Simulating Parametric Thin Shells by Bicubic Hermite Elements

TL;DR

This work introduces the bicubic Hermite element method (BHEM) for elastodynamic simulation and rendering of parametric thin-shell structures. By discretizing with bicubic Hermite patches, BHEM achieves high-order, $\mathcal{C}^1$-continuous representations using far fewer elements than conventional FEM, enabling accurate Kirchhoff–Love shell dynamics with efficient computation. The framework unifies geometry, dynamics, collision handling, and rendering via a single parametric representation, incorporating an implicit Euler solver with augmented Lagrangian constraints and a robust ray--patch intersection algorithm for rendering and collision detection. Comprehensive validations across wrinkles, draping, twisting, and collisions demonstrate high accuracy, strong efficiency, and seamless rendering without remeshing. This approach offers significant potential for realistic, high-fidelity thin-shell simulations in graphics and engineering applications.

Abstract

In this study, we present the bicubic Hermite element method (BHEM), a new computational framework devised for the elastodynamic simulation of parametric thin-shell structures. The BHEM is constructed based on parametric quadrilateral Hermite patches, which serve as a unified representation for shell geometry, simulation, collision avoidance, as well as rendering. Compared with the commonly utilized linear FEM, the BHEM offers higher-order solution spaces, enabling the capture of more intricate and smoother geometries while employing significantly fewer finite elements. In comparison to other high-order methods, the BHEM achieves conforming $\mathcal{C}^1$ continuity for Kirchhoff-Love (KL) shells with minimal complexity. Furthermore, by leveraging the subdivision and convex hull properties of Hermite patches, we develop an efficient algorithm for ray-patch intersections, facilitating collision handling in simulations and ray tracing in rendering. This eliminates the need for laborious remodeling of the pre-existing parametric surface as the conventional approaches do. We substantiate our claims with comprehensive experiments, which demonstrate the high accuracy and versatility of the proposed method.

Figures (23)

• Figure 1: A sheet of square cloth ($30\times30$ patches) drapes on a ball, exhibiting rich wrinkle patterns in the process of reaching a steady state.
• Figure 2: A sheet of square cloth ($30\times30$ patches) drapes on an armadillo model, where the sharp edges of the model are reflected by the cloth deformation.
• Figure 3: A sheet of cloth ($30\times30$ patches) falls on a needle array and then gets pulled away from aside. The bulges on the cloth surface are pushed by the needle tips. This sharp geometry deformation caused by the tiny contact demonstrates well the fine resolution of the patch interpolation.
• Figure 4: The folding process of an oriental paper parasol ($280\times5$ patches), simulated by jointly controlling nodal positions and their first-order derivatives. Bending and subtle wrinkles along the ribs can be observed.
• Figure 5: A surface that is homeomorphic to a rectangle is embedded into a Cartesian grid, of which grid cells are taken as patches. Each BH patch corresponds to an axis-aligned rectangle in the parameter space.