scholarly journals Numerical Solution of Coupled Thermo-Elastic-Plastic Dynamic Problems

2021 ◽  
Vol 8 (4) ◽  
pp. 510-518
Author(s):  
Abduvali Khaldjigitov ◽  
Umidjon Djumayozov ◽  
Dilnoza Sagdullaeva
Keyword(s):  
Deformation Theory ◽  
Recurrence Relations ◽  
Thermal Field ◽  
Dynamic Problems ◽  
Transition Matrices ◽  
Explicit Schemes ◽  
Discrete Equations ◽  
Implicit Schemes ◽  
Implicit Difference Schemes ◽  
Explicit And Implicit Schemes

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.

Numerical Method for Solving the Problems of the Building Structures Dynamics with a Mobile Massive Load

2018 ◽  
Vol 931 ◽  
pp. 72-77
Author(s):  
Leonid N. Panasyuk ◽  
Galina M. Kravchenko ◽  
Vakhtang P. Matua
Keyword(s):  
Direct Methods ◽  
Dynamical Problem ◽  
Building Structures ◽  
Direct Integration ◽  
Dynamic Problems ◽  
Time Step ◽  
Explicit Schemes ◽  
Massive Structure ◽  
The Matrix ◽  
Solving Equations

The article considers the modeling of dynamic processes in buildings and structures. A general formulation of the dynamic problem of a massive load motion on a massive structure is considered. The equation of motion is obtained in the form of a finite element method. The equations solving is performed using direct methods of integrating dynamic problems. Absolutely stable schemes of direct integration are constructed, where the system of solving equations is trivial and the matrix of the system is diagonal. Due to this, the complexity at the time step is as low as in explicit schemes. Therefore, the proposed methods can be considered as explicit absolutely stable schemes of direct integration of a dynamical problem with a variable in time mass. These recommendations are for estimating the accuracy of a numerical solution.

scholarly journals NUMERICAL SOLUTION OF 1D AND 2D THERMOELASTIC COUPLED PROBLEMS

2012 ◽  
Vol 09 ◽  
pp. 503-510
Author(s):  
ABDUVALI KHALDJIGITOV ◽  
AZIZ QALANDAROV ◽  
NIK MOHD ASRI NIK LONG ◽  
ZAINIDIN ESHQUVATOV
Keyword(s):  
Heat Conduction ◽  
Numerical Solution ◽  
Coupled Problems ◽  
Implicit Schemes ◽  
Elasticity Problems ◽  
Explicit And Implicit Schemes ◽  
Solid Liquid ◽  
Good Coincidence

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.

scholarly journals A General Symbolic PDE Solver Generator: Beyond Explicit Schemes

2003 ◽  
Vol 11 (3) ◽  
pp. 225-235 ◽  
Author(s):  
K. Sheshadri ◽  
Peter Fritzson
Keyword(s):  
Finite Difference ◽  
Method Of Lines ◽  
Difference Schemes ◽  
Finite Difference Schemes ◽  
Arbitrary Geometry ◽  
Explicit Schemes ◽  
Implicit Schemes ◽  
The Method Of Lines ◽  
Dependent Variables ◽  
Explicit Finite Difference

This paper presents an extension of our Mathematica- and MathCode-based symbolic-numeric framework for solving a variety of partial differential equation (PDE) problems. The main features of our earlier work, which implemented explicit finite-difference schemes, include the ability to handle (1) arbitrary number of dependent variables, (2) arbitrary dimensionality, and (3) arbitrary geometry, as well as (4) developing finite-difference schemes to any desired order of approximation. In the present paper, extensions of this framework to implicit schemes and the method of lines are discussed. While C++ code is generated, using the MathCode system for the implicit method, Modelica code is generated for the method of lines. The latter provides a preliminary PDE support for the Modelica language. Examples illustrating the various aspects of the solver generator are presented.

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

2021 ◽  
Vol 61 (SI) ◽  
pp. 49-58
Author(s):  
Tomáš Bodnár ◽  
Philippe Fraunié ◽  
Karel Kozel
Keyword(s):  
Numerical Methods ◽  
Finite Difference ◽  
Difference Schemes ◽  
Finite Difference Schemes ◽  
Advection Equation ◽  
One Dimensional ◽  
Implicit Schemes ◽  
Explicit And Implicit Schemes ◽  
Spatial Support ◽  
Modified Equation

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.

scholarly journals Bespoke finite difference schemes that preserve multiple conservation laws

2015 ◽  
Vol 18 (1) ◽  
pp. 372-403 ◽  
Author(s):  
Timothy J. Grant
Keyword(s):  
Partial Differential Equations ◽  
Finite Difference ◽  
Differential Equations ◽  
Conservation Laws ◽  
Kdv Equation ◽  
Finite Difference Schemes ◽  
Local Form ◽  
Systematic Method ◽  
Implicit Schemes ◽  
Explicit And Implicit Schemes

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.

scholarly journals Evaluating the Effect of Numerical Schemes on Hydrological Simulations: HYMOD as A Case Study

Water ◽  
2019 ◽  
Vol 11 (2) ◽  
pp. 329
Author(s):  
Shiyan Zhang ◽  
Khalid Al-Asadi
Keyword(s):  
Hydrological Model ◽  
Optimal Parameter ◽  
Model Performance ◽  
Hydrological Models ◽  
Numerical Schemes ◽  
Order Of Accuracy ◽  
Explicit Schemes ◽  
Implicit Schemes ◽  
The Relationship

The importance of numerical schemes in hydrological models has been increasingly recognized in the hydrological community. However, the relationship between model performance and the properties of numerical schemes remains unclear. In this study, we employed two types of numerical schemes (i.e., explicit Runge-Kutta schemes with different orders of accuracy and partially implicit Euler schemes with different implicit factors) in the hydrological model (HYMOD) to simulate the flow hydrograph of the Leaf River basin from 1948 to 1988. Results computed by different numerical schemes were compared and the relationships between model performance and two scheme properties (i.e., the order of accuracy and the implicit factor) were discussed. Results showed that the more explicit schemes generally lead to the overestimation of flow hydrographs, whereas the more implicit schemes lead to underestimation. In addition, the numerical error tended to decrease with increasing orders of accuracy. As a result, the optimal parameter sets found by low-order schemes significantly deviated from those found by the analytical solution. The findings of this study can provide useful implications for designing suitable numerical schemes for hydrological models.

scholarly journals Multi-core processors use for numerical problems solutions

2020 ◽  
Vol 30 (4) ◽  
pp. 98-105
Author(s):  
S. A. Pryadko ◽  
A. S. Krutogolova ◽  
A. S. Uglyanitsa ◽  
A. E. Ivanov
Keyword(s):  
Heat Transfer ◽  
Numerical Modeling ◽  
Parallel Computing ◽  
Distributed Memory ◽  
Heat Transfer Equation ◽  
Problem Statement ◽  
Implicit Schemes ◽  
Explicit And Implicit Schemes ◽  
Time Required ◽  
Practical Implications

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.

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

2017 ◽  
Vol 7 (2) ◽  
pp. 286-305
Author(s):  
Jingjun Zhao ◽  
Teng Long ◽  
Yang Xu
Keyword(s):  
Differential Equations ◽  
Linear Equations ◽  
Order Conditions ◽  
Linear Differential Equations ◽  
Stability Properties ◽  
Implicit Schemes ◽  
Explicit And Implicit Schemes ◽  
Runge Kutta ◽  
Linear Differential ◽  
Theoretical Results

AbstractExponential 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.

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