Subduction modeling generally requires the tracking of multiple materials
with different properties and with nonlinear viscous and viscoplastic
rheologies. To this end, we implemented a frictional plasticity criterion
that is combined with a viscous diffusion and dislocation creep rheology.
Because

The goal of this paper is primarily to describe and verify our
implementations of complex, multi-material rheology by reproducing the
results of four well-known two-dimensional benchmarks: the indentor
benchmark, the brick experiment, the sandbox experiment and the slab
detachment benchmark. Furthermore, we aim to provide hands-on examples for
prospective users by demonstrating the use of multi-material viscoplasticity
with three-dimensional, thermomechanical models of oceanic subduction,
putting

Earth is a complex dynamic system that deforms on a wide range of spatial and
temporal scales. To obtain realistic predictions of this system from
numerical simulations, it is key to capture the relevant aspects of this
deformation behavior. Here we are concerned with the longer geological
timescales of the subduction of lithospheric plates into the mantle. On such
timescales, rock deformation is mostly nonelastic and characterized by
unrecoverable solid-state creep and brittle-plastic failure

The implementation of plastic yielding into numerical modeling software
entails the definition of a yield criterion that the maximum stress must
satisfy

Meanwhile, many 3-D geodynamical codes offer modeling using complex nonlinear
viscoplastic rheology. Examples of such advanced codes are (in alphabetical
order) CitcomCU

To this list we can now add the recent open-source code

However,

We first present the algorithms underpinning the

A short summary of the governing equations solved by

Similar to the description of temperature, distinct sets of material
parameters are represented by compositional fields that are advected
with the flow. For each field

Definition of symbols.

Abbreviations: df: diffusion; dl: dislocation; vsc: viscous; cp: composite; pl: plastic; vp: viscoplastic.
* Material parameter specified per compositional field. The reference viscosity

The

Deformation of materials on longer timescales is predominantly defined by
brittle fracture or viscous creep in terms of diffusion and dislocation creep
at relatively low stresses

grain boundary or bulk diffusion creep

power-law dislocation creep

plastic yielding.

Plastic yielding (rheology 3) is implemented by locally rescaling the effective viscosity in such a way that the stress does not exceed
the yield stress, also known as the viscosity rescaling method

Both types of viscous creep act simultaneously

The final effective viscosity

Lithospheric geodynamic models often require the specification of materials
with different properties, for example a light and weak upper crust versus a
denser and stronger lithospheric mantle. To provide the functionality needed
for geodynamic modeling, all major material properties of our

The use of multiple compositional fields raises the question of how to
average their properties (in our case viscosity, specific heat, thermal
conductivity, thermal expansivity and density). We have implemented the four
averaging schemes commonly referred to in the literature

The methods above have been shown to affect model results in the context of
subduction:

Characteristics of performed experiments.

* nc: number of compositions. None of the benchmarks include temperature effects in the rheology.

To test and verify our implementation of multi-material viscoplastic
rheologies, we performed four 2-D experiments: the indentor benchmark, the
brick experiment, the numerical sandbox and the slab detachment experiment.
The experiments increase in the number of materials and in the complexity of
the rheology used, as outlined in Table

All experiments were conducted on an in-house computer consisting of 1 Dell
PE-R515 master node and 15 Dell PE-C6145 compute servers made up of

In the indentor benchmark, a rigid indentor “punches”
a rigid-plastic half space. The exact solution to this boundary value problem
is given by slip-line field theory

The angles of the shear bands stemming from the edges of the indented area are 45

The pressure at the surface in the center of the punch (I) and the pressure in triangles ABC and EFG are

The velocity magnitude in areas CDE and ABDC & EDFG is

The numerical setup of the instantaneous indentor benchmark comprises a 2-D
unit square of purely plastic von Mises material, i.e., its yield value

Prandtl's analytical solution of a rigid die indenting a
rigid-plastic half space

The indentor benchmark model parameters.

An estimate of the maximum strain rate and
thus minimum viscosity can be made from

Figure

When using 200 cheap Stokes iterations for the smooth punch, results are not
changed, but wall time is about 1.6 times longer. Using harmonic averaging of
the material properties as discussed in

The indentor benchmark model setup: a unit square with free-slip vertical and no-slip lower boundaries. The punch area has a prescribed
vertical velocity

The punch benchmark results after 500 NIs for a rough punch (left
column) and a smooth punch (right column).

It should be noted that there exists a second end-member solution geometry
for the smooth punch problem: Hill's solution

The indentor experiment performed also shows a trade-off between accuracy in
pressure and velocity measurements and the rigid-plastic-like behavior of
the medium (compare left and right column of Fig.

As brittle failure in rocks is more appropriately described by
pressure-dependent plasticity than by the perfectly plastic deformation

The brick benchmark model parameters.

The background strain rate resulting from
the boundary conditions of

In our instantaneous version of the brick benchmark, a viscous-frictional
plastic medium with a small viscous inclusion at the bottom boundary (Fig.

The angle of internal friction

The brick benchmark model setup after

Figure

Measured shear band angle versus angle of internal friction for
models in tension and compression. Resolution runs from

Measured velocity residual
(

Strain rate norm fields for

To estimate the effect of adaptive mesh refinement on the shear band angles,
we ran additional tests with

Varying the initial viscosity of the viscoplastic medium from

Testing the pressure dependency of our plasticity formulation with the brick
benchmark, shear band angles were found to increase with internal friction
angle

Interestingly, internal angles of friction larger than

However, we have shown that we consistently obtain shear band angles between
Arthur and Coulomb theoretical angles at sufficient resolution and that these
angles verge to Coulomb angles with increasing resolution. This happens despite
the fact that our implementation is relatively basic: it does not include
softening of cohesion or of the internal angle of friction as in

The sandbox experiment setup after

The analog sandbox

The sandbox experiment model parameters.

Parameters are on a sandbox scale, as they are used in the model.
An estimate for the initial strain rate can be made from the boundary
conditions

The results for

Results after

Numerical grid after

The viscosity field (Fig.

The evolution of the numerical sandbox model – a model with AMR, high
viscosity contrasts, large deformation and complex boundary conditions –
compares well with those shown in

Similar to

Slab detachment, or break-off, in the final stage of subduction is often
invoked to explain geophysical and geological observations such as
tomographic images of slab remnants and exhumed ultra-high-pressure rocks
(see for example

The 2-D detachment model geometry is outlined in Fig.

The detachment benchmark model setup of

The detachment benchmark model parameters.

The evolution of the detachment model is shown in Fig.

From comparison of the red and black lines in Fig.

Detachment benchmark model evolution showing the viscosity field
over time. After about

Nondimensional necking width versus time for

The first three benchmarks focused on plastic rheologies. The detachment
benchmark involves modeling of a highly nonlinear, power-law viscosity, which
is often used in subduction modeling. Our observed model evolution compares
well with that of

Lastly, we consider a geodynamical application of the viscoplastic rheology implemented: the spatiotemporal evolution of 3-D subduction. So far,
no

We discuss two models; the first is an adaptation of the free-plate model of

The first model comprises two free plates: an overriding plate (OP) and a
subducting plate (SP) (see Fig.

The second model augments the first with an adjacent plate (AP) separated
from the other plates by a 20

Three-dimensional subduction model setups (also, see Table

Temperature distribution with depth and distance from the ridge for

Viscosity profiles at

Three-dimensional subduction model parameters.

Three-dimensional subduction model parameters (based on

* For a fixed grain size of

Figure

The AMR based on composition and viscosity follows the outline of the plates
as they move through the mantle (Fig.

Model 1 – strain rate Frobenius norm

The SP of model 2 steepens for the first

Model 2 – strain rate Frobenius norm

Viscosity snapshots of

The evolution of model 1 strongly resembles that of

In switching from model 1 to this thermomechanically coupled model 2, we found changes to the setup were necessary to avoid subduction of the plate at locations other than the slab tip (i.e., the sides and back of the plate would subduct as well) due to high mantle temperatures. Therefore, we added the adjacent plate and transform fault. By locating the plate ridges at the left and right vertical boundaries, free motion of the plates perpendicular to the trench is still enabled. Mesh resolution here was reduced over time because the refinement strategy chosen focused on the compositional fields, which moved away from the boundaries. This unfortunately increased the coupling of the OP plate to the left boundary, limiting the plate's ability to move.

The subduction evolution of model 1 and 2 in Figs.

More elaborate models of subduction should incorporate phase changes and
latent heat effects as well as adiabatic and shear heating. This is also
possible with

The four benchmarks shown using our viscoplasticity implementations in

It should be noted that although the rheology described in this paper is
often applied in numerical modeling, more elaborate laws have been proposed.
For example,

Incorporating more realistic nonlinear rheologies such as described in this
paper creates the necessity for additional nonlinear iterations within a
single time step. Also, we have seen that at higher mesh resolutions, more of
such iterations are required to converge the solution. This greatly increases
model run time and therefore it is important to implement a more efficient
nonlinear solving strategy than the Picard iterations currently used by

Nonlinear rheologies also affect the linear solver by introducing large
viscosity gradients. Different strategies to reduce the increased
computational time and under- and overshooting of the numerical approximation of
the resulting pressure gradient are available in ASPECT. For one, one can
reduce the linear tolerance (while making sure the results do not change
significantly), as was shown in Sect.

Numerical modeling of intricate geodynamic processes such as crust and
lithosphere deformation and plate subduction encompasses challenges at
different levels. For one, the 4-D nature of the subduction process requires
state-of-the-art numerical methods to efficiently handle the parallel
computations necessary for such large problem sets. Secondly, models should
incorporate realistic (non)linear rheologies to mimic nature as close as
possible. Thus, the need arises for algorithms that can solve highly nonlinear
equations and deal with large viscosity contrasts effectively. Thirdly,
far-field effects of mantle flow and plate motion cannot be ignored, and
neither can topography building, resulting in a demand for complex boundary
conditions such as open boundaries

In this paper, we have shown that the open-source code

The continued development of

Our simulations were performed with ASPECT version 1.5.0

As discussed in Sect.

The 2-D linear viscous model is composed of three compositions: the mantle,
subducting lithosphere and sticky air to allow for surface topography
build-up and detachment of the lithosphere from the top boundary (Fig.

The evolution of subduction for case 1 is summarized in a plot of the slab
tip depth over time in Fig.

The self-consistent subduction benchmark model parameters.

The self-consistent subduction benchmark model setup. The mantle,
subducting plate and sticky air are represented by three compositional fields
of constant viscosity. The slab geometry of case 1 is indicated in solid
lines; the dashed line outlines the slab tip of case 2 (based on case 3 of

Snapshots of the viscosity field for case 1 are shown in Fig.

Slab tip depth versus model time for four different averaging
methods of the contribution of the compositional fields to viscosity. Colors
indicate the averaging method, while one color goes from light to dark with
local resolution, which varies from

Viscosity field for each averaging method at

Wall time for time step

The infinity norm selects the parameters of the field that is greatest at a specific point. It thus counteracts the numerical diffusion of the compositional boundaries in the calculation of composition-dependent parameters, but unfortunately also sharpens possible viscosity contrasts between the fields. This increases computational time. For more complex models in which wall time is an important factor, we recommend using the geometric averaging method.

With this example of 2-D subduction, we highlight
some of the more recent additions to ASPECT: compositional field
reactions (which can be used to implement phase changes), the true free
surface

Phase changes (changes of one compositional field into another with different
material properties) are implemented by extending our, now compressible,
multicomponent viscoplastic material model with a depth-dependent transition
function

Open boundary conditions are newly implemented as a plug-in to the

The compressible subduction model considers ocean–continent subduction in a
domain of

Compressible subduction model setup: subduction of an oceanic plate
of 80

Compressible subduction model parameters.

Compressible subduction material parameters.

Figure

The change in flow through the left boundary is depicted in Fig.

The phase changes are clearly expressed in the density fields: the density
isocontours in Fig.

Compressible subduction evolution in terms of viscosity. Density is
contoured at

Left boundary in- and outflow for the compressible subduction model. In
red the velocity profile that is prescribed for the first

Figure

AG developed the code implementations and performed the simulations for this paper, with the exception of the 2-D compressible subduction model in the Appendix. This latter model was constructed by CB under the supervision of CT, AG and MF, based on the code of MF (model setup) and AG (open boundary conditions, rheology). CT provided the model output of several benchmarks of his code ELEFANT for comparison and discussion. AG prepared the paper with contributions from CT and WS.

The authors declare that they have no conflict of interest.

The authors declare that they have nothing to disclaim.

We are grateful to the reviewers Dave May and Boris Kaus for their detailed and
constructive reviews. We thank Wolfgang Bangerth, Timo Heister, Rene Gassmöller and
Juliane Dannberg for their precious help concerning the use and fine-tuning of