The hierarchical finite cell method for multi-scale problems in structural mechanics

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Many important problems in structural mechanics are of a multi-scale nature. In problems of this kind, the global behavior of the structure is influenced by phenomena occurring on scales that are orders of magnitude smaller than the dimensions of the structure under investigation. This situation occurs in almost all engineering applications such as civil engineering, naval architecture or mechanical engineering. In order to simulate multi-scale problems reliably in structural mechanics, it is therefore essential to develop efficient numerical methods. In spite of the recent progress made in terms of computing power, performing such simulations based on standard methods, like the finite element method, for all scales is beyond the reach of present-day computer systems. During the first, 2-year period of this research proposal, we tackled several aspects of elastostatic problems connected with multi-scale behavior by further developing the finite cell method (FCM). One of the main achievements we accomplished was to develop a hierarchical refinement strategy for the FCM, which enabled us to compute multi-scale problems efficiently. In the second project period, we wish to extend the hierarchical FCM to three-dimensional problems that crop up in elastodynamics. In this regard, we will address the simulation of wave propagationin heterogeneous materials and composites by developing the FCM further in orderto provide an efficient simulation method. We expect that the new method to be developed will significantly extend the possibilities of wave propagation simulation to support technologies such as structural health monitoring by applying high-frequency loading for detecting damage in engineering structures.