ASPECT can solve for both compressible and incompressible flow, but here we focus on the latter, adopting the Boussinesq approximation and assuming an infinite Prandtl number (i.e., inertial term is omitted). Heat production is not incorporated. This results in the following equations of conservation of momentum (Eq. ), mass (Eq. ) and energy (Eq. ): -∇⋅(2μeffϵ˙(u))+∇P=ρg,∇⋅u=0,∂T∂t+u⋅∇T-∇⋅κ+νh(T)∇T=0, where density ρ=ρ0(1-α(T-T0)). Other symbols are explained in Table . Artificial diffusivity νh is used to prevent oscillations due to the advection of the temperature field. It is calculated according to the entropy viscosity method of , as described in .

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 ci, this formulation introduces an additional advection equation (Eq. ) to the system of Eqs. ()–() described above. As these equations contain no natural diffusion, artificial diffusivity νh is again introduced to stabilize advection: ∂ci∂t+u⋅∇ci-∇⋅(νh(ci))∇ci=0.

ASPECT solves the equations above using the finite element method: the domain is discretized into quadrilateral (in 2-D) or hexahedral (in 3-D) finite elements and the solution (velocity, pressure, temperature and compositional fields) is expanded using Lagrange polynomials as interpolating basis functions. Default settings employ second-order polynomials for velocity, and first-order polynomials for pressure Q2Q1 elements, e.g.,, and second-order polynomials for temperature and composition. Unless stated otherwise, these default polynomial degrees are used in the following. The linearized Stokes system is solved in a procedure involving the iterative Flexible GMRES (FGMRES) solver with an inexact right preconditioner. For details on the construction of the preconditioner, see . A cheap Stokes solve option in which the preconditioner employs only one V cycle is available. The number of such FGMRES iterations before switching to the more expensive preconditioner is set to 0 in this paper, unless stated otherwise. The GMRES method with an incomplete LU decomposition preconditioner is used for the temperature and composition systems. Nonlinearities in the rheology are resolved with Picard-type (fixed-point) iterations, iteratively updating the velocity and pressure, strain rate and viscosity until the relative nonlinear residual for iteration i ||A(xi-1)xi-1-b||2||A(x0)x0-b||2 has fallen below a user-set tolerance (default value of 10-6), or the user-specified maximum number of iterations (NIs) is reached. The initial residual ||A(x0)x0-b||2 is computed with zero velocities and a lithostatic pressure profile calculated at the center horizontal coordinate. x contains the velocity and pressure solutions of the previous iteration, b represents the right-hand side of the Stokes equations and A is the Stokes part of the system matrix.

The ASPECT code is divided into different modules for boundary conditions, initial conditions, mesh refinement etc. Each module comprises of several plug-ins providing different implementations (e.g., constant vs. space- and time-dependent boundary conditions), to which the user can add its own if more functionality is needed. Rheologies are implemented within the so-called Material model module. Plug-ins in this module must provide functions that compute the viscosity, density, thermal conductivity, thermal diffusivity, specific heat and the thermal expansion coefficient at the quadrature points. The solution variables T, P and ci as well as the derived strain rate ϵ˙(u) and the position are available to compute these material properties. This then provides a straightforward way of implementing nonlinear rheologies, which we have taken advantage of.

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 . We thus implement three basis rheologies that can be combined into more complex ones:

grain boundary or bulk diffusion creep

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 Material model plug-in depend on any number of fields, as defined by the user (composition-dependent parameters are denoted with an asterisk in Table ).

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 e.g., for computing the viscosity used in Eq. (13) or (14): μeffvp=∑i=1ncci∑i=1ncciμeffivp,(harmonic)μeffvp=10∑i=1nccilog⁡10(μeffivp)∑i=1ncci,(geometric)μeffvp=∑i=1ncciμeffivp∑i=1ncci,(arithmetic)μeffvp=μkwithk:ck≥ci,i=1,⋯,nc,(infinitynorm) where “nc” is the total number of compositional fields ci in the domain. Note that each field ci is initialized with values on the interval [0,1] and capped values 0≤ci≤1 are used for averaging, as compositional field values may come to slightly exceed this interval over time despite artificial diffusion (Eq. ). The μeffivp value is obtained by evaluating Eqs. () or () using the material constants of composition i. The other material properties are arithmetically averaged or, in case the viscosity averaging method is set to Eq. (), averaged using this infinity norm.

The methods above have been shown to affect model results in the context of subduction: showed that the subduction process can be up to 3 times faster between one averaging method and the other, and the effect of mesh resolution on subduction evolution varies per method as well. Unless stated otherwise, we use the infinity norm rule in this paper; for a discussion of this choice, see Appendix .

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 σy is independent of pressure and remains constant. The material's upper boundary is punched along a distance p by prescribing an inward vertical velocity vp on the otherwise open (stress-free) boundary (see Fig. a). The horizontal component of velocity along p is either set to zero or left free to implement the so-called “rough” and “smooth” punch , respectively, where the smooth punch assumes a frictionless contact between the punched medium and the indentor. Model and numerical parameters of the performed indentor experiments are presented in Table .

Figure  shows the model results for a rough (left column) and smooth (right column) punch. The solutions obtained agree with the analytical solution according to criteria 1–3 listed above. Outside the slip lines, viscosity is uniformly high. The low-viscosity shear bands fit the analytical slip lines well (Fig. a and f) and stem from the edges of the indentor at a 45∘ angle with respect to the top of the medium (Fig. b and g). For a rough punch, measurements of pressure in point I and velocity in points K and L deviate from the analytical solution by about 15 and 1 %, respectively, but the block-like behavior of triangle CDE is evident (Fig. c). The analytical solution is reproduced with errors <0.14% for a smooth punch, but the velocity vectors in Fig. h show some horizontal motion of triangle CDE and the velocity field is more diffuse.

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 increases the velocity error for the smooth punch to ∼1%, but reduces wall time about 4.7 times. Loosening the linear Stokes solver tolerance by 1 order of magnitude to 10-8 reduces the wall time of the rough punch by a factor of 1.6, while keeping the velocity error <1%.

ASPECT successfully reproduces the analytical solution of Prandtl for the rigid-plastic indentor benchmark, a problem with mixed boundary conditions and a nonlinear rigid-plastic rheology with overall viscosity contrasts of 6 orders of magnitude.

It should be noted that there exists a second end-member solution geometry for the smooth punch problem: Hill's solution . Although argues that Hill's solution is probably more correct when considering elasticity theory, and lists Hill's solution for the smooth punch, other numerical studies do not recover this slip-line geometry in 2-D either. In fact, our Prandtl shear band geometry compares well with results of Fig. 6b, smooth, Fig. 1, smooth, Fig. 10, smooth, Figs. 24–27, rough, Fig. 6a, rough, and Fig. 11, rough.

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. ). This same dichotomy is seen however in other studies that performed the experiment. Note, for example, the continuous velocity vectors in . Also, shows that the pressure under the punch improves from a 15% error to a mere 0.5% error when switching from a rough to a smooth footing.

In our instantaneous version of the brick benchmark, a viscous-frictional plastic medium with a small viscous inclusion at the bottom boundary (Fig. ) is either compressed or extended . Strain softening of the cohesion and angle of internal friction of the medium is not incorporated. Compression and extension are prescribed through constant kinematic boundary conditions on the vertical domain walls. The bottom boundary of the 40×10km domain is set to free slip and the top boundary is stress free (material is free to flow in or out). Other domain characteristics and material parameters are given in Table .

The angle of internal friction ϕ (Eq. ) is varied from 0 to 30∘ to test the pressure dependency of the implemented plasticity criterion. It is expected that the resultant shear band angle θ varies with the internal friction angle. Shear band angles are automatically computed from the location of the maximum Frobenius norm of the strain rate ϵ˙:ϵ˙ at x=17.4km, x=19.4km, x=20.6km and x=22.6km .

Figure  depicts the measured shear band angle versus the supplied internal friction angle for 21 runs in both the compressional and tensional regime. The constant and uniform elemental resolution of the runs varies from 256×64 to 1024×256 elements. Lower-resolution runs were performed, but do not resolve the viscous inclusion well (≤ two elements) and are not shown see instead Fig. 12 of. We monitor the residual as a measure of convergence, but the maximum number of nonlinear iterations (NIs; see Sect. ) is fixed at 1000. The red symbols in Fig.  indicate runs for which the residual did not drop below the convergence criterion ϵu=10-4 after 1000 iterations, as is evident from the corresponding red lines in Fig. . In fact, the higher the internal friction angle, the more iterations are needed to reach a particular residual tolerance, and Fig.  also shows that higher internal friction angle runs stall at higher residuals. This coincides with a greater deviation from the theoretical Coulomb solution. Note that in these higher angle runs multiple shear bands are generated and an asymmetry between the right and left shears develops, as shown in Fig. . The spurious shear bands can occur mainly at the top of the domain, as several curved pieces forming one shear band or as complete additional shears. Despite these difficulties, there is a clear trend of measured angles verging from Arthur to Coulomb angles for an increasing resolution. For example, the average deviation from the theoretical Coulomb angle decreases from 2.8 to 0.8∘ in extension when going from a resolution of 256×64 elements to 1024×256 elements.

To estimate the effect of adaptive mesh refinement on the shear band angles, we ran additional tests with 3×333 nonlinear iterations at increasing refinement levels, with refinement based on gradients in the velocity, viscosity or strain rate and different fractions of cells that are refined. These simulations indicate a maximal variation of 5∘ in shear band angle compared to results for a uniform mesh of 512×128 elements.

Varying the initial viscosity of the viscoplastic medium from μmin to μmax for a uniform mesh of 512×128 elements leads to the same shear band angles for well-behaved residual runs (see black lines in Fig. ), while for higher internal angle of friction runs, a variation of maximally 3∘ is found.

Testing the pressure dependency of our plasticity formulation with the brick benchmark, shear band angles were found to increase with internal friction angle ϕ as expected, almost all falling within the theoretical values of Arthur and Coulomb. Moreover, with increasing mesh resolution, the angles approach the Coulomb theoretical angle θ=45±ϕ2 and the variation in error with respect to Coulomb angles decreases. This tendency towards Coulomb angles for higher mesh resolution was also reported by , and because at higher resolution the viscous inclusion is better resolved. Kaus finds that at least 10 to 20 elements are required horizontally within the inclusion to obtain Coulomb angles. This corresponds to our two highest resolutions. recently showed that consistent Coulomb angles can be achieved by an (initially) associated flow law (where ϕ=ψ).

Interestingly, internal angles of friction larger than 15–20∘ lead to irregular convergence behavior that stalls at higher relative residuals; these runs also show shear band angles further away from the theoretical Coulomb angle and multiple additional shears. These additional shears do often coincide with the theoretical angle; see for instance Fig. b. Why these latter shears are not dominant (highest strain rate) would require further investigation. Note that the models of also show multiple shear bands and that these spurious bands and stalling of convergence are shown to be the result of the dynamic pressure dependence of the Drucker–Prager yield criterion by .

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 and , nor does it have a sophisticated guess of the initial stress state or an incremental build-up of the prescribed boundary velocity .

The analog sandbox consists of a basal layer of weak, viscous silicone overlain by brittle sand. The sand is extended by the movement of the right vertical wall and the connected basal plate extending from the wall to, initially, the center of the domain. We model this setup with three compositional fields (Fig. ): (1) a viscous basal layer, (2) an overlying Drucker–Prager dynamic pressure-dependent plastic sand layer and (3) a low-viscous sticky-air layer on top. Extension is driven by a prescribed horizontal velocity on the right vertical boundary and the right half of the lower boundary, mimicking the effect of the moving basal sheet. Basal friction is not taken into account. This approach is appropriate since have shown that the nature of the basal contact is less important than the interaction of the velocity discontinuity (VD in Fig. ) and the silicone. The rest of the bottom boundary has zero velocity (without smoothing of the velocity discontinuity) and this no-slip area increases as the velocity discontinuity moves to the right of the domain. The left boundary is free slip and the top boundary is open. Adaptive mesh refinement (AMR) is applied based on the effective strain rate field or on viscosity and density to obtain a maximum local refinement of 0.39×0.39mm along the shear bands (Fig. ). The model is run until 2cm of extension has occurred, equalling 2880s of model time. Material properties and other model parameters are listed in Table .

The results for 1 and 2cm of extension of the sandbox are presented in Fig. . The evolution of the model closely resembles what was found by . The initially symmetric system forms two conjugate shear zones stemming from the velocity discontinuity imposed by the velocity boundary conditions. With ongoing extension, the silicone layer distributes deformation and the shear bands spread to the edges of the layer. After 1 and 2cm of extension, the angle of the shear bands left and right of the velocity discontinuity (see Fig. d) measure left ∼51, right ∼62 and left ∼54, right ∼60∘. With time the system becomes more and more asymmetric (compare left and right column of Fig. ). The left side of the domain is at rest, while the outer right footwall moves at the prescribed velocity and we observe that the sticky air properly accommodates the movement of the sand.

The viscosity field (Fig. h) is very irregular and displays sharp gradients up to 7 orders of magnitude. Figure  demonstrates viscosity- and density-based AMR: refinement is localized in the low-viscosity shear bands, following the evolution of deformation. Through AMR, the total (velocity, pressure, temperature, composition) number of degrees of freedom (DOFs) is limited to on average ∼475000 DOFs, instead of the ∼1383000 DOFs (∼527000 velocity DOFs) for a uniform resolution, thus decreasing the required computational resources by half (for the same number of cores).

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 and . Although the shear band angles to the right of the velocity discontinuity of 62 and 60∘ after 1 and 2cm of deformation fall just outside the ranges found by and of 45–55 and 45–53∘, respectively, they lie within the theoretical Arthur–Coulomb angles of 54–63∘ for a friction angle of 36∘ (; see also Sect. ). Even when considering that the codes in add strain softening by decreasing the friction angle from 36 to 31∘, for which the range of Arthur–Coulomb angles would be 52.75–60.50∘, their measured angles lie mostly below the Arthur angle. In the previous section, we demonstrated that our shear band angles fall within the Arthur–Coulomb range and verge towards Coulomb angles with increasing mesh resolution. Differences in the measured angle of shear bands are therefore not surprising. A lower-resolution run (maximum of 0.78 mm resolution, not shown) results in angles to the right of the discontinuity of 60 and 55∘.

Similar to , and , we observe an increase in the number of shear bands with resolution as well as a decrease in their width. As explained by , this lack of internal length scale is caused by the singularities in strain rate and pressure deriving from the model setup (e.g., sharp corners of the silicon layer and the discontinuous velocity boundary condition) that are resolved better at higher resolutions, thereby decreasing shear band width.

The 2-D detachment model geometry is outlined in Fig. ; both the lithosphere and the mantle are represented by a compositional field. prescribes a nonlinear viscosity in the subducting lithosphere given by ηL=η0ϵ˙e1n-1, where η0=4.75×1011Pas1n and n=4. Conversion to Eq. () results in the parameters listed in Table . Here we only reproduce the constant mantle viscosity case of , with μmantle=1021Pas. We use a set of 20 tracers placed on the outline of the slab to track the width and depth of necking. The necking width and depth, as well as time, are normalized by the initial slab width of 80km and the characteristic time tc=7.1158×1014s, respectively .

The evolution of the detachment model is shown in Fig. : after about 20Myr, the slab is fully necked and detached. Although a thin line of lithosphere composition is still visible, its value is less than 0.5 and thus the infinite norm ignores this contribution to the viscosity, allowing for full detachment. Upon detachment, slab pull is removed and thus viscosity reaches μmax throughout the remaining lithosphere (Fig. c and d).

From comparison of the red and black lines in Fig. , it can be seen that the width of the necked zone through time agrees very well with the results of . Only after t=0.8, do our results start to deviate due to our use of tracers for measuring the necking width. The two other lines illustrate the effect of using harmonic averaging for the compositional fields' contribution to viscosity (blue line) or for averaging the viscosity over the elements (yellow line): although these averaging methods reduce the wall time by a factor of 3 and 2, respectively, they also result in much faster necking. This agrees with the findings in Appendix A.

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 and other codes . It also demonstrates the effective splitting of a compositional field into two bodies and how the rheology interacts with the compositional fields through localization of deformation at the slab hinge. It should be noted that the particular geometry of the slab with its sharp corners results in a mesh dependence of the solution. Differences in model evolution can also arise from the particular viscosity and material averaging method applied.

The first model comprises two free plates: an overriding plate (OP) and a subducting plate (SP) (see Fig. ). The SP is made up of a crustal layer of non-frictional (von Mises) viscoplastic rheology and a mantle lithospheric layer of constant viscosity (see Table  for actual values). The tip of the SP extends into the mantle for 200km. The OP is of one composition and has a constant viscosity, as does the surrounding mantle. These four compositions are each represented by a compositional field.

The second model augments the first with an adjacent plate (AP) separated from the other plates by a 20 km thick weak zone (WZ). The SP and OP are extended and their thickness is no longer uniform, but based on the temperature field, as if they originate from ridges situated at the left and right vertical domain boundaries. The initial temperature distribution (Fig. ) in the plates is computed according to the age-based plate cooling model, for a mantle temperature Ta of 1593K and surface temperature T0=293K (Eq. 4.2.24 of ; κ=10-6m2s-1). Thickness of the plate is defined at the temperature T for which Ta-TTa-T0=0.1. The maximum thickness of the plate (for time to infinity) is set to 125km, based on . At the trench, both plates have the same thickness as in model 1. The adjacent plate (AP) has a fixed thickness of 100km for an age of ∼60Myr. From a depth of 125km, a linear temperature increase is prescribed everywhere to a bottom temperature of 1771K. The mantle and overriding plate compositions are of nonlinear rheology, with which model 2 differs strongly from model 1. The specific rheological parameters for diffusion and dislocation creep and Drucker–Prager plasticity can be found in Table  and the representative viscosity profiles in Fig. 16.

Figure  depicts the evolution of the subduction system of model 1 over time. The pull of the slab extending into the mantle results in a plastically weakened subduction fault zone through high strain rates. Although this allows for decoupling from the surface, mechanical coupling is strong enough for the OP to move towards the trench. There the OP thickens and a small portion of OP material is entrained with the slab. Within the first ∼7Myr, the velocity of the plates steadily increases about 1 order of magnitude and reaches several centimeters per year. Similar to , poloidal and toroidal flow can be observed close to the slab; flow into the trench alters the shape of the plates. After 8.5Myr, plate velocities drop when the slab tip reaches a depth of 660km (bottom of the domain) and the steep slab starts to bend in to accommodate lateral sliding over the 660km discontinuity. With ongoing subduction, the length of slab being pushed along the 660km discontinuity increases and plate velocities increase to pre-sliding levels. At 35Myr, the plate has completely subducted, lying flat at the 660km discontinuity, while the trench has retreated a total of ∼1000km.

The AMR based on composition and viscosity follows the outline of the plates as they move through the mantle (Fig. ), resulting in a local resolution of roughly 3km while the mantle is resolved with ∼50km elements.

The SP of model 2 steepens for the first 12Myr (Fig. ) until full-fledged subduction starts. Subduction is much slower than in model 1: while in model 1 subduction is completed by 35Myr, rollback of the slab in model 2 only sets in around 50Myr. Simultaneously with slab rollback, the subduction channel is weakened by asthenospheric inflow into the gap between the OP and SP. Formation of this gap is probably initiated by the fact that the OP does not completely release itself from the lateral boundary (see snapshot at 49Myr). Even though the slab reaches the bottom boundary around 50Myr and starts to shallow, it does not lie flat on the 660km boundary, but continues to hover above it. Also note the halo of increased strain rate around the tip of the slab and the small-scale strain rate features in the mantle compared to model 1.