An Improved XFEM for the Poisson Equation with Discontinuous Coefficients

Stąpór, Paweł

doi:10.1515/meceng-2017-0008

Cited by 2 publications

(2 citation statements)

References 17 publications

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“…5a. As can be seen, the approximation of the element allows for a kink in the velocity and temperature fields and a jump in the pressure field along the interface line, without any unphysical oscillations, which can be observed for the standard XFEM approaches, [44].…”

Section: Water Freezing With Natural Convectionmentioning

confidence: 93%

Archive of Mechanical Engineering

Stąpór

2019

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This paper presents an extended finite element method applied to solve phase change problems taking into account natural convection in the liquid phase. It is assumed that the transition from one state to another, e.g., during the solidification of pure metals, is discontinuous and that the physical properties of the phases vary across the interface. According to the classical Stefan condition, the location, topology and rate of the interface changes are determined by the jump in the heat flux. The incompressible Navier-Stokes equations with the Boussinesq approximation of the natural convection flow are solved for the liquid phase. The no-slip condition for velocity and the melting/freezing condition for temperature are imposed on the interface using penalty method. The fractional four-step method is employed for analysing conjugate heat transfer and unsteady viscous flow. The phase interface is tracked by the level set method defined on the same finite element mesh. A new combination of extended basis functions is proposed to approximate the discontinuity in the derivative of the temperature, velocity and the pressure fields. The single-mesh approach is demonstrated using three two-dimensional benchmark problems. The results are compared with the numerical and experimental data obtained by other authors.

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Section: Water Freezing With Natural Convectionmentioning

confidence: 93%

Archive of Mechanical Engineering

Stąpór

2019

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“…The aim of the paper is to reach enhanced gradient predictions, as part of the solution of the Poisson equation which represents the mathematical model of a problem. This aim can be achieved with specific enrichment functions as it is shown in [1], however the stability of solution as well as computational cost is then deteriorated. The alternative approach is based on post-processing techniques which do not influence the size of the problem and stability of the solution.…”

Section: Introductionmentioning

confidence: 99%

Archive of Mechanical Engineering

Stąpór

2019

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In the paper, the extended finite element method (XFEM) is combined with a recovery procedure in the analysis of the discontinuous Poisson problem. The model considers the weak as well as the strong discontinuity. Computationally efficient low-order finite elements provided good convergence are used. The combination of the XFEM with a recovery procedure allows for optimal convergence rates in the gradient i.e. as the same order as the primary solution. The discontinuity is modelled independently of the finite element mesh using a step-enrichment and level set approach. The results show improved gradient prediction locally for the interface element and globally for the entire domain.

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An Improved XFEM for the Poisson Equation with Discontinuous Coefficients

Cited by 2 publications

References 17 publications

Archive of Mechanical Engineering

Archive of Mechanical Engineering

Archive of Mechanical Engineering

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