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Construction of a Practical Computation Code for Heat Convection Problems with Slow Flow

慢流热对流问题实用计算代码的构建

基本信息

批准号：
11554003
负责人：
TABATA Masahisa
金额：
$ 8.13万
依托单位：
KYUSHU UNIVERSITY
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (B)
财政年份：
1999
资助国家：
日本
起止时间：
1999 至 2001
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/en/grant/KAKENHI-PROJECT-11554003/
关键词：
Finite element method Parallel computation Earth's mantle convection Melting glass convection Temperature dependent viscosity Heat convection problems Error estimation constant Accuracy guaranteed computation 一様可解性硝子溶解問題渦輪の速度安定化有限

项目摘要

(1) We have built a finite element scheme for solving numerically heat convection problems with slow flow like Earth's mantle convection in geophysics and melting glass convection in glass product furnaces. We have shown unconditional stability of the scheme and the convergence rate of the finite element solutions. These problems are modeled by Rayleigh-Benard equations with infinite Prandtl number, whose viscosity is strongly dependent on temperature. The obtained scheme is practically useful for three-dimensional problems. In order to reduce computation load we have employed the tetrahedral linear element for every unknown functions, velocity, pressure and temperature, and used stabilized finite element method.(2) We have constructed a computation code for the scheme mentioned above and implemented it on parallel computers. The Earth's mantle convection problem is solved in a spherically symmetric domain. By virtue of this property we have divided the domain into the union of congrue … More nt subdomains, which have allowed us to keep only stiffness matrices in a representative subdomain in solving Stokes equations by a preconditioned iterative method. As a result the required memory has reduced drastically. We could get speeding up of about 20 times in using 24 CPUs of Fujitsu GP7000, a shared memory type computer at Computing and Communications Center, Kyushu University. Using this code, we have studied the viscosity ratio dependency of stationary temperature fields and flow patterns. When the ratio increases, the heads of plumes flatten and the number of plumes increases.(3) We have presented a numerical verification method for solutions of the Navier-Stokes equations, and succeeded in the verification for low Reynolds number problems. Performing accuracy guaranteed computation, we have given a computer aided proof to the existence of bifurcation branches for two-dimensional heat convection problems.(4) Using a code for the convection in a three-dimensional sphere, we have studied the relation between the existence of continents and mantle convection. We have shown numerically that plumes arive under continents in some tens of billion years. Less

(1)本文建立了一个有限元格式，用于数值求解慢流热对流问题，如地球物理学中的地幔对流问题和玻璃制品窑炉中的熔融玻璃对流问题。我们证明了格式的无条件稳定性和有限元解的收敛速度。这些问题是由Rayleigh-Benard方程与无限普朗特数，其粘度是强烈依赖于温度。所得到的格式对三维问题是实用的。为了减少计算量，对速度、压力、温度等未知函数采用四面体线性单元，并采用稳定化有限元法。(2)我们已经为上述方案构造了一个计算程序，并在并行计算机上实现了它。在球对称区域内求解地幔对流问题。根据这个性质，我们将域划分为congrue的并集 ...更多信息 nt子域，这使得我们在用预条件迭代法求解Stokes方程时，只保留一个代表子域中的刚度矩阵。因此，所需的内存大幅减少。我们可以得到约20倍的速度在使用富士通GP 7000，在计算和通信中心，九州大学的共享内存型计算机的24个CPU。利用这个程序，我们研究了稳态温度场和流型的粘度比依赖性。当比值增加时，羽状流头部变平，羽状流数量增加。(3)提出了一种求解Navier-Stokes方程的数值验证方法，并成功地对低雷诺数问题进行了验证。通过保精度计算，给出了二维热对流问题分歧分支存在性的计算机辅助证明。(4)本文利用三维球内对流的数值模拟程序，研究了大陆的存在与地幔对流的关系。我们已经用数字表明，地幔柱在大陆下面存在了大约几百亿年。少