This paper presents a method that modifies commercial engineering-oriented finite element packages for the modelling of Glacial Isostatic Adjustment (GIA) on a self-gravitating, compressible and spherical Earth with 3-D structures. The approach, called the iterative finite element body and surface force (FEMIBSF) approach, solves the equilibrium equation for deformation using the ABAQUS finite element package and calculates potential perturbation consistently with finite element theory, avoiding the use of spherical harmonics. The key to this approach lies in computing the mean external body forces for each finite element within the Earth and pressure on Earth's surface and core-mantle boundary (CMB). These quantities, which drive the deformation and stress perturbation of GIA but are not included in the equation of motion of commercial finite element packages, are implemented therein. The method also demonstrates how to calculate degree-1 deformation directly in the spatial domain and Earth-load system for GIA models. To validate the FEMIBSF method, loading Love numbers (LLNs) for homogeneous and layered earth models are calculated and compared with three independent GIA methodologies: the normal-mode method, the iterative body force method and the spectral-finite element method. Results show that the FEMIBSF method can accurately reproduce the unstable modes for the homogeneous compressible model and agree reasonably well with the Love number results from other methods. It is found that the accuracy of the FEMIBSF method increases with higher resolution, but a non-conformal mesh should be avoided due to creating the so-called hanging nodes. The role of a potential force at the CMB is also studied and found to only affect the long-wavelength surface potential perturbation and deformation in the viscous time regime. In conclusion, the FEMIBSF method is ready for use in realistic GIA studies, with modelled vertical and horizontal displacement rates in a disc load case showing agreement with other two GIA methods within the uncertainty level of GNSS measurements.

The redistribution of past and present ice and ocean loading on Earth's surface causes solid Earth deformation and geoid changes, known as glacial isostatic adjustment. The deformation is controlled by elastic and viscous material parameters, which are inhomogeneous in the Earth. We present a new viscoelastic solid Earth deformation model in ASPECT (Advanced Solver for Problems in Earth's ConvecTion): a modern, massively parallel, open-source finite element code originally designed to simulate convection in the Earth's mantle. We show the performance of solid Earth deformation in ASPECT and compare solutions to TABOO, a semianalytical code, and Abaqus, a commercial finite element code. The maximum deformation and deformation rates using ASPECT agree within 2.6% for the average percentage difference with TABOO and Abaqus on glacial cycle (∼100 kyr) and contemporary ice melt (∼100 years) timescales. This gives confidence in the performance of our new solid Earth deformation model. We also demonstrate the computational efficiency of using adaptively refined meshes, which is a great advantage for solid Earth deformation modeling. Furthermore, we demonstrate the model performance in the presence of lateral viscosity variations in the upper mantle and report on parallel scalability of the code. This benchmarked code can now be used to investigate regional solid Earth deformation rates from ice age and contemporary ice melt. This is especially interesting for low-viscosity regions in the upper mantle beneath Antarctica and Greenland, where it is not fully understood how ice age and contemporary ice melting contribute to geodetic measurements of solid Earth deformation.

We present a finite-element model of post-seismic solid Earth deformation built in the software package Abaqus (version 2018). The model is global and spherical, includes self-gravitation and is built for the purpose of calculating post-seismic deformation in the far field (1/4300gkm) of major earthquakes. An earthquake is simulated by prescribing slip on a fault plane in the mesh and the model relaxes under the resulting change in stress. Both linear Maxwell and biviscous (Burgers) rheological models have been implemented and the model can be easily adapted to include different rheological models and lateral variations in Earth structure, a particular advantage over existing models. We benchmark the model against an analytical coseismic solution and an existing open-source post-seismic model code, demonstrating good agreement for all fault geometries tested. Due to the inclusion of self-gravity, the model has the potential for predicting deformation in response to multiple sources of stress change, for example, changing ice thickness in tectonically active regions.