2003 Fiscal Year Final Research Report Summary
Construction of Numerical Analysis for High-performance Large-Scale Computation
| Project/Area Number |
13304007
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| Research Category |
Grant-in-Aid for Scientific Research (A)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
General mathematics (including Probability theory/Statistical mathematics)
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| Research Institution | Kyushu University |
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Principal Investigator |
TABATA Masahisa Kyushu University, Faculty of Mathematics, Professor, 大学院・数理学研究院, 教授 (30093272)
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| Co-Investigator(Kenkyū-buntansha) |
KANAYAMA Hiroshi Kyushu University, Faculty of Engineering, Professor, 大学院・工学研究院, 教授 (90294884)
USHIJIMA Teruo Univ.of Electro-Communications, Professor, 電気通信学部, 教授 (10012410)
IMAI Hitoshi Tokushima Univ., Faculty of Engineering, Professor, 工学部, 教授 (80203298)
NAKAO Mitsuhiro Kyushu University, Faculty of Mathematics, Professor, 大学院・数理学研究院, 教授 (10136418)
KIKUCHI Fumio Univ.of Tokyo, Graduate School of Mathematical Sciences, Professor, 大学院・数理科学研究科, 教授 (40013734)
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| Project Period (FY) |
2001 – 2003
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| Keywords | finite element method / error estimate / numerical simulation / parallel computation / heat convection problems / moving boundary problems / validated computation / Nedelec's edge elements |
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| Research Abstract |
1.In devising numerical schemes for flow problems, how to approximate the convection torn is a crucial point. Characteristic finite element approximation is based on the approximation of the material derivative, which is the sum of the time derivative term and the convection term. So far finite element schemes of characteristic method of the first-order accuracy in time increment have been used. We have developed a finite element scheme of the second-order accuracy in time increment and obtained the best possible error estimate. This scheme is more robust than the first-order scheme with respect to numerical integration error and can solve flow problems more stably and accurately.
2.We have developed a finite element scheme and established an error estimate for heat convection problems with temperature-dependent viscosity. The viscosity of heat conduction problems such as mantle convection in the Earth and melting glass convection in the furnace is strongly dependent on the temperature.
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The dependence plays an important role in die formation of convection patterns. Our scheme is applicable for the general Rayleigh-Benard problems with temperature-dependent viscosity, thermal conductivity, and thermal expansion coefficient. Using this scheme we have carried out large-scale numerical simulation of Earth's mantle convection in three-dimensional spherical shell and succeeded in obtaining complex heat convection patterns.
3.In the infinite precision computation we have succeeded a large-scale parallel computation using a cluster of high-performance computers with 10CPU and 20GB memory. For one-dimensional boundary-value problems very precise results with precision 4995 digits have been obtained. We have used this system to perform direct numerical simulation of inverse problems and made possible a numerical analysis of inverse problems.
4.Formulating eddy current problems in magnetic vector potential and electric scalar potential, we have solved them using a hierarchical domain decomposition method. This solution has been shown to be effective under the environment of parallel computation. By this method we have carried out large-scale numerical simulation of nonlinear static magnetic problems in magnetic vector potential. Less
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