| The numerical methods, such as the Finite Element Method and the Boundary Element Method, have been successful so far. The formation and existence of the grids of these methods, however, are posing some difficulties in their actual applications. The meshless method, which is still in development now, can work completely or partly without grids, enjoying promises in crack propagation simulation, elasto-plastic and large deformation analysis and impact process. The meshless method, therefore, is much focused in, and becomes an orientation of, the scientific and engineering calculation research.Pioneering achievements have been made in recent years by world experts in meshless method investigation. The Element Free Galerkin Method and the Reproducing Kernel Particle Method, for example, are two new representative meshless numerical methods which have appeared lately. As they don't need any finite element or boundary element grid, they turn out to be more flexible and convenient in numerical calculations in many fields. Targeting these two meshless methods, this study carries out the following relevant research:1 .An in-depth inquiry into the effect on the precision of calculation by weight function, basis function, the size of support domain and the parameters, factors involved in the Element Free Galerkin Method;proposal on the choice of relevant parameters.2. Proposal of the new algorithm of coupling Element Free Galerkin with Finite Element Method. The Element Free Galerkin Method features such advantages as high accuracy, convenient post process, elimination of volume locking, and quick convergence. While a re-division of grids is needed with other methods, it requires no grid upgrading or supplementing. The main idea of the Element Free Galerkin Method, on the other hand, is to base itself on the related node information, different from the node in Finite Element Method which uses only values of unit element. It takes much time to search for the related nodes, with enormous calculations, and is less efficient than the Finite Element Method in case of large-scale calculation. This study combines advantages of both the Element Free Galerkin Method and Finite Element Method, and comes up with a new formation of ramp function, extending the application of the coupling method,which is now still confined to the realization the essential boundary conditions, to the global field at large, with exertion of the merits of the meshless method while keeping the advantage of the Finite Element Method featuring fewer calculations.3. Adoption of the interpolation under the incremental Element Free Galerkin Method;adoption of the incremental stress-strain relation to demonstrate the elasto-plastic constitutive law. Under the precondition of small deformation, the Element Free Galerkin Method for elasto-plastic analysis based on increment constitutive relation is proposed. The penalty method is introduced to modify the energy variation equations, conveniently realizing the essential boundary conditions in EFG method. The principle of virtual work for continuous medium is used to apply the meshless method into steady-state creep simulation, with deduction of related control equation for steady-state creep. Penalty parameters are also used to realize the incompressible conditions and essential boundary conditions of the steady-state creep, with guarantee for the symmetric positive definiteness of the stiff matrix. The Element Free Galerkin Method for geometric nonlinearity analysis is recommended, bringing the Element Free Galerkin Method into issues related to 2-D and axisymmetic geometric nonlinearity. Under the context of small strain and negligence of material nonlinearity, the control equation is deduced for geometric nonlinearity with the Element Free Galerkin Method.4. The Yeoh constitutive model of hyperelastic materials can well display such mechanical behavior as large strain. It is represented by strain energy function. For sake of convenience, the total lagrangian methods are adopted, and the Yeoh constitutive model is introduced into the Reproducing Kernel Particle Method, with proposal of Reproducing Kernel Particle Method for hyperelastic materials as well as study on the calculation method of the Yeoh constitutive model. Elasto-plastic large strain is a constantly encountered problem in projects, which involves both geometric and material nonlinearity. Through the adoption of the Jaumann stress-rate and the strain-rate to describe the constitutive model as well as the adoption of the incremental method, the Reproducing Kernel Particle Method for elasto-plastic large strain analysis is deduced.5. The Taylor impact is important for the research of material dynamics and the impact process. After a study of the Taylor impact of the cylinder projectile, thispaper offers two different kinds of approaches to Taylor impact theory, one being based on suppositions of speed changes in the impact process, and the other being based on the lengths of the projectile after its deformation. With the experiment results, both ways prove feasible. The Reproducing Kernel Particle Method for Taylor impact is also proposed, and with this numerical method, the dynamic behavior of material is analyzed.6. The Reproducing Kernel Particle Method for high-velocity impact process is proposed. The high-velocity impact process, which is quite complicated itself, involves different kinds of nonlinearity. This study adopts the Bodner-Partom constitutive model to describe the large strain and high strain-rate of materials under high-velocity impact, offers the method to determine the parameters of constitutive model, proposes solutions to new interfaces, and uses the velocity collocation method to realize the treatment of the velocity of particles at the interface. The control equation for the calculation of high-velocity impact process by the Reproducing Kernel Particle Method is deduced, and with this method, concrete analysis is given to the impact process of tungsten alloy projectile onto the armour steel target.
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