Development Of The Elasto-Plastic Smoothed Finite Element Methods Program In Julia Language

As one of the most successful numerical methods,the finite element method has gradually revealed some limitations and defects with its wide application in engineering,for example,(1)the finite element method model is too "hard";(2)the stress solution is inaccurate;(3)only the lower bound solution can be given;(4)it is very sensitive to distorted meshes and cannot be applied to large deformation problems.To solve these issues,G.R.Liu et al.proposed the smoothed finite element methods,which is a series of methods designed from the standard finite element method and meshfree methods.The smoothed finite element methods use a Galerkin model based on modified or reconstructed strain fields for cross-element operations to obtain more information from adjacent elements.Much commercial software and open-source programs have been developed for the finite element method,but there is less work on the development of programs for the smoothed finite element methods.Most of the existing programs are written in C++ or Fortran,which are not easy to read and maintain.For programs written in dynamic programming languages such as MATLAB or Python,there are problems in terms of licensing fees or computational efficiency.Thus,it is necessary to develop smoothed finite element methods programs that can consider computational accuracy,computational efficiency,and code readability.In this paper,a smoothed finite element methods program and the parallel algorithms for large-scale meshes are developed based on the open-source,high-performance Julia language.Taking advantages of the Julia language and multi-core computers,a program that balances computational accuracy,computational efficiency,and code readability is implemented.The main research contents and conclusions of this paper can be summarized as follows:1.The ES-FEM program for two-dimensional elastic problems and the FS-FEM program for three-dimensional elastic problems are developed.The parallelizability of the elastic FS-FEM on a multi-core platform is analyzed,and a parallel algorithm is designed for the large-scale meshes model.The global stiffness matrix is computed and assembled in a compressed format to reduce the memory occupation.2.Based on the total theory of plasticity,it is extended to the plastic FS-FEM program using the elastic FS-FEM program.The parallelizability of the plastic FS-FEM on a multi-core platform is analyzed,and a parallel algorithm is designed for the large-scale meshes model.3.A series of numerical experiments and a case study are conducted to verify the computational accuracy of the elastic ES-FEM and FS-FEM,the computational efficiency of the elastic FS-FEM parallel algorithm,the computational accuracy of the plastic FS-FEM,and the computational efficiency of the plastic FS-FEM parallel algorithm.The results show that the computational accuracy of elastic ES-FEM,FS-FEM,and plastic FS-FEM are higher than that of the standard finite element method;the speedup ratios of elastic and plastic FS-FEM parallel algorithms in the case of a 24-core computer with about two million elements are all close to linear speedup ratios,and both serial and parallel algorithms are faster than commercial software.In the case of the mine site,the computation results and trends of plastic FS-FEM are similar to that of the finite difference method,but the computation accuracy still has room for improvement.