Aiming to solve mesh generation,computational stability,accuracy control,and other problems encountered with existing numerical methods,such as the finite element method and the finite volume method,a new numerical co...Aiming to solve mesh generation,computational stability,accuracy control,and other problems encountered with existing numerical methods,such as the finite element method and the finite volume method,a new numerical computational method for continuum mechanics,namely the manifold method based on independent covers(MMIC),is proposed based on the concept of mathematical manifolds,to form partitioned series solutions of partial differential equations.As partitions,the cover meshes have the characteristics of arbitrary shape,arbitrary connection,and arbitrary refinement.They are expected to fundamentally solve the mesh generation problem and can also simulate the precise geometric boundaries of the CAD model and strictly impose boundary conditions.In the selection of series solutions,local analytical solutions(such as series solutions at crack tips and series solutions in infinite domains)or proper forms of complete series can be used to reflect the local or global characteristics of the physical field to accelerate convergence.Various applications are presented.A new method of beam,plate,and shell analysis is proposed.The deformation characteristics of beams,plates,and shells are simulated with polynomial series of suitable forms,and the analysis of curved beams and shells with accurate geometric representation is realized.For the static elastic analysis of two-dimensional structures,a mesh splitting algorithm is proposed,and h-p version adaptive analysis is carried out with error estimation.Thus,automatic computation integrated with CAD is attempted.Adaptive analysis is also attempted for the solution of differential equations of fluids.For the one-dimensional convection-diffusion equation and Burgers equation,calculation results with high precision are obtained in strong convection and shock wave simulations,avoiding nonphysical oscillations.And solving the two-dimensional incompressible Navier-Stokes equation is also attempted.The series solution formula is used to obtain the physical quantity of interest of the material at a space point to eliminate the convection terms.Thus,geometrically nonlinear problems can be analyzed in fixed meshes,and a new method of free surface tracking is proposed.展开更多
基金supported by the Fundamental Research Funds for Central Public Welfare Research Institutes in China(Grant Nos.CKSF2010012/CL,CKSF2013031/CL,CKSF2014054/CL,CKSF2015033/CL,and CKSF2016022/CL)。
文摘Aiming to solve mesh generation,computational stability,accuracy control,and other problems encountered with existing numerical methods,such as the finite element method and the finite volume method,a new numerical computational method for continuum mechanics,namely the manifold method based on independent covers(MMIC),is proposed based on the concept of mathematical manifolds,to form partitioned series solutions of partial differential equations.As partitions,the cover meshes have the characteristics of arbitrary shape,arbitrary connection,and arbitrary refinement.They are expected to fundamentally solve the mesh generation problem and can also simulate the precise geometric boundaries of the CAD model and strictly impose boundary conditions.In the selection of series solutions,local analytical solutions(such as series solutions at crack tips and series solutions in infinite domains)or proper forms of complete series can be used to reflect the local or global characteristics of the physical field to accelerate convergence.Various applications are presented.A new method of beam,plate,and shell analysis is proposed.The deformation characteristics of beams,plates,and shells are simulated with polynomial series of suitable forms,and the analysis of curved beams and shells with accurate geometric representation is realized.For the static elastic analysis of two-dimensional structures,a mesh splitting algorithm is proposed,and h-p version adaptive analysis is carried out with error estimation.Thus,automatic computation integrated with CAD is attempted.Adaptive analysis is also attempted for the solution of differential equations of fluids.For the one-dimensional convection-diffusion equation and Burgers equation,calculation results with high precision are obtained in strong convection and shock wave simulations,avoiding nonphysical oscillations.And solving the two-dimensional incompressible Navier-Stokes equation is also attempted.The series solution formula is used to obtain the physical quantity of interest of the material at a space point to eliminate the convection terms.Thus,geometrically nonlinear problems can be analyzed in fixed meshes,and a new method of free surface tracking is proposed.
