Theory Study On High Precision Meshfree Method In Computational Mechanics

Meshless methods are regarded as next generation of computationalmethods. The key idea of meshless methods is to provide numericalsolutions for integral equations or PDEs with a set of arbitrarilydistributed nodes without using any mesh that provides the connectivityof these nodes. The smoothed particle hydrodynamics (SPH) is a widelyused messless method. However, there are fewer efficient SPH methodsto treat solid‐liquid boundary and the error of SPH approximation couldlead to computational failure in the case that the computation domainincreases or in the case that the derivative order is high.The dissertation proves three lemmas: When particles in the supportdomain of smoothing kernel function are distributed evenly andsymmetrically, the ratio of the real value of linear function to its SPHapproximation is constant, the ratio of the real value of the nthderivative of the polynomial function P?????? to its SPH approximation isconstant and the ratio of the real value of ????P?? to its SPHapproximation is constant. A new kind of high precision SPH methodbased on the lemmas mentioned above is achieved in this dissertation. The dissertation introduces the moving least squares method (MLS)which can make particles in the support domain distribute evenly andsymmetrically. However, the computation can be simplified byintroducing background function instead of MLS. If the backgroundfunction is linear function, the formula deduced in this dissertation isequal to that of Belytschko when calculating 1st derivative.Through the application of SPH approximation to the elasticmechanics equation of variation, it is discovered that the secondderivative of SPH approximation can be obtained by integral action. Bythis method, a new concept of widened‐kernel function is introduced.In this dissertation, the high accuracy SPH approximation is appliedto the elastic mechanics equations and Navier‐Stokes equations. Thecomparisons between the results computed by ANSYS and the methoddeduced in the present study prove that the present method is accurate.However, the precision improvement is limited in CFD because there isno high precision method to deal with boundary conditions.Moreover, the combination of SPH and Dynamic Relaxation (D.R.) isinvestigated. Meanwhile, the boundary conditions are dealt with by thewidened‐kernel function.