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1
Content available 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
Content available 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.
3
Content available 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.
4
Content available Review of Lattice Boltzmann Method Applied to Computational Aeroacoustics
Shao Weidong, Li Jun
Archives of Acoustics
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2019
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Vol. 44, No. 2
215--238
EN
This paper presents the research studies carried out on the application of lattice Boltzmann method (LBM) to computational aeroacoustics (CAA). The Navier-Stokes equation-based solver faces the difficulty of computational efficiency when it has to satisfy the high-order of accuracy and spectral resolution. LBM shows its capabilities in direct and indirect noise computations with superior space-time resolution. The combination of LBM with turbulence models also work very well for practical engineering machinery noise. The hybrid LBM decouples the discretization of physical space from the discretization of moment space, resulting in flexible mesh and adjustable time-marching. Moreover, new solving strategies and acoustic models are developed to further promote the application of LBM to CAA.
5
Content available remote The hp nonconforming mesh refinement in discontinuous Galerkin finite element method based on Zienkiewicz-Zhu error estimation
Jaśkowiec J.
Computer Assisted Methods in Engineering and Science
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2016
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Vol. 23, no. 1
43--67
EN
This paper deals with hp-type adaptation in the discontinuous Galerkin (DG) method. The DG method is formulated in this paper with a non-zero mesh skeleton width, which leads to a version of the method called in this paper the interface discontinuous Galerkin (IDG) method. In this formulation, the mesh skeleton has a finite volume and special finite elements are used for discretization. The skeleton spatial calculations are performed using the finite difference or mid-values formulas which are based on the shape functions of the neighbouring finite elements. The Dirichlet boundary conditions are applied using a nonzero width of the material between the outer boundary and a finite element aligned with the boundary. Next, the paper discusses the mesh refinement of hp type. In the IDG method, the mesh does not have to be conforming. The Zienkiewicz-Zhu (ZZ) error indicator is adapted in the IDG method for the purpose of mesh refinement. The paper is illustrated with two-dimensional examples, in which the mesh refinement for an elliptic problem is performed.
6
Content available remote The discontinuous Galerkin method with higher degree finite difference compatibility conditions and arbitrary local and global basis functions
Jaśkowiec J.
Computer Assisted Methods in Engineering and Science
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2016
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Vol. 23, no. 2/3
109--132
EN
This paper focuses on the discontinuous Galerkin (DG) method in which the compatibility condition on the mesh skeleton and Dirichlet boundary condition on the outer boundary are enforced with the help of one-dimensional finite difference (FD) rules, while in the standard approach those conditions are satisfied by the penalty constraints. The FD rules can be of arbitrary degree and in this paper the rules are applied up to fourth degree. It is shown that the method presented in this paper gives better results in comparison to the standard version of the DG method. The method is based on discontinuous approximation, which means that it can be constructed using arbitrary local basis functions in each finite element. It is quite easy to incorporate some global basis functions in the approximation field and this is also shown in the paper. The paper is illustrated with a couple of two-dimensional examples.
7
Content available remote Discontinuous Galerkin method on reference domain
Jaśkowiec J.
Computer Assisted Methods in Engineering and Science
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2015
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Vol. 22, no. 2
177--204
EN
A reference domain is chosen to formulate numerical model using the discontinuous Galerkin with finite difference (DGFD) method. The differential problem, which is defined for the real domain, is transformed in a weak form to the reference domain. The shape of the real domain results from a considered problem which can be complex. On the other hand, a reference domain can be chosen to be, for example, cube or square, which is convenient for meshing and calculations. Transformation from the reference domain into the real one has to be defined. In this paper, the algorithm for such a transformation is proposed, which is based on second-order differential equations. The paper presents a series of benchmark examples that show both the correctness and flexibility of the proposed algorithms. In the majority of the examples, the reference domain is square when the real domains are, for example, quarter of annulus, circle or full annulus.
8
Content available A discontinuous Galerkin finite element method for dynamic of fully saturated soil
Wrana B.
Archives of Civil Engineering
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2011
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Vol. 57, nr 1
119-134
EN
The fully coupled, porous solid-fluid dynamic field equations with u - p formulation are used in this paper to simulate pore fluid and solid skeleton responses. The present formulation uses physical damping, which dissipates energy by velocity proportional damping. The proposed damping model takes into account the interaction of pore fluid and solid skeleton. The paper focuses on formulation and implementation of Time Discontinuous Galerkin (TOG) methods for soil dynamics in the case of fully saturated soil. This method uses both the displacements and velocities as basic unknowns and approximates them through piecewise linear functions which are continuous in space and discontinuous in time. This leads to stable and third-order accurate solution algorithms for ordinary differential equations. Numerical results using the time-discontinuous Galerkin FEM are compared with results using a conventional central difference, Houbolt, Wilson Fi, HHT-alfa, and Newmark methods. This comparison reveals that the time-discontinuous Galerkin FEM is more stable and more accurate than these traditional methods.
PL
Artykuł podejmuje zagadnjenie analizy rozchodzenia się fal naprężenjowych w gruncie w ujęciu metody elementów skończonych bazując na sformułowaniu rozwiązania ciągłego w przestrzeni i nieciągłego w dziedzinie czasu Galerkina (space and time-discontinous Galerkin TDG finite element method). W tym sformułowaniu zarówno przemieszczenia jak i prędkości są wielkościamj nieznanymi wzajemnie niezależnymi aproksymowanymi ciągłymi funkcjami kształtu w przestrzeni i nieciągłymi funkcjami kształtu w czasie. Do opisu zachowania się gruntu w pełni nasyconego wodą zastosowano sformułowanie u-p w ujęciu metody elementów skończonych. Grunt traktowany jest, jako ośrodek dwufazowy składający się ze szkieletu i wody w porach. Zastosowane sformułowanie uwzględnia tłumienie ośrodka przez uwzględnienie dyssypacji energii proporcjonalnej do prędkości wody względem szkieletu. W artykule przedstawiono porównanie proponowanej metody rozwiązania numerycznego w dziedzinie czasu do metod obecnie stosowanych, takich jak: metoda różnicy centralnej, metoda Houbolta, Wilsona fi, HHT-alfa oraz najczęściej stosowanej metody Newmarka. Z porównania wynika, że proponowana metoda jest metodą stabilną o małym błędzie numerycznego rozwiązania.
9
Content available remote Mathematical description of discontinuous Galerkin method in the theory of thermoelasticity
Węgrzyn-Skrzypczak E., Skrzypczak T.
Prace Naukowe Instytutu Matematyki i Informatyki Politechniki Częstochowskiej
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2010
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Vol. 9, nr 2
235-242
EN
In the paper mathematical description of Discontinuous Galerkin Method (DGM) used in the theory of thermoelasticity is presented. Displacement form of governing equations is introduced as the base of mathematical model. Space discretization methodology for discontinuous finite element method is showed.
10
Content available remote Accuracy of numerical solution of heat diffusion equation
Węgrzyn-Skrzypczak E., Skrzypczak T.
Prace Naukowe Instytutu Matematyki i Informatyki Politechniki Częstochowskiej
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2008
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Vol. 7, nr 1
193-197
EN
Presented paper is focused on results of numerical solution of transient heat conductivity equation in two-dimensional region. Convection term is neglected in mathematical model of the phenomenon. Solutions based on classical Galerkin finite element formulation obtained for girds of different qualities are compared to discontinuous Galerkin method. Spatial discretization of computational domain and order of basis functions are taken into account.
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