基于适合分析T样条的高阶数值流形方法

发布时间：2018-12-30 12:25

【摘要】：数值流形方法是一种非常灵活的数值计算方法,连续体的有限单元方法和块体系统的非连续变形分析方法只是这一数值方法的特例.数值流形方法中高阶位移函数的构造可通过提高权函数的阶次来实现,这种方法往往需要沿单元边界配置适当的边内节点,这些结点的出现增加了前处理的复杂性,特别是对于大型复杂的空间问题.另一方面,在数值流形方法中可通过缩小单元尺寸(h加密)来提高求解精度.当模拟裂纹扩展时,这种细化策略可用来克服裂纹尖端的奇异性.一个传统的解决方案是细化整个网格,但这会导致计算效率的显著降低.将适合分析的T样条(analysis-suitable T-spline,AST)引入数值流形方法中来建立高阶数值流形方法的分析格式,有效的避免了该问题的出现.AST样条基函数具有线性无关,单位分解,局部加密等许多重要性质,使得其非常适合用于工程设计及分析.在引入AST样条后,可通过改变数学覆盖的构造形式建立不同阶次的数值流形方法分析格式;AST样条自身的局部加密性质也使得数值流形方法中的数学网格局部加密更容易实现.算例结果表明:随着AST样条基函数阶次的提高,数值流形方法的计算结果有了明显的改善;基于AST样条基函数的数值流形方法在保持计算精度的前提下降低了自由度的数量.
[Abstract]:Numerical manifold method is a very flexible numerical method. Finite element method for continuum and discontinuous deformation analysis method for block system are only special examples of this numerical method. The construction of higher order displacement function in numerical manifold method can be realized by increasing the order of weight function. This method often needs to configure proper edge nodes along the boundary of the element, and the appearance of these nodes increases the complexity of preprocessing. Especially for large and complex space problems. On the other hand, the solution accuracy can be improved by reducing the element size (h encryption) in the numerical manifold method. When simulating crack growth, this thinning strategy can be used to overcome the singularity of crack tip. One traditional solution is to refine the entire grid, but this results in a significant reduction in computing efficiency. The analysis-suitable T-spline AST is introduced into the numerical manifold method to establish the analytical scheme of the high-order numerical manifold method, which effectively avoids the problem. The AST spline basis function has linear independence and unit decomposition. Local encryption and many other important properties make it very suitable for engineering design and analysis. With the introduction of AST splines, the analysis format of numerical manifold method of different order can be established by changing the structure form of mathematical cover, and the local encryption property of AST spline also makes the local encryption of mathematical mesh in numerical manifold method easier to realize. The numerical results show that the numerical manifold method is improved obviously with the increase of the order of the AST spline basis function, and the numerical manifold method based on the AST spline basis function reduces the number of degrees of freedom on the premise of maintaining the accuracy of the calculation.
【作者单位】：中国科学院武汉岩土力学研究所岩土力学与工程国家重点实验室;中国科学院大学;
【基金】：国家重点基础研究发展计划(973计划)(2014CB047100) 国家自然科学基金(11272330)资助项目
【分类号】：O302

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