High order finite elements for three-dimensional, thin walled nonlinear continua

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This thesis introduces a three-dimensional high-order solid finite element formulation for curved thin and thick-walled, physically nonlinear structures. Utilizing a hexahedral element, it enables an anisotropic Ansatz for the displacement field, allowing individual polynomial degrees for each component and variation in the three local directions. This approach efficiently computes three-dimensional plate and shell-like structures. It examines two nonlinear material models: the deformation theory and the flow theory of plasticity, presenting numerical examples that compare the p-version method with state-of-the-art h-version approximations. Additionally, an alternative approach is proposed that enhances two-dimensional finite element computations with a three-dimensional Ansatz in areas where the reduced approximation significantly deviates from the exact three-dimensional solution. This hp-d method offers a computationally less demanding technique bridging two and three-dimensional finite element approximations. To quantify the discretization error of p-version approximations, an explicit error estimator is applied and investigated for two model problems, demonstrating reliability when appropriate meshes that account for singularities in the exact solution are constructed.