Higher‐order extended FEM for weak discontinuities – level set representation, quadrature and application to magneto‐mechanical problems

Abstract

In this article, we present the application of bilinear and biquadratic extended FEM (XFEM) formulations to model weak discontinuities in magnetic and coupled magneto-mechanical boundary value problems. For properly resolving the location of curved interfaces and the discontinuous physical behaviour, the major part of the contribution is devoted to review and develop methods for level set representation of curved interfaces and numerical integration of the weak form in higher-order XFEM formulations. In order to reduce the complexity of the representation of curved interfaces, an element local approach that allows for an automated computation of the level set values and also improves the compatibility between the level set representation and the integration subdomains is proposed. Integration rules for polygons and strain smoothing are applied in conjunction with biquadratic elements and compared with curved integration subdomains. Eventually, a coupled magneto-mechanical demonstration problem is modelled and solved by XFEM. For demonstration purposes, a magneto-mechanical coupling due to magnetic stresses is considered. Errors and convergence rates are analysed for the different level set representations and numerical integration procedures as well as their dependence on the ratio of material parameters at an interface.

Analytic and numeric solution of a magneto-mechanical inclusion problem

“…XFEM is therefore used to model the heterogeneous material structure of the problem mentioned above. The implemented higher-order XFEM formulation for the solution of stationary magnetic and coupled magneto-mechanical field problems in the context of small deformations is extensively described in [17,31]. In this work, we are using bilinear and biquadratic quadrilateral elements which are labeled Q4 and Q8, respectively.…”

“…In the present contribution, we apply an element local level set representation of the interface. On one hand, this reduces the complexity of computational effort, and on the other hand, it improves the compatibility between the level set representation and the integration subdomains, details are discussed in [17]. Since we model the material by assuming a linear relation for each, the magnetic and the mechanical constitutive behavior according to Eqs.…”

“…The magnetic field and resulting force per unit length acting on a magnetizable cylinder surrounded by vacuum induced by an inhomogeneous magnetic field of a current carrying wire are presented in [8]. This problem is improper to prove the convergence of the implemented numeric solution scheme, as it is already outlined in [17]. In [22,24], a thin-walled cylindrical tube is exposed to a homogeneous magnetic field.…”

“…In Sect. 4, the derived exact solution is applied to study the convergence of an XFEM formulation of the coupled magneto-mechanical problem [17]. The paper is closed by some concluding remarks and an outlook to ongoing research in Sect.…”

Semi-implicit representation of sharp features with level sets

“…Accurate integration requires a particular attention (14). Such an approach has been successfully used in several applications, including magneto-mechanical problems (15). Another approach is to dissociate the computation mesh from the grid on which the level set is defined as adopted by Legrain et al (16).…”