A copper bar of length and uniform thermal conductivity has a circular cross-section…

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Numerical and computational mathematics · Numerical and computational mathematics

Error, Chance and the Machine, Ch Partial differential equations: stability, convergence and solvers

Problem

A copper bar of length L=0.50 mL=0.50\,\mathrm{m} and uniform thermal conductivity k=400 W m−1 K−1k=400\,\mathrm{W\,m^{-1}\,K^{-1}} has a circular cross-section that tapers linearly, with radius r(x)=r0+(r1−r0)x/Lr(x) = r_0 + (r_1-r_0)x/L, from r0=10 mmr_0=10\,\mathrm{mm} at x=0x=0 to r1=5 mmr_1=5\,\mathrm{mm} at x=Lx=L. With u(x)u(x) the temperature above ambient, steady one-dimensional conduction (no heat loss through the sides, no internal source) obeys −ddx ⁣(a(x)dudx)=0,a(x)=kA(x)=kπr(x)2.-\frac{\mathrm{d}}{\mathrm{d}x}\!\left(a(x)\frac{\mathrm{d}u}{\mathrm{d}x}\right)=0,\qquad a(x)=k A(x)=k\pi r(x)^2. Two equal linear (hat-function) finite elements are used, with nodes at x0=0x_0=0, x1=0.25 mx_1=0.25\,\mathrm{m}, x2=0.50 mx_2=0.50\,\mathrm{m}. ((a)) Starting from the weak form ∫0La(x) v′u′ dx=0\int_0^L a(x)\,v'u'\,\mathrm{d}x = 0 for any test function vv with v(0)=v(L)=0v(0)=v(L)=0, show that for a single linear element of length hh the element stiffness matrix is K(e)=aˉeh(1−1−11),aˉe=1h∫x1(e)x2(e)a(x) dx,K^{(e)}=\frac{\bar a_e}{h}\begin{pmatrix}1&-1\\-1&1\end{pmatrix},\qquad \bar a_e=\frac1h\int_{x_1^{(e)}}^{x_2^{(e)}}a(x)\,\mathrm{d}x, the average of a(x)a(x) over the element. ((b)) Evaluate aˉe\bar a_e exactly for each of the two elements of the tapered bar, and hence write down the assembled 3×33\times3 global stiffness matrix KK on the unconstrained node set {x0,x1,x2}\{x_0,x_1,x_2\}.

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