function  true_cubic_result_1d (i_p, Exact)
% Copyright 2000 J.E. Akin. All rights reserved.
% under construction: requires analytic exact solution
%  variable "ar" (analytic result) for "Exact" case number
%  ------------------------------------------------------
%  Matlab graph of i_p-th component value at mesh nodes
%  for a mesh of 4 node cubic line elements
%   If i_p = 0, show RMS value
%  ------------------------------------------------------

% c_x    = x coordinates of nod_per_el line element
% msh_typ_nodes   = connectivity list for elements, nt x nod_per_el
% loop   = corners for nod_per_el line element
% nod_per_el = Nodes per element
% np     = Number of Points
% nt     = Number of elements
% pre_e  = Element items before connectivity list
  pre_e  = 0 ;
% pre_p  = Nodal items before coordinates
  pre_p = 1;
% msh_bc_xyz = Nodal coordinates (with preceeding data)
% t_x    = x coordinates of nod_per_el corners

% UNDER_CONSTRUCTION
  if ( nargin == 0 )
    i_p = 0 ; Exact = 0 ;
  end % if no arguments
  if ( nargin == 1 )
    Exact = 0 ;
  end % if 1 argument
  %b fprintf ('Begin graph for exact case %g \n', Exact)

%    Read coordinate file and connectivity file
%    integer bc code, real xy pairs for np points (pre_p = 1)
  load msh_bc_xyz.tmp ;  

%    Set control data: number of points
  np = size (msh_bc_xyz,1) ;   % number of nodal points  
  fprintf ('Read %g mesh coordinate pairs \n', np)
  ns = size (msh_bc_xyz,2) - pre_p ;   % space dimension
  if ( ns ~= 1)
    error ('This is not a 1D mesh')
  end % if not 1D data
  
%    Set control data: number elements
  load  msh_typ_nodes.tmp ; % nod_per_el nodes per element
  nt = size (msh_typ_nodes,1) ;          % number of elements in mesh 
  nod_per_el = size (msh_typ_nodes,2) - pre_e -1 ; % nodes per elem
  fprintf ('Read %g elements connections \n', nt)
  if ( nod_per_el ~= 4 )
    error ('This is not a mesh of 4 node line elements')
  end % if 

  load node_results.tmp
  nr = size (node_results, 1);
  if ( nr == 0 )
    error ('Error missing file node_results.tmp')
  end % if error
  max_p = size (node_results, 2) ; % number of columns
  fprintf ('Read %g nodal solution values \n', nr)
  fprintf ('     with %g components each \n', max_p)
  if ( i_p > max_p ) 
    fprintf ('Data requested for component i_p = %g \n', i_p)
    error ('i_p > available data')
  end % if error

  H (4)       = 0. ;
  x (np)      = 0.       ;       % pre-allocate array x
  y (np)      = 0.       ;       % pre-allocate array y
  t_nodes (nod_per_el) = 0 ;    % Optional pre-allocation
  t_x (nod_per_el) = 0 ;        % Optional pre-allocation
  t_y (nod_per_el) = 0 ;        % Optional pre-allocation
  c_x (nod_per_el) = 0 ;        % Optional pre-allocation
  c_y (nod_per_el) = 0 ;        % Optional pre-allocation

%                set constants
  loop = [1:nod_per_el] ; % default to sequential order

  % msh_bc_xyz has: pre_p items then: x, y
  x = msh_bc_xyz (1:np, (pre_p+1)) ; % extract x column
   xmax = max (x)      ; xmin = min (x) ;

%  add analytic points
   %b a_inc = (xmax-xmin)/(10*nt) ;
   a_inc = (xmax-xmin)/101     ;
   ax = [xmin:a_inc:xmax] ;                     % analytic points
   [ar] = analytic_1_d_result (i_p, ax, Exact); % analytic results

  if ( i_p >= 1 ) % get FEA answers
    y = node_results(:, i_p) ;
  else % i_p = 0, get root mean sq
    for k = 1:np
      y (k) = sqrt ( sum (node_results (k, 1:max_p).^2)) ;
    end % for k
  end % if get RMS value

  maxa = max (ar)      ; mina = min (ar) ;

%    Cite max, min values
  [V_X, L_X] = max (y) ;
  [V_N, L_N] = min (y) ;
  fprintf ('Max value is %g at node %g \n', V_X, L_X)
  fprintf ('Min value is %g at node %g \n', V_N, L_N)
  null (1:np) = V_N ;

%    Initialize plots
   maxy = max (y)      ; miny = min (y) ;

% finalize axes
   ymax = max ([maxy, maxa])      ; ymin = min ([miny, mina]) ;
   if ( ymax == ymin )
     if ( abs (ymax) > 0 )
       ymax = ymax + abs (ymax)/20. ;
       ymin = ymin - abs (ymin)/20. ;
     end % if
   end % if

%b ymin=-0.7
   clf                               % clear graphics
   axis ([xmin, xmax, ymin, ymax])   % set axes

   hold on                           % hold image for plots
   xlabel (['X, Node at 45 deg (', int2str(nod_per_el), ...
     ' per element), Element at 90 deg'])
   if ( i_p >= 1 )
     title(['Exact (dash) & Cubic FEA Component\_', int2str(i_p),':   ', ...
            int2str(nt),' Elements, ', int2str(np),' Nodes (', ...
            int2str(nod_per_el), ' per Element)'])
     ylabel (['Component ', int2str(i_p), ' (max = ', ...
     num2str(V_X), ', min = ', num2str(V_N), ')'])
   else % i_p = 0, get root mean sq
     title(['Exact (dash) & Cubic FEA RMS\_value:  ', int2str(nt), ...
            ' Elements, ', int2str(np),' Nodes (', ...
            int2str(nod_per_el), ' per Element)'])
     ylabel (['Solution RMS Value (max = ', ...
     num2str(V_X), ', min = ', num2str(V_N), ')'])
   end % if get RMS value

%      add anal plot
     plot (ax, ar, 'r--')

%    Loop over all elements
  for it = 1:nt  ;

%      Extract element connectivity 
    t_nodes = msh_typ_nodes (it, (pre_e+2):(nod_per_el+pre_e+1)); 

%      Skip point elements, if any
    if ( all (t_nodes) ) % then valid line

%      Extract element coordinates & values
      t_x  = x (t_nodes) ;         % x at those nodes, only
      t_y  = y (t_nodes) ;         % y at those nodes, only
      plot (t_x, t_y, 'ko')        % plot nodal value symbols

%      Plot the element number
      x_bar = sum (t_x' )/nod_per_el ;
      t_text = sprintf ('   (%g)', it);  % offset # from pt
      text (x_bar, V_N, t_text, 'Rotation', 90) % incline
      plot (x_bar, V_N, 'k+')
      %b text (x_bar, 0.0, t_text, 'Rotation', 90) % incline
      %b plot (x_bar, 0.0, 'k+')

%      Plot this element (if non-sequential use loop)
    %  % c_x  = t_x (loop) ;       % x for nod_per_el line element
    %  % c_y  = t_y (loop) ;       % values at nodes
    %  % plot (c_x, c_y)           % plot nod_per_el lines
    %  plot (t_x, t_y)           % plot nod_per_el lines

%      Loop over local points on the cubic polynomial element
      n_poly = ceil (100 / nt)
      for k = 1: (n_poly + 1) % points in parametric space
%          get element parametric interpolation functions
         R     = (k - 1)/n_poly ;   % on 0 to 1
         X     = 2*R - 1 ;          % on -1 to 1
%        X = LOCAL COORDIATE OF POINT, -1 <= X <= +1
%        LOCAL NODE COORD. ARE    1----2---3----4
         H (1) = (1 - X)*(3*X + 1)*(3*X - 1)/16 ;  % shape functions
         H (2) = 9*(1 + X)*(X - 1)*(3*X - 1)/16 ;  % shape functions
         H (3) = 9*(1 + X)*(1 - X)*(3*X + 1)/16 ;  % shape functions
         H (4) = (1 + X)*(3*X + 1)*(3*X - 1)/16 ;  % shape functions
         x_el (k) = H * t_x ;  % true x value
         y_el (k) = H * t_y ;  % true y value
      end % for k
      plot (x_el, y_el)
    end % if zero in connectivity
  end % for over all elements

% plot node points on axis
  for i = 1:np
    t_text = sprintf ('   %g', i);  % offset # from pt
    text (x(i), null(i), t_text, 'Rotation', 45) % incline
  end % for all
  plot (x, null, 'k*')
  grid
 
% label max min points 
  % plot (x(L_X), y(L_X), 'kx')
  % plot (x(L_N), y(L_N), 'ko')
  v_text = sprintf ('---min') ;
  text  (x(L_N), V_N, v_text) ;
  v_text = sprintf ('---max') ;
  text  (x(L_X), V_X, [v_text]) ;

%       -depsc -tiff            % for an eps version
  %b print ('-dpsc', ['true_L4_result_', int2str(i_p), '_graph'])
  hold off
  %b v_text = ['Created true_L4_result_', int2str(i_p), '_graph.ps'] ;
  %b fprintf (1,'%s', v_text) ; fprintf (1, ' \n' )
% end of true_cubic_result_1d