Isogeometric shape design optimization of heat conduction problems

An isogeometric shape sensitivity analysis method is developed for heat conduction problems using the adjoint variable method. When the isogeometric method is extended to isogeometric shape optimization, the geometric properties of design are embedded in NURBS basis functions for CAD and response analysis so that the design parameterization is much easier than that in finite element based method without subsequent communications with CAD description since the perturbation of control points naturally results in shape changes. Thus, exact geometric models can be used in both response and shape sensitivity analyses, where normal vector and curvature are continuous over the whole design space so that enhanced shape sensitivity can be expected, especially for the heat conduction problems that include various types of boundary conditions. Moreover, the NURBS basis functions conveniently provide a smooth and non-local design velocity field for the shape design optimization, where computation is not easy in the finite element based method. Through numerical examples, the developed isogeometric sensitivity is verified to demonstrate excellent agreements with finite difference one. Also, it turns out that the proposed method works very well in various shape optimization problems.