A background decomposition method for domain integration in weak-form meshfree methods

• A background decomposition method for evaluation of domain integrals is presented.
• The method uses quadtree and octree techniques for 2D and 3D problems, respectively.
• The method is useful for cases with irregularly and non-uniformly distributed nodes.
• The method can be used for adaptive analysis.

An efficient technique is presented for evaluation of a domain integral in which the integrand is defined by its values at a discrete set of nodes with highly varying density. The proposed technique uses quadtree and octree techniques for 2D and 3D domains, respectively, so that the background of the integration domain can be divided into a few partitions with different grades of nodal density. The integrals over all partitions are then evaluated and added together to get the value of the whole-domain integral. Some numerical examples are given to show the accuracy and efficiency of the presented method.