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FEATool Multiphysics
v1.17.1
Finite Element Analysis Toolbox
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FEATool Multiphysics
v1.17.1
Finite Element Analysis Toolbox
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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
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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
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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
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