Finite elements
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.