Upcoming Event: Oden Institute Seminar

Mathematical Preconditions for Predictive Meshfree Solutions in Extreme Event Simulation

Dr. Mike Hillman, Principal Scientist, Karagozian & Case, Inc.

Abstract

The mechanics of extreme events span fracture, localization, shocks, and fluid-like plastic flow, necessitating a departure from traditional Lagrangian finite elements for effective solutions. While meshfree methods offer a flexible alternative, I will argue in this talk that they must be rigorously formulated for truly predictive simulations. This requires moving beyond colloquial definitions of accuracy and stability and instead clearly specifying mathematical criteria. Specifically: achieving completeness, the best approximation, variational consistency, and convergence, while precluding material, kernel, rank, and tensile instabilities. Emphasis is placed on parameter-free, natural forms of stabilization.
 

Many meshfree and particle methods employ pseudo-derivatives where the Gauss–Green divergence theorem is inapplicable. I will discuss how their use directly prevents achieving the above properties. Next, I will cover the importance of strain smoothing, showing that it is an essential ingredient in satisfying variational consistency, capturing strain localization, and precluding tensile instability. Finally, results will be presented for applications including close-in explosions and fragment-impact problems.

Biography

Dr. Michael C. Hillman is a Principal Scientist at Karagozian & Case, specializing in nonlinear computational solid mechanics and extreme event simulation. An NSF CAREER awardee and formerly a tenured Associate Professor at Penn State, his work focuses on industrial-strength Galerkin meshfree methods. His frameworks of VCI, NSNI, and RKPD have proven foundational in the field and are implemented in production codes across the defense and commercial sectors, including LS-DYNA, SIERRA, and DoD solid codes.