Derive the stiffness matrix for heat transfer using shape functions for a four noded quadrilateral elements?
Added by James B.
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To derive the stiffness matrix for heat transfer using shape functions for a four-noded quadrilateral element, we will follow these steps: Show more…
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Consider the heat transfer problem with initial and Dirichlet boundary conditions and function. The analytic solution to this problem is. Write an ID Galerkin code to solve this problem. Use N = 11 nodes, but make the number of nodes general. Use the ID Lagrange basis functions. Keep your code general so that different f(x,t) and/or boundary conditions can be implemented. Use 2nd order Gaussian quadrature to numerically integrate the f(x,t) (right-hand side) weak integral. - Build the elemental mass and stiffness matrices by mapping from the X-space to xi-space, integrating in parent space, and assembling to global as done in class. (1) Solve first by using a forward Euler time derivative discretization with a time-step of. Plot the results at the final time. Increase the time-step until you find the instability. What dt does this occur at? How does the solution change as N decreases? (2) Solve the same problem with the same time-steps using implicit backward Euler. What happens as the time-step is equal to or greater than the spatial step size? Explain why.
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