3D numerical simulation of an anisotropic bead type thermistor and multiplicity of solutions
We perform some 3D numerical experiments for the approximation of the solutions to a bead type thermistor problem. We consider the case of a diagonal anisotropic diffusion matrix whose jth entry is of the form |∂u/∂xj|pj−2∂u/∂xj, u being the temperature inside the thermistor and the exponents pj, 1≤...
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| Published in: | Mathematics and computers in simulation Vol. 220; pp. 640 - 672 |
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Elsevier B.V
01.06.2024
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| ISSN: | 0378-4754, 1872-7166 |
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| Abstract | We perform some 3D numerical experiments for the approximation of the solutions to a bead type thermistor problem. We consider the case of a diagonal anisotropic diffusion matrix whose jth entry is of the form |∂u/∂xj|pj−2∂u/∂xj, u being the temperature inside the thermistor and the exponents pj, 1≤j≤3, lie in the interval (1,+∞). We first show some existence results for different notions of solutions, prove a maximum principle for each type of solution, and study certain symmetry properties for these solutions in a bead type thermistor. These properties lead us to the introduction of a symmetric solution and we show the existence of such a solution.
We have developed a numerical algorithm for the computation of the numerical solutions in a bead type thermistor. This algorithm combines a fixed-point technique with a standard finite element method (FEM). Some numerical tests have shown the existence of non-symmetric solutions and this leads to multiple many solutions (at least three). We discuss the numerical results obtained for different values of the exponents pj and the applied voltage on different meshes. |
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| AbstractList | We perform some 3D numerical experiments for the approximation of the solutions to a bead type thermistor problem. We consider the case of a diagonal anisotropic diffusion matrix whose jth entry is of the form |∂u/∂xj|pj−2∂u/∂xj, u being the temperature inside the thermistor and the exponents pj, 1≤j≤3, lie in the interval (1,+∞). We first show some existence results for different notions of solutions, prove a maximum principle for each type of solution, and study certain symmetry properties for these solutions in a bead type thermistor. These properties lead us to the introduction of a symmetric solution and we show the existence of such a solution.
We have developed a numerical algorithm for the computation of the numerical solutions in a bead type thermistor. This algorithm combines a fixed-point technique with a standard finite element method (FEM). Some numerical tests have shown the existence of non-symmetric solutions and this leads to multiple many solutions (at least three). We discuss the numerical results obtained for different values of the exponents pj and the applied voltage on different meshes. |
| Author | Lahrache, Manar Ortegón Gallego, Francisco Rhoudaf, Mohamed |
| Author_xml | – sequence: 1 givenname: Manar orcidid: 0000-0003-0720-6629 surname: Lahrache fullname: Lahrache, Manar email: manar.lahrache@edu.umi.ac.ma organization: Département de Mathématiques, Faculté des Sciences, Université Moulay Ismaïl, BP 11201 Zitoune, Meknès, Morocco – sequence: 2 givenname: Francisco orcidid: 0000-0002-9821-8002 surname: Ortegón Gallego fullname: Ortegón Gallego, Francisco email: francisco.ortegon@uca.es organization: Departamento de Matemáticas, Facultad de Ciencias del Mar y Ambientales, Universidad de Cádiz, Campus del Río San Pedro, 11510 Puerto Real, Spain – sequence: 3 givenname: Mohamed surname: Rhoudaf fullname: Rhoudaf, Mohamed email: m.rhoudaf@umi.ac.ma organization: Département de Mathématiques, Faculté des Sciences, Université Moulay Ismaïl, BP 11201 Zitoune, Meknès, Morocco |
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| Keywords | Thermistor problem Finite element method Non-uniformly elliptic system Anisotropic Sobolev space Numerical solution |
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| Title | 3D numerical simulation of an anisotropic bead type thermistor and multiplicity of solutions |
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