Jianguo Xin
Abstract
We present the implementation of well-conditioned hierarchical bases for one-dimensional, triangular and tetrahedral elements in finite element FEMLAB software. Using the domain mesh information provided by FEMLAB, we found an easy way to maintain the continuity of solution across the interelement boundaries. The conditionings of the global stiffness matrices of several standard problems are compared with the Lagrange bases and are smaller for all cases.
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Index Terms
- Implementation of hierarchical bases in FEMLAB for simplicial elements
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Reviews
Jingping Long
Hierarchical shape function bases are essential for the efficiency of p -version refinement, in which the polynomial order is increased to improve the accuracy. A good basis should not result in stiffness matrices with large condition numbers. When the mesh consists of quadrilateral or hexahedral elements, such bases can be constructed from a one-dimension orthogonal basis using a tensor product. The construction is not trivial if the mesh has triangular or tetrahedral elements. Implementation is also sometimes challenging. In this paper, the authors discuss the implementation of a certain hierarchical shape function basis that is expected to give significantly smaller condition numbers for certain problems when the polynomial order is high. The main contribution of this paper is the introduction of an ordering convention that automatically guarantees the inter-element continuity. The authors didn't elaborate on how the error decreases as the polynomial order increases. Also, the domains of the given examples are suitable for quadrilateral or hexahedral meshes, thus the need for triangular and tetrahedral meshes is not convincing. Further, only condition numbers are compared. Since the paper focuses on implementation, readers expect to see a superior convergence rate in some realistic, real-world example, which the paper lacks. Online Computing Reviews Service
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