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.

中文翻译：

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浅冰架广义插值质点法。 1：浅架近似和冰厚演化


我们开发了一种用于冰流浅架近似（SSA）的广义插值材料点方法（GIMPM）。 GIMPM 可被视为有限元方法的粒子版本，此处用于求解动量平衡和冰厚度演化方程的浅架近似。我们引入了域边界处粒子分裂和积分的新颖数值方案，以准确模拟冰架的扩展。所提出的 GIMPM-SSA 框架的优点包括历史或内部状态变量的有效平流，没有扩散误差，在子元素尺度上自动跟踪冰锋和接地线，以及基于有限元惯例的弱公式。具有最小额外计算成本的元素方法。我们使用一维和二维基准示例展示了 GIMPM 的数值精度和稳定性。我们还将 GIMPM 与标准材料点法 (sMPM) 和 sMPM 的重新加权形式的准确性进行了比较。我们发现，对于使用 sMPM 进行的 SSA 模拟，网格交叉误差非常严重，而 GIMPM 成功地缓解了该误差。虽然通过实施简单的材料点重新加权方案可以合理地减少 sMPM 中的网格交叉误差，但这种方法不如 GIMPM 准确。因此，我们证明 GIMPM-SSA 框架对于冰盖演化的模拟是可行的，并且能够实现历史或状态变量（例如冰厚度或损坏）的边界跟踪和无误差平流。