Numerical solution for the linear transient heat conduction equation using an Explicit Green’s Approach

Mansur, Webe João; Vasconcellos, Carlos Alexandre Bastos de; Zambrozuski, N.J.M.; Filho, Otto Corrêa Rotunno

doi:10.1016/j.ijheatmasstransfer.2008.07.036

Cited by 33 publications

(17 citation statements)

References 21 publications

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“…In fact, the optimal value is that for which the vertical line in Figure 3 , i.e. equal substeps [13,15], the stability constraint is 4 i t λ ∆ ≤ and iii) for the optimal 1 α , the stability constraint is increased to …”

Section: Time Stability Regionmentioning

confidence: 99%

The Explicit Green’s Approach with stability enhancement for solving the bioheat transfer equation

Loureiro¹,

Mansur²,

Wrobel³

et al. 2014

International Journal of Heat and Mass Transfer

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The aim of this paper is to propose a strategy for performing a stability enhancement into the Explicit Green's Approach (ExGA) method applied to the bioheat transfer equation. The ExGA method is a time-stepping technique that uses numerical Green's functions in the time domain; these functions are here computed by the FEM.Basically, a new two nonequal time substeps procedure is proposed to compute Green's functions at the first time step. This is accomplished by adopting the standard explicit Euler scheme and an optimized procedure to yield the best stability constraint, allowing a reduction into the number of time steps without loss of accuracy. In addition, the concept of local numerical Green's functions is introduced and explored aiming at reducing the computational effort of nodal Green's functions calculation. Two examples are presented in order to show the potentialities of the proposed methodology, one to illustrate the accuracy and another applied to skin burn simulations.

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Section: Time Stability Regionmentioning

confidence: 99%

The Explicit Green’s Approach with stability enhancement for solving the bioheat transfer equation

Loureiro¹,

Mansur²,

Wrobel³

et al. 2014

International Journal of Heat and Mass Transfer

Self Cite

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“…where [C] is the capacitance matrix, [K] is the thermal conductance matrix, and {F} is a vector of equivalent nodal heat loads [3]. Applying the Laplace transform to (7) (Boyce and DiPrima [8]), and following the procedure presented by Mansur et al [3], the final expression of the vector temperature in terms of Green's matrix is:…”

Section: Numerical Green's Functionmentioning

confidence: 99%

“…Applying the Laplace transform to (7) (Boyce and DiPrima [8]), and following the procedure presented by Mansur et al [3], the final expression of the vector temperature in terms of Green's matrix is:…”

Section: Numerical Green's Functionmentioning

confidence: 99%

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Numerical Green’s function for a two-dimensional diffusion equation

Vasconcellos

Ferro

Mansur

et al. 2009

Mesh Reduction Methods

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This paper presents a novel form to calculate Green's function by using a numerical method. In this paper, Green's function is calculated for the twodimensional diffusion equation. The numerical Green's function is defined as Green's matrix that represents the domain of the problem to be solved in terms of the physical properties and geometrical characteristic. Green's matrix is the basis of the numerical method called 'Explicit Green's Approach' (ExGA) that allows explicit time marching with a time step larger than the one required by other methods found in the literature, without losing precision. The method uses Green's matrix which is determined numerically by the Finite Element Method (FEM). The paper presents one application in heat conduction and another in groundwater flow, demonstrating that the results are quite accurate when compared to analytical solutions and to other numerical solutions.

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“…Loureiro [14] reported many successful applications of the Explicit Green's Approach using both implicit and explicit methods, such as the Newmark family of algorithms to integrate the dynamic equation originated from the finite element method spatial discretization. After that, also in a finite element context, Mansur et al [19] and Vasconcellos [24] extended the ExGA to heat conduction problems where the Green's function was computed by the traditional CrankNicolson method.…”

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confidence: 99%

A new family of time integration methods for heat conduction problems using numerical green’s functions

Loureiro

Mansur

2009

Comput Mech

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This paper is concerned with the formulation and numerical implementation of a new class of time integration schemes applied to linear heat conduction problems. The temperature field at any time level is calculated in terms of the numerical Green's function matrix of the model problem by considering an analytical time integral equation. After spatial discretization by the finite element method, the Green's function matrix which transfers solution from t to t + t is explicitly computed in nodal coordinates using efficient implicit and explicit Runge-Kutta methods. It is shown that the stability and the accuracy of the proposed method are highly improved when a sub-step procedure is used to calculate recursively the Green's function matrix at the end of the first time step. As a result, with a suitable choice of the number of sub-steps, large time steps can be used without degenerating the numerical solution. Finally, the effectiveness of the present methodology is demonstrated by analyzing two numerical examples.

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Numerical solution for the linear transient heat conduction equation using an Explicit Green’s Approach

Cited by 33 publications

References 21 publications

The Explicit Green’s Approach with stability enhancement for solving the bioheat transfer equation

The Explicit Green’s Approach with stability enhancement for solving the bioheat transfer equation

Numerical Green’s function for a two-dimensional diffusion equation

A new family of time integration methods for heat conduction problems using numerical green’s functions

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