The Shifted Boundary Method (SBM) in Isogeometric Analysis (IGA)
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The advent of Isogeometric Analysis (IGA) has revolutionized computational mechanics by bridging the gap between Computer-Aided Design (CAD) and Computer-Aided Engineering (CAE). This study introduces a pioneering integration of the Shifted Boundary Method (SBM) within the IGA framework, applying it to a Poisson problem with Dirichlet boundary conditions. To accomplish this, the research effectively combines both body-fitted and unfitted approaches, specifically utilizing SBM. A significant achievement of this work is the implementation of a fully penalty-free approach in applying SBM within IGA, negating the need for penalty calibration in Dirichlet boundary terms. This research demonstrates how order elevation, coupled with SBM through higher-order Taylor expansions, consistently achieves optimal p+1 convergence, where p is the order of the IGA basis functions. This advancement showcases the synergy between IGA’s adaptability and SBM’s effective boundary condition handling. Additionally, analysing the condition number of the problem matrix reveals that SBM, when integrated with IGA, effectively circumvents the "small cut-cell problem", a common issue in numerical methods with embedded boundaries. This highlights SBM’s robustness and broad applicability in various engineering scenarios. In summary, this integration of SBM in IGA marks a significant stride in computational mechanics, offering enhanced accuracy and efficiency in handling complex boundary problems. It opens new possibilities for advanced numerical techniques, promising further advancements and innovative applications in the field.
