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Numerical solution of the heat advection equation in a two-dimensional domain using the discontinuous Galerkin method

Węgrzyn-Skrzypczak Ewa

Journal of Applied Mathematics and Computational Mechanics

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2023

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Vol. 22, nr 3

57--68

EN

This paper presents a numerical solution of the heat advection equation in a two dimensional domain using the Discontinuous Galerkin Method (DGM). The advection equation is widely used in heat transfer problems, particularly in the field of fluid dynamics. The discontinuous Galerkin method is a numerical technique that allows for the solution of partial differential equations using a piecewise polynomial approximation. In this study, DGM is applied to the heat advection equation and its effectiveness in solving the problem is investigated. The findings of this study suggest that the Discontinuous Galerkin Method is a promising approach for solving heat transfer problems in a two-dimensional domain.

2

Numerical modeling of the solidification process with consideration of shrinkage cavities formation and the influence of solid phase content on the feeding of the casting

Węgrzyn-Skrzypczak Ewa, Skrzypczak Tomasz

Journal of Applied Mathematics and Computational Mechanics

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2023

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Vol. 22, nr 2

75--86

EN

The paper focuses on the numerical modeling of the three-dimensional solidification process of steel using the finite element method (FEM). The model includes and discusses the formation of shrinkage cavities and the influence of the solid phase content on the feeding of the casting through the riser. The analysis assumed a critical value of the solid phase content, at which the transport of liquid phase from the riser to the casting is interrupted. The results of numerical simulation are presented to investigate the influence of this factor on the final quality of the casting. The model neglects the fluid motion in the liquid and solid-liquid regions and replaces the influence of the mold with appropriate boundary conditions.

3

The comparison of results obtained from the continuous and discontinuous Galerkin Method for the thermoelasticity problem

Węgrzyn-Skrzypczak Ewa

Journal of Applied Mathematics and Computational Mechanics

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2021

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Vol. 20, nr 3

77--88

EN

The presented paper is focused on the comparison of the Continuous and Discontinuous Galerkin Methods in terms of thermoelasticity for a cubic element. For this purpose, a numerical model of the phenomenon was built using both methods together with the Finite Element Method (FEM). The comparison of the results of numerical simulation obtained with the use of an original computer program based on the derived final set of FEM equations for both methods is presented.

4

Investigation of the influence of liquid phase motion on the solidification process in the three-dimensional region

Węgrzyn-Skrzypczak Ewa

Journal of Applied Mathematics and Computational Mechanics

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2021

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Vol. 20, nr 1

103--111

EN

In this paper, the numerical model of solidification process with the motion of the liquid phase is presented. The mathematical description of the considered problem is based on the heat conduction equation with convective term and the Navier-Stokes equations with continuity equation. The numerical model uses the Finite Element Method (FEM). The simulations of the solidification process with or without the fluid motion effect are presented and discussed.

5

Analysis of the three-dimensional thermoelasticity problem with the use of the continuous Galerkin method

Węgrzyn-Skrzypczak Ewa

Journal of Applied Mathematics and Computational Mechanics

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2020

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Vol. 19, nr 3

111--121

EN

The paper focuses on the mathematical and numerical modelling of the thermoelasticity problem in the three-dimensional region. The governing equations of the mathematical model are a set of equilibrium equations. The numerical model uses continuous Galerkin formulation together with the Finite Element Method (FEM). Both models are discussed in detail. The final set of FEM equations is derived. The example of numerical calculations obtained with the use of an original computer program is presented. The main goal of presented paper is to develop the alternative model to the one based on the Discontinuous Galerkin Method (DGM).

6

Discontinuous Galerkin method for the three-dimensional problem of thermoelasticity

Węgrzyn-Skrzypczak Ewa

Journal of Applied Mathematics and Computational Mechanics

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2019

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Vol. 18, nr 4

115--126

EN

The paper is focused on the mathematical and numerical approaches for the thermoelasticity problem in the three-dimensional domain. The mathematical description of considered problem is based on the second order differential equations of elasticity with the term describing thermal deformations. The numerical model uses the discontinuous Galerkin method which is widely used to solve the problems of hydrodynamics. The presented paper shows the possibility of using the mentioned method to solve the problem of thermomechanics.

7

Analytical validation of the Finite Element Method model for Laplace equation

Węgrzyn-Skrzypczak Ewa, Skrzypczak Tomasz

Journal of Applied Mathematics and Computational Mechanics

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2019

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Vol. 18, nr 3

97--106

EN

The presented paper is focused on the comparison of the numerical solution of the Laplace equation in a two-dimensional space with the results obtained with the use of the analytical method. The results of the numerical model are computed on the base of the Finite Element Method. The analytical solution of the considered equation is obtained using the Fourier series. Finally the results of both methods are compared in order to verify the accuracy of numerical implementation.