Isogeometric Shape Design Sensitivity Analysis Using Mixed Transformation Method for Kronecker Delta Property

The isogeometric method is very effective in shape design optimization due to its effectiveness through the easy design parameterization and accurate sensitivities considering the higher order geometric terms. Due to non-interpolatory property of the NUBRS basis functions, however, the treatment of essential boundary condition is not as straightforward in the isogeometric analysis as in the finite element analysis. Taking advantages of the transformation method developed in meshfree methods, we investigate the isogeometric shape sensitivity analysis with the treatment of essential boundary conditions. Using the property that isogeometric basis functions do not depend on design changes, the transformed shape sensitivity equation is developed and verified for the problem having the essential boundary conditions. Numerical costs to construct the transformed basis function are not as much as the meshfree methods due to the NURBS property that only boundary nodes have their supports on the boundary. Through demonstrative numerical examples having the essential boundary conditions, the effectiveness of proposed design sensitivity analysis is verified.