Numerical Solution of Coupled Thermo-Elastic-Plastic Dynamic Problems

The article considers a numerical method for solving a two-dimensional coupled dynamic thermoplastic boundary value problem based on deformation theory of plasticity. Discrete equations are compiled by the finite-difference method in the form of explicit and implicit schemes. The solution of the explicit schemes is reduced to the recurrence relations regarding the components of displacement and temperature. Implicit schemes are efficiently solved using the elimination method for systems with a three diagonal matrix along the appropriate directions. In this case, the diagonal predominance of the transition matrices ensures the convergence of implicit difference schemes. The problem of a thermoplastic rectangle clamped from all sides under the action of an internal thermal field is solved numerically. The stress-strain state of a thermoplastic rectangle and the distribution of displacement and temperature over various sections and points in time have been investigated.

MODIFIED EQUATION FOR A CLASS OF EXPLICIT AND IMPLICIT SCHEMES SOLVING ONE-DIMENSIONAL ADVECTION PROBLEM

This paper presents the general modified equation for a family of finite-difference schemes solving one-dimensional advection equation. The whole family of explicit and implicit schemes working at two time-levels and having three point spatial support is considered. Some of the classical schemes (upwind, Lax-Friedrichs, Lax-Wendroff) are discussed as examples, showing the possible implications arising from the modified equation to the properties of the considered numerical methods.

Multi-core processors use for numerical problems solutions

Problem statement. The use of programming technologies on modern multicore systems is an integral part of an enterprise whose activities involve multitasking or the need to make a large number of calculations over a certain time. The article discusses the development of such technologies aimed at increasing the speed of solving various issues, for example, numerical modeling.Objective. Search for alternative ways to increase the speed of calculations by increasing the number of processors. As an example of increasing the calculation speed depending on the number of processors, the well-known heat-transfer equation is taken, and classical numerical schemes for its solution are given. The use of explicit and implicit schemes is compared, including for the possibility of parallelization of calculations.Results. The article describes systems with shared and distributed memory, describes their possible use for solving various problems, and provides recommendations for their use.Practical implications. Parallel computing helps to solve many problems in various fields, as it reduces the time required to solve partial differential equations.

A New Class of Difference Methods with Intrinsic Parallelism for Burgers–Fisher Equation

This paper proposes a new class of difference methods with intrinsic parallelism for solving the Burgers–Fisher equation. A new class of parallel difference schemes of pure alternating segment explicit-implicit (PASE-I) and pure alternating segment implicit-explicit (PASI-E) are constructed by taking simple classical explicit and implicit schemes, combined with the alternating segment technique. The existence, uniqueness, linear absolute stability, and convergence for the solutions of PASE-I and PASI-E schemes are well illustrated. Both theoretical analysis and numerical experiments show that PASE-I and PASI-E schemes are linearly absolute stable, with 2-order time accuracy and 2-order spatial accuracy. Compared with the implicit scheme and the Crank–Nicolson (C-N) scheme, the computational efficiency of the PASE-I (PASI-E) scheme is greatly improved. The PASE-I and PASI-E schemes have obvious parallel computing properties, which show that the difference methods with intrinsic parallelism in this paper are feasible to solve the Burgers–Fisher equation.

First and second order dual phase lag equation. Numerical solution using the explicit and implicit schemes of the finite difference method

In the paper the different variants of the dual phase lag equation (DPLE) are considered. As one knows, the mathematical form of DPLE results from the generalization of the Fourier law in which two delay times are introduced, namely the relaxation time τq and the thermalization one τT. Depending on the order of development of the left and right hand sides of the generalized Fourier law into the Taylor series one can obtain the different forms of the DPLE. It is also possible to consider the others forms of equation discussed resulting from the introduction of the new variable or variables (substitution). In the paper a thin metal film subjected to a laser pulse is considered (the 1D problem). Theoretical considerations are illustrated by the examples of numerical computations. The discussion of the results obtained is also presented.

Exponential Additive Runge-Kutta Methods for Semi-Linear Differential Equations

Exponential additive Runge-Kutta methods for solving semi-linear equations are discussed. Related order conditions and stability properties for both explicit and implicit schemes are developed, according to the dimension of the coefficients in the linear terms. Several examples illustrate our theoretical results.

Surface Hardening of Parts Arc

The paper presents the results of studies aimed at justifying and developing the ways and means of cylindrical parts made of hardenable alloys surface hardening by heating of the electric arc between the inert electrode and the element. The shapes of electrode sharpening and the position of the electrode relative to the element are substantiated. The sequence of surface hardening operations is revealed and the critical rate of cooling is determined and its value is proved with numerical experiments. It was necessary to develop a program for calculating temperature fields in details, with a difference-differential scheme obtained from a combination of explicit and implicit schemes being developed for the first time. A number of investigations were conducted to determine the mechanical properties, the residual stresses, the fatigue resistance, the hardness and the microstructure of hardened layers. As a result, the surface hardening technology is recommended for reconditioning and manufacturing cylindrical machine elements.

Bespoke finite difference schemes that preserve multiple conservation laws

Conservation laws provide important constraints on the solutions of partial differential equations (PDEs), therefore it is important to preserve them when discretizing such equations. In this paper, a new systematic method for discretizing a PDE, so as to preserve the local form of multiple conservation laws, is presented. The technique, which uses symbolic computation, is applied to the Korteweg–de Vries (KdV) equation to find novel explicit and implicit schemes that have finite difference analogues of its first and second conservation laws and its first and third conservation laws. The resulting schemes are numerically compared with a multisymplectic scheme.

NUMERICAL SOLUTION OF 1D AND 2D THERMOELASTIC COUPLED PROBLEMS

Study the heats propagation in a solid, liquid continuums is an actual problems. The liquid continuum may be considered as a biomaterial. The present investigation is devoted to the study of 1D and 2D dynamic coupled thermo elasticity problems. In case of coupled problems the motion and heat conduction equations are considered together. For numerical solution of thermo elasticity problems an explicit and implicit schemes are constructed. The explicit and implicit schemes by using recurrent formulas and the "consecutive" methods are solved. Comparison of two results shows a good coincidence.