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Subsection 4.3.4: Scheme Up Subsection 4.3.4: Scheme Subsubsection 4.3.4.2: Finite Element Integration Rule

#### 4.3.4.1 Gauss-Kronrod Trapezoidal Rule

The scheme type for the Gauss-Kronrod trapezoidal rule is “fibers-3d-gkt” for unconstrained and uncoupled continuous fiber distributions. This integration rule should only be used with 3D fiber density distributions. This scheme automatically limits the range of integration to fibers that are in tension [33]. A Gauss-Kronrod quadrature rule is employed for integration across latitudes of the unit sphere. A trapezoidal rule is used for integration across longitudes. The following material parameters need to be defined:
 Number of integration points across latitudes [ ] Number of integration points across longitudes [ ]
The parameter <nph> must be one of the values 7, 11, 15, 19, 23 and 27. The parameter <nth> may be any number greater than 0. Odd values for <nth> produce more accurate results than even values. A recommended combination is nph=7 and nth=31. The total number of integration points is nph ×nth. Increasing values of require increasing computational time.
Example:
<scheme type="fibers-3d-gkt">
<nph>7</nph>
<nth>31</nth>
</scheme>


Subsection 4.3.4: Scheme Up Subsection 4.3.4: Scheme Subsubsection 4.3.4.2: Finite Element Integration Rule