مهندسی عمران شریف (Nov 2020)
A COMPARATIVE INVESTIGATION ON STIFFNESS MATRIX CONDITIONS FOR FEA, XFEA, AND IGA IN FRACTURE PROBLEM
Abstract
Fracture mechanics is a vast and growing open research field of sciences which concerned with the study of crack propagation. Stiffness matrix is an inherent feature of a numerical method. Its condition has a great influence on numerical calculation and the stability of the solution. Since the application of numerical methods such as the standard finite element method, the extended finite element method and the isogeometric analysis approach in the problems of fracture mechanics has been approved, in this contribution, a theoretical comparison between stiffness matrices is conducted. For this purpose, three computer codes are prepared to make a comparison on the stiffness matrix geometry and mathematical properties such as matrix sparsity, stiffness index, bandwidth, number of independent rows and columns, zero and non-zero elements, symmetric/ nonsymmetric and hermitian/ nonhermitian. The 8-node singular element is used in the framework of the finite element method. And in the case of extended finite element, the principles of enrichment of the interpolation functions of finite element and the application of the partition of unity method is considered. Also in the isogeometric analysis approach repetition of two different control points between two patches can create a discontinuity and also demonstrates a singularity in the stiffness matrix. In addition, the NURBS of order 3 are utilized as the basis functions to approximate the geometry and the solution. By comparing the stress distribution, the accuracy of the calculations and the smoothness of the results are investigated. It is found that, stiffness matrix obtained from the isogeometric analysis method is non-diagonal in the fracture problems. Extended finite element stiffness matrix in comparison with the other methods possess a better numerical conditions.
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