The XFEM for nonlinear thermal and phase change problems

Stąpór, Paweł

doi:10.1108/hff-02-2014-0052

Cited by 9 publications

(7 citation statements)

References 15 publications

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“…Thus, the first iteration is modified. Instead of the incremental method, the total approach is utilized, Stąpór (2015). In this case, the tangent matrix is reduced to:…”

Section: Time Approximation and The Newton-raphson Methodsmentioning

confidence: 99%

“…Liu et al (2014) use level set method and XFEM for solving the problem of thermal conduction in particulate composites with various imperfect interfaces. A one-dimensional physically non-linear phase change problem is considered by Stąpór (2015). In this paper, the XFEM is adapted to a two-dimensional case where unequal densities of the phases are taken into account.…”

Section: Introductionmentioning

confidence: 99%

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A two-dimensional simulation of solidification processes in materials with thermo-dependent properties using XFEM

Stąpór

2016

International Journal of Numerical Methods for Heat &Amp; Fluid Flow

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Purpose – The purpose of this paper is to carry out a finite element simulation of a physically non-linear phase change problem in a two-dimensional space without adaptive remeshing or moving-mesh algorithms. The extended finite element method (XFEM) and the level set method (LSM) were used to capture the transient solution and motion of phase boundaries. It was crucial to consider the effects of unequal densities of the solid and liquid phases and the flow in the liquid region. Design/methodology/approach – The XFEM and the LSM are applied to solve non-linear transient problems with a phase change in a two-dimensional space. The model assumes thermo-dependent properties of the material and unequal densities of the phases; it also allows for convection in the liquid phase. A non-linear system of equations is derived and a numerical solution is proposed. The Newton-Raphson method is used to solve the problem and the LSM is applied to track the interface. Findings – The robustness and utility of the method are demonstrated on several two-dimensional benchmark problems. Originality/value – The novel procedure based on the XFEM and the LSM was developed to solve physically non-linear phase change problems with unequal densities of phases in a two-dimensional space.

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“…Thus, the first iteration is modified. Instead of the incremental method, the total approach is utilized, Stąpór (2015). In this case, the tangent matrix is reduced to:…”

Section: Time Approximation and The Newton-raphson Methodsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

A two-dimensional simulation of solidification processes in materials with thermo-dependent properties using XFEM

Stąpór

2016

International Journal of Numerical Methods for Heat &Amp; Fluid Flow

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“…Zuo et al [13] obtained the thermal field in concrete with cooling pipe using the XFEM. Stapór [14] solved nonlinear transient problems with a phase change using the XFEM. The XFEM is a very effective numerical method for modeling the discontinuity, however, it has several drawbacks, for instance, discretization errors exist for complex-shaped structures; only C 0 -continuity of shape function exists; and mesh generation is still required.…”

Section: Introductionmentioning

confidence: 99%

Adaptive Extended Isogeometric Analysis for Steady-State Heat Transfer in Heterogeneous Media

Fang¹,

Yu²,

Yang³

2021

Computer Modeling in Engineering &Amp; Sciences

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Steady-state heat transfer problems in heterogeneous solid are simulated by developing an adaptive extended isogeometric analysis (XIGA) method based on locally refined non-uniforms rational B-splines (LR NURBS). In the XIGA, the LR NURBS, which have a simple local refinement algorithm and good description ability for complex geometries, are employed to represent the geometry and discretize the field variables; and some special enrichment functions are introduced into the approximation of temperature field, thus the computational mesh is independent of the material interfaces, which are described with the level set method. Similar to the approximation of temperature field, a temperature gradient recovery technique for heterogeneous media is proposed, and based on the Zienkiewicz-Zhu recovery technique a posteriori error estimator is defined to automatically identify the locally refined regions. The convergence and performance properties of the developed method are verified by using three numerical examples. The numerical results show that (1) The convergence speed of the adaptive local refinement is faster than that of the uniform global refinement; (2) The convergence rate of the high-order basis functions is faster than that of the low-order basis functions; and (3) The existing inclusions change the local distributions of the temperature, and the extreme values of the temperature gradients take place around the inclusion interfaces.

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“…Merle & Dolbow [5] and Chessa et al [6] apply the XFEM to solve phase-change problems. The analysis of the one-dimensional physically non-linear phase-change problem is considered in [7].…”

Section: Introductionmentioning

confidence: 99%

An Improved XFEM for the Poisson Equation with Discontinuous Coefficients

Stąpór¹

2017

Archive of Mechanical Engineering

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AN IMPROVED XFEM FOR THE POISSON EQUATION WITH DISCONTINUOUS COEFFICIENTSDiscontinuous coefficients in the Poisson equation lead to the weak discontinuity in the solution, e.g. the gradient in the field quantity exhibits a rapid change across an interface. In the real world, discontinuities are frequently found (cracks, material interfaces, voids, phase-change phenomena) and their mathematical model can be represented by Poisson type equation. In this study, the extended finite element method (XFEM) is used to solve the formulated discontinuous problem. The XFEM solution introduce the discontinuity through nodal enrichment function, and controls it by additional degrees of freedom. This allows one to make the finite element mesh independent of discontinuity location. The quality of the solution depends mainly on the assumed enrichment basis functions. In the paper, a new set of enrichments are proposed in the solution of the Poisson equation with discontinuous coefficients. The global and local error estimates are used in order to assess the quality of the solution. The stability of the solution is investigated using the condition number of the stiffness matrix. The solutions obtained with standard and new enrichment functions are compared and discussed.

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The XFEM for nonlinear thermal and phase change problems

Cited by 9 publications

References 15 publications

A two-dimensional simulation of solidification processes in materials with thermo-dependent properties using XFEM

A two-dimensional simulation of solidification processes in materials with thermo-dependent properties using XFEM

Adaptive Extended Isogeometric Analysis for Steady-State Heat Transfer in Heterogeneous Media

An Improved XFEM for the Poisson Equation with Discontinuous Coefficients

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