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Convergence behaviour of the enriched scaled boundary finite element method
Author(s) -
Bremm Sophia,
Hell Sascha,
Becker Wilfried
Publication year - 2019
Publication title -
international journal for numerical methods in engineering
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.421
H-Index - 168
eISSN - 1097-0207
pISSN - 0029-5981
DOI - 10.1002/nme.6162
Subject(s) - singularity , gravitational singularity , mathematics , mathematical analysis , finite element method , boundary value problem , discretization , boundary (topology) , rate of convergence , scaling , convergence (economics) , geometry , physics , computer science , computer network , channel (broadcasting) , economics , thermodynamics , economic growth
Summary In this work, a very efficient numerical solution of three‐dimensional boundary value problems of linear elasticity including stress singularities is discussed, focussing on its convergence behaviour. For the employed scaled boundary finite element method, a discretization is only needed at the boundary, while the solution is considered analytically in a scaling coordinate. This presents a major advantage for two‐dimensional problems, when the scaling center is placed at a stress singularity. Unfortunately, three‐dimensional problems usually do not only include point singularities but also line singularities, which results in singular gradients in the boundary coordinates and thereby diminishes the method's original advantages. To alleviate this drawback, this work discusses an enrichment of the separation of variables representation with analytical asymptotic near fields of the line singularities. In contrast to previous works, besides the near‐field functions with λ =0.5, also those with λ =1.5 were determined and used for enrichment. This leads to a high accuracy and it is shown that this approach is required to recover the convergence properties of smooth boundary value problems without singularities when using quadratic Lagrange shape functions. In order to recover the convergence rates for higher order shape functions, near‐field functions with higher singularity exponent have to be included for enrichment.