Two Kinds Of Numerical Methods With Low Requirements For Meshes

With the computer technology rapidly developing,many efficient numerical methods have been provided to solve practical engineering problems,like finite element methods(FEM).In order to overcome this difficulty,smoothed finite element methods(S-FEMs)on lower-quality meshes and some meshless approaches were established.They have been widely used in various complicated fields of mechanics for solids,heat transfer,structural acoustics and so on.The chapter 2 is to develop some basic mathematic theories for G~sspaces of functions that rely on weakened weak(W2)formulations,upon which S-FEMs and the smoothed point interpolation methods(S-PIMs)are based to well deal with problems on low-quality or heavily distorted meshes.In the thesis based on the G~s hspace theory founded by Liu etc.,the concepts of a G~sspace,which is unnecessarily depending on options of shape functions,and the norms are introduced in terms of mathematics.G~snorms are not only less than those in H~1 spaces,but also convergent,which is the theoretical basis of numerical solutions obtained by W2 forms.In addition,the equivalency between norms and semi-norms in G~sspace is further developed to ensure the stability of W2 numerical schemes from G~sspaces.These theoretical results will greatly benefit the future study of numerical methods in the G spaces.In the chapter 3,we apply an unsymmetric strong-form meshless collocation methods(intrinsic Kansa methods),intrinsically on smooth,closed,connected and complete Riemannian manifolds with codimensions dco? 1,to solve elliptic partial differential equations.The proposed methods make use of oversampling strong-form collocations and the least-squares minimization.By restricting some standard kernels in embedding spaces to manifolds,ours methods are as easy to implement as their domain-type analogies.We employ transformed differential operators handled either analytically or approximately,which allows collocations solely on the manifold.Under some essential smooth assumptions,we mainly prove that the proposed formulation is high-order convergent.Finally,in order to verify the theoretical results in these two chapters,we separately apply the corresponding methods to numerical cases.(1)First one numerical example is presented by using typical S-FEM models known as the NS-FEM and?S-FEM to examine the properties of G~sspaces,in comparison with the standard FEM with weak formulation.And another application of NS-FEM is that modified procedure offers a viable practical computational means to effectively compute the lower bounds of eigenvalues in an another 2D solid mechanics problem.(2)On arbitrarily codimensional manifolds,numerical results are compared to verify convergence rates between numerical and theoretical collocation settings,as well as two proposed intrinsic Kansa methods.Lastly,we apply the analytic method to solve shallow water equations on various manifolds.