# 统计代写|主成分分析代写Principal Component Analysis代考|Galerkin Finite Element Method for 1D Steady-State Heat Conduction

In summary, the Galerkin finite element method has been applied to solve the problem of one-dimensional steady-state heat conduction within a solid body. The process starts with the conservation of energy principle leading to the heat conduction equation (5.59): kxd2Tdx2+Q=0k_x \frac{d^2T}{dx^2} + Q = 0k x ​

dx 2

d 2 T ​ +Q=0, where kxk_xk x ​ is the thermal conductivity, TTT is the temperature, and QQQ is the internal heat generation rate.

For the finite element approach, a two-node linear element is used to approximate the temperature distribution (5.60), with the interpolation functions given by Equation 5.20. Upon substitution into the heat conduction equation, the residual integrals (5.61) are obtained.

Integration by parts and substitution yield the element equations (5.63) and (5.64), which are arranged in matrix form (5.65) with a conductance matrix [k], nodal force vector {f_Q} due to internal heat generation, and a gradient boundary condition vector {fg} representing heat flux at the element ends.

For the example problem involving a composite circular rod made of aluminum and copper sections, the conductance matrices [kal] and [kcu] are computed for each material based on their respective thermal conductivities. Applying boundary conditions—fixed temperature at one end and specified heat flux at the other—and assembling the global system of equations, the steady-state temperature distribution within the rod is determined.

By solving the system of equations (either directly or through a triangularization process), the temperatures at each node are found. The heat flow at the right end (node 5) is then calculated using the gradient boundary condition. Despite minor discrepancies due to computational round-off errors, the finite element solution is exact for this particular problem, confirming that the heat flow at the right end matches the prescribed inflow at the left end when computed with sufficient accuracy.

### MATLAB代写

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