Seminar Series: Stabilized Extended B-spline Material Point Method for Multi-field Problems

"Stabilized Extended B-spline Material Point Method for Multi-field Problems"

Abstract:
Material point method (MPM), a hybrid Eulerian-Lagrangian particle-based continuum method, is a promising alternative to the traditional mesh based finite element method due to its robustness in handling extreme deformations and no-slip contact without additional algorithmic treatment. The use of B-spline shape functions in MPM effectively mitigates the classical numerical artifacts typically associated with linear shape functions, such as artificial fractures and cell crossing instabilities. Implicit MPM is preferred in modeling highly deformable materials like elastomers, hydrogels and biological tissues often characterized and used under quasi-static conditions. However, the B-spline MPM approach faces challenges with the ill-conditioning of the stiffness matrix in implicit models, due to fewer material points at boundaries, and complexities in imposing boundary conditions. Furthermore, modeling at quasi compressible limit, for coupled problems such as capturing large deformations along with water transport in hydrogels, require development of inf-sup stable mixed methods [1]. Here, we resolve these challenges using a subdivision stabilized mixed formulation, and further extending the subdivision-technique to extended B-spline technique. The extended B-spline shape functions addresses the ill-conditioning, where the displacement and pressure (or chemical potential) fields near the detected physical boundaries are interpolated to the interior ones using polynomials. The subdivision-based extended B-spline method achieves an oscillation-free, inf-sup stable mixed discretization in modeling nearly incompressible materials, such as elastomers or hydrogels, under instantaneous loading. Through the use of the penalty method, essential boundary conditions are weakly imposed directly on the Lagrangian particles, instead of the Eulerian background grid where the solution is obtained. The stability and accuracy of our mixed B-spline MPM is tested at extreme deformations and verified by comparing our result with benchmark problems, including incompressible cylindrical elastomer and hydrogels with a cavity inside under pressure loading. These examples showcase the effectiveness of developed particle-based methods in modeling practical soft material applications without any numerical instabilities.

[1] Madadi, A. Dortdivanlioglu, B. A subdivision-stabilized B-spline mixed material point method. Computer Methods in Applied Mechanics and Engineering (2024).

Presented by Ashkan Ali Madadi as part of the USACM Student Chapter Seminar Series on 31 January 2024

Bio:
Ashkan Ali Madadi is currently pursuing his Ph.D. in the MUSE program of the Civil Engineering department and a researcher in the DOE Center for Materials for Water and Energy Systems at the University of Texas at Austin, specializing in the computational mechanics of soft polymeric materials. He completed his M.Sc. and B.Sc. in Civil Engineering at Sharif University of Technology in 2020 and 2017, respectively. During his master’s program, Ashkan concentrated on the multiscale modeling of nanomaterials using coarse-grained molecular dynamics. In his ongoing doctoral research, Ashkan is exploring the Material Point Method (MPM) — a hybrid Eulerian-Lagrangian, particle-based continuum method for studying multiphysics problems. This method is recognized as a promising alternative to traditional mesh-based finite element methods, especially for its robustness in handling extreme deforma- tions and facilitating no-slip contact without the need for additional algorithmic treatments. Ashkan’s current work primarily focuses on resolving numerical challenges in particle based modeling using MPM to enhance its stability and accuracy. These foundational improvements are crucial for his subsequent application of MPM in modeling and improving the mechanics of ultra-filtration membranes. Ashkan aims to advance our understanding and development of polymeric membranes, showcasing the practical implications of his research in computational mechanics.