Element formulation

A C₁-continuous tetrahedral element requires a complete 9^th-order polynomial (binomial(2^dimension+1 + dimension over 2^dimension+1)), which has in 3D binomial(12 over 9)=220 monomials, see Ženı́šek, A. (1973). “Polynomial approximation on tetrahedrons in the finite element method”. In: Journal of Approximation Theory 7.4, pp. 334–351.
Taking the field value Φ and its gradient ∇Φ as 4 degrees of freedom (DOF), this requires 220/4=55 nodes. This is too much.
Therefore, we consider an incompatible mode element, requiring C₁-continuity only at specific nodes

a 3D 3^rd-order polynomial has 20 monomials
To be compatible with ordinary C3D4 meshes we interpolate 4 pseudo-DOF (pDOF) at 4 points: the function values at the face centers

Element formulation sketch

It is important that only adjacent nodes are involved when interpolating the pDOF (red arrows). This ensures common pDOF-values at adjacent elements
we have then 16 DOF + 4 pDOF = binomial(3+3 over 3), which means we can adopt a complete 3^rd-order polynomial
Numerical integration is done with rules from Jinyun, Y. (1984). “Symmetric gaussian quadrature formulae for tetrahedronal regions”. In: Computer Methods in Applied Mechanics and Engineering 43, pp. 349–353. and Cools, R. (2003). “An encyclopaedia of cubature formulas”. In: Journal of Complexity 19.3. Oberwolfach Special Issue, pp. 445–453. We use 29 integration points with positive weights, the integration is exact up to 6^th order polynomials

a 3D 5^th-order polynomial has 56 monomials
To be compatible with ordinary C3D4 meshes we interpolate 40 pseudo-DOF (pDOF) at 10 points: at the six corner centers and the 4 face centers we interpolate the function value (1) and its gradient (3)

Element formulation sketch

It is important that only adjacent nodes are involved when interpolating the pDOF (red arrows). This ensures common pDOF-values at adjacent elements
we have then 16 DOF + 10*4 pDOF = binomial(5+3 over 5), which means we can adopt a complete 5^th-order polynomial
Note that the higher order element is more sensitive to extreme aspect ratios
Numerical integration is done with rules from Jinyun, Y. (1984). “Symmetric gaussian quadrature formulae for tetrahedronal regions”. In: Computer Methods in Applied Mechanics and Engineering 43, pp. 349–353. and Cools, R. (2003). “An encyclopaedia of cubature formulas”. In: Journal of Complexity 19.3. Oberwolfach Special Issue, pp. 445–453. We use 29 integration points with positive weights, the integration is exact up to 6^th order polynomials