A new energy and momentum conserving algorithm for the non‐linear dynamics of shells

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Abstract
A numerical time‐integration scheme for the dynamics of non‐linear elastic shells is presented that simultaneously and independent of the time‐step size inherits exactly the conservation laws of total linear, total angular momentum as well as total energy. The proposed technique generalizes to non‐linear shells recent work of the authors on non‐linear elastodynamics and is ideally suited for long‐term/large‐scale simulations. The algorithm is second‐order accurate and can be immediately extended with no modification to a fourth‐order accurate scheme. The property of exact energy conservation induces a strong notion of non‐linear numerical stability which manifests itself in actual simulations. The superior performance of the proposed scheme method relative to conventional time‐integrators is demonstrated in numerical simulations exhibiting large strains coupled with a large overall rigid motion. These numerical experiments show that symplectic schemes often regarded as unconditionally stable, such as the mid‐point rule, can exhibit a dramatic blow‐up in finite time while the present method remains perfectly stable.