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| Discontinuous patch tests in Hansbo and Hansbo’s type of methods |
| LING Dao sheng1,2, SHI Ji sen1,2, ZHANG Ru ru1,2, WANG Yun gang3 |
| 1.MOE Key Laboratory of Soft soils and Geoenvironmental Engineering, Zhejiang University, Hangzhou 310058, China;2. Institute of Geotechnical Engineering, Zhejiang University, Hangzhou 310058, China;3. Zhejiang University City College, Hangzhou 310015, China |
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Abstract The discontinuous patch test designed by Dolbow etc. was conducted with the augmented finite element method in order to study the computational precision and stability of the Hansbo & Hansbo's type of methods. The test results show that the meshes which adopted 3 node triangular mathematical element pass the discontinuous patch test in machine precision, whereas the meshes which adopted 4 node quadrilateral mathematical element generally fail, and the computational precision is highly depended on the geometrical shape of mathematical element and the integration methods of stiffness matrix of the cracked elements. Further theoretical analysis reveals that the geometrical separation of mathematical and physical element introduces the nonlinear natural coordinate transformation from physical element to mathematical element and the Jacobi determinant of physical element. The integrand of element stiffness matrix is more complicated than that of standard finite element. The numerical integration accuracy was reduced, and the failure of the discontinuous patch test was induced. The theoretical and numerical analysis show that adopting triangular or parallelogram mathematical elements within the potential discontinuous deformation region, and subdividing each cracked physical element into fine integrating subdomain are reasonable and effective ways to ensure the Hansbo & Hansbo's type of methods to pass the discontinuous patch test in high precision．
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Published: 01 November 2015
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Hansbo类有限单元法的非连续分片试验
为了研究Hansbo和Hansbo类有限单元法的数值分析精度和稳定性,基于增强有限元法,对该类有限单元法进行非连续分片试验.结果表明,采用3节点三角形数学单元的网格能够以机器精度通过分片试验,采用4节点四边形数学单元的网格一般不能以机器精度通过分片试验,且计算精度与数学单元形状和刚度矩阵积分方法有关.理论分析证明,数学单元与物理单元分离引入数学单元与物理单元母单元间的非线性坐标变换和不同于数学单元的物理单元Jacobi行列式,导致单元刚度矩阵被积函数比常规有限单元复杂,降低数值积分精度.在非连续变形区域采用三角形或平行四边形数学单元、或将含裂纹物理单元精细划分成积分子域是保证Hansbo和Hansbo类方法高精度通过非连续分片试验的合理和有效方法.
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