Abstract: In order to achieve the optimal design of heterogeneous objects with the
characteristic of spatially graded layout, the concept of two-phase graded finite elements is
established. Lagrangian shape functions are employed to interpolate the volume fractions of the
constituent materials. Adaptive lower bounds of the design variables are introduced to control the
local gradients in the nodal neighborhoods. Method of moving asymptotes is used to solve the
optimization problem for specific functionalities and objectives. A metal clip is analyzed and
optimized to verify the feasibility and robustness of this approach.

Key words: graded finite elements, heterogeneous object, adaptive lower bounds, method of
moving asymptotes