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An Excel program that uses tri-diagonal matrix algorithm (TDMA) is proposed to calculate one dimensional unsteady heat conduction using implicit method. The program is unified analysis method to calculate symmetric flat plate, solid cylinder, sphere, double-sided flat plate, hollow cylinder, and spherical shell geometries. Three different boundary conditions and arbitrary heat generation rates can be applied. The unsteady numerical analysis is compared with the exact solution to verify the accuracy. Furthermore, the effectiveness of the program is demonstrated by solving various engineering examples using this Excel program.

This article presents an algorithm for solving the unsteady problem of one-dimensional coupled thermoelastic diffusion perturbations propagation in a multicomponent isotropic half-space, as a result of surface and bulk external effects. One-dimensional physico-mechanical processes, in a continuum, have been described by a local-equilibrium model, which included the coupled linear equations of an elastic medium motion, heat transfer, and mass transfer. The unknown functions of displacement, temperature, and concentration increments were sought in the integral form, which was a convolution of the surface and bulk Green’s functions and external effects functions. The Laplace transform on time and the Fourier sine and cosine transforms on the coordinate were used to find the Green’s functions. The obtained Green’s functions was analyzed. Test calculations were performed on the examples of some technological processes.

This article considers an unsteady elastic diffusion model of Euler⁻Bernoulli beam oscillations in the presence of diffusion flux relaxation. We used the model of coupled elastic diffusion for a homogeneous orthotropic multicomponent continuum to formulate the problem. A model of unsteady bending for the elastic diffusive Euler⁻Bernoulli beam was obtained using Hamilton’s variational principle. The Laplace transform on time and the Fourier series expansion by the spatial coordinate were used to solve the obtained problem.

Weak Galerkin (WG method) is a numerical method based on the weak functions and weak partial derivatives, which has been widely developed in recent decades. In the implementation, different degrees of freedom (DOFs) are distributed in the element interior and boundary respectively, which leads to a great impact on the computational complexity. In our work, we propose a new reduced-order weak galerkin finite element (RO-WG) method with seldom DOFs, where the proper orthogonal decomposition (POD) technique is adopted to save CPU time. The parabolic equation is considered for a test problem. Some numerical examples are given to prove the superiority of this method, where the CPU time can be reduced a lot. [ABSTRACT FROM AUTHOR]