A geometry projection method for topology optimization of frames with structural shapes: A Geometry Projection Method

We present a topology optimization method based on the geometry projection technique for the design of frames made of structural shapes. An equivalent-section approach is formulated that represents the cross-section of the structural shapes as a homogeneous rectangular section. The accuracy of this approach is demonstrated by comparison to analyses performed using body-fitted meshes of the original sections for different loads and boundary conditions. A novel geometric representation is also introduced to represent the equivalent section as a cuboid. Like offset solids, this representation is endowed with an explicit expression for the computation of the signed distance to the boundary of the primitive and of its sensitivities, allowing for an efficient implementation. An overlap constraint is imposed via the formulation of auxiliary primitives associated to the structural members, which guarantees the resulting designs do not exhibit impractical intersections of primitives that would preclude their construction. The efficacy and efficiency of the method is demonstrated via 2D and 3D design examples. The examples demonstrate that the proposed method renders optimal designs and exhibits good convergence. They also illustrate the ability to design structures with far lower optimal volume fractions than those typically employed in continuum topology optimization techniques.