Shells - thin curved structures such as eggshells, insect wings, and pressure vessels - are ubiquitous in nature and engineering. They are slender yet structurally capable, supporting many times their own weight. They can also undergo dramatic geometric transitions, such as buckling, snap-through, and bistability. This combination of strength and geometric nonlinearity stems from the interplay between bending and stretching in thin geometries. As a result, simulating shells is numerically demanding. Finite-element methods are the standard tool, but thin shells require high order elements that adds substantial formulation and implementation complexity. An alternative approach to address some of these problems comes from the computer graphics community. Discrete differential geometry methods discretize the shell on a triangulated mesh using only nodal positions as degrees of freedom. This sidesteps the need for high order elements in finite element analysis. However, existing formulations are typically developed for visual realism rather than mechanical accuracy, and lack the physically meaningful energy formulations required for engineering design. This thesis develops a discrete Kirchhoff-Love shell framework that combines a triangulated-mesh discretization with the mechanical rigor required for engineering design. The formulation employs a Koiter energy with standard engineering material constants, supports a spontaneous-curvature field for active materials, and provides analytic gradients and Hessians that enable Newton-Raphson equilibrium solving, adjoint sensitivity analysis, and energy-landscape path-finding. The framework is verified against four canonical shell benchmarks. The framework is first applied to photoactive liquid crystal elastomer shells, governed by a spontaneous-curvature evolution law that couples illumination intensity and direction to surface geometry. Simulations capture a flat sheet that bifurcates to a cylindrical configuration under uniform illumination, resulting from the Gauss curvature coupling between bending and stretching. A second example shows a thin active sheet that reorients to track a moving light source, analogous to a sunflower. Tape springs are thin curved strips used as deployable booms in spacecraft; they fold compactly via snap-through buckling and recover a stiff deployed state on release. The framework is then applied to simulate their forward bending response in opposite-sense and equal-sense loading modes. The opposite-sense results match analytical predictions for the propagation moment and localized fold geometry. Building on the tape spring analysis, a multi-equilibrium topology optimization method computes adjoint sensitivities through the deployed and folded states simultaneously. A parameter sweep over volume fraction, design region, and folded-state moment threshold reveals non-intuitive topologies, including hourglass designs that single-state optimizers cannot access. Finally, the nudged elastic band method is applied within the discrete-shell framework to compute the minimum-energy path for the eversion of a bistable spherical cap. The path passes through a non-axisymmetric saddle, illustrating the energetic favorability of asymmetric eversion, which stems from the interplay between Gauss curvature and stretch. Together, these applications establish the framework as a versatile, mechanically grounded platform for simulating the nonlinear mechanics of thin shells.
Sathvik R. Sanagala (Thu,) studied this question.