FEATool  v1.9Finite Element Analysis Toolbox
sf_line_H3.m File Reference

Description

SF_LINE_H3 Third order 1D C1 Hermite shape functions for lines.

[ VBASE, NLDOF, XLDOF, SFUN ] = SF_LINE_H3( I_EVAL, N_SDIM, N_VERT, I_DOF, XI, AINVJAC, VBASE ) Evaluates C1 Hermite shape functions on 1D line elements with value and first derivatives defined in the nodes. 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       [2,4]                  Number of local degrees of freedom on
vertices, edges, faces, cell interiors,
and vertices without boundary conditions
xLDof       [2,n_ldof]             Local coordinates of local dofs
sfun        string                 Function name of called shape function
```
sf_line_P3

Code listing

``` nLDof = [2 0 0 0;
2 0 0 0];
xLDof = [1 0 1 0;
0 1 0 1];
sfun  = 'sf_line_H3';

switch i_eval    % Evaluation type flag.

case 1   % Evaluation of function values.

switch i_dof   % Basis function to evaluate.

case 1
vBase = 3*xi(1)^2 - 2*xi(1)^3;
case 2
vBase = 3*xi(2)^2 - 2*xi(2)^3;
case 3
vBase = ( xi(1)^2 - xi(1)^3 ) .* aInvJac(:,3);
case 4
vBase = ( xi(2)^3 - xi(2)^2 ) .* aInvJac(:,3);
end

case 2   % Evaluation of first derivatives.

switch i_dof   % Basis function derivative to evaluate.
case 1
dNdxi1 = -6*xi(1)*(xi(1) - 1);
dNdxi2 =  0;
case 2
dNdxi1 =  0;
dNdxi2 = -6*xi(2)*(xi(2) - 1);
case 3
dNdxi1 = -xi(1)*(3*xi(1) - 2) .* aInvJac(:,3);
dNdxi2 =  0;
case 4
dNdxi1 =  0;
dNdxi2 =  xi(2)*(3*xi(2) - 2) .* aInvJac(:,3);
end

vBase = aInvJac(:,1) .* dNdxi1 + aInvJac(:,2) .* dNdxi2;

case 22   % Evaluation of second derivatives.

switch i_dof   % Basis function derivative to evaluate.
case 1
dNdxi1 =  6*(2*xi(1)-1) ./ aInvJac(:,3);
dNdxi2 =  0;
case 2
dNdxi1 =  0;
dNdxi2 = -6*(2*xi(2)-1) ./ aInvJac(:,3);
case 3
dNdxi1 =  6*xi(1) - 2;
dNdxi2 =  0;
case 4
dNdxi1 =  0;
dNdxi2 =  6*xi(2) - 2;
end

vBase = aInvJac(:,1) .* dNdxi1 + aInvJac(:,2) .* dNdxi2;

otherwise
vBase = 0;

end```