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1

Complementing Wolfgang answer, if you enumerate your nodes as $(0, 0), (1, 0), (0, 1), (\alpha_1, 0), (1 - \alpha_3, \alpha_3), (0, \alpha_2)$ you get the following basis functions: \begin{align} &\operatorname{N_{0}}{\left(x,y \right)} = 1 + \frac{y^{2}}{\alpha_{2}} + \frac{y \left(- \alpha_{2} - 1\right)}{\alpha_{2}} + \frac{x^{2}}{\alpha_{1}} + \frac{...


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Almost all of the statements about the convergence of the finite element are only about the finite element space, not about what specific basis you choose for it. As a consequence, you are for example free to use an equidistant set of nodes to generate basis functions for the usual $P_k$ of $Q_k$ elements, or a set of nodes that correspond to Chebyshev or ...


2

The number of nodal points has nothing to do with the number of integration points. Where I went wrong was that I did not consider the integration at the points $\xi = 0$ and $\eta = 0$, because I thought they only applied to 9-node elements. I get the area 4 square units when I account for the integration point at the center of the element, weighted by $\...


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Broadly speaking the PDE in structural engineering is the PDE of elasticty or elastoplasticity. This can be small or large deformation depending on whether you are solving static problems or problems in which large deformation is expected, such as shock and blast. For elements such as beams, shells, plates, rods you can derive specialized pdes starting from ...


2

To have an itroduction you can have a look at this book where it is explained what you are asking (if I have understood correctly the question). In chapter 1 you can see an example of axially loaded rod solved with the PVW and in parallel discretized through the FEM and the solutions are compared. Later it continues with problems with a bigger dimensionality ...


0

I assume we have a simplex $K$ with a face $F \subset \partial K$ and that $K\subset \mathbb{R}^d, F\subset\mathbb{R}^{d-1}$ are both full-dimensional. I show in the following only the scaling $|F|\leq ch^{d-1}|\hat{F}|$ but note that $|K|\leq ch^{d}|\hat{K}|$ follows similarly. First define the linear affine and inavertible map $\psi:\hat{F}\rightarrow F$ ...


3

The derivatives of the basis functions with respect to the physical space are given by \begin{equation} \phi_{,x}=\phi_{,\xi}\xi_{,x}+\phi_{,\eta}\eta_{,x} \\ \phi_{,y}=\phi_{,\xi}\xi_{,y}+\phi_{,\eta}\eta_{,y} \\ \phi_{,z}=\phi_{,\xi}\xi_{,z}+\phi_{,\eta}\eta_{,z} \end{equation} Hence, you need $\xi_{,x}$, $\xi_{,y}$, $\xi_{,z}$, $\eta_{,x}$, $\eta_{,y}$, $\...


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Advantages to MoM/BEM: BEM naturally incorporates boundaries at infinity, while FEM needs to truncate the domain at some point. This can either be a feature of a bug. The truncation means that FEM naturally gets to model periodic structures very easily, but not open boundaries; vice versa for BEM. Computational effort for BEM usually depends on the ...


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