The three-dimensional meshfree numerical manifold method based on parallel computing

The numerical manifold method, extensively utilized in numerical computations, faces significant challenges in generating complex manifold elements, particularly for three-dimensional applications. To overcome this challenge, the meshfree numerical manifold method is developed by integrating the moving least-squares method into the numerical manifold method, effectively bypassing the need for meshing complex geometric objects. However, the implementation of the moving least-squares method introduces computational efficiency issues. To mitigate these, parallel computing methods have been incorporated, resulting in a tenfold increase in the speed of assembling the stiffness matrix with central processing unit parallelism, and a twentyfold increase with graphics processing unit parallelism. The static mechanical system equations for the meshfree numerical manifold method are derived using the Galerkin method. The method’s effectiveness and accuracy are then validated through a series of numerical experiments. The experiments demonstrated that the meshfree numerical manifold method achieves a high precision with minimal nodes and integration points. Additionally, positioning nodes outside the domain significantly improves computational accuracy at the boundaries.