Quasi-convex Reproducing Kernel Particle Method For Thermo-mechanical Coupling Problems

Since the end of last century,the reproducing kernel particle method was a new meshless method developed by the smooth particle hydrodynamics method.The key is to reproduce kernel approximation function through constructing the kernel function the approximation function of field variables.By satisfying the reproducing condition of the traditional kernel approximation,the constructed trial function obtains the smoothness of not less than the reproducing kernel function and higher precision.The reproducing kernel particle method can eliminate the instability on the boundary of the problem domain and improve the calculation accuracy,so that the reproducing kernel particle method is applied widely to the nonlinear large deformation problems,discontinuous problems,structural dynamics,composite structures,fracture mechanics and other fields.To sovel the problem that the reproducing kernel particle method does not have convex approximation property,the improvement on reproducing condition was proposed by Wang Dongdong,and then the quasi-convex-reproducing kernel approximation was proposed.Without increasing the calculation time and decreasing numerical precision,shape function of this method has higher order positive property.The main contents of each chapter are as follows: In the first chapter,the research background,significance and innovations are introduced in detail,and the research status and existing problems of meshless methods are briefly introduced.In Chapter 2,the theory of reproducing kernel approximation is introduced firstly.By relaxing its reconstruction conditions,the quasi-convex reproducing kernel approximation theory proposed by Wang Dongdong is derived and the related formula is derived.At the same time,several weight function selection methods are introduced.In Chapter 3,the properties of orthotropic materials and functionally graded materials are introduced.Then,the quasi-convex reproducing kernel approximation is combined with the Galerkin integral weak form and the penalty function method is used to process the essential boundary conditions.The quasi-convex reproducing kernel particle method for composite elastic mechanics problem is established,and the corresponding MATLAB program is compiled.Through several numerical examples,the accuracy and effectiveness of the proposed method on composite elastic mechanics problem are verified.Besides,the selection of calculation parameters and the random distribution are analyzed.In Chapter 4,the decoupling simplification of the steady-state thermo-mechanical coupling problem is analyzed and discussed firstly.Then,the quasi-convex reproducing kernel approximation is combined with the Galerkin integral weak form and the penalty function method is used to process the essential boundary conditions.The quasi-convex reproducing kernel particle method for steady-state heat conduction problem and steady-state thermal-mechanical coupling problem is established,and the corresponding MATLAB program is compiled.Through several numerical examples,the accuracy and effectiveness of the proposed method on steady-state thermal-mechanical coupling problem are verified.Besides,the selection of calculation parameters and the random distribution are analyzed.In Chapter 5,the characteristics and time integration schemes of transient thermo-mechanical coupling problem are introduced firstly.Then,the quasi-convex reproducing kernel approximation is combined with the Galerkin integral weak form and the penalty function method is used to process the essential boundary conditions.The quasi-convex reproducing kernel particle method for transient heat conduction problem and transient thermal-mechanical coupling problem is established,and the corresponding MATLAB program is compiled.Through several numerical examples,the accuracy and effectiveness of the proposed method on transient thermal-mechanical coupling problem are verified.Besides,the selection of calculation parameters and the random distribution are analyzed.Chapter 6 summarizes the contents of this paper,points out the innovations and shortcomings of the article,and looks forward to the future development prospects.