Verification of ansys and matlab Heat Conduction Results Using an “Intrinsically” Verified Exact Analytical Solution

Abstract

A two-dimensional (2D) transient thermal conduction problem is examined, and numerical solutions to the problem generated by ansys and matlab, employing the finite element (FE) method, are compared against an “intrinsically” verified analytical solution. Various grid densities and time-step combinations are used in the numerical solutions, including some as recommended by default in the ansys software, including coarse, medium, and fine spatial grids. The transient temperature solutions from the analytical and numerical schemes are compared at four specific locations on the body, and time-dependent error curves are generated for each point. Additionally, tabular values of each solution are presented for a more detailed comparison. Two different test cases are examined for the various numerical solutions using selected grid densities. The first case involves uniform constant heating on a portion of one surface for a long duration, up to a dimensionless time of 30. The second test case still involves uniform constant heating but for a dimensionless time of one, immediately followed by an insulated condition on that same surface for another duration of one dimensionless time unit. Although the errors at large times for both ansys and matlab are extremely small, the errors found within the short-duration test are more significant, in particular when the heating is suddenly set “on.” Surprisingly, very small errors occur when the heating is suddenly set “off.” The solution developed using the matlab differential equation solver is found to have errors an order of magnitude larger than those generated using ansys with a similar mesh size and same FE type (quadratic triangular). This occurs not only during the transient but also for steady-state problems. However, the matlab computational efficiency is superior. Also, when using ansys, at early times the numerical errors are very sensitive to the time-step chosen rather than the spatial discretization. Therefore, the time-steps have to be very small near a singularity.