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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.

We studied regenerating bilayered tissue toroids dissected from Hydra vulgaris polyps and relate our macroscopic observations to the dynamics of force-generating mesoscopic cytoskeletal structures. Tissue fragments undergo a specific toroid-spheroid folding process leading to complete regeneration towards a new organism. The time scale of folding is too fast for biochemical signalling or morphogenetic gradients, which forced us to assume purely mechanical self-organization. The initial pattern selection dynamics was studied by embedding toroids into hydro-gels, allowing us to observe the deformation modes over longer periods of time. We found increasing mechanical fluctuations which break the toroidal symmetry, and discuss the evolution of their power spectra for various gel stiffnesses. Our observations are related to singlecell studies which explain the mechanical feasibility of the folding process. In addition, we observed switching of cells from a tissue bound to a migrating state after folding failure as well as in tissue injury. We found a supra-cellular * This paper is dedicated to Malcolm Steinberg.

This paper presents an adaption of periodic boundary conditions (BC), which is termed tessellation BC. While periodic BC restrict strain localization zones to obey the periodicity of the microstructure, the proposed tessellation BC adjust the periodicity frame to meet the localization zone. Thereby, arbitrary developing localization zones are permitted. Still the formulation is intrinsically unsusceptible against spurious localization. Additionally, a modification of the Hough transformation is derived, which constitutes an unbiased criterion for the detection of the localization zone. The behavior of the derived formulation is demonstrated by various examples and compared with other BC. It is thereby shown that tessellation BC lead to a reasonable dependence of the effective stress on the localization direction. Furthermore, good convergence of stiffness values with increasing size of the representative volume element is shown as well as beneficial characteristics in use with strain softening material.

To analyse a composite material with magnetically switchable stiffness properties, the coupling between the magnetic and the mechanical field has to be considered. Here the mechanical as well as the magnetic problem are computed using the eXtended Finite Element Method (XFEM). After a brief introduction of the framework, some results are presented focusing on the ability to calculate correct loads and displacements as well as on convergence rates. (© 2012 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)