The Immersed Boundary Conformal Method for Kirchhoff-Love and Reissner-Mindlin shells

TL;DR

The paper addresses the challenge of simulating thin-shell structures within immersed domains by introducing the Immersed Boundary Conformal Method (IBCM), which creates conformal boundary layers to enable strong Dirichlet enforcement and supports local refinement. It integrates high-degree B-spline spaces with Nitsche-based coupling to connect boundary layers to interior patches while using high-order quadrature for cut-element integration, yielding stable and accurate solutions for Kirchhoff-Love and Reissner-Mindlin shells. The methodology is framed within a unified differential-geometry-based shell theory, accommodating laminated orthotropic materials and allowing mixed KL/RM coupling to model localized phenomena such as boundary effects and cracks. Numerical tests demonstrate optimal convergence rates, preserved coercivity, and effective handling of complex geometries, including conformal cylinder interfaces and damaged cylinders. The approach offers a flexible, accurate, and scalable framework for complex shell analyses in embedded geometries with potential applications in aerospace, automotive, and mechanical engineering.

Abstract

This work utilizes the Immersed Boundary Conformal Method (IBCM) to analyze Kirchhoff-Love and Reissner-Mindlin shell structures within an immersed domain framework. Immersed boundary methods involve embedding complex geometries within a background grid, which allows for great flexibility in modeling intricate shapes and features despite the simplicity of the approach. The IBCM method introduces additional layers conformal to the boundaries, allowing for the strong imposition of Dirichlet boundary conditions and facilitating local refinement. In this study, the construction of boundary layers is combined with high-degree spline-based approximation spaces to further increase efficiency. The Nitsche method, employing non-symmetric average operators, is used to couple the boundary layers with the inner patch, while stabilizing the formulation with minimal penalty parameters. High-order quadrature rules are applied for integration over cut elements and patch interfaces. Numerical experiments demonstrate the efficiency and accuracy of the proposed formulation, highlighting its potential for complex shell structures modeled through Kirchhoff-Love and Reissner-Mindlin theories. These tests include the generation of conformal interfaces, the coupling of Kirchhoff-Love and Reissner-Mindlin theories, and the simulation of a damaged shell.

Figures (16)

• Figure 1: Step-by-step construction of the internal patch and boundary layers with corresponding discretizations. (a) Untrimmed parametric domain $\hat{\Pi}_0$ and associated boundary. (b) Construction of the parametric domain through trimming the regions $\hat{\Pi}_1$ and $\hat{\Pi}_2$ delimited by the corresponding trimming curves $\partial\hat{\Pi}_1$ and $\partial\hat{\Pi}_2$. (c) Resulting parametric $\Omega$ domain and associated boundary $\partial\Omega$. (d) Offset curves $\partial\hat{\Pi}'_1$ and $\partial\hat{\Pi}'_2$ and associated trimming regions $\hat{\Pi}'_1$ and $\hat{\Pi}'_2$. (e) Resulting internal domain $\hat{\Omega}_0$ and associated boundary layers $\hat{\Omega}_1$ and $\hat{\Omega}_2$ with correspondent boundaries $\partial\hat{\Omega}_0$, $\hat{\Omega}_1$, and $\partial\hat{\Omega}_2$, respectively. (f) Discretization of the internal patch and boundary layers, and interfaces $\hat{\Gamma}_1$ and $\hat{\Gamma}_2$. (g) Untrimmed surface $\Pi_0= \hat{\bm{\mathcal{F}}}(\hat{\Pi}_0)$ and correspondent boundary $\partial\Pi_0 = \hat{\bm{\mathcal{F}}}(\partial\hat{\Pi}_0)$ with superimposted discretization. (h) Trimmed surface $\Omega=\hat{\bm{\mathcal{F}}}(\hat{\Omega})$ and correspondent boundary $\partial\Omega=\hat{\bm{\mathcal{F}}}(\partial\hat{\Omega})$ with superimposted discretization. (i) Internal surface $\Omega_0 = \hat{\bm{\mathcal{F}}}(\hat{\Omega}_0)$ and correspondent boundary layers $\Omega_1 = \hat{\bm{\mathcal{F}}}(\hat{\Omega}_1)$ and $\Omega_2 = \hat{\bm{\mathcal{F}}}(\hat{\Omega}_2)$ with superimposed discretization and interfaces $\Gamma_1=\hat{\bm{\mathcal{F}}}(\hat{\Gamma}_1)$ and $\Gamma_2=\hat{\bm{\mathcal{F}}}(\hat{\Gamma}_2)$
• Figure 2: Geometry and discretization for the laminated plate in the first set of tests. (a) First refinement level for the single patch discretization with corresponding types of boundary conditions and applied loads. (b) First refinement level for the IBCM-based discretization with corresponding types of boundary conditions and applied loads. Third (c) and fourth (d) refinement levels for the IBCM-based discretization.
• Figure 3: Convergence curves for the plate shown in Fig.\ref{['fig:RES - LamGeo']} in $L^2$ error norm and $H^1$ error seminorm for a Reissner-Mindlin theory. The curves are obtained for four different polynomial orders $p=1,2,3,4$ and three thickness values $\tau=100,10,1$ [mm]. Two discretization are taken into account, a single trimmed patch as shown in Fig.\ref{['fig:RES - LamGeom a']} and a IBCM-based one as shown in Fig.\ref{['fig:RES - LamGeom b']}.
• Figure 4: Convergence curves for the plate shown in Fig.\ref{['fig:RES - LamGeo']} in $L^2$ error norm and $H^1$ and $H^2$ error seminorms for a Kirchhoff-Love theory. The curves are obtained for three different polynomial orders $p=2,3,4$ and three thickness values $\tau=100,10,1$ [mm]. Two discretization are taken into account, a single trimmed patch as shown in Fig.\ref{['fig:RES - LamGeom a']} and a IBCM-based one as shown in Fig.\ref{['fig:RES - LamGeom b']}.
• Figure 5: Contour plots of the laminated plate in Section \ref{['ssec:Res - lam']} with superimposed discretization. Displacement magnitude (a), representative component of the membrane force $N^{11}$ (b), representative component of the bending moment $M^{11}$ (c) for the Kirchhoff-Love theory, and correspondents contour plots for the Reissner-Mindlin theory (d), (e), (f), respectively. It is worth mentioning that the noticeable difference between (c) and (f) is due to the different manufactured solution adopted in this test. In particular assigning also a manufactured rotation field in the Reissner-Mindlin discretization affects in this case only the bending response.