## 1 Introduction

### 1.1 Overview of Numerical Techniques for Long-Term Tectonic Modeling

[2] Numerical simulation of the long-term (>10,000 years) evolution of geological structures at the crustal to lithospheric scales are denoted long-term tectonic modeling (LTM). Challenges arise in LTM because the geological structures of interest involve localized deformation at plate boundaries such as narrow fault and shear zones (e.g., ≤1 km) within a domain of a much larger scale (e.g., ≥100 km). To mechanically describe materials exhibiting strain localization, nonlinear rheologies are commonly used, which include power-law viscosity [e.g., *Montési*, 2003; *Behn et al*., 2002], damage [e.g., *Lyakhovsky et al*., 1993, 2012], or strain-weakening plasticity based on Mohr-Coulomb models for frictional materials [e.g., *Poliakov et al*., 1994; *Poliakov and Buck*, 1998]. The materials that compose the Earth's lithosphere are brittle when temperature and confining pressure are low but exhibit ductility when any of these thermodynamic variables are high [e.g., *Jaeger and Cook*, 1976; *Scholz*, 1988]. LTM should address these severe transitions as well as variable stiffness within each of the brittle and ductile regimes. Simulation techniques for LTM must also account for large amounts of deformation whether localized or distributed.

[3] Assessing the predictive power of new rheologies is another fundamental issue in LTM since no agreement has been reached in the scientific community on the first-order structure of the Earth's continental lithosphere. For example, the geodynamics community is presently addressing whether loads are supported by a strong crust and mantle lithosphere separated by a weak fluid-like lower crust [e.g., *Watts and Burov*, 2003] or by a strong upper crustal layer sitting on top of weaker lower crust and mantle lithosphere [e.g., *Scholz*, 2002; *Bourne et al*., 1998]. Many studies focused on lithospheric rheology have shown that the metamorphic and magmatic reactions involving hydrous fluids as well as the polymineralic nature of rocks can control the rheological behavior of the lithosphere through time and space [e.g., *Handy*, 1990; *Lavier and Manatschal*, 2006; *Ranalli*, 1997]. This rheological evolution is not taken into account by the monomineralic rheological flow laws usually used in geodynamic modeling [e.g., *Ranalli*, 1995; *Kohlstedt et al*., 1995]. LTM simulators may be used as test beds for new rheological models for the flow of rocks, as exemplified in studies of rifting [e.g., *Huismans and Beaumont*, 2002; *Lavier and Manatschal*, 2006]. In the long run, for LTM simulations to be able to answer these open questions about the structure of the continental lithosphere, preference will be given to numerical techniques that can implement these complex nonlinear rheologies with ease.

[4] The available numerical techniques for LTM can be largely divided into two groups. The first group models the material response as elastoviscoplastic (EVP), where the brittle behavior is modeled by strain-weakening elastoplasticity and/or damage, while the ductile behavior is modeled by Maxwell viscoelasticity [e.g., *Albert et al*., 2000; *Poliakov and Buck*, 1998; *Gerya and Yuen*, 2007; *Popov and Sobolev*, 2008]. This material description naturally represents elastic compressibility, strain weakening, and confining pressure dependence. The second group uses viscoplastic (VP) models, ignoring the elastic response of the Earth's lithosphere entirely. These models treat lithospheric motion as a viscous flow where the material response is represented using a nonlinear viscosity. The lithosphere is modeled as a high viscosity region where the brittle behavior is mimicked by a yield stress varying with an internal variable [e.g., *Tackley*, 2000; *Čžková et al*., 2002; *Billen and Hirth*, 2007; *Fullsack*, 1995; *van Hunen et al*., 2001; *Braun et al*., 2008; *Dabrowski et al*., 2008]. There are also hybrid models which consider deviatoric components of elasticity on top of the viscoplastic model [hence, partial EVP (pEVP)] [e.g., *Gerya and Yuen*, 2007; *Moresi et al*., 2003; *OzBench et al*., 2008; *Kaus*, 2010].

[5] Whether the motion is described in a Lagrangian or an Eulerian framework can be another classification criterion for existing techniques for LTM although this distinction is not always clear. It is important to track free surfaces or phase boundaries in the physical domain, for instance, to study surface processes or interactions between the lithosphere and the mantle [e.g., *Kaus et al*., 2010; *Duretz et al*., 2011]. These tasks are an inherent part of the Lagrangian framework in the sense that no extra operations are needed. However, the large deformation involved in LTM requires remeshing in the Lagrangian framework because the numerical approximation degrades as the mesh distortion becomes severe. Similar limitations are faced by Arbitrary Eulerian-Lagrangian methods [e.g., *Fullsack*, 1995; *Moresi et al*., 2003; *Braun et al*., 2008; *Popov and Sobolev*, 2008]. This remeshing process needs to be handled with care. Remeshing violates the fundamental premise of the Lagrangian description of motion; that is, the material points are attached to the deforming mesh. Practically, remeshing causes numerical diffusion of the advected variables as in a naively implemented advection in the Eulerian framework. The numerical diffusion, however, can be efficiently remedied by introducing particles and letting them carry phase information as well as history-dependent variables [2005; 2011; 2012; 2008; 1995; 2003; 2007; 2008; 2010]. The role that particles play and the computational complexity vary among different algorithms. Although without need for remeshing, the Eulerian framework usually needs extra operations involving particles or level sets to define and keep sharp internal and external boundaries [e.g., *Gerya and Yuen*, 2007; *Braun et al*., 2008]. There is also a wealth of literature describing Eulerian formulations of elastoplasticity [*Duddu et al*., 2009, 2012; *Plohr and Sharp*, 1988, 1992; *Demarco and Dvorkin*, 2005; *Trangenstein and Colella*, 1991; *Trangenstein*, 1994; *Miller and Colella*, 2001, 2002]. These methodologies avoid remeshing but need to solve a larger system of equations at each time step when compared with standard Lagrangian formulations and still need special interface tracking algorithms.

[6] An inspection of the available techniques used in LTM reveals a mixed use of explicit and implicit constitutive updates. The implementation of new rheologies is relatively straightforward when an explicit constitutive update is used because it does not involve subiteration within a time step even for nonlinear constitutive models. As a result, a numerical technique that adopts explicit updates can simulate both linear and nonlinear rheologies with equal ease. This is particularly true in the case of strain-weakening elastic-plastic models, where the constitutive update is more complicated than in the effective viscosity approach often used in VP or pEVP models. This is a desirable feature of a numerical technique for LTM which, as explained earlier, should work as a test bed for new rheologies. This validation process can be applied to newly proposed models on benchmark problems as well as to established models when new data becomes available. A drawback of an explicit constitutive update is that the time step size is restricted by stability requirements. In contrast, implicit constitutive updates may involve numerically expensive iterations, and the implementation of a novel nonlinear rheology can be a challenge by itself because finding a tangent stiffness operator is not always straightforward. Nevertheless, the resulting time-stepping techniques control the time step size by accuracy requirements, not stability; thus, significantly larger time steps can be taken in the models adopting implicit constitutive updates.

### 1.2 Need for Compressible Elasticity in Long-Term Tectonic Modeling

[7] A fundamental difference between the available implementations of the EVP approach and the others lies in how to account for elastic (reversible) deformation. The VP approach, for instance, ignores elasticity by assuming that elastic stresses can be relaxed by flow mechanisms such as creep over a Maxwell timescale (e.g., ∼1 My for shear modulus of 30 GPa and viscosity of 10^{24} Pa ·s). Nevertheless, there is a broad range of situations in which elastic stresses are important in the overall force balance. Well-known examples include the bending of oceanic lithosphere when subducted or loaded by an island chain. These model problems can be described in the framework of the thin-plate (slender-body) approximations. In particular, the bending of an oceanic lithosphere is accurately represented by the bending of an infinite or semiinfinite thin elastic plate [e.g., *Watts*, 2001; *Turcotte and Schubert*, 2002].

[8] Aware of the need to account for elastic stresses, some researchers came up with the pEVP models where only the deviatoric component of elastic stresses are brought into the incompressible VP model [e.g., *Moresi et al*., 2003; *OzBench et al*., 2008; *Kaus*, 2010]. Nevertheless, the imposition of the incompressibility constraint is still too restrictive for arbitrary motions, leading to simulation responses that are overly stiff for some loading patterns.

[9] Additionally, the volumetric component of deformation, elastic or inelastic, is important in many practical instances as evidenced, for example, by the measured Poisson's ratio (*ν*) of common rocks in the vicinity of 0.3 rather than 0.5 [e.g., *King and Christensen*, 1996]; nonisochoric phase transformations like that of peridotite to serpentine in the hydrated mantle [e.g., *Hyndman and Peacock*, 2003; *Hetényi et al*., 2011]; permanent volume change during brittle deformation of rocks [e.g., *Brace et al*., 1966], and thermal expansion and contraction of rocks [e.g., *Korenaga*, 2007; *Choi et al*., 2008; *Schrank et al*., 2012].

[10] To demonstrate the need for nonisochoric deformation in LTM, we analyze the flexure of a finite-length elastic plate as an example. Through a back-of-the-envelope calculation, we can show that an elastically incompressible plate would be overly stiff compared to a reasonably compressible counterpart. In the thin-plate theory, the maximum displacement (*w*_{max}) of a fixed-length plate by force loading is inversely proportional to the flexural rigidity (*D*) [2002]:

*D* is defined as *E**H*^{3}/12(1−*ν*^{2}), where *E* is the Young's modulus, *H* is the plate's thickness, and *ν* is Poisson's ratio. The following holds for two plates with different Poisson's ratios if everything else is the same:

According to this relationship, elastic flexure can be underestimated in the incompressible (*ν*=0.5) case to be only about 67% of the flexure for *ν*=0.25, a more relevant value for rock modeling.

[11] In the case of an infinitely long thin plate, the difference in flexure due to different Poisson's ratios would be less noticeable because the flexure scales with [2002]. However, the assumption of infinitely long plate does not always hold. Compounded with the rheological complexity such as layered structures and poorly understood polymineralic rheologies, continental plates have finite width and length delimited by major boundary faults. Thus, assuming the lithosphere to be elastically incompressible in LTM bears a potential error other than those associated with numerical approximation.

[12] The previous considerations in addition to the fact that the Lagrangian framework can handle free boundaries naturally and an explicit constitutive update allows for an easy implementation of complex rheologies provide critical motivations for the development of a new solver for LTM based on the explicit Lagrangian EVP approach that allows for unstructured meshing, which is the fundamental contribution reported in this work.

### 1.3 Need for a New Explicit Lagrangian Elastoviscoplastic Solver

[13] The combination of an explicit constitutive update, the Lagrangian reference frame, and the EVP material model has been implemented in a family of codes following the Fast Lagrangian Analysis of Continua (FLAC) algorithm [1988]. Termed geoFLAC, hereafter, these specific implementations of the generic FLAC algorithm [e.g., *Poliakov et al*., 1993] require a structured quadrilateral mesh which severely limits the meshing flexibility needed for adequately capturing strain localization with a locally refined mesh. Additionally, each quadrilateral is decomposed into two sets of overlapping linear triangles that guarantee a symmetrical response to loading but leads to redundant computations. GeoFLAC uses an explicit scheme for the time integration of the momentum equation in the dynamic form as well as for the constitutive update. All of these features bring both advantages and disadvantages and thus deserve critical assessment when inherited. For instance, the explicit time integration and stress update require small time step sizes to ensure stability, increasing the computational cost of the solution. On the other hand, the explicit schemes allow for the simple implementation of arbitrarily complex nonlinear rheologies. In spite of this ambivalence, we put more weight on the relative ease with implementing rheologies, which are almost always nonlinear in LTM. As another example, we believe that the structured quadrilateral mesh of geoFLAC can be replaced with other types of mesh for improved flexibility and performance.

[14] Through such critical evaluations of the FLAC algorithm and its implementation, we distilled a new code, DynEarthSol2D, as an extension and simplification of the geoFLAC algorithm for the EVP material model. An implementation of this methodology is released to the public with the publication of this paper and is named DynEarthSol2D (available for download at http://bitbucket.org/tan2/dynearthsol2). The most notable improvement is the removal of the restrictions on meshing. As a result, we can solve problems on unstructured triangular meshes while keeping the simple explicit material update that made geoFLAC dominant in the field. The use of the state-of-the-art mesh generation tools for triangulations allows for the following: (i) adaptive mesh refinement in regions of highly localized deformation, (ii) high quality of the mesh is maintained by adjusting nodal connectivity, (iii) simple mesh refinement and unrefinement to keep the size of the computational problem in a narrow band, without seriously compromising the quality of the simulation results, and (iv) easier and more faithful tracking of curvilinear boundary, such as the Moho.

[15] The rest of the paper is structured as follows. We first describe the key components of the proposed algorithm in detail, including the newly adopted techniques like the conservative mapping via a local supermesh construction. Results from relevant benchmark tests are presented next to verify our implementation as well as to demonstrate the algorithm's versatility and excellent performance. Finally, conclusions and future work are discussed.