Description

SF_LINE_P4 1D Fourth order Lagrange shape functions for lines (P4).

[ VBASE, NLDOF, XLDOF, SFUN ] = SF_LINE_P4( I_EVAL, N_SDIM, N_VERT, I_DOF, XI, AINVJAC, VBASE ) Evaluates conforming fourth order P4 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: 1-5            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 3];
 xLDof = [1 0 3/4 1/2 1/4;
          0 1 1/4 1/2 3/4];
 sfun  = 'sf_line_P4';


 switch i_eval    % Evaluation type flag.

   case 1   % Evaluation of function values.

     switch i_dof   % Basis function to evaluate.

       case 1
         vBase = (32*xi(1)^4)/3 - 16*xi(1)^3 + (22*xi(1)^2)/3 - xi(1);
       case 2
         vBase = (32*xi(1)^4)/3 - (80*xi(1)^3)/3 + (70*xi(1)^2)/3 - (25*xi(1))/3 + 1;
       case 3
         vBase = - (128*xi(1)^4)/3 + (224*xi(1)^3)/3 - (112*xi(1)^2)/3 + (16*xi(1))/3;
       case 4
         vBase = 64*xi(1)^4 - 128*xi(1)^3 + 76*xi(1)^2 - 12*xi(1);
       case 5
         vBase = - (128*xi(1)^4)/3 + 96*xi(1)^3 - (208*xi(1)^2)/3 + 16*xi(1);
     end

   case 2   % Evaluation of first derivative.

     switch i_dof   % Basis function derivative to evaluate.

       case 1
         dNdxi1 = ((8*xi(1) - 3)*(16*xi(1)^2 - 12*xi(1) + 1))/3;
       case 2
         dNdxi1 = ((8*xi(1) - 5)*(16*xi(1)^2 - 20*xi(1) + 5))/3;
       case 3
         dNdxi1 = - (512*xi(1)^3)/3 + 224*xi(1)^2 - (224*xi(1))/3 + 16/3;
       case 4
         dNdxi1 = 4*(2*xi(1) - 1)*(32*xi(1)^2 - 32*xi(1) + 3);
       case 5
         dNdxi1 = - (512*xi(1)^3)/3 + 288*xi(1)^2 - (416*xi(1))/3 + 16;
     end

     vBase = aInvJac(:,1) * dNdxi1;

   case 22   % Evaluation of second derivatives.

     switch i_dof   % Basis function derivative to evaluate.

       case 1
         dNdxi1 = 128*xi(1)^2 - 96*xi(1) + 44/3;
       case 2
         dNdxi1 = 128*xi(1)^2 - 160*xi(1) + 140/3;
       case 3
         dNdxi1 = - 512*xi(1)^2 + 448*xi(1) - 224/3;
       case 4
         dNdxi1 = 768*xi(1)^2 - 768*xi(1) + 152;
       case 5
         dNdxi1 = - 512*xi(1)^2 + 576*xi(1) - 416/3;
     end

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

   otherwise
     vBase = 0;

 end