Shape-free Finite Element Method Based On Analytical Trial Functions

During the past sixty years, the finite element method (FEM) has become one of thedominant numerical simulation techniques in the fields of both engineering and science.Although the contribution of FEM is huge, it doesn’t mean that it is already flawless. Inmany cases, the classic isoparametric element models are extremely sensitive to meshdistortions, which leads to poor accuracy and other numerical problems (e.g., elementcan not work). Many scholars are making or have made their efforts to improve theability of FEM to overcome the sensitivity problem to severe mesh distortionsNevertheless, there are still more or less limitations in exisiting element models, whichmeans this problem has not been solved from the outset.The objective of this thesis is to develop formulations of so called shape-free finiteelement method. These elements employ the fundamental analytical solutions of theelasticity as the trial functions (analytical trial function method), and can perform verywell in various severely distorted meshes. That is to say, their shape can be very freebecause the accuracy of both the stresses and the displacements will not be influencedby the element shapes. Hence, the pressure on mesh generation is greatly relaxedbecause good precision can be still obtained even severely distorted meshes are used.The main contributions of this thesis are as follows:First, based on the existing achievements, the fundamental analytical solutions ofthe plane anisotropic and3D isotropic elasticity are systematically derived. They formthe complete and independent low-order polynomial series in Cartesian coordinates,which are suitable for being taken as the trial functions of the finite elements.Second, by using the fundamental analytical solutions of the plane anisotropicelasticity as the analytical trial functions, the existing plane isotropic8-and12-nodehybrid stress-function elements HSF-Q8and HSF-Q12are generalized to the planeanisotropic models. Thus, two new elements, SF8(15) and SF12(23), are successfullydeveloped. They exhibit excellent performance on avoiding the sensitivity problem tomesh distortions. So long as the element edges keep straight, they can perform very wellin various convex and concave quadrilateral distortions. This is a kind of shape-freefinite elements.Third, the unsymmetric technique together with the analytical trial function method and the generalized conforming theory are combined to develop a new unsymmetricplane8-node element US-ATFQ8, which is superior to existing unsymmetric elementUS-QUAD8and the hybrid stress-function element HSF-Q8. The new elementUS-ATFQ8possesses all the advantages of both US-QUAD8and HSF-Q8. On the onehand, it successfully solved the problems of the interpolation failure and the rotationdependency existing in element US-QUAD8; on the other hand, it also perfoms well incurved-edge and mid-side node distortions which element HSF-Q8can not do.Especially, it is astonished that, besides the first-and the second-order displacementproblems, the new element US-ATFQ8can still capture the exact solutions of thethird-order displacement problems, which is never to see in publications for other8-node models. It can be said that the new element US-ATFQ8is a ture shape-freeelement.Finally, according to the successful experience for plane problem, a3D hexahedral20-node hybrid stress-function element HSF-3D20and a3D hexahedral20-nodeunsymmetric element US-ATF3D20are constructed. However, due to the complexity ofthe3D problems, some new technical difficulties inevitably occur. So, currently, theperformances of the new models are not ideal. Further investigations are required tosolve these problems.