Simultaneous solution of ropes and membranes on all level sets within a bulk domain

Abstract

AbstractWe formulate a mechanical model in the context of the finite strain theory for implicitly defined ropes and membranes, which applies simultaneously to all level sets of a level‐set function over a bulk domain. Further, a numerical method to solve this model is introduced. Tangential operators, which occur in the geometrically non‐linear model, are defined in the framework of the tangential differential calculus (TDC). The weak form of the governing PDE, which is required for the numerical simulation, is formulated using the co‐area formula. For the numerical analysis, the bulk domain is discretized with higher‐order finite elements which are conforming to the boundary but do not have to be aligned to the level sets within the domain. The resulting numerical method may be called Bulk Trace FEM. Higher‐order convergence studies of numerical examples, based on the residual error, the stored energy error, or the length/area of the deformed structures, ensure the validity of this method and confirm optimal convergence rates.