Investigation of linear dependence problem of three-dimensional partition of unity-based finite element methods
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2025-04-30Title of Paper:
Investigation of linear dependence problem of three-dimensional partition of unity-based finite element methodsJournal:
Computer Meth in Appl Mech and EngngSummary:
A known problem of partition of unity-based generalized finite element methods is the linear dependence of the approximation space, which leads to singular stiffness matrix. Up to now, the linear dependence
problem has not been fully understood and an efficient way to alleviate it is not available. In our previous paper ‘‘Prediction of rank deficiency in partition of unity-based methods with plane triangular or quadrilateral meshes’’ [Comput. Methods Appl. Mech. Engrg. 200 (2011) 665–674], the origin of the
linear dependence problem was first dissected and then a method was proposed to reliably predict the rank deficiency of the linearly dependent global approximations of two-dimensional partition of unitybased
generalized finite element methods. This paper extends the previous work to three-dimensional cases. The linear dependence problem is first investigated at an element level and then extended to the whole mesh. Derivation of general formulations in a three-dimensional setting is undoubtedly more
challenging than a two-dimensional setting because of the complicated element topology. The principle of the increase of rank deficiency is once more applied. The methodology of summing up the added rank
deficiency of each element as that of the whole mesh is further proved to be valid in three-dimensional cases. This work together with the previous work is regarded as the essential step to successfully and completely solve the linear dependence problem in partition of unity-based finite element methods.Co-author:
X.M. An, Z.Y. Zhao, H.H. Zhang, L.X. LiVolume:
233-236Page Number:
137- 151Translation or Not:
NoDate of Publication:
2012-10-18
