A new and efficient meshfree method to solve partial differential equations: Application to three-dimensional transient heat transfer problems

The paper presents the average radial particle method (ARPM), a new mesh-free technique for solving partial differential equations (PDEs). Here, we use the ARPM to solve 3D transient heat transfer problems. ARPM numerically approximates spatial derivatives by discretizing the domain by particles such that each particle is only affected by its direct neighbors. One feature that makes ARPM different is using a representative neighboring particle whose average variable value, like temperature, is used to approximate first and second spatial derivatives. ARPM has several advantages over other numerical methods. It is highly efficient, with a time requirement of only 0.6 µs per particle per step. It makes conducting rapid simulations with half a million particles in one minute possible. It is also distinct from other methods because it does not suffer from boundary or surface effects. Besides, the ARPM application is straightforward and could be easily integrated into software packages. Additionally, ARPM has lower convergence requirements for both time and space. The method's effectiveness is validated through five problems with different configurations and boundary conditions, demonstrating its accuracy and efficiency.