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Combining finite element methods for the incompressible Stokes equations with particle-in-cell methods is an important technique in computational geodynamics that has been widely applied in mantle convection, lithosphere dynamics and crustal-scale modelling. In these applications, particles are used to transport along properties of the medium such as the temperature, chemical compositions or other material properties; the particle methods are therefore used to reduce the advection equation to an ordinary differential equation for each particle, resulting in a problem that is simpler to solve than the original equation for which stabilization techniques are necessary to avoid oscillations.

On the other hand, replacing field-based descriptions by quantities only defined at the locations of particles introduces numerical errors. These errors have previously been investigated, but a complete understanding from both the theoretical and practical sides was so far lacking. In addition, we are not aware of systematic guidance regarding the question of how many particles one needs to choose per mesh cell to achieve a certain accuracy.

In this paper we modify two existing instantaneous benchmarks and present two new analytic benchmarks for time-dependent incompressible Stokes flow in order to compare the convergence rate and accuracy of various combinations of finite elements, particle advection and particle interpolation methods. Using these benchmarks, we find that in order to retain the optimal accuracy of the finite element formulation, one needs to use a sufficiently accurate particle interpolation algorithm. Additionally, we observe and explain that for our higher-order finite-element methods it is necessary to increase the number of particles per cell as the mesh resolution increases (i.e. as the grid cell size decreases) to avoid a reduction in convergence order.

Our methods and results allow designing new particle-in-cell methods with specific convergence rates, and also provide guidance for the choice of common building blocks and parameters such as the number of particles per cell. In addition, our new time-dependent benchmark provides a simple test that can be used to compare different implementations, algorithms and for the assessment of new numerical methods for particle interpolation and advection. We provide a reference implementation of this benchmark in aspect (the ‘Advanced Solver for Problems in Earth’s ConvecTion’), an open source code for geodynamic modelling.

 

Abstract This paper provides an overview of the new features of the finite element library deal.II , version 9.5. 

Abstract. Geodynamical simulations over the past decades have widely beenbuilt on quadrilateral and hexahedral finite elements. For thediscretization of the key Stokes equation describing slow, viscousflow, most codes use either the unstable Q1×P0 element, astabilized version of the equal-order Q1×Q1 element, ormore recently the stable Taylor–Hood element with continuous(Q2×Q1) or discontinuous (Q2×P-1)pressure. However, it is not clear which of these choices isactually the best at accurately simulating “typical” geodynamicsituations. Herein, we provide a systematic comparison of all of theseelements for the first time. We use a series of benchmarks that illuminate differentaspects of the features we consider typical of mantle convectionand geodynamical simulations. We will show in particular that the stabilizedQ1×Q1 element has great difficulty producing accuratesolutions for buoyancy-driven flows – the dominant forcing formantle convection flow – and that the Q1×P0 element istoo unstable and inaccurate in practice. As a consequence, webelieve that the Q2×Q1 and Q2×P-1 elementsprovide the most robust and reliable choice for geodynamical simulations,despite the greater complexity in their implementation and thesubstantially higher computational cost when solving linearsystems. 

The traditional workflow in continuum mechanics simulations is that a geometry description —for example obtained using Constructive Solid Geometry (CSG) or Computer Aided Design (CAD) tools—forms the input for a mesh generator. The mesh is then used as the sole input for the finite element, finite volume, and finite difference solver, which at this point no longer has access to the original, “underlying” geometry. However, many modern techniques—for example, adaptive mesh refinement and the use of higher order geometry approximation methods—really do need information about the underlying geometry to realize their full potential. We have undertaken an exhaustive study of where typical finite element codes use geometry information, with the goal of determining what information geometry tools would have to provide. Our study shows that nearly all geometry-related needs inside the simulators can be satisfied by just two “primitives”: elementary queries posed by the simulation software to the geometry description. We then show that it is possible to provide these primitives in all of the frequently used ways in which geometries are described in common industrial workflows, and illustrate our solutions using a number of examples. 

Abstract This paper provides an overview of the new features of the finite element library deal.II, version 9.3. 

Abstract This paper provides an overview of the new features of the finite element library deal.II, version 9.2. 

SUMMARY Mantle convection and long-term lithosphere dynamics in the Earth and other planets can be treated as the slow deformation of a highly viscous fluid, and as such can be described using the compressible Navier–Stokes equations. Since on Earth-sized planets the influence of compressibility is not a dominant effect, density deviations from a reference profile are at most on the order of a few percent and using the full governing equations poses numerical challenges, most modelling studies have simplified the governing equations. Common approximations assume a temporally constant, but depth-dependent reference profile for the density (the anelastic liquid approximation), or drop compressibility altogether and use a constant reference density (the Boussinesq approximation). In most previous studies of mantle convection and crustal dynamics, one can assume that the error introduced by these approximations was small compared to the errors that resulted from poorly constrained material behaviour and limited numerical accuracy. However, as model parametrizations have become more realistic, and model resolution has improved, this may no longer be the case and the error due to using simplified conservation equations might no longer be negligible: while such approximations may be reasonable for models of mantle plumes or slabs traversing the whole mantle, they may be unsatisfactory for layered materials experiencing phase transitions or materials undergoing significant heating or cooling. For example, at boundary layers or close to dynamically changing density gradients, the error arising from the use of the aforementioned compressibility approximations can be the dominant error source, and common approximations may fail to capture the physical behaviour of interest. In this paper, we discuss new formulations of the continuity equation that include dynamic density variations due to temperature, pressure and composition without using a reference profile for the density. We quantify the improvement in accuracy relative to existing formulations in a number of benchmark models and evaluate for which practical applications these effects are important. Finally, we consider numerical aspects of the new formulations. We implement and test these formulations in the freely available community software aspect, and use this code for our numerical experiments.