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1

A quick note on the .stl for 3D modelling: As noted in the documentation here (Limitations at the bottom) for 3D models MATLAB cannot recognise multi-domains from 3D STL files. If two objects share common points, each is stored in a separate cell, essentially making objects disconnected analytically with no shared mesh/common interface between the two. ...

1

Double quotes(" ") represent strings in python, and hence the argument of the function Expression() is a string literal rather than a floating-point number, i.e., ln(5). To correctly specify ln(5) as an argument, remove the double-quotes.

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No, you can't lump the $K$ matrix: that would not be a consistent approximation to the second-order differential operator it is supposed to represent. But if you're trying to be a bit more formal, just write out what that lumped mass matrix would actually be: Most rows of the matrix (corresponding to nodes not next to the boundary) would simply add up to ...

0

Defining current density in this system can be done by considering the average current density within the winding region, $J_{0}$ (then subdividing this into elements). For winding height h, Inner cross section $L_{1}$, Outer cross section $L_{2}$ with N turns carrying current I: $J_{0} = \frac{IN}{h(L_{2} - L_{1})/2}$. I note this expression does not ...

3

This is not the best way to make these calculations. First map the cell $K$ 4---------3 | | | K | | | 1---------2 to unit cell [0,1]x[0,1] by $$\xi = (x - x_1)/\Delta x, \qquad \eta = (y - y_1)/\Delta y$$ The the shape functions are $$N_1(\xi,\eta) = (1-\xi)(1-\eta), \qquad N_2(\xi,\eta) = \xi (1-\eta), \qquad \ldots,$$ Then $$... 2 I think the easiest way to calculate the shape functions is to go through the Lagrange polynomials. The pressure p in element e reads$$p^e(x,y)\approx\sum_{i=1}^{i=4}N_i(x,y)p_i.$$Let's start with the shape function of the first node N_1(x,y). We want to generate a function with the properties$$N_1(x_1,y_1)=1,\qquad{}N_1(x_{i\neq{}1},y_i{\neq{}1})...

2

You can arrive at the Jacobian analytically it just takes a few steps So assuming we have our typical FE field values: $$u_i = \sum_j \phi^j u_i^j$$ Where $i$ represents coordinate direction, $\phi$ our shape function, and $j$ index to DOF. Start with our stabilization parameter, assuming you are using Euclidian norm:  \tau = \sqrt{\alpha \lVert u \...

7

$s_k$ is the "approximate Newton" search direction. So in essence, when they say Choose $s_k$ such that $\|F(x_k)+F'(x_k)s_k\| \le \eta_k \|F(x_k)\|$ they are saying: Solve the Newton system $F'(x_k)s_k = - F(x_k)$ inexactly for $s_k$ until the norm of the residual $F(x_k)+F'(x_k)s_k$ is smaller than the norm of the right hand side $-F(x_k)$ by a ...

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