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. Plain Language Summary: Ice shelves largely govern the evolution of the Antarctic ice sheet by buttressing grounded ice flow into the ocean. This buttressing is sensitive to changes in ice thickness, upstream ice flow, ice front position, contact with bedrock, fracture‐induced weakening, and calving. The current generation of ice flow models are particularly ill‐suited for capturing the processes associated with fracture and boundary tracking because they solve equations using exclusively mesh‐based methods. For large‐deformation ice flow, these methods produce diffusion errors when advecting history variables (e.g., fracture variables) and often rely on overly approximate or cumbersome schemes to track the ice front position. Here, we introduce an ice shelf flow model based on the material point method, a particle variation of the finite element method where boundaries are naturally tracked and advection errors are alleviated. A mesh is only needed when solving the momentum equations, where the particles serve as moving integration points. As part of our implementation, we introduce new schemes for particle splitting and integration at domain boundaries, and we test several shape functions for mapping between the particles and the grid. We demonstrate the accuracy and stability of the method with a series of benchmark examples. Key Points: Our material point method for ice shelf flow enables error‐free advection of history variables, such as damage, and ice front trackingThe method can be readily implemented into existing finite element software (here, Elmer/Ice), and is suitable for large‐scale applicationThe method is verified against analytical solutions for steady‐state flow and front evolution, and tested on an idealized marine ice sheet [ABSTRACT FROM AUTHOR]