几种有限体元格式及其在辐射热传导问题中的应用
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摘要
有限体积法由于能保持某些物理量(如质量、能量等)的局部守恒性,它已成为一种广泛应用于科学与工程计算领域的重要数值方法.有限体元(FVE)法是一种重要的有限体积法,虽然对它已有大量的研究工作,但仍存在许多需要研究的问题.ICF问题描述的是一种聚变等离子体流体力学问题,其中三温辐射热传导方程的数值求解在该问题的数值模拟中占有主要工作量,因此,对该类问题的有限体元格式及其快速算法进行研究,是一项具有重要理论和实际应用价值的工作.本文针对定常扩散和二维辐射热传导等问题构造和分析了几种有限体元格式,获得了一批算法和理论结果.
针对定常扩散问题,当扩散系数连续时,在一致四边形网格上,首次从理论上严格证明了逐点意义下等参双线性有限体元解的渐近展式和超收敛结果；当扩散系数间断时,在含混合单元的四边形网格上,构造了两种等参双线性有限体元格式；数值实验验证了理论结果的正确性,表明了混合网格上的第二种格式具有饱和收敛阶.针对线性抛物问题,设计了相应的等参双线性有限体元格式,数值结果表明有限体元解在逐点意义下存在超收敛性.
针对连续系数和间断系数两种情形的定常扩散问题,在非结构四边形网格上,分别建立了两种保对称有限体元格式,对前一情形,当网格拟一致时,从理论上证明了有限体元解在L2和H1范数下均具有饱和收敛阶；针对一类具有非局部边界的二维椭圆问题,设计了一种保对称有限体元格式,并得到其最优L2模估计.数值实验验证了理论结果的正确性,表明了新格式具有一些特点：对扭曲网格、间断系数问题的较强适应性,对流函数在正交网格单元中心点的超逼近性,混合网格上的第二种格式具有饱和收敛阶.
针对连续系数和间断系数两种情形的定常扩散问题,分别在四边形网格上建立了相应的二阶混合有限体元格式,通过建立并从理论上证明了分层基下该有限体元刚度矩阵和二次有限元刚度矩阵的谱等价关系,设计了一种高效GAMG预条件子；数值实验验证了理论结果的正确性和新预条件子的高效性与稳定性,表明了混合网格上的第二种格式具有饱和收敛阶.
针对柱坐标系下多介质输运管单温辐射热传导模型问题,首先设计并分析了相应的分片线性保对称有限体元格式,接着将上述的二阶混合有限体元格式应用于该问题的数值求解,数值实验表明：经新格式计算得到的热传导过程与实际物理现象相符,且具有较小的能量守恒误差.
针对含和不含混合单元两种情形对应的轴对称辐射热传导问题以及含混合单元情形的球截面辐射热传导问题,在Euler四边形网格上建立了相应的保对称有限体元格式和等参双线性有限体元格式,数值结果表明：经这些格式计算得到的温度分布和辐射热传导过程比较合理,且能量守恒误差较小,从而验证了这些格式在求解该类问题时的有效性.
Since the Finite Volume Method can preserve local conservation of cer-tain physical quantities, such as mass, energy and so on, it has become one of the important numerical methods in scientific and engineering computa-tion. The Finite Volume Element (FVE) Method is an important kind of the Finite Volume Methods. Although there existed many literatures about the FVE Method, there still have many problems to study. The Inertial Con-finement Fusion (ICF) problem is one type of fluid mechanic problems about fusional plasma. The efficient numerical methods for the two dimensional three temperatures (2D3T) radiation heat conduction equations are crucial for stimulating ICF problem, because it dominates the CPU time during nu-merical solving. Hence, it is a meaningful work to construct efficient FVE scheme and to design fast linear solver for the radiative heat conduction problem. Serval FVE schemes are systemically discussed on the stationary diffusion problem and radiative heat conduction problem in this dissertation, respectively.
The pointwise asymptotic expansions of the isoparametric bilinear finite volume element is firstly derived and proved on uniform grids for the station-ary diffusion problem with smooth variant coefficients. Two corresponding finite volume element schemes are proposed on the quadrilateral grids with mixed cells. Numerical results confirm the theoretical results, and show the saturated convergent order of the second new scheme on mixed grids and the superconvergence in the sense of pointwise for parabolic problem.
Firstly, two preserving-symmetry finite volume element (SFVE) schemes for the stationary diffusion problem are established on unstructure quadrilat-eral grids. The saturated order of error in both L2-norm and H1-norm for the discrete solutions under quasi-uniform partition is proved when the diffusion coefficient is smooth. Secondly, two SFVE schemes are similarly obtained on the mixed-quadrilateral grids. Finally, one SFVE scheme is designed for the two dimensional elliptic problem with nonlocal boundary conditions, and the optimal error estimates in the L2 norm are derived. Numerical experiments verify the theoretical results and show similar traits about the schemes on mixed grids.
Based on the hierarchical basis, the spectral equivalence is firstly estab-lished for two kinds of stiff matrices from the quadratic finite element and second order mixed-type finite volume element method, then a new precondi-tioner for GMRES is proposed. Two second-order mixed-type finite volume element schemes are constructed on mixed quadrilateral grids. Numerical results confirm our theoretic analysis, show the efficiency and robustness of our preconditioner, and show similar traits about the schemes on mixed grids as above.
A SFVE scheme and a second order mixed-type finite volume element scheme are proposed for the two dimensional radiative heat conduction prob-lem about cylindrical transport pipe. The energy conservative error is an-alyzed for the SFVE scheme. Numerical experiments show that the energy conservative error of new scheme is small, and numerical simulation results are consistent with the actual physical phenomena.
The new schemes on mixed grids above for the stationary diffusion prob-lem are applied to solving radiative heat conduction problems about 2D3T. Numerical results show that the energy conservative errors of new schemes are small and acceptable, and the heat conduction process is consistent with the actual physical phenomena. All numerical results indicte that our new FVE schemes are effective and robust.
针对定常扩散问题,当扩散系数连续时,在一致四边形网格上,首次从理论上严格证明了逐点意义下等参双线性有限体元解的渐近展式和超收敛结果；当扩散系数间断时,在含混合单元的四边形网格上,构造了两种等参双线性有限体元格式；数值实验验证了理论结果的正确性,表明了混合网格上的第二种格式具有饱和收敛阶.针对线性抛物问题,设计了相应的等参双线性有限体元格式,数值结果表明有限体元解在逐点意义下存在超收敛性.
针对连续系数和间断系数两种情形的定常扩散问题,在非结构四边形网格上,分别建立了两种保对称有限体元格式,对前一情形,当网格拟一致时,从理论上证明了有限体元解在L2和H1范数下均具有饱和收敛阶；针对一类具有非局部边界的二维椭圆问题,设计了一种保对称有限体元格式,并得到其最优L2模估计.数值实验验证了理论结果的正确性,表明了新格式具有一些特点：对扭曲网格、间断系数问题的较强适应性,对流函数在正交网格单元中心点的超逼近性,混合网格上的第二种格式具有饱和收敛阶.
针对连续系数和间断系数两种情形的定常扩散问题,分别在四边形网格上建立了相应的二阶混合有限体元格式,通过建立并从理论上证明了分层基下该有限体元刚度矩阵和二次有限元刚度矩阵的谱等价关系,设计了一种高效GAMG预条件子；数值实验验证了理论结果的正确性和新预条件子的高效性与稳定性,表明了混合网格上的第二种格式具有饱和收敛阶.
针对柱坐标系下多介质输运管单温辐射热传导模型问题,首先设计并分析了相应的分片线性保对称有限体元格式,接着将上述的二阶混合有限体元格式应用于该问题的数值求解,数值实验表明：经新格式计算得到的热传导过程与实际物理现象相符,且具有较小的能量守恒误差.
针对含和不含混合单元两种情形对应的轴对称辐射热传导问题以及含混合单元情形的球截面辐射热传导问题,在Euler四边形网格上建立了相应的保对称有限体元格式和等参双线性有限体元格式,数值结果表明：经这些格式计算得到的温度分布和辐射热传导过程比较合理,且能量守恒误差较小,从而验证了这些格式在求解该类问题时的有效性.
Since the Finite Volume Method can preserve local conservation of cer-tain physical quantities, such as mass, energy and so on, it has become one of the important numerical methods in scientific and engineering computa-tion. The Finite Volume Element (FVE) Method is an important kind of the Finite Volume Methods. Although there existed many literatures about the FVE Method, there still have many problems to study. The Inertial Con-finement Fusion (ICF) problem is one type of fluid mechanic problems about fusional plasma. The efficient numerical methods for the two dimensional three temperatures (2D3T) radiation heat conduction equations are crucial for stimulating ICF problem, because it dominates the CPU time during nu-merical solving. Hence, it is a meaningful work to construct efficient FVE scheme and to design fast linear solver for the radiative heat conduction problem. Serval FVE schemes are systemically discussed on the stationary diffusion problem and radiative heat conduction problem in this dissertation, respectively.
The pointwise asymptotic expansions of the isoparametric bilinear finite volume element is firstly derived and proved on uniform grids for the station-ary diffusion problem with smooth variant coefficients. Two corresponding finite volume element schemes are proposed on the quadrilateral grids with mixed cells. Numerical results confirm the theoretical results, and show the saturated convergent order of the second new scheme on mixed grids and the superconvergence in the sense of pointwise for parabolic problem.
Firstly, two preserving-symmetry finite volume element (SFVE) schemes for the stationary diffusion problem are established on unstructure quadrilat-eral grids. The saturated order of error in both L2-norm and H1-norm for the discrete solutions under quasi-uniform partition is proved when the diffusion coefficient is smooth. Secondly, two SFVE schemes are similarly obtained on the mixed-quadrilateral grids. Finally, one SFVE scheme is designed for the two dimensional elliptic problem with nonlocal boundary conditions, and the optimal error estimates in the L2 norm are derived. Numerical experiments verify the theoretical results and show similar traits about the schemes on mixed grids.
Based on the hierarchical basis, the spectral equivalence is firstly estab-lished for two kinds of stiff matrices from the quadratic finite element and second order mixed-type finite volume element method, then a new precondi-tioner for GMRES is proposed. Two second-order mixed-type finite volume element schemes are constructed on mixed quadrilateral grids. Numerical results confirm our theoretic analysis, show the efficiency and robustness of our preconditioner, and show similar traits about the schemes on mixed grids as above.
A SFVE scheme and a second order mixed-type finite volume element scheme are proposed for the two dimensional radiative heat conduction prob-lem about cylindrical transport pipe. The energy conservative error is an-alyzed for the SFVE scheme. Numerical experiments show that the energy conservative error of new scheme is small, and numerical simulation results are consistent with the actual physical phenomena.
The new schemes on mixed grids above for the stationary diffusion prob-lem are applied to solving radiative heat conduction problems about 2D3T. Numerical results show that the energy conservative errors of new schemes are small and acceptable, and the heat conduction process is consistent with the actual physical phenomena. All numerical results indicte that our new FVE schemes are effective and robust.
引文
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