Study on Discrete Null-Field Equation Methods for Bounded Simply Connected Domains: Better Locations of Source Nodes
The discrete null-field equation method (DNFEM) was proposed based on the null-field equation (NFE) of Green’s representation formulation, where only disk domains were discussed. However, the study of the DNFEM for bounded simply connected domains <i>S</i> is essential for practical appl...
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2024-12-01
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| author | Li-Ping Zhang Zi-Cai Li Hung-Tsai Huang Ming-Gong Lee Alexander L. Kazakov |
| author_facet | Li-Ping Zhang Zi-Cai Li Hung-Tsai Huang Ming-Gong Lee Alexander L. Kazakov |
| author_sort | Li-Ping Zhang |
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| description | The discrete null-field equation method (DNFEM) was proposed based on the null-field equation (NFE) of Green’s representation formulation, where only disk domains were discussed. However, the study of the DNFEM for bounded simply connected domains <i>S</i> is essential for practical applications. Since the source nodes must be located outside of a solution domain <i>S</i>, the first issue in computations is how to locate them. It includes two topics—Topic I: The source nodes must be located not only outside <i>S</i> but also outside the exterior boundary layers. The width of the exterior boundary layers is derived as <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>O</mi><mo>(</mo><mn>1</mn><mo>/</mo><mi>N</mi><mo>)</mo></mrow></semantics></math></inline-formula>, where <i>N</i> is the number of unknowns in the DNFEM. Topic II: There are numerous locations for source nodes outside the exterior boundary layers. Based on the sensitivity index, several better choices of pseudo-boundaries are studied for bounded simply connected domains. The advanced study of Topics I and II needs stability and error analysis. The bounds of condition numbers (Cond) are derived for bounded simply connected domains, and they are similar to those of the method of fundamental solutions (MFS). New error bounds are also provided for bounded simply connected domains. The thorough study of determining better locations of source nodes is also valid for the MFS and the discrete boundary integral equation method (DBIEM). The development of algorithms based on the NFE lags far behind that of the traditional boundary element method (BEM). Some progress has been made by following the MFS, and reported in this paper. From the theory and computations in this paper, the DNFEM may become a competent boundary method in scientific/engineering computing. |
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| institution | Kabale University |
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| language | English |
| publishDate | 2024-12-01 |
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| series | Mathematics |
| spelling | doaj-art-718485e86c1c43499c85c684ac4ae4872025-01-10T13:18:20ZengMDPI AGMathematics2227-73902024-12-0113113110.3390/math13010131Study on Discrete Null-Field Equation Methods for Bounded Simply Connected Domains: Better Locations of Source NodesLi-Ping Zhang0Zi-Cai Li1Hung-Tsai Huang2Ming-Gong Lee3Alexander L. Kazakov4School of Mathematical Sciences, Qufu Normal University, Qufu 273165, ChinaDepartment of Applied Mathematics, National Sun Yat-Sen University, Kaohsiung 80424, TaiwanDepartment of Data Science and Analytics, I-Shou University, Kaohsiung 84001, TaiwanDepartment of Tourism and Leisure/Ph.D. Program in Engineering Science, Chung Hua University, Hsin-Chu 30012, TaiwanMatrosov Institute for System Dynamics and Control Theory, Siberian Branch of the Russian Academy of Sciences, 134, Lermontov St., Irkutsk 664033, RussiaThe discrete null-field equation method (DNFEM) was proposed based on the null-field equation (NFE) of Green’s representation formulation, where only disk domains were discussed. However, the study of the DNFEM for bounded simply connected domains <i>S</i> is essential for practical applications. Since the source nodes must be located outside of a solution domain <i>S</i>, the first issue in computations is how to locate them. It includes two topics—Topic I: The source nodes must be located not only outside <i>S</i> but also outside the exterior boundary layers. The width of the exterior boundary layers is derived as <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>O</mi><mo>(</mo><mn>1</mn><mo>/</mo><mi>N</mi><mo>)</mo></mrow></semantics></math></inline-formula>, where <i>N</i> is the number of unknowns in the DNFEM. Topic II: There are numerous locations for source nodes outside the exterior boundary layers. Based on the sensitivity index, several better choices of pseudo-boundaries are studied for bounded simply connected domains. The advanced study of Topics I and II needs stability and error analysis. The bounds of condition numbers (Cond) are derived for bounded simply connected domains, and they are similar to those of the method of fundamental solutions (MFS). New error bounds are also provided for bounded simply connected domains. The thorough study of determining better locations of source nodes is also valid for the MFS and the discrete boundary integral equation method (DBIEM). The development of algorithms based on the NFE lags far behind that of the traditional boundary element method (BEM). Some progress has been made by following the MFS, and reported in this paper. From the theory and computations in this paper, the DNFEM may become a competent boundary method in scientific/engineering computing.https://www.mdpi.com/2227-7390/13/1/131null-field equationdiscrete null-field equation methodinterior boundary layersexterior boundary layersconformal mappingmethod of fundamental solutions |
| spellingShingle | Li-Ping Zhang Zi-Cai Li Hung-Tsai Huang Ming-Gong Lee Alexander L. Kazakov Study on Discrete Null-Field Equation Methods for Bounded Simply Connected Domains: Better Locations of Source Nodes Mathematics null-field equation discrete null-field equation method interior boundary layers exterior boundary layers conformal mapping method of fundamental solutions |
| title | Study on Discrete Null-Field Equation Methods for Bounded Simply Connected Domains: Better Locations of Source Nodes |
| title_full | Study on Discrete Null-Field Equation Methods for Bounded Simply Connected Domains: Better Locations of Source Nodes |
| title_fullStr | Study on Discrete Null-Field Equation Methods for Bounded Simply Connected Domains: Better Locations of Source Nodes |
| title_full_unstemmed | Study on Discrete Null-Field Equation Methods for Bounded Simply Connected Domains: Better Locations of Source Nodes |
| title_short | Study on Discrete Null-Field Equation Methods for Bounded Simply Connected Domains: Better Locations of Source Nodes |
| title_sort | study on discrete null field equation methods for bounded simply connected domains better locations of source nodes |
| topic | null-field equation discrete null-field equation method interior boundary layers exterior boundary layers conformal mapping method of fundamental solutions |
| url | https://www.mdpi.com/2227-7390/13/1/131 |
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