Numerical Solution of Thermal Phenomena in Welding Problems
We present a novel finite element method to solve the thermal variables in welding problems. The mathematical model is based on the enthalpy formulation of the energy conservation law, which is simultaneously valid for the solid, liquid, and mushy regions. Both isothermal and non-isothermal melting...
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MDPI AG
2023-07-01
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| Series: | Mathematics |
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| Online Access: | https://www.mdpi.com/2227-7390/11/13/3009 |
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| author | Mario Freire-Torres Manuel Colera Jaime Carpio |
| author_facet | Mario Freire-Torres Manuel Colera Jaime Carpio |
| author_sort | Mario Freire-Torres |
| collection | DOAJ |
| description | We present a novel finite element method to solve the thermal variables in welding problems. The mathematical model is based on the enthalpy formulation of the energy conservation law, which is simultaneously valid for the solid, liquid, and mushy regions. Both isothermal and non-isothermal melting models are considered to relate the enthalpy with the temperature. Quadratic triangular elements with local anisotropic mesh adaptation are employed for the space discretization of the governing equation, and a second-order backward differentiation formula is employed for the time discretization. The resulting non-linear discretized system is solved with a simple Newton algorithm with two versions: the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>θ</mi></semantics></math></inline-formula>-Newton algorithm, which considers the temperature as the main unknown variable, as in most works in the literature, and the <i>h</i>-Newton algorithm, which considers the enthalpy, which is the main novelty of the present work. Then, we show via numerical experiments that the <i>h</i>-Newton method is robust and converges well to the solution, both for isothermal and non-isothermal melting. However, the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>θ</mi></semantics></math></inline-formula>-method can only be applied to the case of non-isothermal melting and converges only for a sufficiently large melting temperature range or sufficiently small time step. Numerical experiments also confirm that the method is able to adequately capture the discontinuities or sharp variations in the solution without the need for any kind of numerical dissipation. |
| first_indexed | 2024-03-11T01:34:07Z |
| format | Article |
| id | doaj.art-5be82230442e4dfda6781890c481462b |
| institution | Directory Open Access Journal |
| issn | 2227-7390 |
| language | English |
| last_indexed | 2024-03-11T01:34:07Z |
| publishDate | 2023-07-01 |
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| series | Mathematics |
| spelling | doaj.art-5be82230442e4dfda6781890c481462b2023-11-18T17:04:31ZengMDPI AGMathematics2227-73902023-07-011113300910.3390/math11133009Numerical Solution of Thermal Phenomena in Welding ProblemsMario Freire-Torres0Manuel Colera1Jaime Carpio2Departamento de Ingenieria Energetica, E.T.S. Ingenieros Industriales, Universidad Politecnica de Madrid, 28006 Madrid, SpainDepartamento de Ingenieria Energetica, E.T.S. Ingenieros Industriales, Universidad Politecnica de Madrid, 28006 Madrid, SpainDepartamento de Ingenieria Energetica, E.T.S. Ingenieros Industriales, Universidad Politecnica de Madrid, 28006 Madrid, SpainWe present a novel finite element method to solve the thermal variables in welding problems. The mathematical model is based on the enthalpy formulation of the energy conservation law, which is simultaneously valid for the solid, liquid, and mushy regions. Both isothermal and non-isothermal melting models are considered to relate the enthalpy with the temperature. Quadratic triangular elements with local anisotropic mesh adaptation are employed for the space discretization of the governing equation, and a second-order backward differentiation formula is employed for the time discretization. The resulting non-linear discretized system is solved with a simple Newton algorithm with two versions: the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>θ</mi></semantics></math></inline-formula>-Newton algorithm, which considers the temperature as the main unknown variable, as in most works in the literature, and the <i>h</i>-Newton algorithm, which considers the enthalpy, which is the main novelty of the present work. Then, we show via numerical experiments that the <i>h</i>-Newton method is robust and converges well to the solution, both for isothermal and non-isothermal melting. However, the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>θ</mi></semantics></math></inline-formula>-method can only be applied to the case of non-isothermal melting and converges only for a sufficiently large melting temperature range or sufficiently small time step. Numerical experiments also confirm that the method is able to adequately capture the discontinuities or sharp variations in the solution without the need for any kind of numerical dissipation.https://www.mdpi.com/2227-7390/11/13/3009energy conservation lawwelding processmelting processsolidification processesisothermal phase-change processnon-isothermal phase-change process |
| spellingShingle | Mario Freire-Torres Manuel Colera Jaime Carpio Numerical Solution of Thermal Phenomena in Welding Problems Mathematics energy conservation law welding process melting process solidification processes isothermal phase-change process non-isothermal phase-change process |
| title | Numerical Solution of Thermal Phenomena in Welding Problems |
| title_full | Numerical Solution of Thermal Phenomena in Welding Problems |
| title_fullStr | Numerical Solution of Thermal Phenomena in Welding Problems |
| title_full_unstemmed | Numerical Solution of Thermal Phenomena in Welding Problems |
| title_short | Numerical Solution of Thermal Phenomena in Welding Problems |
| title_sort | numerical solution of thermal phenomena in welding problems |
| topic | energy conservation law welding process melting process solidification processes isothermal phase-change process non-isothermal phase-change process |
| url | https://www.mdpi.com/2227-7390/11/13/3009 |
| work_keys_str_mv | AT mariofreiretorres numericalsolutionofthermalphenomenainweldingproblems AT manuelcolera numericalsolutionofthermalphenomenainweldingproblems AT jaimecarpio numericalsolutionofthermalphenomenainweldingproblems |