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Theory Manual Version 3.6
 Subsection 4.1.3: Tetrahedral Elements Up Section 4.1: Solid Elements Section 4.2: Shell Elements 

4.1.4 Quadratic Tetrahedral Elements

FEBio implements a 10-node quadratic tetrahedral element. It has four corner nodes and six nodes located at the midpoint of the edges. The shape functions in terms area coordinates are given below. The area coordinates relate to the isoparametric coordinates as follows. The shape functions follow. The following integration rules are implemented for this element type.
4-point Gauss rule
r s t w
0.58541020 0.13819660 0.13819660 0.041666667
0.13819660 0.58541020 0.13819660 0.041666667
0.13819660 0.13819660 0.58541020 0.041666667
0.13819660 0.13819660 0.13819660 0.041666667
8-point Gauss rule
r s t w
0.01583591 0.328054697 0.328054697 0.023087995
0.328054697 0.01583591 0.328054697 0.023087995
0.328054697 0.328054697 0.01583591 0.023087995
0.328054697 0.328054697 0.328054697 0.023087995
0.679143178 0.106952274 0.106952274 0.018578672
0.106952274 0.679143178 0.106952274 0.018578672
0.106952274 0.106952274 0.679143178 0.018578672
0.106952274 0.106952274 0.106952274 0.018578672
11-point Gauss-Lobatto rule
r s t w
0 0 0 0.002777778
1 0 0 0.002777778
0 1 0 0.002777778
0 0 1 0.002777778
0.5 0 0 0.011111111
0.5 0.5 0 0.011111111
0 0.5 0 0.011111111
0 0 0.5 0.011111111
0.5 0 0.5 0.011111111
0 0.5 0.5 0.011111111
0.25 0.25 0.25 0.088888889
FEBio also implements a 15-node quadratic tetrahedral element.
figure ../Figures/FigQuadraticTetrahedralElements.png

Quadratic tetrahedral elements available in FEBio. Left, a 10-node quadratic tet. Right, a 15-node quadratic tet.
The following integration rules are implemented for this element type.
8-point Gauss rule [A]  [A] 
Note that weights sum up to one and not to the volume of the tet in the natural coordinate system (i.e. 1/6).
r s t w
0.0158359099 0.3280546970 0.3280546970 0.138527967
0.3280546970 0.0158359099 0.3280546970 0.138527967
0.3280546970 0.3280546970 0.0158359099 0.138527967
0.3280546970 0.3280546970 0.3280546970 0.138527967
0.6791431780 0.1069522740 0.1069522740 0.111472033
0.1069522740 0.6791431780 0.1069522740 0.111472033
0.1069522740 0.1069522740 0.6791431780 0.111472033
0.1069522740 0.1069522740 0.1069522740 0.111472033
11-point Gauss rule
r s t w
0.25 0.25 0.25 -0.01315555556
0.071428571428571 0.071428571428571 0.071428571428571 0.007622222222
0.785714285714286 0.071428571428571 0.071428571428571 0.007622222222
0.071428571428571 0.785714285714286 0.071428571428571 0.007622222222
0.071428571428571 0.071428571428571 0.785714285714286 0.007622222222
0.399403576166799 0.100596423833201 0.100596423833201 0.024888888889
0.100596423833201 0.399403576166799 0.100596423833201 0.024888888889
0.100596423833201 0.100596423833201 0.399403576166799 0.024888888889
0.399403576166799 0.399403576166799 0.100596423833201 0.024888888889
0.399403576166799 0.100596423833201 0.399403576166799 0.024888888889
0.100596423833201 0.399403576166799 0.399403576166799 0.024888888889
15-point Gauss rule
r s t w
0.25 0.25 0.25 0.030283678097089
0.333333333333333 0.333333333333333 0.333333333333333 0.006026785714286
0.000000000000000 0.333333333333333 0.333333333333333 0.006026785714286
0.333333333333333 0.000000000000000 0.333333333333333 0.006026785714286
0.333333333333333 0.333333333333333 0.000000000000000 0.006026785714286
0.090909090909091 0.090909090909091 0.090909090909091 0.011645249086029
0.727272727272727 0.090909090909091 0.090909090909091 0.011645249086029
0.090909090909091 0.727272727272727 0.090909090909091 0.011645249086029
0.090909090909091 0.090909090909091 0.727272727272727 0.011645249086029
0.433449846426336 0.066550153573664 0.066550153573664 0.010949141561386
0.066550153573664 0.433449846426336 0.066550153573664 0.010949141561386
0.066550153573664 0.066550153573664 0.433449846426336 0.010949141561386
0.066550153573664 0.433449846426336 0.433449846426336 0.010949141561386
0.433449846426336 0.066550153573664 0.433449846426336 0.010949141561386
0.433449846426336 0.433449846426336 0.066550153573664 0.010949141561386
 Subsection 4.1.3: Tetrahedral Elements Up Section 4.1: Solid Elements Section 4.2: Shell Elements