A mesh free method using rectangular pre-solved domains using Kronecker products

ABSTRACT

In this paper, a mesh free algorithm using large pre-solved domain is developed. Using the largest rectangle inside an arbitrary domain, a pre-solved rectangular domain is established using Kronecker product and graph theory rules. This pre-solved domain is efficiently inserted into the mesh free formulation of partial differential equations (PDEs) and engineering problems to reduce the computational complexity and execution time of the solution. The general solution of the pre-solved rectangular domain is formulated for second-order shape functions. The efficiency of the present algorithm depends on the relative size of the large rectangular domain and the main domain; however, the method remains as efficient as a standard method for even small relative sizes. For adaptive procedures with nonuniform density of distributed points in the domain, smaller (e.g. sub-maximal) rectangular domain can be used. The application of the method is demonstrated using some examples.

KEYWORDS:

• Efficient mesh free method
• graph theory
• Kronecker product
• large pre-solved domain
• partial differential equations

Notes

The star point of a cloud is the node for which the function u (e.g., u_h(x)) is calculated. A cloud is known with its star point and the maximum value of the weight function in a cloud occurs in the star point.