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--- docs:classes:fieldsymmetry [2009/02/03 07:57]
gibson
+++ docs:classes:fieldsymmetry [2010/02/02 07:55] (current)
@@ Line 14: / Line 14: @@
 Let G be the group generated by these symmetries. Specific choice of boundary conditions at the walls
 might constrain the symmetry group of a specific flow to a subgroup G. But for generality,the FieldSymmetry
-class can represent any symmetry in G. Any element of G is of the form
+class can represent any symmetry in G. The FieldSymmetry class parameterizes symmetries σ ⊂ G symmetries
+as follows. Any element of the symmetry group is defined by six parameters
+<latex> $ \begin{align*}
+  \sigma &= (s_x, s_y, s_x, a_x, a_z, s)\\
+  s_x, s_y, s_z, s &= \pm 1\\
+  a_x, a_z &\in [-0.5, 0.5)
+\end{align*} $ </latex>
+with the action of σ on a velocity field u as
+<latex>
+  \sigma [u, v, w](x,y,z) = s (s_x u, s_y v, s_z w)(s_x x + a_x L_x, s_y y, s_z z + a_z L_z)
+</latex>
+The following is a brief overview of FieldSymmetry functionality. For a complete description,
+see the header file {{:librarycode:symmetry.h}}.
+===== Constructors / Initialization =====
+In C++ code, elements of the symmetry group can initialized as follows,
+where %%sx,sy,sz%% are of type %%int%% and %%ax, az%% are of type %%Real%%.
+<code c++>
+  FieldSymmetry sigma0(sx, sy, sz, ax, az, s);
+  FieldSymmetry sigma1(sx, sy, sz, ax, az);  // the s argument s defaults to 1
+  FieldSymmetry sigma2(sx, sy, sz);          // s defaults to 1; ax,az to 0
+  FieldSymmetry tau(ax, az);                 // pure translation: s,sx,sy,sz default to 1
+  FieldSymmetry identity;                    // the identity: s defaults to 1; ax,az to 0; sx,sy,sz to 1
+</code>
+===== Operations =====
+FieldSymmetries act on each other and velocity fields as follows
+<code c++>
+  FieldSymmetry sigma4 = sigma2 * sigma1;     // group multiplication
+  sigma4 *= sigma3;                           // sigma4 now equals sigma3*sigma2*sigma1 ---note order!
+  FlowField u("u");                           // read velocity field u from disk
+  FlowField v = sigma1 * u;                   // Make new field v = sigma1 u
+  v *= sigma2;                                // v now equals sigma2 * sigma1 * u
+</code>
+FieldSymmetries can be saved to / read from disk...
+<code c++>
+  sigma1.save("sigma1");                      // saves to ASCII file sigma1.asc
+  FieldSymmetry sigma7("sigma1");             // make a new symmetry element sigma7 == sigma1
+</code>
+...compared to each other...
+<code c++>
+  if (sigma1 != identity)
+    ...
+  else if (sigma1 == sigma7)
+    ...
+</code>
+===== ASCII IO =====
+==== FieldSymmetry ====
+The FieldSymmetry uses ASCII input-output. The storage format is
+  s sx sy sz ax az
+Thus, the following C++ channelflow code
+<code c++>
+FieldSymmetry sigma(-1, -1, -1, 0.3, 0.1);
+sigma.save("sigma");
+</code>
+produces the ASCII file ''sigma.asc'' with contents
+-1 -1 -1 0.3 0.1
+which can then be read back into a channeflow program with
+<code c++>
+FieldSymmetry sigma("sigma");
+</code>
+Note that  the order of parameters in the ASCII file is different than the order in the ''FieldSymmetry'' constructor: the overall multiplicative sign ''s'' goes first in the file and last in the C++ constructor. I apologize for this. The reasons for the difference are are historical. The next release of channelflow will have order (s, sx, sy, sz, ax, az) for both.
+==== SymmetryList ====
+The ''SymmetryList'' class is a essentially an array of ''FieldSymmetry'' objects with simple ASCII IO methods. The ASCII format is
+   % N
+   s0 sx0 sy0 sz0 ax0 az0
+   s1 sx1 sy1 sz1 ax1 az1
+   ...
+where N is the number of symmetries listed in the file. Thus the file ''S.asc'' with contents
+   % 2
+  1  1 -1  0.5  0.0
+-1 -1  1  0.5  0.5
+represents the symmetries σ0 = (1, 1, 1, -1, 0.5, 0.0) and σ1 = (1, -1, -1,  1,  0.5,  0.5). These are the generators of the S
+[[docs:math:symmetry#isotropy_groups_of_known_solutions|S symmetry group]]. The generators can be loaded into channelflow, used, and saved as follows
+<code c++>
+SymmetryList S("S");      // load generators from ASCII file
+FlowField foo = S[0](u);  // apply (1,  1,  1, -1, 0.5, 0.0) to u
+FlowField bar = S[1](u);  // apply (1, -1, -1,  1, 0.5, 0.5) to u
+S.save("Q");              // save generators into another file
+SymmetryList P(4);        // Create another symmetry group
+P[0] = FieldSymmetry(1,1,1, 0.2, 0.0);
+P[1] = etc.;
+</code>

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