to the boundaries
(surfaces in <IMG BORDER=0 ALIGN=BOTTOM ALT="" SRC="img98.gif">, lines and points in <IMG BORDER=0 ALIGN=BOTTOM ALT="" SRC="img12.gif"> or <IMG BORDER=0 ALIGN=BOTTOM ALT="" SRC="img98.gif">). These numbers allows for the treatment &quot;by block&quot; 
of the boundary conditions, the loads, or to define the curved lines (or surfaces).
<OL><LI> <b> Reference numbers:</b>  <BR>
<P>
In the figure <A HREF="node15.html#figint1">1.1</A>, the body is clamped at boundary <b>AF</b>, and pulled along boundary <b>CD</b>. It is 
composed of two materials, with a circular hole and a half-circle.
<P>
In this configuration there are two sub-domains:
<P>
<OL><LI> the copper domain <b>ABEF</b>, and
<LI> the iron domain <b>BCDE</b>,
</OL>
<P>
and at least four reference numbers:
<OL><LI> to define the circles which form the boundary of the hole and half-circle, and the line which partitions the domain
      into two sub-domains,
<LI> to define the fixed portion of the boundary, and
<LI> to define the edge <b>DC</b> where the load is applied.
</OL>
<P>
<b> Remark:</b> Lines <b>AC</b> and <b>FD</b> do not have reference numbers assigned as no particular conditions applies 
there. <BR>
<P>
<P><A NAME=1054>&#160;</A><IMG BORDER=0 ALIGN=BOTTOM ALT="" SRC="img99.gif">
<BR><STRONG>Figure 1.1:</STRONG> <i> Clamped body with applied load</i><A NAME=figint1>&#160;</A><BR>
<P>
<P>
<LI> <b> Nodes and points:</b> <BR>
<P>
Consider the two elements illustrated in figure <A HREF="node15.html#figint2">1.2</A>: a straight P2  triangle and  a curved P2 triangle.
A straight element is an element whose points are the vertices. A curved element is an element whose points are
the vertices plus some other characteristics. In the case of a P2 triangle, the straight  element
is defined by 3, and the curved element by 6 points, both having 6 nodes.
<P>
<P><A NAME=1063>&#160;</A><IMG BORDER=0 ALIGN=BOTTOM ALT="" SRC="img100.gif">
<BR><STRONG>Figure 1.2:</STRONG> <i> P2 triangular elements</i><A NAME=figint2>&#160;</A><BR>
<P></OL> 
 </DL>
<P>
<P>
<P>
<H2><A NAME=SECTION04113000000000000000>1.1.3 Relation between the mathematical and computer operators</A></H2>
  <A NAME=secint_op>&#160;</A>
<P>
<P>
<P>
So far we have defined various mathematical operators to solve a typical problem.
 With each of these mathematical  operators we will now associate a computer module, to solve the problem
numerically. The data structures (D.S.) provides the interfaces between the
modules. In this section we will mention the modules and data structures associated with each step.
<P>
<DL COMPACT><P>
<P>
<DT>Step 1:
<DD> <b> Mesh of the domain</b> <BR>
<P>
This step is the simplest with respect to the mathematical operator. The discretization must respect
the general rules of the finite element mesh. Let <IMG BORDER=0 ALIGN=MIDDLE ALT="" SRC="img101.gif"> and <IMG BORDER=0 ALIGN=MIDDLE ALT="" SRC="img102.gif"> be elements of <IMG BORDER=0 ALIGN=MIDDLE ALT="" SRC="img25.gif"> then:
<P>
<UL><LI>
<TABLE COLS=2>
<COL ALIGN=LEFT><COL ALIGN=LEFT>
<TR><TD VALIGN=BASELINE ALIGN=LEFT NOWRAP>
<IMG BORDER=0 ALIGN=BOTTOM ALT="" SRC="img103.gif">
</TD><TD VALIGN=BASELINE ALIGN=LEFT NOWRAP>
<P><IMG BORDER=0 ALIGN=BOTTOM ALT="" SRC="img104.gif"><P></TD></TR>
</TABLE>
<P>
<LI> The angles must not be too small.
<P>
</UL>
<P>
The corresponding implementation is not trivial (see for example [<A HREF="node65.html#mod_plg">George-1986</A>]) and, even if all the 
necessary operators are present
in the library, it proves labour costly. A well adapted mesh is an essential condition for good results.
<P>
The mesh generation technique adopted in MODULEF has a &quot;tree&quot; (top-down, bottom-up) structure, 
where the domain is partitioned into 
geometrically simple sub-domains. These  sub-domains are then manipulated by simple operators (symmetry, rotation,
translation, etc.) and then glued together to obtain the final mesh. This procedure is implemented by the module
<b> APNOPO</b> for a two-dimensional domain. The mesh generation technique, as well as the different modules
 available, are described in detail in  [<A HREF="node65.html#mod_104"><A NAME=tex2html37 HREF="../Guide3/welcome.html">MODULEF User Guide - 3</A></A>].
<P>
The resulting mesh is stored in the D.S. <b> NOPO</b>. This D.S. contains the mesh description: 
the elements, the nodes 
(with or without vertices), as well as the coordinates at the vertices, that have been generated.
<P>
At this stage  the nodes have been numbered. However, the interpolation has not been introduced, 
which will be discussed in the next step.
<P>
<b> Example:</b>
<P>
<IMG BORDER=0 ALIGN=BOTTOM ALT="" SRC="img105.gif">
  \
<IMG BORDER=0 ALIGN=BOTTOM ALT="" SRC="img106.gif">
<P>
<P>
<P>
A plot of the mesh is generated by module <b> TRNOPO</b> <BR>[4][<A HREF="node65.html#mod_96"><A NAME=tex2html41 HREF="../Guide6/welcome.html">MODULEF User Guide - 6</A></A>], which will be discussed in the 
next section.
<P>
<P>
<P>
<DT>Step 2:
<DD><b> Choice of finite element</b> <BR>
<P>
In this step the interpolation of functions are taken into account. This procedure is implemented by module
<b> COMACO</b> [<A HREF="node65.html#mod_13">13</A>]. Two cases may occur:
  <OL><LI> The chosen finite element exists in one of the following libraries: 
    <DL COMPACT><DT>THER
<DD> Thermal elements  [<A HREF="node65.html#mod_100">100</A>]  
     <DT>ELAS
<DD> Linear elastic elements [<A HREF="node65.html#mod_101">101</A>]
     <DT>ELNL
<DD> Nonlinear elements [<A HREF="node65.html#mod_97">97</A>]
<P>
 </DL> 
  <LI> The chosen element does not exist, in which case 
    <UL><LI> it can either be incorporated in one of the following libraries: <b> THER</b>, <b> ELAS</b>, <b> ELNL</b>, 
           <b> FLUI</b>, or <b> PERS </b> [<A HREF="node65.html#mod_95">95</A>] 
           (this operation is described in [<A HREF="node65.html#mod_95">95</A>]), after which we return to the first case, or
     <LI> it can be directly described in <b> COMACO</b> using data cards.
    </UL></OL>
<P>
The results of this step are found in two data structures: <b> MAIL</b> and <b> COOR</b>. The D.S. <b> MAIL</b> 
contains the description of the mesh (nodes, points, etc.) and of the interpolation (number and name of
the variational unknowns, mnemonics, etc.). The D.S. <b> COOR</b> contains the coordinates of the points
(vertices or not).
<P>
<b> Note:</b> The D.S. <b> COOR</b> does not contain the coordinates of the nodes on exit from <b> COMACO</b>. 
Consider again the examples of the elements &quot;straight P2 triangle&quot; and &quot;curved P2 triangle&quot;: In the first 
example the D.S. <b> COOR</b> contains the coordinates of the vertices. In the second example, the D.S. 
<b> COOR</b> contains the coordinates of the vertices and those of the mid-side points.
<P>
<P>
<P>
<DT>Step 3:
<DD><b> Entering of physical data, and creation of element matrices and arrays</b> <BR>
<P>
The purpose of this step is to construct the element matrices and arrays, <IMG BORDER=0 ALIGN=MIDDLE ALT="" SRC="img75.gif">, <IMG BORDER=0 ALIGN=MIDDLE ALT="" SRC="img76.gif">, and <IMG BORDER=0 ALIGN=MIDDLE ALT="" SRC="img107.gif">. 
This calculation 
requires the knowledge of the physical data which describes the materials, loads, temperatures, etc.
In the present example, this corresponds to <b> f</b> and <b> a</b> which appear on the right- and left-hand-side
of the equation. All physical data must be known at the time of activation of the module calculating the 
element matrices and arrays.
<P>
There are two possibilities:
  <OL><LI> For the general case (data not constant), the data is provided by a subroutine or array. The module
         <b> COFORC</b> [<A HREF="node65.html#mod_14">14</A>] constructs the D.S. <b> FORC</b>, which indicates how the necessary data 
         will be provided in order to calculate the element force vectors. Module <b> COMILI</b> [<A HREF="node65.html#mod_14">14</A>]
         constructs the 
         D.S. <b> MILI</b>, which indicates how the necessary information (physical properties) will be provided
         in order to be able to calculate the element matrices. The element matrices and arrays are 
         constructed by module <b> THENEW</b> [<A HREF="node65.html#mod_14">14</A>].
   <LI> When the material characteristics, loads or heat sources are constant by sub-domain or by boundary section
         characterized by reference number, module <b> THERCT</b> [<A HREF="node65.html#mod_14">14</A>] is used for the
         thermal case and <b> ELASCT</b> [<A HREF="node65.html#mod_14">14</A>] in elasticity, instead of <b> COFORC</b>, 
         <b> COMILI</b>, and <b> THENEW</b>.   
  </OL>
<P>
In all the cases, it is possible to provide these characteristics by sub-domain and/or reference number, which
minimizes the amount of data. The results of this step are found in the D.S. <b> TAE</b>, containing the 
element matrices and  arrays corresponding to all the elements in the mesh.
<P>
<b> Remarks:</b> 
  <OL><LI> This step changes according to the nature of the problem. The modules <b> THENEW</b>, <b> THERCT</b> and
<b> ELASCT</b> performs this step for the cases of thermal and linear elasticity problems. These modules can be 
adapted for the solution of other problems.
<P>
Other modules implement this step for some specific problems (ex: <BR>[2]
 <b> COTAE</b> in nonlinear elasticity with large 
deformations [<A HREF="node65.html#mod_97">97</A>]; <b> BIHAP1</b>, <BR>[4] <b> BIHAP2</b> for the biharmonic problem [<A HREF="node65.html#mod_93">93</A>], etc.).
  <LI> In order to treat a new problem, it is often sufficient to construct the D.S. <b> TAE</b> from the D.S.
<b> MAIL</b> and D.S. <b> COOR</b>, and from the material characteristics.
  </OL>
<P>
<P>
<P>
<DT>Step 4:
<DD><b> Solution of the linear problem </b><BR>
<P>
This step can be decomposed into several sub-steps. The sub-steps vary according to the desired method of solution, 
however the basic principles remain the same. The MODULEF library contains a large number of solution methods for