A local solution approach for adaptive hierarchical refinement in isogeometric analysis

Adaptively refining the state variables and fast solving the equations are two key issues in isogeometric analysis. Currently existing research more focuses on the first issue while the efficiency issue is not elaborated when the overall equations are solved at each refinement step. This article develops an approach to promote the efficiency by locally solving the equations that govern the refined variables at each refinement step. Here, hierarchical refinement of the state variables in NURBS is adopted and transformational matrix that transforms the basis functions from the current level to the refined level is utilized to handle the relations between the equations at different levels. By dividing the coefficient matrix into blocks and estimating the error of state variables for different blocks, we represent the global solution method in a hierarchical structure and separately construct and solve equations to obtain the solution for each level in the structure. We validate the proposed method by some steady heat-transfer and structural analysis problems in domains with different dimensionality. We have compared the numerical result accuracy as well as the time cost of the proposed method with those of current available refinement approaches, and it turns out to be very promising.