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COMPUTATIONAL METHODS FOR ENGINEERING SCIENCE
Edited by: B.H.V. Topping
Strategies for Incorporating Material Discontinuities into Finite Element Formulations
R. Darvizeh and K. Davey
School of MACE, The University of Manchester, United Kingdom
R. Darvizeh, K. Davey, "Strategies for Incorporating Material Discontinuities into Finite Element Formulations", in B.H.V. Topping, (Editor), "Computational Methods for Engineering Science", Saxe-Coburg Publications, Stirlingshire, UK, Chapter 7, pp 167-192, 2012. doi:10.4203/csets.30.7
Keywords: material discontinuities, finite elements, non-physical variables.
Researchers have investigated the modelling of material-discontinuity phenomena using different numerical approaches such as the extended finite element method (XFEM), molecular dynamics simulation, mesh free methods, adaptive mesh refinement, and the discontinuous Galerkin FEM (DGFEM). The focus of the work presented here is restricted to FE-based methods developed to capture material discontinuities. The two most common FE-based techniques which are used for the modelling of material discontinuities are the discontinuous Galerkin finite element method (DGFEM) and the generalized or extended finite element method (G/XFEM). Both DGFE and XFE methods are based on the general framework of the standard finite element method with some extensions (i.e. XFEM) or violations (i.e. DGFEM). However, an alternative approach exists (introduced here), which lies completely within the framework of the continuous Galerkin FEM (CGFEM) without violation or extension for approximation of a discontinuous solution.
A recent development for the modelling of material discontinuities is discussed in the paper. The method presented in this paper is founded on the non-physical variable concept and is known as the non-physical finite element method (NFEM). The NFEM involves replacing a discontinuous physical-field variable by a limiting-continuous non-physical variable, which is continuous over the domain and has a source like behaviour at the place of discontinuity. A non-physical variable is mathematically defined and related to its physical counterpart but possesses limiting continuity and substitutes for the discontinuous physical field variables in an equivalent form of the governing equations. This approach permits the discretisation of the non-physical variable with the polynomial basis standard to the CGFEM. The strategy of the non-physical method was originally devised for the modelling of material discontinuities in solidification problems [1, 2] involving a strong discontinuity in enthalpy and a weak discontinuity in temperature. The work presented here extends previous works by providing a general framework for the non-physical method to facilitate modelling of strong discontinuities in all the state variables and velocity arising with material discontinuities. The approach is founded on the transport form of the governing conservation laws. The advantage of the non-physical methodology is that it permits the precise annihilation of discontinuous behaviour in the governing finite element equations by means of a distribution like source term at the discontinuity location.
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