凸二次函数优化问题在快速多极边界元法中的应用

利用快速多极边界元法(FMM-BEM)求解大规模工程问题最终结为稀疏线性方程组的求解,因此,采用更好的方法求解线性方程组可以提高边界元法的计算效率.本文利用最优化数值技术处理,将稀疏线性方程组的求解等价为求解一个凸二次函数极小化的问题,并利用最优化理论及相关数学理论证明了其解的存在唯一性,为该理论的形成和发展奠定了理论基础.     

The bending of elastic circular ring of non-homogeneous and variable cross section under the actions of arbitrary loads

On the basis of the stepped reduction method suggested in [1], we investigate the problem of the bending of elastic circular ring of non-homogeneous and variable cross section under the actions of arbitrary loads. The general solution of this problem is obtained so that it can be used for the calculations of strength and rigidity of practical problems such as arch, tunnel etc. In order to examine results of this paper and explain the application of this new method, an example is brought out at the end of this paper. Circular ring and arch are commonly used structures in engineering. Timoshenko, S.^[2], Barber, J. R.^[3], Tsumura Rimitsu^[4] et al. have studied these problems of bending, but, so far as we know, it has been solely restricted to the general solution of homogeneous uniform cross section ring. The only known solution for the problems with variable cross section ones has been solely restricted to the solution of special case of flexural rigidity in linear function of coordinates. On account of fundamental equations of the non-homogeneous variable cross section problem being variable coefficients, it is very difficult to solve them. In this paper, we use the stepped reduction method suggested in [1] to transform the variable coefficient differential equation into equivalent constant coefficient one. After introducing virtual internal forces, we obtain general solution of an elastic circular ring with non-homogeneity and variable cross section under the actions of arbitrary loads.     

The Space-Time Finite Element Method for Parabolic Problems

ＩｎｔｒｏｄｕｃｔｉｏｎＴｈｅｅｑｕａｔｉｏｎｓｗｅｃｏｎｓｉｄｅｒｅｄａｒｅａｓｆｏｌｌｏｗｓｕｔ-Δｕ =ｆ(ｕ) ,　　Ω× [0 ,Ｔ] ,ｕ｜ Ω =0 ,　　　　　 Ω× [0 ,Ｔ] ,ｕ( · ,0 ) =ｕ0 ,Ω ,( 1 )ｗｈｅｒｅΩ ∈Ｒ2 ,ｔｈｅｆｕｎｃｔｉｏｎｆ(ｕ)ｓａｔｉｓｆｉｅｓ｜ｆ(ｕ)｜≤ｃ｜ｕ｜ ,　　 ｕ∈Ｃ(Ω) . ( 2 )Ａｎｄｆ(ｕ)ｉｓＬｉｐｓｃｈｉｔｚｃｏｎｔｉｎｕｏｕｓ,ｉ.ｅ.ｉｔｓａｔｉｓｆｉｅｓ｜ｆ(ｕ) -ｆ(ｖ) ｜≤Ｌ｜ｕ-ｖ｜ ,　　 ｕ ,ｖ∈Ｃ(Ω) ,( 3 )ｗｈｅｒｅＬｉｓＬｉｐｓｃｈｉｔｚｃｏｎｓｔａ…     

速多极边界元法解的存在唯一性

本文对求解3维弹性摩擦接触问题的快速多极边界元法(FM- BEM)在数学理论上作了深入探讨.首先,利用向量和子空间理论找出快速优化广义极小残余算法(GMRES(m) )求解边界元方程组所满足的代数条件,使对工程用FM- BEM解的研究转化为对代数问题的讨论,然后,分三步证明了FM- BEM解的存在唯一性,为FM- BEM求解弹性摩擦接触工程问题提供强有力的数学支撑.     

抛物方程的时空有限元方法

讨论了一类半线性抛物方程的自适应有限元方法,即空间连续、时间间断的时空有限元方法。利用有限元方法和有限差分方法相结合的技巧,不对时空网格施加限制条件,证明弱解的存在唯一,并且给出了时间最大模、空间L2模,即L∞(L2)模的误差估计,同时给出了数值分析结果,并对理论结果作了验证。