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This paper proposes a density-based topology optimization method for natural convection problems using the lattice Boltzmann method (LBM). As the LBM can be developed as a completely explicit scheme, its attractive features over the traditional ones, such as the finite element method, are (1) suitability for solving unsteady flow problems and (2) scalability for large-scale parallel computing. We develop an LBM code for solving unsteady natural convection problems and provide its sensitivity analysis based on the so-called adjoint lattice Boltzmann method. Notably, the adjoint equation is derived from the discrete particle velocity Boltzmann equation and can be solved similarly to the original LBM concerning unsteady natural convection problems. We first show that the proposed method can produce similar results to the previous work in a steady-state natural convection problem. We then demonstrate the efficacy of the proposed method through 2D numerical examples concerning unsteady natural convection. As a large-scale problem, we tackle a 3D unsteady natural convection problem on a parallel supercomputer. All the developed codes written in C++ are available at . [ABSTRACT FROM AUTHOR]

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.

We consider the problem of determining the stress-strain state of an orthotropic multicomponent cylinder affected by unsteady surface elastic diffusive perturbations. The coupled system of elastic diffusion equations in the polar coordinate system is used as a mathematical model. Diffusion relaxation effects, implying finite rates of diffusion flux propagation, are taken into account. The solution to this problem is sought in the integral form and is represented as convolutions of Green's functions with functions defining surface elastodiffusive perturbations. We use the Laplace transform by time and Fourier series expansion in Bessel functions of the first kind to find Green's functions. The Laplace transform inversion is done analytically due to residues and operational calculus tables. An analytical solution to the problem is obtained. A numerical study of the interaction of mechanical and diffusion fields in a continuous orthotropic cylinder is performed. We used three-component material as an example. The cylinder is under pressure, which is uniformly distributed over it surface. We use three-component material as an example. [ABSTRACT FROM AUTHOR]

In this paper, the unsteady vibrations of an ortotropic rectangular Kirchhoff–Love plate are studied. In general formulation, the plate is under the action of longitudinal and transverse forces, bending and torque moments. It also set a diffusion fluxes density and a diffusion fluxes relaxation. The coupled elastic diffusion orthotropic multicomponent continuum model has been used to formulate the problem. The d'Alembert variational principle has been used to obtain the plate transverse elastic diffusion vibrations equations from the model. [ABSTRACT FROM AUTHOR]

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.

A finite difference method for a time-dependent singularly perturbed convection–diffusion–reaction problem involving two small parameters in one space dimension is considered. We use the classical implicit Euler method for time discretization and upwind scheme on the Shishkin–Bakhvalov mesh for spatial discretization. The method is analysed for convergence and is shown to be uniform with respect to both the perturbation parameters. The use of the Shishkin–Bakhvalov mesh gives first-order convergence unlike the Shishkin mesh where convergence is deteriorated due to the presence of a logarithmic factor. Numerical results are presented to validate the theoretical estimates obtained. [ABSTRACT FROM PUBLISHER]

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]