Fundamental Study On Meshless Numerical Integration Method

The meshless method is one of domain methods in engineering with widespread application at present, because of its high computation precision. So it receives the value to calculate worker's favor. But this method faces a major issue is the usage of common integration, which limits its superiority. Therefore the meshless integral question theory urgently waited to study. This article is in the foundation of works of predecessor. It bases on the meshless method numerical integration fundamental research , some beneficial explorations are done by applying each kind of integral theory in the meshless numerical computation domain, including the meshless method numerical integration mathematics foundation, the existing each meshless numerical integration method and so on. The prime task induction is as follows:1. Thoroughly discussing several commonly used meshless method interpolation theory and proving the unified question between each meshless method. Carrying on the induction of the moving least squares method, the radial basis function, reproducing kernel function method, the nature neighbor interpolation, Partition of Unity method and so on..2. Researching numerical integration theory and further inducing the numerical integration method which each meshless method uses. Summarizing these method general character and carrying on the classification of them. Dividing three big kinds: background cells integration, finite cells integration and point integration.3. Through analysis the integral error of the meshless method, discovering reasons of each kind of errors and inducing the general character of the meshless integral error. After the view of the nature neighbor method integral error, through comparing the actual effect of various integrals methods, finally choosing the Monte Carlo integral method to solve these errors, also giving the steps of the use of Monte Carlo integral method in the nature neighbor method, as well as in other meshless methods..4. Through the massive examples, confirming the computation validity of the Monte Carlo integral in nature neighbor method.Through the above several aspect exploration, obtaining the main conclusions: The meshless method shape function is a rational formula and using the dynamic interpolation, which causes the integral errors; In view of nature neighbouring method, when points are lots, the Monte Carlo integration computing precision is higher than the Hammer integration.In summary, the Monte Carlo method can be an effective solution to solve the errors which caused by the shape function of meshless method, resulting error by inequality between shape function supported region and integral region and the unifiedlessness of the dynamic interpolation shape function formula. A series of examples and theory solution contrast prove the correctness of this article in both theory and method .