Goal-Adaptive Meshing of Isogeometric Kirchhoff-Love Shells

TL;DR

This work develops a goal-oriented adaptive meshing framework for isogeometric Kirchhoff-Love shells by integrating Dual-Weighted Residual (DWR) error estimators with Truncated Hierarchical B-splines (THB-splines) and mesh admissibility. It extends the DWR approach to both nonlinear shell problems and eigenvalue problems (modal and buckling) and uses enriched dual spaces for accurate local error contributions, driven by user-defined goal functionals. The method exploits THB-splines for local refinement and employs a refined/coarsening strategy with admissible meshes plus a quasi-interpolation transfer to reuse solutions across meshes. Numerical benchmarks including linear static, modal, buckling, and nonlinear post-buckling (snap-through and wrinkling) demonstrate high estimator accuracy and efficient allocation of degrees of freedom, indicating strong potential for industrial shell simulations.

Abstract

Mesh adaptivity is a technique to provide detail in numerical solutions without the need to refine the mesh over the whole domain. Mesh adaptivity in isogeometric analysis can be driven by Truncated Hierarchical B-splines (THB-splines) which add degrees of freedom locally based on finer B-spline bases. Labeling of elements for refinement is typically done using residual-based error estimators. In this paper, an adaptive meshing workflow for isogeometric Kirchhoff-Love shell analysis is developed. This framework includes THB-splines, mesh admissibility for combined refinement and coarsening and the Dual-Weighted Residual (DWR) method for computing element-wise error contributions. The DWR can be used in several structural analysis problems, allowing the user to specify a goal quantity of interest which is used to mark elements and refine the mesh. This goal functional can involve, for example, displacements, stresses, eigenfrequencies etc. The proposed framework is evaluated through a set of different benchmark problems, including modal analysis, buckling analysis and non-linear snap-through and bifurcation problems, showing high accuracy of the DWR estimator and efficient allocation of degrees of freedom for advanced shell computations.

Figures (23)

• Figure 1: A typical flowchart for an adaptive meshing routine. The classical solution stepping depicts a process without adaptive meshing. Here, a solution is obtained by the solve and the solution is advanced (e.g. in time or load step) and recomputed. The adaptive meshing step denotes the additional operations for mesh adaptivity and the Addition for solution stepping includes an additional transfer step in case the adaptive meshing method is applied to solution. The Estimate block provides an error estimation with local contributions per element or per degree of freedom (DoF). The Mark block contains a marking rule that marks regions for refinement based on a specific rule. The Refine block transforms the current mesh to a new mesh, where regions are refined and coarsened based on the marking rule. The block Transfer transfers the previous solution to the new mesh, so that it can be used to recompute the present interval on a modified mesh. This recomputation is performed again in the Solve block and follows through the subsequent blocks, until an adaptivity criterion is reached. For example, a criterion that requires the total error in the mesh to be within certain bounds.
• Figure 2: Principles of refinement for different spline bases. The top figures represent the basis on level $\mathbcal{B}^0$, optionally with refined basis functions coloured blue. Bottom pictures represent refined bases. left) uniform refinement (hence $\mathbcal{B}^1$; middle) HB-refinement; right) THB-refinement, with truncated basis functions coloured yellow. Note that the refinement basis functions are from $V^1$. The unrefined unique knot vector in all cases is $\Xi = \{0,1/8,2/8,\dots,7/8,1\}$ and the degree of the basis is 2. The bases are generated in G+SmoJuttler2014.
• Figure 3: Recursive marking strategy on the marked element of level $\ell$ on the initial mesh represented in (a). As a first step, the support extension of the marked element is obtained (b), from which the parents that are active on level $\ell-1$ define the $\mathcal{T}$-neighborhood of the marked cell (c). Starting the same procedure on the marked cells of level $\ell-1$, the support extension can again be obtained (d) with their corresponding parents on level $\ell-2$, marking the $\mathcal{T}$-neighborhood of the marked elements of level $\ell-1$ (e). The complete recursive marking from the marked element in (a) is depicted in (f).
• Figure 4: Given the mesh from \ref{['fig:Neighborhoods_ref']}-(f), the coarsening neighborhoods are evaluated in (a)-(c) of this figure. The cell marks the cell of level $\ell$ that is marked for coarsening to its parent and the cells mark cells that are marked for refinement. The ring around the cell marked for coarsening depicts the region that should be checked for the coarsening neighborhood. That is, it defines the region that should not contain cells of level $\ell+1$ (for $\mathcal{N}_c$) or cells of level $\ell$ that are marked for refinement (for $\mathcal{N}'_c$). The cells for which $\mathcal{N}_c=\emptyset$ are marked in (d) and the cells with $\mathcal{N}'_c=\emptyset$ are marked in (e). The final mesh after refinement and coarsening is depicted in (f). The coarsened elements that satisfy $\mathcal{N}_c\cup\mathcal{N}'_c=\emptyset$ are marked as coarsened elements.
• Figure 5: A graphical summary of the adaptive meshing flowchart from \ref{['fig:flowchart']} used in the present work. The equations which are used in each step are indicated in the blocks. The adaptive meshing iterations are performed within each solution step until the total error $\Delta\mathbcal{L}$ is contained in the interval $[\text{tol}_r,\text{tol}_c]$, $\text{tol}_r<\text{tol}_c$, following the tolerances in \ref{['subsec:labeling']}. In case of convergence, the solution is advanced, e.g. with an arc-length iteration. \ref{['alg:flowchart2']} provides an algorithm corresponding to this flow-chart.