Session: 02-04: Modeling and Simulation of Active Materials

Paper Number: 92672

92672 - A Comparison of Meshfree and Finite Element Based Magnetostatic Modeling 

Conventional numerical modeling of magnetic problems often utilizes grid-based methods such as finite element method (FEM) or finite volume method. Mesh dependency of these approaches introduce challenges for micromagnetics problems which require mesh sizes on the order of the exchange length (~5-10 nm) to model domain walls and other spin textures. This mesh size restricts models to nano and microscale dimensions (e.g., less than ~1 cubic micron volume) on standard desktop computers, rendering larger geometries practically unrealizable. While mesh refinement can enable larger models, capturing domain wall movement is a large deformation problem which leads to mesh distortion issues, and extensive remeshing. The purpose of this work is to explore alternate computational techniques that may overcome mesh restrictions and potentially enable multiscale modeling.

This research takes advantage of the past two decades of progress in the field of meshfree methods by utilizing the Reproducing Kernel Particle Method (RKPM) [1] to solve magnetic problems. RKPM is a Lagrangian, Moving Least Squares-based method which discretizes the geometry using a scattering of nodes. A key advantage of the RK approximation is the ability to decouple the treatment of continuity and completeness that are necessary for model convergence. This is achieved by constructing compactly supported window (shape) functions that are enriched with a polynomial basis to ensure partition of unity. This decoupling along with the wavelet-like properties of nodal window functions make RKPM ideal for multiscale problems. Also, to keep the technique truly meshfree, this research adopts direct nodal integration with gradient based stabilization using Stabilized Conforming Nodal Integration (SCNI) [2]. 

As the initial part of the path to an eventual meshfree micromagnetics solver, this study focuses on validating the RKPM technique with SCNI for several magnetostatics problems. Multiple constitutive behaviors are considered including: (i) a uniformly magnetized permanent magnet (ii) constant susceptibility and (iii) nonlinear MH behavior.  Comparative analysis of the RKPM results with FEM results are presented based on accuracy and computational cost (i.e., run time). The focus of the findings is to illustrate the viability of meshfree approach as the complexity in computation varies. 

References

[1] Huang,T., Wei, H., Chen, J. S., Hillman, M., “RKPM2D: an open-source implementation of nodally integrated reproducing kernel particle method for solving partial differential equations” Computational Particle Mechanics, (2019).

[2] Chen, J. S.,Wu, C. T., and Yoon, S. “A stabilized conforming nodal integration for Galerkin mesh-free methods.” Int. J. Numer. Meth. Eng., 50(2), (2001).

Presenting Author: John Domann Virginia Tech