We present a new, open-source viscoelastic solid earth deformation model, Elmer/Earth. Using the multi-physics finite-element package Elmer, a model to compute viscoelastic material deformation has been implemented into the existing linear elasticity solver routine. Unlike approaches often implemented in engineering codes, our solver accounts for the restoring force of buoyancy within a system of layers with depth-varying density. It does this by directly integrating the solution of the system rather than by applying stress-jump conditions in the form of Winkler foundations on inter-layer boundaries, as is usually needed when solving the minimization problem given by the stress divergence in commercial codes.
We benchmarked the new model with results from a commercial finite-element engineering package (ABAQUS, v2018) and another open-source code that uses viscoelastic normal mode theory, TABOO, using a flat-earth setup loaded by a cylindrical disc of 100

Reconstructing ice-sheet history and predicting ice-sheet response to changes in climate are imperative for accurately predicting future ice-mass loss and hence sea-level rise. An important component of ice-sheet evolution is the isostatic response of the solid earth that occurs as a result of changes in the mass of the ice sheet. Over glacial cycles the waxing and waning of ice sheets causes the underlying earth to deform as the ice loading at the surface grows and shrinks. This deformation occurs both instantaneously as an elastic response and over longer timescales as the viscous mantle flows back to previously glaciated regions in order to regain gravitational equilibrium. How fast or slowly the earth deforms depends on the underlying mantle viscosity, and, although typically thought to occur over several thousands of years

This isostatic response of the bedrock can strongly influence ice-sheet dynamics. Deformation of the earth changes the elevation of the ice sheet which in turn affects the surface temperature and the rate of accumulation or ablation. Solid earth deformation also alters the gradient of the bedrock on which the ice sheet rests, particularly at the periphery, altering the internal forces as well as the driving stress and therefore the flow of the ice sheet

Including the isostatic response of bedrock in an ice-sheet model is therefore crucial to obtaining accurate predictions of ice-sheet mass balance, and there are several methods which can be used. Computing the isostatic response with a self-gravitating viscoelastic spherical earth is the most accurate, but most computationally expensive, method. Several simple approximations are often made using models with a combinations of local lithosphere or elastic lithosphere with diffusive asthenosphere or relaxing asthenosphere

A further improvement to an ice-sheet model can be made by coupling a model of solid earth deformation to the ice-sheet model. Studies have demonstrated that the feedback between the two systems can have large impacts on ice-sheet evolution

The implementation of the viscoelastic rheology and additional force terms, to a large extent, follows the one suggested by

The linearized equation of motion for solid earth deformation

Elmer/Earth is based on the open-source finite-element package Elmer

Many commercial codes lack an implementation of the second term in Eq. (

Here we take advantage of the accessibility of the source code of Elmer by including this term in the weak formulation that uses the viscoelastic stress. The second term in Eq. (

Discretization of the time derivatives for stress and pressure (in the case of incompressible material) is implemented by the first-order implicit difference

Equation (

Benchmark tests are performed in order to validate the new implementation of Elmer/Earth in
comparison to two other codes: ABAQUS and TABOO. We force the models with changing surface load, representing an idealized ice loading experiment. Specific geometry, earth structure and ice loading for the benchmarking case are described in Sect.

We use the finite-element software package ABAQUS (

TABOO is an open-source post-glacial rebound calculator

In order to test and compare the newly built Elmer/Earth model, a simple benchmark case has been set up for each of the models presented in Sect.

For the flat-earth approximation, the three-dimensional model domain stretches 4000 km in each horizontal direction from the centre of the ice load. This distance is 80 times the diameter of the test load, which is more than sufficient to allow mantle deformation below the ice load

Geometry construction and meshing for Elmer/Earth simulations was achieved using the open-source software Gmsh

Top and side view of the reference run Elmer/Earth mesh (

The earth structure used for the benchmarking case is one that is included as part of the TABOO package and is summarized in Table

The viscosity of the upper and lower mantle is set to

For the benchmark case we compute the deformation caused by an instantaneously imposed ice load at

Cross section of the reference run with Elmer/Earth (

The temporal evolution of the vertical displacement of the reference Elmer/Earth run (

Temporal evolution of vertical displacement of the reference Elmer/Earth run (

Properties of the different layers in the flat-earth model benchmark. Vertical distances are with respect to earth's centre. The ABAQUS reference model uses a material model with a constant Poisson ratio of 0.49 throughout the whole domain.

For all runs of Elmer/Earth presented in Sects.

Comparing the results of the benchmarking exercise with two models that use different methods gives us confidence in the implementation of the new Elmer code. Figure

Comparison of results for deformation at the load centre (0

The displacement curves for all three models over major parts of the simulation agree to within an order of 10

Difference in deformation of Elmer/Earth relative to ABAQUS and TABOO at the load centre (0

The small differences between the results could be caused by several factors. Mesh differences between Elmer/Earth and ABAQUS are the likely cause of some small differences with ABAQUS having a regular grid mesh and Elmer having a finer mesh at the centre of the disc. There seems to be a correlation of the resolution in the centre with the displacement in both FEM-based models. It seems that the ABAQUS model setup does not provide enough horizontal mesh resolution at the centre, where the load is applied. This is confirmed by results obtained with

The deformation calculated by TABOO is less than Elmer/Earth and ABAQUS at each location. This may be due to the fundamental differences in the computation methods employed by the TABOO code, implementing normal mode methods rather than finite-element methods. Furthermore, TABOO computes deformation on a self-gravitating solid earth, whereas ABAQUS and Elmer do not include self-gravitation, which would result in some differences between these models. Nevertheless, the differences observed in the displacement curves are still within an acceptable tolerance.

In order to obtain some insight into parallel performance as well as the dependency on the mesh resolution of Elmer/Earth, three meshes with different resolutions and mesh partitions (4, 16 and 32) have been created (see Table

Parameters of the meshes and their partitions used for Elmer/Earth test runs.

Identical numerical parameters and methods, as described in Sect.

Tests were performed on the Linux cluster

We want to emphasize that we only studied a limited set of problem sizes or computing resource configurations, and only single runs (no statistics) were performed. Results presented in the following thus have to be interpreted in view of the limitations. All runs performed are summarized in Table

Timings of different scalability test runs. All timings are given in seconds.

A comparison of a simulation performed with 16 cores (single compute node) with

On the other hand, if looking at strong scalability (i.e., increasing core numbers while reducing load/core), doubling computational resources from 16 cores (single compute node) to 32 cores (inter-nodal) for the fixed-size smaller problem (

Despite applying the same solution method, it is not really possible to compare the performance of Elmer/Earth to ABAQUS, since the latter was run on a different platform using a regular mesh of 10

We further studied the accuracy and consistency of Elmer/Earth results with respect to spatial and temporal discretization sizes. To that end, we ran the same numerical setup on all three meshes given in Table

Vertical deformation at the centre (0

Results are depicted in Fig.

We presented a newly implemented viscoelastic addition to the linear elasticity solver of the open-source finite-element package Elmer and its application to a flat-earth model. Robust projection of future ice-sheet change depends on coupled solid earth and ice dynamic processes at high spatial resolution, and Elmer/Earth provides a new open-source capability in conjunction with the existing ice-sheet model Elmer/Ice

For the time being, Elmer/Earth is a so-called flat-earth model

We benchmarked Elmer/Earth with another FEM code, ABAQUS, as well as a spherical viscoelastic normal mode code, TABOO, and these comparisons show good agreement in the range of deviation in solution method as well as numerical approaches.

Scaling figures presented in Sect.

Elmer (version 8.4) is available for download under GitHub (

The animation (

TZ helped developing and implementing the model setup and performing the computations in Elmer. GAN contributed to the design of the benchmark setup and performed the computations in ABAQUS and TABOO. JR implemented the altered model equations in the source code of Elmer. MAK conceived the study and consulted in the model implementation and contributed to the design of the benchmark test. All authors contributed to the paper.

The authors declare that they have no conflict of interest.

Development of the viscoelastic model was supported under the Australian Research Council's Special Research Initiative for Antarctic Gateway Partnership (Project ID SR140300001) and Discovery Project DP170100224. Part of the work of Thomas Zwinger was enabled by a visiting scientist scholarship from UTAS. This research was undertaken with the assistance and resources from the National Computational Infrastructure (NCI Australia), an NCRIS-enabled capability supported by the Australian Government. We want to express our gratitude to Peter Råback (CSC) for solving a problem with post-processing of Elmer/Earth data and Fredrik Robertsén (CSC) for the discussion on scalability test results. We are grateful to Giorgio Spada for making TABOO open source. We want to thank the two reviewers and the editor for constructive suggestions to improve the quality of this paper.

This research has been supported by the Australian Research Council (grant nos. SR140300001 and DP170100224).

This paper was edited by Thomas Poulet and reviewed by PingPing Huang and Surendra Adhikari.