Minimum time-step size for diffusion problem in FEM analysis

Thomas, Hywel Rhys; Zhou, Zheng

doi:10.1002/(sici)1097-0207(19971030)40:20<3865::aid-nme246>3.0.co;2-c

Cited by 42 publications

(47 citation statements)

References 9 publications

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“…The effect of time step size on the solution of transient heat transfer problems using the finite element method has been previously investigated in Thomas and Zhou (1997). However, it is important to note that the penalty formulation was unaffected at the tested time step of ∆x 2 100 ρcp k s (figure 7), suggesting that the Lagrange multiplier is more sensitive to this phenomenon.…”

Section: Phase 1d Problemmentioning

confidence: 95%

A XFEM Lagrange Multiplier Technique for Stefan Problems

Martin¹,

Chaouki²,

Robert³

et al. 2016

Frontiers in Heat and Mass Transfer

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The two dimensional phase change problem was solved using the extended finite element method with a Lagrange formulation to apply the interface boundary condition. The Lagrange multiplier space is identical to the solution space and does not require stabilization. The solid-liquid interface velocity is determined by the jump in heat flux across the i nterface. Two methods to calculate the jump are used and c ompared. The first is based on an averaged temperature gradient near the interface. The second uses the Lagrange multiplier values to evaluate the jump. The Lagrange multiplier based approach was shown to be more robust and precise.

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Section: Phase 1d Problemmentioning

confidence: 95%

A XFEM Lagrange Multiplier Technique for Stefan Problems

Martin¹,

Chaouki²,

Robert³

et al. 2016

Frontiers in Heat and Mass Transfer

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“…Physically unrealisable results such as drying of soil near to an influx boundary can be obtained. To control numerical oscillation, Karthikeyan et al (2001) showed that a simple minimum time-step criteria, developed by Thomas and Zhou (1997) for heat diffusion problems with constant material properties, were also adequate for ensuring non-oscillatory solutions in realistic 2-D seepage flow regimes occurring within unsaturated porous media. The criteria were carefully verified using a range of highly non-linear geo-hydrologic properties commonly encountered in practice.…”

Section: Introductionmentioning

confidence: 99%

Numerical simulation of Richards equation in partially saturated porous media: under-relaxation and mass balance

2007

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Numerical simulation of Richards equation in unsaturated soil is known to be difficult because of the highly non-linear material properties involved. An extensive study was conducted to clarify the role of mass balance and under-relaxation on the rate in which the correct solution is approached. Both finite element and finite difference techniques were considered. For the former, the hbased formulation was compared with a massconservative mixed form. The conductivity function (K) was under-relaxed in two ways while the capacity function (C) is computed following the standard mass or non-mass conservative schemes recommended in literature. For fairly coarse discretisation, it was found that large errors were produced when K was under-relaxed by evaluating it at the average of heads from current and previous time step (UR1), regardless of the numerical scheme used. Maintaining global mass balance is found to have little impact on the accuracy. All numerical schemes that underrelaxed K by computing it at the average of two most recent iterations in the current time step (UR2) converged quicker to the correct solution with increasing discretisation, although more iterations per time step than UR1 is needed to achieve a stable solution. An important practical ramification is that it appears to be possible to achieve reasonably accurate and oscillation-free results using fairly coarse discretisation by making only minor modifications (namely, using UR2 for conductivity function) to the h-based finite element formulation and applying some minimum time step criteria.

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“…This is due to a violation of a discrete maximum principle. The principle further gives a The results are also verified in [57], where a minimum time step-size for the backward Euler method is derived. The phenomenon is different to the spatial oscillations which can be observed when using a reduced quadrature rule, as found in [39].…”

Section: Remarkmentioning

confidence: 73%

“…To avoid the loss of accuracy, a minimum time step criterion is derived. Also [57,62] investigate the effect of time step-size on the accuracy of time integration, in particular with regard to the minimum time step criterion. Kujawski and Wiberg [37] investigated a different class of time integration methods.…”

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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Minimum time-step size for diffusion problem in FEM analysis

Cited by 42 publications

References 9 publications

A XFEM Lagrange Multiplier Technique for Stefan Problems

A XFEM Lagrange Multiplier Technique for Stefan Problems

Numerical simulation of Richards equation in partially saturated porous media: under-relaxation and mass balance

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

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