一类抛物型方程有限元算法的计算准则
Criteria of Finite Element Algorithm for a Class of Parabolic Equation
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摘要: 用有限元法分析瞬态温度场,很有可能得到“振荡”和“超界”的计算结果.这两种现象不符合热传导规律.为解决此问题,我们提出时间单调性和空间单调性的概念,推导出三维无源热传导方程的数值解的时间单调性的几组充分条件.对某些特殊边值问题,使用规则单元网格,可以得到合理结果时Δt/Δx2的上下界公式.文中还研究了空间单调性.最后我们还讨论了集中质量阵的算法.针对以热传导方程为代表的这一类抛物型方程的有限元算法,我们创造性地给出几组计算准则.Abstract: In finite element analysis of transient temperature field, it is quite notorious that the numerical solution may quite likely oscillate and/or exceed the reasonable scope, which violates the natural law of heat conduction. For this reason, we put forward the concept of lime monotony and spatial monotony, and then derive several sufficient conditions for nionotonic solutions in lime dimension for 3-D passive heal conduction equations with a group of finite difference schemes. For some special boundary conditions and regular element meshes, the lower and upper bounds for Δt/Δx2 can be obtained from those conditions so that reasonable numerical solutions are guaranteed. Spatial monotony is also discussed. Finally, the lumped mass method is analyzed.We creatively give several new criteria for the finite element solutions of a class of parabolic equation represented by heal conduction equation.
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[1] 孔祥谦,《有限单元法在传热学中的应用》,第二版,科学出版社,北京(l1986), 169-180. [2] 515科研组,瞬变温度场问题的有限元解法和最大模原理,计算数学,4,2 (1982), 113-120. [3] 黄玉霞,带第三类边界条件的热传导方程的守恒型格式和最大模原理,计算数学,5,2 (1983),220-224. [4] Rank,E.C.Ratz,and H.Werner,On the impertance of the diserele maximum principle in transient analysis using finite element method.Int.J.Numer.Meth.Eng.,19,12(1983),1771-1782. [5] Zienkiewicz, O, C,著,尹泽勇、江伯南译,唐立民、刘迎曦校,《有限元法》,科学出版社,北京(1986), 600-637. [6] Householder, A. S.著,孙家旭等,《数值分析中的矩阵论》,科学出版社,北京(1986),9-32. [7] 张柔雷、佘颖禾,有限元法求解暂态温度场中阶跃现象的分析,南京工学院学报,3 (1983),102-107. [8] 纪峥,关于瞬态温度场有限元分析中采用协调或集中质量热容阵的探讨,计算结构力学及其应用,3, 2 (1986), 35-41. [9] 欧阳华江、肖丁,一维热传导方程有限元解的单调性准则,计算结构力学及其应用.(待发表). [10] 欧阳华江,混凝土长期变形的基本理论及其应用,大连理工大学博士论文(1989),29-68. -
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