Implementation of an X-FEM Solver for the Classical Two-Phase Stefan Problem

Bernauer, Martin K.; Herzog, Roland

doi:10.1007/s10915-011-9543-x

Cited by 24 publications

(27 citation statements)

References 43 publications

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“…In this work, two methods are used. The first is the penalization method Chessa et al (2002); Bernauer and Herzog (2011);Zabaras et al (2006), which applies the melting temperature on the interface by multiplying (1b) by a very large penalization parameter β (eq. 11) and including it in the weak form (8) for intersected elements only.…”

Section: Finite Element Formulationmentioning

confidence: 99%

“…The absence of nodes on the interface means that we cannot apply specified values directly and an additional constraint must be added in the finite element formulation to apply the appropriate boundary condition on the interface. Bernauer and Herzog (2011) and Lagrange multiplier method Moes et al (2006); Ji and Dolbow (2004); Babuska (1973). The penalty method requires the definition of a free numerical parameter to be determined by trial and error.…”

Section: Introductionmentioning

confidence: 99%

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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: Finite Element Formulationmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

A XFEM Lagrange Multiplier Technique for Stefan Problems

Martin¹,

Chaouki²,

Robert³

et al. 2016

Frontiers in Heat and Mass Transfer

View full text Add to dashboard Cite

show abstract

“…The mass flux interface boundary condition is imposed using the penalty method Chessa et al (2002); Bernauer and Herzog (2011). This technique multiplies the residual form of equation (4c) by a very large penalization parameter β and introduces it in the finite element formulation of the momentum equation.…”

Section: Stokes Problemmentioning

confidence: 99%

“…Equation (22) is first order hyperbolic and must be stabilized to minimize the presence of oscillations in the solution Chessa et al (2002); Bernauer and Herzog (2011). The GLS method is used here Hughes et al (1989).…”

Section: Frontiers In Heat Andmentioning

confidence: 99%

“…Numerous extended finite element models for the solutions of the classical (diffusive) Stefan problem are found in the literature Chessa et al (2002); Bernauer and Herzog (2011); Merle and Dolbow (2002); Ji et al (2002). More complex models involving convection with constant density have also been developed using different numerical techniques Zabaras et al (2006); Vynnycky and Kimura (2007); Brent et al (1988).…”

Section: Introductionmentioning

confidence: 99%

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Modelling of Phase Change With Non-Constant Density Using Xfem and a Lagrange Multiplier

Martin¹,

Chaouki²,

Robert³

et al. 2017

Frontiers in Heat and Mass Transfer

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A two phase model for two-dimensional solidification problems with variable densities was developed by coupling the Stefan problem with the Stokes problem and applying a mass conserving velocity condition on the phase change interface. The extended finite element method (XFEM) was used to capture the strong discontinuity of the velocity and pressure as well as the jump in heat flux at the i nterface. The melting temperature and velocity condition were imposed on the interface using a Lagrange multiplier and the penalization method, respectively. The resulting formulations were then coupled using a fixed point iteration a lgorithm. Three examples were investigated and the results were compared to numerical results coming from a commercial software using ALE techniques to track the solid/liquid interface. The model was able to reproduce the benchmark simulations while maintaining a sharp phase change interface and conserving mass.

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A Nitsche‐based cut finite element solver for two‐phase Stefan problems

Tchinda Ngueyong,

Urquiza,

Martin

2023

Numerical Meth Engineering

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We present a cut finite element method based on ghost penalty stabilization technique for solving two‐phase Stefan problems. The essential interfacial constraints are weakly enforced using the symmetric variant of Nitsche's method. To track the interface efficiently, the level‐set technique is employed and the front location is updated at each time step by solving a transport equation. Because this is a convection problem, we utilize the continuous interior penalty method to stabilize the finite element formulation. According to the Stefan condition, the normal velocity of the interface is proportional to the jump in the interfacial flux. To accurately approximate this quantity, we use a recent post‐processing technique that combines a ghost penalty regularization and the domain integral method on cut elements. We validate our algorithm with several numerical examples that demonstrate optimal convergence.

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Implementation of an X-FEM Solver for the Classical Two-Phase Stefan Problem

Cited by 24 publications

References 43 publications

A XFEM Lagrange Multiplier Technique for Stefan Problems

A XFEM Lagrange Multiplier Technique for Stefan Problems

Modelling of Phase Change With Non-Constant Density Using Xfem and a Lagrange Multiplier

A Nitsche‐based cut finite element solver for two‐phase Stefan problems

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