A Generalized Interpolation Material Point Method for Shallow Ice Shelves. 1: Shallow Shelf Approximation and Ice Thickness Evolution

We develop a generalized interpolation material point method (GIMPM) for the shallow shelf approximation (SSA) of ice flow. The GIMPM, which can be viewed as a particle version of the finite element method, is used here to solve the shallow shelf approximations of the momentum balance and ice thickness evolution equations. We introduce novel numerical schemes for particle splitting and integration at domain boundaries to accurately simulate the spreading of an ice shelf. The advantages of the proposed GIMPM-SSA framework include efficient advection of history or internal state variables without diffusion errors, automated tracking of the ice front and grounding line at sub-element scales, and a weak formulation based on well-established conventions of the finite element method with minimal additional computational cost. We demonstrate the numerical accuracy and stability of the GIMPM using 1-D and 2-D benchmark examples. We also compare the accuracy of the GIMPM with the standard material point method (sMPM) and a reweighted form of the sMPM. We find that the grid-crossing error is very severe for SSA simulations with the sMPM, whereas the GIMPM successfully mitigates this error. While the grid-crossing error can be reasonably reduced in the sMPM by implementing a simple material point reweighting scheme, this approach it not as accurate as the GIMPM. Thus, we illustrate that the GIMPM-SSA framework is viable for the simulation of ice sheet-shelf evolution and enables boundary tracking and error-free advection of history or state variables, such as ice thickness or damage.

Automated summary

The Generalized Interpolation Material Point Method (GIMPM) is developed for solving the shallow shelf approximation (SSA) of ice flow, introducing novel numerical schemes for accurate simulation of ice shelf spreading. The GIMPM-SSA framework allows for efficient advection of variables, automated boundary tracking, and error-free advection, demonstrating numerical accuracy and stability in benchmark examples. Comparisons with the standard material point methods show that GIMPM successfully mitigates grid-crossing errors, making it a viable option for ice sheet-shelf evolution simulations.