A boundary element formulation for the heat equation with dissipative and heat generation terms

Pettres, Roberto; Lacerda, Luiz Alkimin de; Carrer, J. A. M.

doi:10.1016/j.enganabound.2014.11.005

Cited by 16 publications

(5 citation statements)

References 9 publications

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“…The BEM formulation used in order to validate the FDM-T approach has been used in several studies with numerical and analytical validations. Recent works can be motioned such as Pettres, Lacerda, & Carrer (2015) applied the BEM formulation for the pure diffusion problem, and the numerical results were validated with the analytical solution and linear regression was applied to evaluate the correlation between the solutions. Also, Carrer, Cunha, & Mansur (2017) present a study for two-dimensional diffusion-advection equation using BEM for non-isotropic material.…”

Section: Numerical Examplesmentioning

confidence: 99%

“…To ensure the confiability of results for this approach, the Boundary Element Method is used to compare and validated the results. The BEM has been successfully applied in the analysis of diffusion-advection equation in the last few years e. g. (Carrer, et al, 2017, Pettres, et al, 2015and Cunha, et al, 2013, as pure diffusion problems see Carrer, et al (2012), Ochiai, et al (2006 and Pettres, et al (2015). Numerical methods and theory of solutions of the problems for differential equations using FDM have been studied extensively by many researchers (Liu, Cheng, & Li Xiaoli, 2018), (Čiegis & Suboč, 2018) and so on.…”

Section: Introductionmentioning

confidence: 99%

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A finite difference method formulation based on the time integral domain for the solution of Transient diffusion-advection problem

Oliveira

Thoaldo²,

Dias³

2022

J. Eng. Exact Sci.

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This paper is concerned with the development of a formulation of the Finite Difference Method, in the solution of the transient diffusion-advection equation. In this approach, a weighting function with respect to time is used in the fundamental differential equation. By assuming a linear variation in some time interval, a time integration is performed. This integration reduces the order of time derivative in the basic equation and the initial conditions can be taken into account directly in consequence. Three pollutant transport problems are presented in order to verify the stability and accuracy of the proposed approach. The results for this approach are compared with the Boundary Element Methods results. The comparisons of numerical results show a good agreement between the furnished results for the proposed approach and the boundary element method results assumed here as reference results. By analyzing the results one can conclude that the formulation presented in this paper is capable of producing accurate results for the problem.

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Section: Numerical Examplesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

A finite difference method formulation based on the time integral domain for the solution of Transient diffusion-advection problem

Oliveira

Thoaldo²,

Dias³

2022

J. Eng. Exact Sci.

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“…) is a Dirac's delta, 𝑋 __ and 𝑠 are the localization and magnitude of source, respectively, represents the beginning and reproduction of the cancer cells in an empty domain, or healthy tissue, for example. Particularly, if 𝜏 = 0 it is found the diffusion equation with dissipative term and heat generation that was studied by (Pettres et al, 2015).…”

Section: __mentioning

confidence: 99%

“…Comment on the singular integrationssame as in the classical 2D BEM formulation. For more details, see (Pettres et al, 2015). The domain integrals were calculated using the twodimensional Gauss quadrature method as described in (Pettres, 2021) and illustrate by Figure 2 that represent the location of a triangular cell and some Gaussian points and weights.…”

Section: D-bem Formulationmentioning

confidence: 99%

A second dynamic population invasion study from reactive telegraph equation and boundary element formulation – a numerical assay about tumour growth in vitro

Pettres

Cirilo

2022

J. Biotechnol. Biodivers.

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This paper is a continuation of a study already carried out on the use of the reactive-telegraph equation to analyse problems of population dynamics based on a formulation of the boundary element method (BEM). In this paper, the numerical model simulates the evolution of a tumour as a problem of population density of cancer cells from different reactive terms coupled to the reactive-telegraph equation to describe the growth and distribution of the population, similar to the two-dimensional in vitro tumour growth experiment. The mathematical model developed is called D-BEM, uses a time independent fundamental solution and the finite difference method is combined with BEM to approximate the time derivative terms and the Gaussian quadrature is used to calculate the domain integrals. The solution of the system nonlinear of equations is based on the Gaussian elimination method and the stability of the proposed formulation was verified. As the telegraph equation has a wave behaviour, a phase change phenomenon that can lead to the appearance of negative population density may occur, an algorithm was developed to guarantee the solution's positivity. Important results were obtained and demonstrate the effect of the delay parameter on tumour growth. In one of the tested cases, the results indicated an oscillatory behaviour in the tumour growth when the delay parameter assumed increasing values. The results of numerical simulations that sought to represent tumour growth, as well as the entire formulation of the boundary elements are presented below.

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“…For these kind of problems, the fundamental solution of simplified homogeneous, linear heat conduction problem may still be employed with a price of appearance of domain integrals within the boundary integral equations. One approach could be use of internal cells [24] for the computation of the domain integrals which would deteriorate the boundary only merit of BEM. Alternatively, the domain integrals can be transformed into boundary integrals through a set of interpolation functions and particular solutions.…”

Section: Introductionmentioning

confidence: 99%

Isogeometric and NURBS-enhanced boundary element formulations for steady-state heat conduction with volumetric heat source and nonlinear boundary conditions

Gumus¹,

Baranoğlu²,

Çetin³

2022

Engineering Analysis with Boundary Elements

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A boundary element formulation for the heat equation with dissipative and heat generation terms

Cited by 16 publications

References 9 publications

A finite difference method formulation based on the time integral domain for the solution of Transient diffusion-advection problem

A finite difference method formulation based on the time integral domain for the solution of Transient diffusion-advection problem

A second dynamic population invasion study from reactive telegraph equation and boundary element formulation – a numerical assay about tumour growth in vitro

Isogeometric and NURBS-enhanced boundary element formulations for steady-state heat conduction with volumetric heat source and nonlinear boundary conditions

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