ABSTRACT
As discussed in Chapter 3, the standard displacement FEM has a lot of good properties but shows some shortcomings, including the stress accuracy issues [1,2]. Therefore, efforts have been made to overcome these shortcomings and to improve the accuracy of the solution, including the S-FEM models presented in the previous chapters and the mixed FEM models [3-9] based on the mixed variational principles. All these efforts mainly focus on solution accuracy improvements. Obtaining the exact solution (at least in a norm) using a discrete numerical method is, however, a much more fascinating and attractive idea in the area of computational methods. Some interesting efforts have been made recently in Liu’s group aiming at obtaining the exact solution in a norm using discrete models [1,2,10]. The so-called alpha finite element method using four-node quadrilateral elements (αFEM-Q4) has been developed for the purpose of finding a nearly exact solution in strain energy using coarse meshes [1]. The αFEM-Q4 gives a novel idea that works in the framework of FEMQ4, by simply scaling the gradient of strains using a factor α ∈ [0, 1]. Because the change needed is minor, the coding of αFEM-Q4 is almost exactly the same as the standard FEM-Q4. In addition, the resultant strain energy function for the αFEM-Q4 model has a very simple polynomial form in terms of α. Based on such a simple function of strain energy curves, a general procedure of αFEM-Q4 has been suggested to obtain nearly exact or best possible solutions, using meshes with the same aspect ratio. An exact-α approach is devised for overestimation problems and a zero-α approach for underestimation problems. The αFEM-Q4 has clearly opened a new window of opportunity to obtain numerical solutions that are exact at least in a norm. However, the original αFEM-Q4 based on quadrilateral elements cannot provide the exact solution to all elasticity problems. Furthermore, the αFEM-Q4 requires a quadrilateral mesh that cannot be generated in a fully automated manner for complicated domains.
