Enrichment terms for line singularities in second-order elliptic boundary value problems in 3D heterogeneous media: Application to heat conduction problems

Abstract

A novel enrichment technique is proposed for the solution quality enhancement near weak singularities along straight lines in second-order elliptic boundary value problems. The considered 3D media can generally be heterogeneous. Enrichment is applied through construction of proper 3D singular functions for heterogeneous media. An advantage over similar approaches is that the singular bases are constructed without any knowledge of the analytical singularity order. To this end, the governing PDE is considered in a cylindrical fictitious domain whose axis lies along the singular line, over which the 3D equilibrated singular basis functions are developed. Combined with the smooth solution, the total solution undergoes imposition of the boundary conditions, so that the proper singular functions are automatically built. The singular solution series is primarily made by a combination of first kind Chebyshev polynomials and trigonometric functions, subjected to weighted residual imposition of the PDE to extract the equilibrated singular functions. Meanwhile, the cumbersome 3D integrals break into algebraic combinations of 1D predefined integrals, so that no numerical quadrature will be needed. The bases are tested as a boundary method in the solution of multiple 3D problems including weak singularities, to show their accuracy and efficiency. The proposed singular bases may be implemented in enriched techniques such as XFEM.