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# 数学代写|有限元方法代写finite differences method代考|ENGR7961 Serendipity type elements

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## 数学代写|有限元代写Finite Element Method代考|Serendipity type elements

The method used in constructing the Lagrange type of elements is very systematic. However, the Lagrange type of elements is not very widely used, due to the presence of the interior nodes. Serendipity type elements are created by inspective construction methods. We intentionally construct high order elements without interior nodes.

Consider the eight-node element shown in Figure 7.17a. The element has four corner nodes and four mid-side nodes. The shape functions in the natural coordinates for the quadratic rectangular element are given as
$$\begin{array}{ll} N_j=\frac{1}{4}\left(1+\xi_j \xi\right)\left(1+\eta_j \eta\right)\left(\xi_j \xi+\eta_j \eta-1\right) \quad \text { for corner nodes } j=1,2,3,4 \ N_j=\frac{1}{2}\left(1-\xi^2\right)\left(1+\eta_j \eta\right) & \text { for mid-side nodes } j=5,7 \ N_j=\frac{1}{2}\left(1+\xi_j \xi\right)\left(1-\eta^2\right) & \text { for mid-side nodes } j=6,8 \end{array}$$
where $\left(\xi_j, \eta_j\right)$ are the natural coordinates of node $j$. It is very easy to observe that the shape functions possess the delta function property. The shape function is constructed by simple inspections making use of the shape function properties. For example, for the corner node 1 (where $\xi_1=-1, \eta_1=-1$ ), the shape function $N_1$ has to pass the following three lines as shown in Figure 7.18 to ensure its vanishing at remote nodes:
\begin{aligned} 1-\xi=0 \Rightarrow \text { vanishes at nodes } 2,6,3 \ 1-\eta=0 \Rightarrow \text { vanishes at nodes } 3,4,7 \ -\xi-\eta-1=0 \Rightarrow \text { vanishes at nodes } 5,8 \end{aligned}

## 数学代写|有限元代写Finite Element Method代考|ELEMENTS WITH CURVED EDGES

Using high order elements, elements with curved edges can be used in the modelling. Two relatively frequently used higher order elements of curved edges are shown in Figure 7.20(a). In formulating these types of elements, the same mapping technique used for the linear quadrilateral elements (Section 7.4) can be used. In the physical coordinate system, elements with curved edges, as shown in Figure 7.20 (a), are first formed in the problem domain. These elements are then mapped into the natural coordinate system using Eq. (7.67). The elements mapped in the natural coordinate system will have straight edges, as shown in Figure $7.20(\mathrm{~b})$

Higher order elements of curved edges are often used for modelling curved boundaries. Note that elements with excessively curved edges may cause problems in the numerical integration. Therefore, more elements should be used where the curvature of the boundary is large. In addition, it is recommended that in the internal portion of the domain, elements with straight edges should be used whenever possible. More details on modelling issues will be discussed intensively in Chapter 11 .

## 数学代写|有限元代写Finite Element Method代考|Serendipity type elements

$N_j=\frac{1}{4}\left(1+\xi_j \xi\right)\left(1+\eta_j \eta\right)\left(\xi_j \xi+\eta_j \eta-1\right) \quad$ for corner nodes $j=1,2,3,4 N_j=\frac{1}{2}\left(1-\xi^2\right)\left(1+\eta_j \eta\right) \quad$ for mid-side nodes

$1-\xi=0 \Rightarrow$ vanishes at nodes $2,6,31-\eta=0 \Rightarrow$ vanishes at nodes $3,4,7-\xi-\eta-1=0 \Rightarrow$ vanishes at nodes 5,8

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