统计代写|主成分分析代写Principal Component Analysis代考|Solving Heat Conduction with Quadratic Elements

The passage describes the process of solving a heat conduction problem using a higher-order (quadratic) three-node line element instead of the linear, two-node elements previously used. The objective is to showcase that the fundamental process of formulating the element equations remains the same and that the assembly process for the global system of equations is also unchanged. Additionally, it highlights that the results are comparable even though a higher-degree interpolation function is employed.

The problem involves a composite rod made of aluminum and copper, divided into two elements. Each element has three nodes with equally spaced coordinates. Using the Galerkin finite element method and considering heat conduction along the rod without internal heat sources, the temperature distribution across the rod needs to be determined based on the boundary conditions and heat flux applied at node 1.

Here’s a summary of the steps followed:

Element Conductance Matrix: The conductance matrix k_ij is computed for each element by integrating the product of the thermal conductivity, cross-sectional area, and the derivative of the shape functions (interpolation functions) over the element length. The derivatives of the quadratic shape functions are calculated and substituted into these integrals to obtain the conductance matrix for both the aluminum and copper portions of the rod.

Assembly: The individual element conductance matrices are assembled to form the global conductance matrix [K]. Since the flux continuity is maintained at the interface between elements due to the properties of the interpolation functions, there is no overlap except at the shared node.

System Equations: With the known gradient term at node 1 (fg1) and the specified temperature T5, the system of equations is constructed. The nonhomogeneous boundary condition at node 5 is incorporated by modifying the relevant equations.

Solution: Using Gaussian elimination (or any other suitable solver), the system of equations is solved to determine the nodal temperatures T1 to T4. Once these temperatures are found, the heat flux at node 5 (q5) can be calculated from the last equation.

It is noted that despite using quadratic interpolation functions, the solution for the nodal temperatures in this example coincides with the exact solution—a linear temperature distribution in each half of the bar. This is because the quadratic interpolation functions inherently contain the linear terms that represent the exact solution in this case.

Lastly, the quadratic field variable representation in terms of nodal variables is explicitly formulated by matching coefficients with the interpolation functions. It is demonstrated that in the context of the Example 7.1 with equally spaced nodes, the coefficient a2 associated with the quadratic term is indeed zero, reflecting the linear nature of the temperature distribution in this specific scenario.

MATLAB代写

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