Integration Points
For several tasks, especially for finite elements and contact, different integration rules are used, which are summarized here. The interval of all integration rules is \(\in [-1,1]\), thus giving a total sum for integration weights of 2. The points \(\xi_{ip}\) and weights \(w_{ip}\) for Gauss rules read:
The following table collects some typical input parameters for nodes, objects and markers:
type/order
|
point 0
|
point 1
|
point 2
|
point 3
|
|---|---|---|---|---|
Gauss 1
|
0
|
|||
Gauss 3
|
\(-\sqrt{1 / 3}\)
|
\(\sqrt{1 / 3}\)
|
||
Gauss 5
|
\(-\sqrt{3 / 5}\)
|
0
|
\(\sqrt{3 / 5}\)
|
|
Gauss 7
|
\(-\sqrt{3 / 7 + \sqrt{120} / 35}\)
|
\(-\sqrt{3 / 7 - \sqrt{120} / 35}\)
|
\(\sqrt{3 / 7 - \sqrt{120} / 35}\)
|
\(\sqrt{3 / 7 + \sqrt{120} / 35}\)
|
type/order
|
weight 0
|
weight 1
|
weight 2
|
weight 3
|
Gauss 1
|
2
|
|||
Gauss 3
|
1
|
1
|
||
Gauss 5
|
\(5 / 9\)
|
\(8 / 9\)
|
\(5 / 9\)
|
|
Gauss 7
|
\(1 / 2 - 5 / (3 \sqrt{120})\)
|
\(1 / 2 + 5 / (3*\sqrt{120})\)
|
\(1 / 2 + 5 / (3*\sqrt{120})\)
|
\(1 / 2 - 5 / (3*\sqrt{120})\)
|
The points \(\xi_{ip}\) and weights \(w_{ip}\) for Lobatto rules read:
type/order
|
point 0
|
point 1
|
point 2
|
point 3
|
|---|---|---|---|---|
Lobatto 1
|
-1
|
1
|
||
Lobatto 3
|
-1
|
0
|
1
|
|
Lobatto 5
|
-1
|
\(-\sqrt{1/5}\)
|
\(\sqrt{1/5}\)
|
1
|
type/order
|
weight 0
|
weight 1
|
weight 2
|
weight 3
|
Lobatto 1
|
1
|
1
|
||
Lobatto 3
|
\(1/3\)
|
\(4/3\)
|
\(1/3\)
|
|
Lobatto 5
|
\(1/6\)
|
\(5/6\)
|
\(5/6\)
|
\(1/6\)
|
Further integration rules can be found in the C++ code of Exudyn, see file BasicLinalg.h.