<p>We present a new numerical method to simulate the structural patterns emerging from the long-term large-deformation tectonic flows in both two and three spatial dimensions.  The domains of different material properties are each represented by a level set function discretized on a Eulerian mesh with the discontinuous Galerkin method. The level sets are advected by a velocity field provided by a coupled Stokes flow solver. Our method accurately captures the material interface by the adaptive mesh refinement, reduces the computational expenses compared to the traditional particle-in-cell method and offers straightforward handling of geometric splitting and merging.  Under the unified finite element framework, our method promises the flexibility in the choice of mesh geometry as well as the potential for extending to complex rheology.  With passive tracers geat and around areas of interest, the finite strain of the flow field can be integrated through any time interval within the total simulation time.  The strain ellipsoids thus obtained offers the possibility for ground-truthing the simulated deformation patterns with the field structural analysis.  Our results demonstrate identical physical behaviour when compared with established structural geology and geodynamic benchmarks.</p> <p>The style of the crustal dynamics on the Archean Earth has been subject to controversy on whether a vertical tectonic style in the form of Rayleigh-Taylor instability, induced by an inverted density profile, prevails in the early history of the Earth and if so, how the transition to the present-day plate tectonics, characterized by dominantly horizontal movement, is manifested in the rock record.  Equipped with our modelling scheme, we construct numerical models to simulate the lithological distributions and deformation patterns resulted from a synchronous operation of vertical tectonism and horizontal shearing. The latter can be viewed as a possible result of some far-field tectonic boundary condition (e.g. oblique convergence).  Many aspects of the simulation in terms of the map pattern, foliation/lineation trend and strain distribution compare favorably with the field observations in Neoarchean granitoid-greenstone terranes in the Superior Province as well as worldwide.  Therefore, it is concluded that the vertical and horizontal tectonism are not mutually exclusive tectonic regimes  The symbiosis of both tectonic processes is a viable mechanism for establishing the crustal architecture and the deformation pattern we see today in many Neoarchean terranes and might represent a transition from the former to the latter in the Neoarchean.</p>

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We formulate a numerical framework to model the structural patterns emerged from the long-term highly viscous tectonic flow for both two and three spatial dimensions by coupling the discontinuous Galerkin level set method with a finite element Stokes-like flow solver. Our formulation, implemented with adaptive mesh refinement near the material interface, allows for accurate interface capturing and automatic handling of topological splitting and merging. Compared to particle-in-cell family of methods, the level set formulation has the advantage of retaining information on the interface geometry, less memory requirement and the savings of computational expense on the two-way particle-mesh information transfer. Furthermore, our formulation discretizes the level set in the same finite element framework as the flow solver, thus enabling us to fully exploit the advantages of the finite element method such as the flexibility of mesh geometry and the ease of handling anisotropic materials. In order to track the finite deformation in the modelling domain, passive tracer particles are generated at and around locations of interest, whose deformation can be accumulated through arbitrary time interval within the total modelled time span, thus offering a fully dynamical approach for modelling non-steady and inhomogeneous structural patterns. The material distribution and the finite deformation pattern generated from the numerical model can be directly compared with the geological map patterns and the field structural analyses, thus offering the possibility of ground-truthing the modelling results by field evidence.

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