[★★★★★] Linear Analysis of A Single Element

In this example, we show a very basic analysis using isogeometric analysis and compare the result with the traditional finite element analysis.

The model can be downloaded here linear-analysis-of-a-single-element.zip.

Geometry

We analyze a single quadrilateral element with arbitrary lengths of edges. For traditional finite element analysis that uses isoparametric mapping, it is required to define four nodes that represent the vertexes of the quadrilateral. For example,

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1 2 3 4 5	`# FEA node 1 0 0 node 2 .5 0 node 3 .75 1 node 4 0 1`

For isogeometric analysis, the geometry is controlled by the control nodes, which form control polygon in higher dimensions. Since NURBS is used, for each control node, a weight shall be assigned. This means for an \(n\) dimensional problem, the control nodes shall have size of \(n+1\). In this 2D example, we use the same node command to define control nodes, the size shall be handled properly.

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1 2 3 4 5	`# IGA node 1 0 0 1 node 2 .5 0 1 node 3 .75 1 1 node 4 0 1 1`

Here we use unity for all four nodes. For the geometry (linear) considered, NURBS falls back to B-Spline, which is sufficient.

Element

In FEA, each element is defined by the nodes ordered anticlockwise. So to define a CP4 element with unit thickness and material model with tag \(1\), the following command can be used.

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1 2	`# FEA element CP4 1 1 2 3 4 1 1`

In IGA, a different strategy is used. Each patch that consists of a number of elements is considered as a giant element, hence the number of control nodes is not known in advance. Meanwhile, the knot vectors may have various lengths. In suanPan, a key-value style definition is used. Please refer to the corresponding page for more details of PatchQuad element. In this example, the following command is used. Please note the order of control nodes is different from that in FEA.

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1	`element PatchQuad 1 -node 1 2 4 3 -material 1 -thickness 1 -knotx 0 0 1 1 -knoty 0 0 1 1`

Other Parts

The rest of the model is pretty simple, as long as BCs and loads are defined, the model is ready to be analyzed.

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1 2 3 4 5 6	`fix 1 1 1 fix 2 2 1 2 cload 1 0 1 2 3 4 step static 1 analyze exit`