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Abstract: Surface subdivision gains popularity in surface design owing to its flexible applications for geometry with complicated topology and simple computational scheme, and the geometric partial differential equation (GPDE) method is an advanced technology for constructing high-quality smooth surfaces. In this paper, we composite these two ingredients to form a unified method for freeform surface design. We choose the mean curvature flow and Willmore flow as our driven GPDEs, and the finite element method coupled with a hybrid Loop and Catmull–Clark subdivision algorithm as the numerical simulation method. This research presents a novel technique to evaluate the finite element basis functions and the first attempt for constructing the GPDE subdivision surface with hybrid control meshes consisting of triangles and quadrilaterals. Numerical experiments show that the construction method is efficient and robust, yielding high-quality hybrid subdivision surfaces. [Copyright &y& Elsevier]

Abstract: Beams are high aspect ratio structural members that are used extensively in civil, automotive, aerospace, and MEMS applications. In all such applications, one must typically analyze and optimize the beams through computer simulations. Standard 3D finite element analysis (FEA) of beams can be used in such simulations; it is however prone to errors, and is computationally expensive for thin structures. Therefore, a common strategy is to carry out a dimensionally reduced 1D beam analysis. Unfortunately, 1D beam analysis is hard to automate and integrate with 3D CAD. In this paper, we propose an alternate “algebraic reduction” method that combines the generality of 3D FEA, and the computational efficiency of 1D beam analysis. This is achieved via a dual-representation framework where the geometry of the beam is captured via a 3D finite element mesh, while the physics is captured via a 1D beam model. The proposed method is formally established, and supported through numerical experiments. [Copyright &y& Elsevier]