We present a semi-brittle rheology and explore its potential for simulating glacier and ice sheet deformation using a numerical model, DynEarthSol3D (DES), in simple, idealized experiments. DES is a finite-element solver for the dynamic and quasi-static simulation of continuous media. The experiments within demonstrate the potential for DES to simulate ice failure and deformation in dynamic regions of glaciers, especially at quickly changing boundaries like glacier termini in contact with the ocean. We explore the effect that different rheological assumptions have on the pattern of flow and failure. We find that the use of a semi-brittle constitutive law is a sufficient material condition to form the characteristic pattern of basal crevasse-aided pinch-and-swell geometry, which is observed globally in floating portions of ice and can often aid in eroding the ice sheet margins in direct contact with oceans.

Accurate prediction of global sea level rise depends critically on numerical models' ability to project the removal of ice from the margins of ice sheets and glaciers under climate change scenarios – especially those in contact with oceans. In the past 5 years numerical models have largely risen to the challenge of simulating the continent-scale, steady-state viscous flow of ice, leading to the development of the latest class of ice sheet models that represent ice physics across many flow regimes and in three spatial dimensions. These are often nonlinearly viscous, thermo-mechanical models that solve the so-called full-Stokes (FS) equations (e.g., Martin et al., 2004; Gagliardini and Zwinger, 2008; Larour et al., 2012). Models based on shallow ice (SIA) and shallow shelf (SSA) approximations of the FS equations are also in wide use and simulate ice flow well in most areas (e.g., Winkelmann et al., 2011; Lipscomb et al., 2013).

Despite recent advances many pertinent questions in glaciology remain that could potentially be addressed best from a computational perspective, particularly with regard to calving. However representing the smaller-scale physics at the heart of this particular problem (i.e., the fracture of crystalline material) often imposes too large a computational cost to remain a tractable problem for many models. Thus, both the FS and SIA/SSA formulations often employ parametrizations for the most physically complicated aspects of their systems. In particular, the failure of ice within many ice sheet models is often treated using linear elastic fracture mechanics (e.g., Larour et al., 2004), as a rheologically more flexible, time-dependent scalar damage field (e.g., Duddu and Waisman, 2012), or a mixture of the two (Krug et al., 2014, 2015).

Ice rheology has been studied using both geophysical observations and
laboratory experiments (Budd and Jacka, 1989; Sammonds et al., 1998; Goldsby
and Kohlstedt, 2001; Mahrenholtz and Wu, 1992). Over short timescales ice
behaves elastically before yielding or flowing viscously. Over long
timescales ice behaves as a viscous fluid for which the viscosity is nonlinearly
dependent on both temperature and effective stress (Glen, 1955). The
resulting constitutive law is called Glen's flow law in the glaciological
literature and can be written as

Most ice flow numerical models simulate the long-term (hundreds to thousands of years), large-scale behavior of ice sheets using a nonlinear viscous formulation to calculate the stress tensor (e.g., Larour et al., 2012). Indeed, for simulating long-term flow of ice sheets this is an excellent approximation as the Maxwell viscoelastic (VE) stress relaxation timescale (time to dissipate elastic stresses) is on the order of a hours to days – depending on local material properties that affect the ice viscosity and shear modulus (MacAyeal and Sergienko, 2013). When simulating ice rupture, however, these models often employ failure criteria developed with elastic underpinnings. For example, linear elastic fracture mechanics has been a popular and largely accurate criterion for simulating ice fracture when compared to in situ crevasse measurements (e.g., van der Veen, 1998; Rist et al., 1999; Mottram and Benn, 2009; Luckman et al., 2012; Krug et al., 2014). Particle-based numerical models also show a great deal of promise for the simulation of tidewater glacier calving on an interannual timescale, where ice failure occurs via the breakage of elastic bonds. Bassis and Jacobs (2013) recently modeled the retreat of Helheim Glacier using tightly packed particles that interacted elastically and broke once a threshold stress was achieved, often due to the influences of basal topography and buoyancy in floating portions. Astrom et al. (2013) also implemented a particle-based model that included the effects of viscosity by allowing particles joined by elastic beams to form new bonds with nearby particles when stresses were below brittle failure. Thus even though long-term ice flow is approximated well by purely viscous formulations, efforts to simulate ice failure typically incorporate some measure of elastic ice behavior.

While it is true that the Maxwell viscoelastic relaxation time is on the order of hours to days (very short timescales), the consideration of elastic stresses may prove illuminating and useful in understanding terminus retreat. This retreat depends on a long history of failure accumulation – accrued over timescales orders of magnitude much larger than the Maxwell relaxation time – as well as a time-dependent forcing by the ocean on the floating ice (e.g., Bindschadler et al., 2011). For example, calving via the detachment of large, tabular icebergs is an important end-member of observed calving styles (Amundson and Truffer, 2010). The fractures that determine the size of very large icebergs – such as the “loose tooth” at the terminus of the Amery Ice Shelf, and thus the calving rate in these locations – are exactly those features that result from ductile and brittle deformation over yearly to decadal timescales (Bassis et al., 2008). Thus while the elastic component of stress relaxes away over long-term simulations, the fractures resulting from the elastic component of stress remain and affect the ice dynamics (e.g., the complete disintegration of the Larsen B Ice Shelf, examined by Glasser and Scambos, 2008).

In this paper we employ a Lagrangian finite-element model with explicit time integration that allows for both the elastic and viscous components of ice deformation to be taken into account, while simulating ice failure on unstructured meshes. We examine how failure zones form and propagate in an advecting ice slab as it loses contact with the underlying bedrock and begins to float while relaxing the rheological assumption of ice as a purely nonlinear viscous material. The model exploration here is not meant to be wide-reaching and exhaustive; rather, it is presented as a tool to aid in the exploration of how the failure of ice impacts its flow and how rheological assumptions result in different qualitative expressions of ice flow and failure.

DES (DynEarthSol3D) is a robust, adaptive, two- and three-dimensional finite-element model that solves the momentum balance and heat equation in Lagrangian form using unstructured meshes.

While many FS models neglect acceleration and formulate ice flow as a static
problem, momentum conservation in DES takes the full dynamic form:

The temperature field of the ice is modeled using the following heat
equation:

Schematic of one time step in DES.

The governing equations are discretized using an unstructured mesh composed
of triangular (2-D) or tetrahedral (3-D) elements. The approximate
displacement

In DES we make use of both stress and velocity Dirichlet boundary conditions.
Subaerial ice is subject to a traction-free boundary condition, or

The following is a recapitulation of the presentation of DES's constitutive formulation presented by Choi et al. (2013). The interested reader is directed to that work and its references that describe the well-established field of the numerical modeling of large-strain continuum problems involving material failure and elasticity. DES can accumulate large strains by adding up small-strain increments, and the use of small strains at every time step is justified by the relatively small time step size.

The user has the choice in DES of evaluating the stress field as either linear elastic (with the option of employing a Mohr–Coulomb (MC) failure threshold) or linear Maxwell viscoelastic (with no associated failure threshold). We use the former (often termed elastoplastic, EC) to approximate the brittle, rupture-prone behavior of ice. In tectonophysics, for example, elastoplasticity is often understood as the formation or activation of faults. We take a similar interpretation of this rheology as applied to ice: while in our experiments ice is initially rupture-free, the brittle failure of ice approximated by this rheology indicates both the initiation of zones of failure and enables the re-activation of older zones of failure.

For the ductile, rupture-free behavior of ice we use Maxwell viscoelasticity with no associated failure threshold. Finally, as ice in nature both simultaneously flows and ruptures, we combine the two stress evaluations in a third constitutive framework which we call semi-brittle. This case simply calculates both the brittle (elastoplastic) and ductile (viscoelastic) stresses at each point in time in the domain and, depending on a strain rate threshold (Schulson and Duval, 2009), selects the ductile stress if the local strain rate is below the threshold and the brittle stress if the strain rate is above the threshold. Many numerical and material parameters play a role in these stress evaluations, and we have tuned those parameters to match laboratory-derived strain- and strain-rate-vs.-time curves (Appendix A).

In DES the updated stress tensor in the momentum equation is calculated
using the strain rate and strain tensors. These are determined by the
constitutive relation. For the ductile (Maxwell viscoelastic) rheology,
viscosity is determined by Glen's flow law:

In the case of a Mohr–Coulomb material, it is convenient to express the
yield function for the tensile failure as

Frictional materials generally follow a non-associative flow rule, meaning
the direction of plastic flow in the principal stress space is not the same
as the direction of the vector normal to the yield surface. The plastic flow
potential for tensile failure can be defined as

DES is formulated as a finite-element method with explicit time integration, and the order of calculations can be seen in Fig. 1. The advantage of using this method is that the computational cost of each time step is small (compared to implicit methods where advancing by one large time step involves the solving of large, ill-conditioned linear systems) and the implementation of nonlinear rheologies is simple.

The use of the explicit time integration means that the time step is limited
to very small values, on the order of

In addition to the dynamic time-stepping routine, several other numerical
techniques are employed that distinguish this model from implicit
finite-element schemes commonly used to solve FS systems. DES solves the dynamic
momentum balance equation, Eq. (2), by damping the inertial forces at each
time step, giving rise to the quasi-static (i.e., static with time-dependent
boundary conditions) solution. Originally proposed by Cundall (1989), this
variant of dynamic relaxation applies forces at each node in the domain
opposing the direction of the node's velocity vector:

Schematic of experiments.

The linear triangular elements used in DES are known to suffer volumetric locking when subject to incompressible deformations (e.g., Hughes, 2000). Because we model phenomena that require incompressible plastic and viscous flow, we use an anti-volumetric-locking correction based on the nodal mixed discretization methodology (Detournay and Dzik, 2006; De Micheli and Mocellin, 2009). The technique simply averages the volumetric strain rate over a group of neighboring elements and then replaces each element's volumetric strain rate with the averaged one. Choi et al. (2013) describes this technique in greater detail.

Finally, DES makes use of adaptive remeshing. Based on the quality constraints selected by the user, DES assesses the mesh quality at fixed step intervals and remeshes if elements are found in violation (e.g., if a triangular element contains an angle smaller than some input threshold). New nodes may be inserted into the mesh (or old ones deleted), and the mesh topology can be changed through edge flipping. The nodes are provided to the Triangle library (Shewchuk, 1996) to construct a new triangulation of the domain. After the new mesh is created, the boundary conditions, derivatives of shape functions, and mass matrix are recalculated. When deformation is distributed over a large region or the whole domain, remeshing may result in a new mesh quite different from the old one. Because of this possibility the fields associated with nodes (e.g., velocity and temperature) are linearly interpolated from the old mesh to the new. For data associated with elements (e.g., strain and stress) DES uses an approximate conservative mapping described in detail by Ta et al. (2015).

Comparisons of purely brittle

Observations have shown that the bending that occurs as ice transitions from resting on land to floating in water (the grounding line) promotes the failure of ice from the bottom up, called basal crevasses, that often appear with characteristic regularity in spacing, persisting within the ice for long distances and eventually promoting the calving of ice (Bindschadler et al., 2011; Glasser and Scambos, 2008; Logan et al., 2013; McGrath et al., 2012; James et al., 2014; Murray et al., 2015). The main motivation of using DES to understand this phenomenon (or a simplified version thereof) is its rheological flexibility. That is, a wide array of phenomena in nature may be explored in DES by relaxing the assumption that ice is a purely nonlinear viscous fluid and by examining how ice deformation differs if ice is assumed to be ductile, brittle, or some mixture of the two.

To distill the effects that certain constitutive choices have on the time-dependent ice deformation, we divide this section according to how different constitutive models available within DES simulate the ductile, brittle, and semi-brittle deformation of ice in two different geometrically simple experiments: experiment 1 – a tilted, planar, pseudo-rigid box being advected through a bending fulcrum (Fig. 2a) – and experiment 2, a flat wedge undergoing a transition from frozen or freely slipping to buoyantly floating with an added 1 m diurnal tidal signal (Fig. 2b). Both experiments are initialized with the same temperature configuration, and the temperatures on the boundaries are fixed throughout the experiments. These experiments, while extremely simple, are designed in order to examine the effect that bending has on these relatively complicated rheologies. They are not meant as realistic, glacier-like scenarios but rather are idealized scenarios designed solely for the purpose of understanding the range of deformation behaviors for different constitutive formulations. Their purposeful simplicity allows us to examine the effect that bending has on ice and to attribute deformation and flow patterns solely to the choice in rheology.

For experiment 1, we set the ice thickness to 1000 m and prescribe the
inflow, basal, and outflow side velocities to be 300 m yr

Figure 3 shows the effective stress (a), strain rate (c), and viscosity (e) after 20 years of model time for purely brittle ice. Purely brittle ice experiences stress 1 to 2 orders of magnitude higher than purely ductile ice (Fig. 3a and b). For the brittle case, stresses are highest (about 1 MPa) on the basal surface and in a thin vertical line where the surface bends at the grounding line, whereas in the ductile ice case we observe only a slight increase in stress at the grounding line and nowhere else. Overall we observe in both experiments lower-viscosity ice at the grounding line and higher ice viscosity upstream and downstream of the grounding line (Fig. 3e and f); however for the ductile case there is a much wider zone of low-viscosity ice – by as much as 2 orders of magnitude smaller than the brittle ice. This means that the strains associated with deformation are more localized for the brittle rheology (limited to about 1 or 2 elements in width, making for a very small deformational process zone) and conversely very diffuse for the ductile rheology. The comparative weakness and low effective viscosity of the ductile rheology are reflected by the surface topography: the left-hand side of the domain in the ductile simulation shows a depression at the ice surface, where the ice is essentially slumping toward the right-hand side of the domain under the force of gravity. Additionally, the velocity boundary condition imposed at the left side of the domain introduces an artifact in the flow that is expressed as an artificially steep surface depression: while the boundary nodes' velocities are prescribed, the nearby interior nodes relax and flow downhill under the force of gravity.

Numerical and material parameters used in model runs whose values remained constant.

Simulation of brittle ice slab being advected down the inclined
plane throughout the model time. Pink vertical line denotes the change in the
angle of applied velocity boundary conditions from 3

Because these simulations are intended to only compare ductile and brittle
approximations for ice flow, the ductile ice does not fail (no yield envelope
has been provided for this stress calculation). The brittle ice can and does
fail, however, as dictated by the Mohr–Coulomb threshold, with a pattern
shown in Fig. 4. Reasonable values for the yield envelope properties were
selected from the literature and are listed in Table 1 (Bassis and Jacobs,
2013; Fish and Zaretsky, 1998; Sammonds et al., 1998). The plastic strain (or
amount of strain a failed element undergoes once it has reached yield stress)
for the brittle ice shows a very regular, localized pattern. We executed the
same experiments with applied velocities of 600 and 900 m yr

Effective stresses in

To ensure that the kinematic velocity conditions are not contaminating the
stress field in experiment 1 (Fig. 2a), we executed the purely brittle and
ductile ice according to the setup for experiment 2 in Fig. 2b, with a flat,
freely slipping or frozen bed. We maintain a static grounding line, and due
to numerical constraints on DES's remeshing algorithm, we cannot maintain the
initial thickness gradient; thus, the driving stress decreases throughout the
simulation, leading to a model time of approximately 3 years. The geometry of
the domain is shown in Fig. 2b, where the thickness of the left side is
1050 m, decreasing linearly over 50 km to 900 m on the right. We found
that this initial thickness gradient produces a driving stress with
reasonable terminus velocities matching those of glaciers with ice shelves
in Antarctica (Rignot et al., 2011). We employed the geometric setup in
experiment 1 because purely brittle ice initialized as shown in Fig. 2b does
not flow; it remains static (Fig. 5a and c). Figure 5 shows the effective
stress and viscosity of brittle and ductile ice after 1 year of model time.
Again, the brittle ice experiences stresses an order of magnitude higher than
the ductile ice (Fig. 5a and b) and a much smaller process zone of high
stress at the grounding line, and the resulting viscosity field after 1 year
of model time (Fig. 5c and d) shows a large difference: 2 orders of magnitude at
the grounding line and as much as 4 or 5 upstream. The brittle ice remains
essentially in its initial configuration at the end of 1 year, having
accumulated a maximum of 10

Here we investigate the partitioning of viscous or ductile ice flow and brittle failure under boundary conditions that promote the formation of basal crevasses at glacier grounding lines – areas of fast ice movement and flexure. We suggest this may represent an advance from previous damage-centric models where damage is estimated either in static snapshots throughout time (Borstad et al., 2016) for entire ice sheets or in completely time-dependent but small-strain conditions, as in Duddu et al. (2013) (i.e., the domain was not characterized by strains exceeding 100 % with advecting ice).

In the glaciological literature there is evidence for a transition from
ductile flow to brittle failure depending on the applied strain rate:
specimens of ice experiencing low strain rate flow in a ductile manner, with
viscosities adhering to Eq. (7), and those straining faster than a
laboratory-observed value of 10

Simulation of semi-brittle wedge of ice after 6 months of model time,
undergoing a jump in boundary conditions from freely slipping on the basal
side (left of grounding line – dashed arrow) to floating in the ocean (right
of grounding line). The horizontal velocity

We execute DES with semi-brittle ice according to the setup in experiment 2 for two different mesh sizes: 100 and 50 m (computational resources did not allow for 25 m resolution). No ice melting is applied to these boundaries as this effect is the subject of future work.

Figure 6 shows the velocity, effective stress, strain rate, and viscosity at
6 months of model time. Up until this time in the simulation the ice reaches a
maximum velocity of about 2 km a

Plastic strain over 2.5 years of model time. Ice fails in tension at the surface near the terminus with regularity at the grounding line where hydrostatic stress is applied (in pink). Ice forms boudin-like features after accommodating a large amount of strain.

We also determine the distribution of ice failure for the semi-brittle rheology (Fig. 7). Ice at the surface is regularly and heavily broken as the yield strength there is the lowest (this is the case for all frictional materials in the vertical plane). As the floating portion of the ice extends further past the grounding line, the ice thins, allowing for necking at the grounding line and at other places where the ice has failed in the floating tongue. This thinning as ice begins to float is a feature of marine-terminating ice sheets and is accentuated in nature by intense basal melting. Toward the end of the simulation the floating tongue has accumulated so much strain that it begins to form undulated pinch-and-swell structures (Fig. 7). We term this characteristic pinch-and-swell geometry boudins, where we count 18 boudins that have a mean spacing of 530 m and a standard deviation of 150 m at the end of the 3-year model run. While the basal crevasses form sequentially – that is, failed ice to the right of the domain is older than that to the left – these features develop more fully into the characteristic boudin-like shape all at the same time in the model. Once the ice has lost all its driving stress, the ice begins to thicken just beyond the grounding line, which is a consequence of the boundary conditions and the lack of true bedrock below the ice. As in the purely brittle ice in experiment 2, simulations with a frozen bed resulted in almost no deformation at all.

Geometries of the mesh for a resolution of

Computational limitations prevented the simulation of experiment 2 under quartered resolution, and so Fig. 8 shows the result of the same semi-brittle, freely slipping ice after 8 months of model time for halved resolution. Boudins develop again here, although at much shorter wavelength than for a resolution of 100 m: they have a mean spacing of 250 m with a standard deviation of 83 m, approximately half that of the coarser experiment. Gone, however, is the rather jagged, undulating surface shown in Fig. 8a; instead, Fig. 8b shows that mesh refinement produces a much smoother surface in ice that was initially floating in the domain (and did not traverse the grounding line). Experiments with 100 m resolution took less than 1 h to complete, and those with 50 m resolution were completed in approximately 24 h. Lastly, to ensure that the unrealistic initial geometry of experiment 2 does not contaminate results, we ran experiments where ice was held in its initial geometry before flowing out across the grounding line, to allow the effect of elastic shocks to decay. Model results were the same for both coarser and finer resolution: boudins formed with the same regularity and pattern.

The experiments performed in this study are not meant to be exhaustive and
wide-reaching; rather, they were performed to show how a semi-brittle
ice-like material responds to very idealized initial and boundary conditions.
Because we do not actually simulate fractures – ice in DES is represented as
a continuum material – we must assume that at some level of plastic strain
the ice in a simulation is considered broken. Appendix A shows calibration
experiments wherein we determined that an accumulated plastic strain value of
0.03 is sufficient to consider semi-brittle ice to have ruptured. Zones of
intense, vertical localization in these experiments can be considered to have
ruptured for plastic strain values

Schematic of how semi-brittle deformation could proceed in nature,
through space and time.

One main result from these experiments is the observation that boudins may form as a consequence of semi-brittle rheology. This observation may appear to be complicated by the fact that the boudin size or spacing scales with mesh size. Failure in ice is marked by localized strain, and computational strain localization is well known to be mesh dependent under rate-independent plasticity, which is the brittle rheology implemented in DES. In this sense then ice failure in DES scales with element size, and this is consistent with and predicted by rate-independent plasticity. While it may be unsatisfying to the reader that these geometric features' sizes are sensitive to resolution, we emphasize that they are a qualitative (rather than quantitative) observation, resulting from this specific semi-brittle rheology and jump in boundary condition from slipping to floating. Future work needs to be devoted to examining the convergence of this feature against increased resolution. Our results, however, would be inconsistent with those developed in a theoretical perturbation formalism by Bassis and Ma (2015), wherein the dominant wavelength for boudin spacing was on the order of ice thickness (our experiments here show wavelengths of approximately half to quarter ice thickness). These differences might reflect the primary assumptions underpinning the two approaches – one is viscous (although it allows for a brittle limit), and the other is semi-brittle. Further, the formulation developed by Bassis and Ma (2015) permits a central role for basal melting within basal crevasses, an undoubtedly crucial feature that DES does not implement in a sophisticated way at present. Care should be taken in extrapolating results here to real glaciers; these experiments are performed only as an initial exploration of the potential for this kind of rheological framework to aid in understanding patterns of flow and failure seen in nature. Certainly, that DES lacks the implementation of Neumann boundary conditions indicates that studies where constant ice fluxes must be maintained are best left to other numerical models at the moment. More work must also be conducted to understand the competing effects of viscoelastic damping and brittle failure propagation.

From very simple model runs we learned there may indeed be a ductile-to-brittle transition in ice that is likely very difficult to capture in many numerical models. Figure 9 indicates how we imagine failure and subsequent deformation occurring in floating ice masses in nature: at grounding lines, both an increase in ice velocity (due to loss of retaining frictional forces applied by bedrock contact) and an application of bending moment (due to tides and the equilibrated response that beams and plates exhibit when they are partially supported by fluid) lead to high stresses and strain rates that initiate ice failure from the bottom up (Fig. 9a). As ice accelerates into open ocean, it thins, promoting further crack propagation, which can be further widened by intrusions of warm, buoyant meltwater (Fig. 9b – not explored in these experiments). Further, thinning, stretching, and ice melting when simulated with a semi-brittle rheology like the one presented here can lead to ice geometries that are like those seen in nature (Bindschadler et al., 2011; Luckman et al., 2012). Since the location and size of basal crevasses can directly impact calving rates by propagating upward through the full thickness of the ice (Logan et al., 2013), understanding their evolution and growth may be critical to predicting calving occurrence and terminus position.

Future work with DES must explore the utility of a semi-brittle ice rheology in more realistic scenarios and with the inclusion of a freely varying grounding line (e.g., one that evolves based on ice thickness) and basal melting – two ice dynamic processes incorporated in other numerical models and known to be critical processes in glacier and ice sheet retreat. At present this study has shown that the assumption of a semi-brittle ice rheology can reproduce the brittle rupture of ice, general ice flow characteristics, and idealized patterns of failure in simple situations and may be recommended as a tool through which future studies of ice failure related to calving and ice dynamics can be conducted.

The model DynEarthSol3D is freely available under the terms of the MIT/X
Windows System license from the following URL:

In an effort to ensure that flow and deformation as represented in DES are
reasonable, we calibrated the numerical and material parameters of
semi-brittle ice to match strain- and strain-rate-vs.-time experiments
performed on laboratory-derived ice (Fig. A1). We essentially followed the
same exercise as in Duddu and Waisman (2012), in which material parameters
are calibrated against deformation curves derived by Mahrenholtz and
Wu (1992). While this exercise has its limitations (laboratory-grown ice does
not necessarily represent natural ice), it is our attempt at reproducing ice
behavior to the best of DES's ability. Ice in both the laboratory setting and
DES was isothermal at

Calibration experiments for semi-brittle ice based on
laboratory-derived data presented in Mahrenholtz and Wu (1992).

eriments, and ice is subjected to three
different stresses: 0.93, 0.82, and 0.64 MPa. Figure A1a and b show the
initial mesh and final rupture of the semi-brittle ice for an applied stress
of 0.82 MPa. Figure A1b shows the rupture of semi-brittle ice after
approximately 150 h: from this we see that the accumulated plastic strain
(overlaid in grey) is

In Choi et al. (2013), DES performed the benchmark tests for a range of
material or rheological behaviors to validate and verify this numerical
method. These tests included (1) flexure of a finite-length elastic plate,
(2) thermal diffusion of a half-space cooling plate, (3) stress build-up in
a Maxwell viscoelastic material, (4) Rayleigh–Taylor instability, and (5)
Mohr–Coulomb oedometer test. Thus DES has been verified and validated and is
already in use in fields relating to crustal deformation (Ta et al., 2015).
Despite this prior exercise in verification and validation demonstrating that
DES's numerics are well understood, we executed DES according to a benchmark
test presented by Pattyn et al. (2008) in the ISMIP-HOM study in the spirit
of presenting DES as a numerical model suitable for the community of
glaciologists. All experiments in Pattyn et al. (2008) were designed to be
isothermal, and many employ boundary conditions that DES unfortunately cannot
accommodate due to its entirely mobile mesh (e.g., periodic boundary
conditions). However, experiment E (Haut Glacier d'Arolla) calls for boundary
conditions that DES can easily employ, in two tests: first, a completely frozen
bed everywhere in the domain, and, second, a completely frozen bed except for

DES with semi-brittle ice executed according to experiment E
presented by Pattyn et al. (2008).

Liz C. Logan, Luc L. Lavier, and Ginny A. Catania helped design the experiments performed herein. Liz C. Logan coded and executed these experiments, and prepared this manuscript. Eunseo Choi and Eh Tan developed the code in large part, which was modified by Liz C. Logan and Luc L. Lavier for the experiments in this paper.

This work was funded by NSF grant ARC-0941678 and the King Abdullah University of Science and Technology. The ice modeling was performed at the University of Texas, Institute for Geophysics; the University of Memphis; and Academia Sinica in Taiwan. The authors gratefully acknowledge A. Vieli, one anonymous reviewer, and Jeremy Bassis for extremely helpful comments that improved this paper. Edited by: A. Vieli Reviewed by: J. Bassis and one anonymous referee