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

Loureiro, Felipe dos Santos; Mansur, Webe João

doi:10.1007/s00466-009-0389-0

Cited by 9 publications

(4 citation statements)

References 20 publications

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“…We next consider a two‐dimensional rectangular slab with initial temperature of unity which is uniform over the entire domain 30. One the left boundary ( x = 0), the Dirichlet boundary condition for the temperature is set to zero (cooled side) while all other boundaries are insulated (null heat flux) as depicted in Figure 14(a).…”

Section: Numerical Examplesmentioning

confidence: 99%

Design of order-preserving algorithms for transient first-order systems with controllable numerical dissipation

Masuri

Sellier

Zhou

et al. 2011

Int. J. Numer. Meth. Engng.

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SUMMARYUsing a new design procedure termed as Algorithms by Design, which we have successfully introduced in our previous efforts for second-order systems, alternatively, we advance in this exposition, the design and development of a computational framework that permits order-preserving second-order time accurate, unconditionally stable, zero-order overshooting behavior, and features with controllable numerical dissipation and dispersion via a family of algorithms for effectively solving transient first-order systems. The key feature is the incorporation of a spurious root to introduce controllable numerical dissipation while preserving second-order accuracy (order-preserving feature) resulting in a two-root system, namely, the principal root ( 1∞ ) and a spurious root ( 2∞ ). In contrast to the classical Trapezoidal family of algorithms which are the most popular, the present framework has the same order of computational complexity, but a higher payoff that is a significant advance to the field for tackling a wide class of applications dealing with first-order transient systems. We also present the special case with selection of 1∞ = 1 and any 2∞ leading to the design of a family of generalized single-step single-solve [GS4-1] algorithms recovering the Crank-Nicolson method at one end ( 2∞ = 1) and the Midpoint Rule at the other end ( 2∞ = 0) and anything in between, all of which have spectral radius features resembling that of the Crank-Nicolson method. More interestingly, with the particular choice of 1∞ = 2∞ = 0, the developed framework additionally inherits L-stable features. We illustrate the successful design of the developed GS4-1 framework using two simple illustrative numerical examples.

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Section: Numerical Examplesmentioning

confidence: 99%

Design of order-preserving algorithms for transient first-order systems with controllable numerical dissipation

Masuri

Sellier

Zhou

et al. 2011

Int. J. Numer. Meth. Engng.

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“…Mancuso et al [42] apply continuous Galerkin formulation for time integration of linear heat conduction problems and obtain A-stable integration schemes of higher order. Loureiro and Mansur [40] use SDIRK method to compute a numerical Green's function matrix for the solution of linear heat conduction problems. In the recent study of [24] singly diagonally implicit Runge-Kutta (SDIRK) schemes are compared with backward difference formula of order two (BDF2).…”

Section: Introductionmentioning

confidence: 99%

Experimental validation of high-order time integration for non-linear heat transfer problems

et al. 2011

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An accurate prediction of the temperature distribution in space and time plays an important role in many industrial applications, in particular when phase transformations are involved. In this article the thermo-physical properties of steel 51CrV4 (SAE 6150) are determined and used in numerical simulations. For the simulation of the temperature field a semi-discrete approach is used, consisting of a finite element approximation in space and a high order RungeKutta integration in time. Several adaptive high-order time integration method (stiffly accurate diagonally implicit Runge-Kutta methods) are applied and their computational efficiency is investigated. The theoretical rates of convergence are achieved for all problems, including the non-linear case. Whereas the second order accurate method of Ellsiepen with time adaptive step-size control proves to be most efficient. Further, the influence of the material model on the simulation results is studied and the numerical results are verified by experiments. The best correlation of the simulation and experimental data is achieved using temperature-dependent parameters.

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“…This is indeed the so-called particular or steady-state solution studied by Loureiro and Mansur [15,16], for which the time shape function concerned with the heat load vector is represented exactly into the final solution. The key feature now is to replace the analytical Green's matrix by its numerical counterpart, i.e.,…”

Section: Appendix Amentioning

confidence: 99%

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

Cited by 9 publications

References 20 publications

Design of order-preserving algorithms for transient first-order systems with controllable numerical dissipation

Design of order-preserving algorithms for transient first-order systems with controllable numerical dissipation

Experimental validation of high-order time integration for non-linear heat transfer problems

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

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