UC San Diego

Isogeometric analysis (IGA) has emerged as a powerful approach in the field of structural analysis, benefiting from the seamless integration between the computer-aided design (CAD) geometry and the analysis model by employing non-uniform rational B-splines (NURBS) or other types of splines as basis functions. The spline basis functions naturally satisfy the $C^1$ continuity making it particularly suitable for the approximation of the Kirchhoff--Love shell formulation for thin shells. Despite its advantages, the application of IGA to structural analysis and design optimization of complex shell CAD geometries consisting of multiple non-matching spline patches remains challenging due to arbitrary surface intersections. To achieve the streamlined design-analysis-optimization workflow, effective coupling of intersecting shell patches in structural analysis and special handling to maintain intersections during design optimization is essential.

In this work, a shell coupling algorithm is developed to directly analyze complex shell structures represented as collections of untrimmed NURBS patches. Shell patches are modeled mechanically as Kirchhoff--Love shells and discretized isogeometrically. Coupling of non-matching patches uses a penalty-based formulation, employing a series of topologically 1D, geometrically 2D quadrature meshes. The quadrature meshes act as integration domains for the penalty energy to preserve displacement and rotational continuities at patch intersections. For design optimization, a free-form deformation (FFD)-based formulation is proposed for shape and thickness optimization of non-matching shell structures, ensuring compatibility of design variables at patch intersections throughout the optimization process. Lagrange extraction is employed to link control points associated with the B-spline FFD block and shell patches, and the extraction operators are also used to represent NURBS functions by Lagrangian bases in IGA analysis. Analytical sensitivities are derived to facilitate efficient gradient-based optimization algorithms. Additionally, a novel method for shape optimization of non-matching isogeometric shells incorporating intersection movement is introduced, allowing shell patches to move independently during shape updates. This flexibility is achieved by an implicit state function, with analytical sensitivities derived for the relative movement of shell patches. The differentiable intersections expand the design space and overcome challenges associated with large mesh distortion when optimal shapes involve significant movement of patch intersections in physical space. Shell patches are represented by the NURBS bases during the optimization process, enabling efficient integration of analysis and design models, and allowing for the multilevel design concept.

The proposed structural analysis algorithm and design optimization methods are validated through various benchmark problems and applied to practical aircraft wings, demonstrating their effectiveness for complex shell structures.