FEATool Multiphysics  v1.16.4 Finite Element Analysis Toolbox
sf_line_P3.m File Reference

## Description

SF_LINE_P3 1D Third order Lagrange shape functions for lines (P3).

[ VBASE, NLDOF, XLDOF, SFUN ] = SF_LINE_P3( I_EVAL, N_SDIM, N_VERT, I_DOF, XI, AINVJAC, VBASE ) Evaluates conforming third order P3 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: 1-4            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

sflag3, sf_line_H3

# Code listing

 nLDof = [2 0 0 2];
xLDof = [1 0 2/3 1/3;
0 1 1/3 2/3];
sfun  = 'sf_line_P3';

switch i_eval    % Evaluation type flag.

case 1   % Evaluation of function values.
xi = xi(1);

switch i_dof   % Basis function to evaluate.

case 1
vBase = xi*(2 - 3*xi)*(1 - 3*xi)/2;
case 2
vBase = (1 - xi)*(2 - 3*xi)*(1 - 3*xi)/2;
case 3
vBase = 9*xi*(1-xi)*(3*xi - 1)/2;
case 4
vBase = 9*xi*(1-xi)*(2 - 3*xi)/2;
end

case 2   % Evaluation of first derivative.
xi = xi(1);

switch i_dof   % Basis function derivative to evaluate.
case 1
dNdxi = (27*xi^2)/2 - 9*xi + 1;
case 2
dNdxi = 18*xi - (27*xi^2)/2 - 11/2;
case 3
dNdxi = 36*xi - (81*xi^2)/2 - 9/2;
case 4
dNdxi = (81*xi^2)/2 - 45*xi + 9;
end

vBase = aInvJac(:,1)*dNdxi;

case 22   % Evaluation of second derivatives.
xi = xi(1);

switch i_dof   % Basis function derivative to evaluate.
case 1
dNdxi = 27*xi - 9;
case 2
dNdxi = 18 - 27*xi;
case 3
dNdxi = 36 - 81*xi;
case 4
dNdxi = 81*xi - 45;
end

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

otherwise
vBase = 0;

end