An embedded domain method for non‐linear structural membranes

Abstract

An embedded domain method for structural membranes in large displacement theory is outlined. The membrane is immersed in a three-dimensional background mesh with level-set data at the nodes. The mechanical model for the implicitly defined membrane is formulated using the Tangential Differential Calculus (TDC). The embedded domain method has to properly consider the numerical integration and boundary conditions within the background elements cut by the membrane. Furthermore, stabilization is required to address linear dependencies and conditioning issues.

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

“…\end{equation}$$The divergence theorem for the embedded manifolds in Ω is $$$$\begin{align} \int _{\Omega }w\cdot \mathrm{div}_{\Gamma }\bm{v}\cdot {\left\Vert \nabla \phi \right\Vert} \,\mathrm{d}\Omega = & -\int _{\Omega }\nabla _{\bm{x}}^{\Gamma }w\cdot \bm{v}\cdot {\left\Vert \nabla \phi \right\Vert} \,\mathrm{d}\Omega +\int _{\Omega }\varkappa \cdot w\cdot {\left(\bm{v}\cdot \bm{n}\right)}\cdot {\left\Vert \nabla \phi \right\Vert} \,\mathrm{d}\Omega \\ & +\int _{\partial \Omega }w\cdot {\left(\bm{v}\cdot \bm{q}\right)}\cdot {\left(\bm{q}\cdot \bm{m}\right)}\cdot {\left\Vert \nabla \phi \right\Vert} \,\mathrm{d}\partial \Omega .\nonumber\end{align}$$$$The expression $$\Vert \nabla \phi \Vert$$ comes from the co‐area formula [3–6, 12]. For a more detailed description see the contribution [12] of these conference proceedings and [6].…”

“…For the definition of the BVP, some vector fields are required: along the boundaries $$\partial \Omega _{\bm{X}}$$ and $$\partial \Omega _{\bm{x}}$$ unit normal vectors M and m , respectively, N and n are the respective unit normal vectors on the level sets $$\Gamma _{\!\!\bm{X}}^c$$ and $$\Gamma _{\!\bm{x}}^c$$ in the bulk domains, and Q and q the conormal vectors on the boundaries. More details of these vector‐fields can be found in the literature [6] and [12].…”

“…Undeformed reference configuration in blue and deformed spatial configuration in red, (A) for ropes and (B) for membranes, based on Figure 7 in the literature [6]. …”

“…Furthermore, a FEM variant for the solution of this mechanical model is developed. Therefore, the co‐area formula [3–5] is used to formulate an appropriate weak form [6]. Similar approaches for other PDEs on manifolds, for example, the Laplace‐Beltrami operator, conservation and diffusion problems, can be found in the literature [3, 7, 8] and references therein.…”

“…Similar approaches for other PDEs on manifolds, for example, the Laplace‐Beltrami operator, conservation and diffusion problems, can be found in the literature [3, 7, 8] and references therein. To get deeper insight into the work discussed here, we refer the interested reader to the literature [6].…”