Malek, Alaeddin | Search NASA

DOE OSTI2018

Solving macroscopic and microscopic pin-fin heat sink problems by adapted spectral method

In this paper, simulation of heat transfer in a heat sink with macroscopic and microscopic scales when one pin-fin is added to the system is investigated by the proposed spectral method. In the microscopic problems, heat transfer model uses dual-phase lag formulas in contrast with macroscopic problems when Fourier law is used to formulate the governing equation. In macroscopic problem, the results are compared with COMSOL multiphysics software results and a good agreement between the results are shown. In microscopic problems, 3D Gaussian heat source is used and boundary conditions obey the Newton law. Comparisons show the efficiency of the current method, while the results are compared with existed literature. It is shown that: by adding one pin fin to the heat sink problem, temperature of the system will decrease. Numerical results are given and graphs of the temperature for various kinds of pin-fin heat sink problems are depicted.

97 MATHEMATICS AND COMPUTING↗

NASA NTRS1994

Pseudospectral collocation methods for fourth order differential equations

Collocation schemes are presented for solving linear fourth order differential equations in one and two dimensions. The variational formulation of the model fourth order problem is discretized by approximating the integrals by a Gaussian quadrature rule generalized to include the values of the derivative of the integrand at the boundary points. Collocation schemes are derived which are equivalent to this discrete variational problem. An efficient preconditioner based on a low-order finite difference approximation to the same differential operator is presented. The corresponding multidomain problem is also considered and interface conditions are derived. Pseudospectral approximations which are C1 continuous at the interfaces are used in each subdomain to approximate the solution. The approximations are also shown to be C3 continuous at the interfaces asymptotically. A complete analysis of the collocation scheme for the multidomain problem is provided. The extension of the method to the biharmonic equation in two dimensions is discussed and results are presented for a problem defined in a nonrectangular domain.

Malek, Alaeddin↗