FEATool  v1.9
Finite Element Analysis Toolbox
sf_line_P5.m File Reference

Description

SF_LINE_P5 1D Fifth order Lagrange shape functions for lines (P5).

[ VBASE, NLDOF, XLDOF, SFUN ] = SF_LINE_P5( I_EVAL, N_SDIM, N_VERT, I_DOF, XI, AINVJAC, VBASE ) Evaluates conforming fifth order P5 Lagrange shape functions on 1D line elements with values defined in the nodes and center. XI are Barycentric coordinates.

Input       Value/[Size]           Description
-----------------------------------------------------------------------------------
i_eval      scalar:  1             Evaluate function values
                    >1             Evaluate values of derivatives
n_sdim      scalar: 1              Number of space dimensions
n_vert      scalar: 2              Number of vertices per cell
i_dof       scalar: 1-6            Local basis function to evaluate
xi          array [2,1]            Local coordinates of evaluation point
aInvJac     [n,3]                  Inverse of transformation Jacobian
vBase       [n]                    Preallocated output vector
                                                                                  .
Output      Value/[Size]           Description
-----------------------------------------------------------------------------------
vBase       [n]                    Evaluated function values
nLDof       [4]                    Number of local degrees of freedom on
                                   vertices, edges, faces, and cell interiors
xLDof       [2,n_ldof]             Local coordinates of local dofs
sfun        string                 Function name of called shape function
See also
sf_line_P1

Code listing

 nLDof = [2 0 0 4];
 xLDof = [1 0 4/5 3/5 2/5 1/5;
          0 1 1/5 2/5 3/5 4/5];
 sfun  = 'sf_line_P5';


 switch i_eval    % Evaluation type flag.

   case 1   % Evaluation of function values.

     switch i_dof   % Basis function to evaluate.

       case 1
         vBase = (625*xi(1)^5)/24 - (625*xi(1)^4)/12 + (875*xi(1)^3)/24 - (125*xi(1)^2)/12 + xi(1);
       case 2
         vBase = - (625*xi(1)^5)/24 + (625*xi(1)^4)/8 - (2125*xi(1)^3)/24 + (375*xi(1)^2)/8 - (137*xi(1))/12 + 1;
       case 3
         vBase = - (3125*xi(1)^5)/24 + (6875*xi(1)^4)/24 - (5125*xi(1)^3)/24 + (1525*xi(1)^2)/24 - (25*xi(1))/4;
       case 4
         vBase = (3125*xi(1)^5)/12 - 625*xi(1)^4 + (6125*xi(1)^3)/12 - (325*xi(1)^2)/2 + (50*xi(1))/3;
       case 5
         vBase = - (3125*xi(1)^5)/12 + (8125*xi(1)^4)/12 - (7375*xi(1)^3)/12 + (2675*xi(1)^2)/12 - 25*xi(1);
       case 6
         vBase = (3125*xi(1)^5)/24 - (4375*xi(1)^4)/12 + (8875*xi(1)^3)/24 - (1925*xi(1)^2)/12 + 25*xi(1);
     end

   case 2   % Evaluation of first derivative.

     switch i_dof   % Basis function derivative to evaluate.

       case 1
         dNdxi1 = (3125*xi(1)^4)/24 - (625*xi(1)^3)/3 + (875*xi(1)^2)/8 - (125*xi(1))/6 + 1;
       case 2
         dNdxi1 = - (3125*xi(1)^4)/24 + (625*xi(1)^3)/2 - (2125*xi(1)^2)/8 + (375*xi(1))/4 - 137/12;
       case 3
         dNdxi1 = - (15625*xi(1)^4)/24 + (6875*xi(1)^3)/6 - (5125*xi(1)^2)/8 + (1525*xi(1))/12 - 25/4;
       case 4
         dNdxi1 = (15625*xi(1)^4)/12 - 2500*xi(1)^3 + (6125*xi(1)^2)/4 - 325*xi(1) + 50/3;
       case 5
         dNdxi1 = - (15625*xi(1)^4)/12 + (8125*xi(1)^3)/3 - (7375*xi(1)^2)/4 + (2675*xi(1))/6 - 25;
       case 6
         dNdxi1 = (15625*xi(1)^4)/24 - (4375*xi(1)^3)/3 + (8875*xi(1)^2)/8 - (1925*xi(1))/6 + 25;
     end

     vBase = aInvJac(:,1) * dNdxi1;

   case 22   % Evaluation of second derivatives.

     switch i_dof   % Basis function derivative to evaluate.

       case 1
         dNdxi1 = (125*(5*xi(1) - 2)*(10*xi(1)^2 - 8*xi(1) + 1))/12;
       case 2
         dNdxi1 = -(125*(5*xi(1) - 3)*(10*xi(1)^2 - 12*xi(1) + 3))/12;
       case 3
         dNdxi1 = - (15625*xi(1)^3)/6 + (6875*xi(1)^2)/2 - (5125*xi(1))/4 + 1525/12;
       case 4
         dNdxi1 = (15625*xi(1)^3)/3 - 7500*xi(1)^2 + (6125*xi(1))/2 - 325;
       case 5
         dNdxi1 = - (15625*xi(1)^3)/3 + 8125*xi(1)^2 - (7375*xi(1))/2 + 2675/6;
       case 6
         dNdxi1 = (15625*xi(1)^3)/6 - 4375*xi(1)^2 + (8875*xi(1))/4 - 1925/6;
     end

     vBase = -aInvJac(:,1) ./ aInvJac(:,3) * dNdxi1;

   otherwise
     vBase = 0;

 end