CAJ | 학술논문

　　基于 Neumann 级数展开的 Monte-Carlo 随机有限元在涉及几何构形存在随机性的问题时，需要对网格进行重新划分，需要极大的计算量。为解决该问题，提高运算效率，提出一种新的计算裂纹问题的随机方法。该方法结合了扩展有限元法与随机有限元法的优点，通过对扩展有限元控制方程进行 Neumann 展开，可方便地处理几何构形的随机性，不需重新划分网格。该方法具有计算量小，计算效率高的优点，并能保持较高的计算精度。利用矩阵级数理论讨论了该方法的收敛性。最后通过数值算例验证了该方法的有效性。
　　기우 Neumann 급수전개적 Monte-Carlo 수궤유한원재섭급궤하구형존재수궤성적문제시，수요대망격진행중신화분，수요겁대적계산량。위해결해문제，제고운산효솔，제출일충신적계산렬문문제적수궤방법。해방법결합료확전유한원법여수궤유한원법적우점，통과대확전유한원공제방정진행 Neumann 전개，가방편지처리궤하구형적수궤성，불수중신화분망격。해방법구유계산량소，계산효솔고적우점，병능보지교고적계산정도。이용구진급수이론토론료해방법적수렴성。최후통과수치산례험증료해방법적유효성。
The introduction of the full paper discusses relevant matters and then proposes the method mentioned in the title.Sections 1 and 2 present the stochastically extended finite element method based on Neumann expansions ;this method is the main and core result of our research .Section 1 introduces the extended finite element method as applied to crack propagation.Section 2 presents the method mentioned in the title.The Neumann expansion of the control equation of extended finite element method is employed in this method so as to deal with the randomness of geometric configuration conveniently without remeshing .Convergence of this method is discussed using the matrix theory in section 3.The validity and efficiency of this method are verified with numerical examples in section 4. The calculation results, presented in Fig 2, and their analysis show that this method has the advantage of high com -putational efficiency; also it maintains excellent computation accuracy .