Ring plane headers in docs (#497)

examples/ringPlaneStrain/ringPlaneStrain.m

@@ -18,9 +18,7 @@
 global Re
 global Ri
 Re = 200 ; Ri = 100 ;
-
 %md## Linear analysis
-
 %md### Analytic solution
 %md
 %md The solution displacement field is extracted from chapter 4 of  (Timoshenko and Goodier, Theory of Elasticity, 3rd edition). The Navier's equation, imposing no temperature variation, no volumetric forces, and considering a radial dispalcement field leads to:
@@ -216,18 +214,18 @@
 %md
 %md### Verification
 %mdThe numerical and analytic solutions are compared for the external surface (since all the elements on the same surface have the same analytic solution):
-
+%
 global Y
-
+%
 Y = 2*sigmaY0 / sqrt(3) ;
 % p0 = Y/2 * (1-a^2/b^2)  ; % Yielding pressure
 p0 = Y/2 * (1-Ri^2/Re^2)  ; % Yielding pressure
-
+%
 pressure_vals = loadFactorsMat(:,3)*p ;
-
+%
 cvals = zeros(length(pressure_vals),1) ;
 ubAna = zeros(length(pressure_vals),1) ;
-
+%
 % Plastic front value
 for i = 1:length(cvals)
 	p = pressure_vals(i) ;
@@ -240,7 +238,7 @@
 	end
 	cvals(i) = val ;
 end
-
+%
 % Analytic radial displacement at outer surface
 for i = 1:length(cvals)
 	p = pressure_vals(i) ;
@@ -253,32 +251,29 @@
 		ubAna(i) = Y*c^2/(E*Re) * (1-nu^2) ;
 	end	
 end
-
-% Plot parameters
+%md
+%md### Plots
+% plot parameters
 lw = 2.0 ; ms = 11 ; plotFontSize = 10 ;
 fig = figure;
 hold on, grid on
-
+% node to plot the solution
 node = 5 ;
 dofX = node * 6 - 5 ;
 ubNum = matUs(dofX, :) ; 
-
+%
 plot(ubNum, pressure_vals, 'b-o', 'linewidth', lw,'markersize', ms)
 plot(ubAna, pressure_vals, 'g-x', 'linewidth', lw,'markersize', ms)
-
+%
 legend ({'FEM', 'Analytic',}, 'location', 'east');
 labx = xlabel('u_b'); laby = ylabel('p') ;
 tit = title('p-u_b');
 set(labx, 'fontsize', plotFontSize*.8);
 set(laby, 'fontsize', plotFontSize*.8);
 set(tit, 'fontsize', plotFontSize);
-
-% Check solution
-% analyticCheckTolerance = 1e-2 ;
-% verifBoolean = ( ( ubNum(end) - ubAna(end) ) < analyticCheckTolerance ) ;
-
-% verifboolean de ambos analisis
-%md The numerical solution is verified: 
+%md
+%md### Verificaation
+%md The numerical solution is verified for both cases: 
 analyticCheckTolerance = 1e-2 ;
 verifBoolean = ( ( numericalRi - analyticValRi ) < analyticCheckTolerance ) && ...
                ( ( numericalRe - analyticValRe ) < analyticCheckTolerance ) && ...