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FEATool Multiphysics
v1.17.2
Finite Element Analysis Toolbox
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FEATool Multiphysics
v1.17.2
Finite Element Analysis Toolbox
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SF_SIMP_P2 Quadratic Lagrange shape function for simplices (P2).
[ VBASE, NLDOF, XLDOF, SFUN ] = SF_SIMP_P2( I_EVAL, N_SDIM, N_VERT, I_DOF, XI, AINVJAC, VBASE ) Evaluates conforming quadratic P2 Lagrange shape functions on simplices with values defined in the nodes. XI 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: 1-3 Number of space dimensions
n_vert scalar: 2-4 Number of vertices per cell
i_dof scalar: 1-n_ldof Local basis function to evaluate
xi [n_sdim+1] Local coordinates of evaluation point
aInvJac [n,n_sdim+1*n_sdim] 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 [n_sdim,n_ldof] Local coordinates of local dofs
sfun string Function name of called shape function
switch n_sdim
case 1
nLDof = [ 2 0 0 1 ];
xLDof = [ 1 0 0.5; ...
0 1 0.5 ];
case 2
nLDof = [ 3 3 0 0 ];
xLDof = [ 1 0 0 0.5 0 0.5; ...
0 1 0 0.5 0.5 0 ; ...
0 0 1 0 0.5 0.5 ];
case 3
nLDof = [ 4 6 0 0 ];
xLDof = [ 1 0 0 0 0.5 0 0.5 0.5 0 0 ; ...
0 1 0 0 0.5 0.5 0 0 0.5 0 ; ...
0 0 1 0 0 0.5 0.5 0 0 0.5; ...
0 0 0 1 0 0 0 0.5 0.5 0.5 ];
end
sfun = 'sf_simp_P2';
% Evaluation type flag.
if( i_eval==1 ) % Evaluation of function values.
if( i_dof<=n_vert ) % Node dofs.
vBase = xi(i_dof)*(2*xi(i_dof)-1);
else % Edge dofs.
i = mod(i_dof-n_vert-1,min(n_vert,3))+1;
j = mod(i,min(n_vert,3))+1;
if ( i_dof>7 )
j = 4;
end
vBase = 4*xi(i)*xi(j);
end
elseif( i_eval>=2 && i_eval<=n_sdim+1 ) % Evaluation of first derivatives.
if( i_dof<=n_vert ) % Node dofs.
i_col = i_dof + (i_eval-2)*n_vert;
dNdxi = 4*xi(i_dof)-1;
vBase = aInvJac(:,i_col)*dNdxi;
else % Edge dofs.
i = mod(i_dof-n_vert-1,min(n_vert,3))+1;
j = mod(i,min(n_vert,3))+1;
if ( i_dof>7 )
j = 4;
end
i_col = i + (i_eval-2)*n_vert;
j_col = j + (i_eval-2)*n_vert;
dNdxii = 4*xi(j);
dNdxij = 4*xi(i);
vBase = aInvJac(:,i_col)*dNdxii + aInvJac(:,j_col)*dNdxij;
end
elseif( any(i_eval==[22 23 24 32 33 34 42 43 44]) ) % Evaluation of second derivatives.
i_eval1 = floor(i_eval/10);
i_eval2 = i_eval - 10*i_eval1;
if( i_dof<=n_vert ) % Node dofs.
d2Ndxi2 = 4;
i_eval = floor(i_eval/10);
i_col = i_dof + (i_eval1-2)*n_vert;
j_col = i_dof + (i_eval2-2)*n_vert;
vBase = aInvJac(:,i_col).*aInvJac(:,j_col)*d2Ndxi2;
else % Edge dofs.
d2Ndxii2 = 0;
d2Ndxiidxij = 4;
d2Ndxijdxii = d2Ndxiidxij;
d2Ndxij2 = 0;
i = mod(i_dof-n_vert-1,min(n_vert,3))+1;
j = mod(i,min(n_vert,3))+1;
if( i_dof>7 )
j = 4;
end
i_col = i + (i_eval1-2)*n_vert;
j_col = j + (i_eval1-2)*n_vert;
k_col = i + (i_eval2-2)*n_vert;
l_col = j + (i_eval2-2)*n_vert;
vBase = aInvJac(:,k_col).*( aInvJac(:,i_col)*d2Ndxii2 + aInvJac(:,j_col)*d2Ndxiidxij ) + ...
aInvJac(:,l_col).*( aInvJac(:,i_col)*d2Ndxijdxii + aInvJac(:,j_col)*d2Ndxij2 );
end
else
vBase = 0;
end