On a Semilinear Parabolic Problem with Four-Point Boundary Conditions

This paper studies a semilinear parabolic equation in 1D along with nonlocal boundary conditions. The value at each boundary point is associated with the value at an interior point of the domain, which is known as a four-point boundary condition. First, the solvability of a steady-state problem is addressed and a constructive algorithm for finding a solution is proposed. Combining this schema with the semi-discretization in time, a constructive algorithm for approximation of a solution to a transient problem is developed. The well-posedness of the problem is shown using the semigroup theory in C-spaces. Numerical experiments support the theoretical algorithms.

Analysis free surface of nonlinear seepage using the MQRBF method

In order to solve the characteristics of low accuracy and slow efficiency in traditional numerical solution the free surface problem, the multiquardatic radial base function collocation method(MQ RBF) is used to analyze the constant seepage and unsteady seepage of the homogeneous earth dam. Computation of transient problem of free surface of earth dam by the linear derivation of Richards equation. The results show that the calculation accuracy of the MQRBF is higher than that of the traditional numerical method. The solution process does not involve numerical integral calculation and grid reorganization, which greatly reduces the calculation amount. Compared with the Trefftz method, it has the advantage of solving boundary values and internal values at the same time. It is not limited by the solution of the Laplace equation, and its application is wider and simpler.

Iterative Multiscale Approach for Heat Conduction With Radiation Problem in Porous Materials

This paper describes a thermal homogenization approach to the application of a multiscale formulation for heat conduction with radiation problems in a porous material. The suggested formulation enables to evaluate the effective macroscopic thermal conductivity, based on the microscopic properties such as porosity, and can also provide the microscopic radiosity heat flux, based on the macroscopic temperature gradient field. This is done by scaling up and down between the microscopic and macroscopic models according to the suggested methodology. The proposed methodology involves a new iterative upscaling procedure, which uses heat conduction at macroscopic problem and heat transfer by conduction and radiation at microscopic problem. This reduces the required computational time, while maintaining the required level of accuracy. The suggested multiscale formulation has been verified by comparing its results with reference finite element (FE) solutions of porous (filled with air) materials examples; the results shows excellent agreement (up to 5% discrepancy) with reference solutions. The efficiency of the suggested formulation was demonstrated by solving a full-scale engineering transient problem.

Highly-conductive energetic coherent interfaces subject to in-plane degradation

The presence of an interface can influence the thermo-mechanical response of a body. This influence is especially pronounced at small scales where the interface area to bulk volume ratio significantly increases. Since the thermo-mechanical properties of an interface can differ from those of the bulk, within interface continuum theory an interface is endowed with its own thermo-mechanical energetic structure. To date, the effects of interface in-plane damage on the thermo-mechanical response of a highly conductive interface have not been accounted for. Therefore in this contribution the computational aspects of thermo-mechanical solids containing highly conductive interfaces subject to in-plane degradation are studied. To this end, the equations governing a fully non-linear transient problem are given. They are solved using the finite element method. The results are illustrated through a series of three-dimensional numerical examples for various interfacial parameters. In particular a comparison is made between the results of the intact and the degraded highly conductive interface formulation.