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While the interest in higher-order models in physics and mechanics grows, their numerical simulation still poses a challenge, especially for arbitrary shaped three-dimensional domains. This contribution presents the mathematical framework as well as the application to different problems in the field of material science, fracture mechanics and diffusion problems. All models under consideration require at least C 1 continuity, which prevents the application of standard finite element analysis and local mesh refinements. Introducing isogeometric analysis (IGA) for the discretization in a finite element framework enables us to deal with these requirements. Moreover, a general hierarchical refinement scheme based on a subdivision projection is presented here for one, two and three dimensional simulations. This technique allows to enhance the approximation space using finer splines on each level but preserves the …