Articles | Volume 13, issue 1
https://doi.org/10.5194/se-13-229-2022
© Author(s) 2022. This work is distributed under
the Creative Commons Attribution 4.0 License.
the Creative Commons Attribution 4.0 License.
https://doi.org/10.5194/se-13-229-2022
© Author(s) 2022. This work is distributed under
the Creative Commons Attribution 4.0 License.
the Creative Commons Attribution 4.0 License.
On the choice of finite element for applications in geodynamics
Department of Earth Sciences, Utrecht University, Utrecht, the Netherlands
Wolfgang Bangerth
Department of Mathematics, Department of Geosciences, Colorado State University, Fort Collins, CO, USA
Related authors
Erik van der Wiel, Cedric Thieulot, and Douwe J. J. van Hinsbergen
Solid Earth, 15, 861–875, https://doi.org/10.5194/se-15-861-2024, https://doi.org/10.5194/se-15-861-2024, 2024
Short summary
Short summary
Geodynamic models of mantle convection provide a powerful tool to study the structure and composition of the Earth's mantle. Comparing such models with other datasets is difficult. We explore the use of
configurational entropy, which allows us to quantify mixing in models. The entropy may be used to analyse the mixed state of the mantle as a whole and may also be useful to validate numerical models against anomalies in the mantle that are obtained from seismology and geochemistry.
Cedric Thieulot and Wolfgang Bangerth
EGUsphere, https://doi.org/10.5194/egusphere-2024-1668, https://doi.org/10.5194/egusphere-2024-1668, 2024
Short summary
Short summary
One of the main numerical methods in geodynamics is the finite-element method. Many types of elements have been used in the past decades in hundreds of publications. They usually fall under two categories: quadrilaterals and triangles. For the first time we compare results obtained with the most used elements of each type on a series of geodynamical benchmarks and draw conclusions as to which are the best ones and which are to be preferably avoided.
Rene Gassmöller, Juliane Dannberg, Wolfgang Bangerth, Elbridge Gerry Puckett, and Cedric Thieulot
Geosci. Model Dev., 17, 4115–4134, https://doi.org/10.5194/gmd-17-4115-2024, https://doi.org/10.5194/gmd-17-4115-2024, 2024
Short summary
Short summary
Numerical models that use simulated particles are a powerful tool for investigating flow in the interior of the Earth, but the accuracy of these models is not fully understood. Here we present two new benchmarks that allow measurement of model accuracy. We then document that better accuracy matters for applications like convection beneath an oceanic plate. Our benchmarks and methods are freely available to help the community develop better models.
Iris van Zelst, Cedric Thieulot, and Timothy J. Craig
Solid Earth, 14, 683–707, https://doi.org/10.5194/se-14-683-2023, https://doi.org/10.5194/se-14-683-2023, 2023
Short summary
Short summary
A common simplification in subduction zone models is the use of constant thermal parameters, while experiments have shown that they vary with temperature. We test various formulations of temperature-dependent thermal parameters and show that they change the thermal structure of the subducting slab. We recommend that modelling studies of the thermal structure of subduction zones take the temperature dependence of thermal parameters into account, especially when providing insights into seismicity.
Barend Cornelis Root, Josef Sebera, Wolfgang Szwillus, Cedric Thieulot, Zdeněk Martinec, and Javier Fullea
Solid Earth, 13, 849–873, https://doi.org/10.5194/se-13-849-2022, https://doi.org/10.5194/se-13-849-2022, 2022
Short summary
Short summary
Several alternative gravity modelling techniques and associated numerical codes with their own advantages and limitations are available for the solid Earth community. With upcoming state-of-the-art lithosphere density models and accurate global gravity field data sets, it is vital to understand the differences of the various approaches. In this paper, we discuss the four widely used techniques: spherical harmonics, tesseroid integration, triangle integration, and hexahedral integration.
Iris van Zelst, Fabio Crameri, Adina E. Pusok, Anne Glerum, Juliane Dannberg, and Cedric Thieulot
Solid Earth, 13, 583–637, https://doi.org/10.5194/se-13-583-2022, https://doi.org/10.5194/se-13-583-2022, 2022
Short summary
Short summary
Geodynamic modelling provides a powerful tool to investigate processes in the Earth’s crust, mantle, and core that are not directly observable. In this review, we present a comprehensive yet concise overview of the modelling process with an emphasis on best practices. We also highlight synergies with related fields, such as seismology and geology. Hence, this review is the perfect starting point for anyone wishing to (re)gain a solid understanding of geodynamic modelling as a whole.
Melchior Schuh-Senlis, Cedric Thieulot, Paul Cupillard, and Guillaume Caumon
Solid Earth, 11, 1909–1930, https://doi.org/10.5194/se-11-1909-2020, https://doi.org/10.5194/se-11-1909-2020, 2020
Short summary
Short summary
This paper presents a numerical method for restoring models of the subsurface to a previous state in their deformation history, acting as a numerical time machine for geological structures. The method relies on the assumption that rock layers can be modeled as highly viscous fluids. It shows promising results on simple setups, including models with faults and non-flat topography. While issues still remain, this could open a way to add more physics to reverse time structural modeling.
Menno Fraters, Cedric Thieulot, Arie van den Berg, and Wim Spakman
Solid Earth, 10, 1785–1807, https://doi.org/10.5194/se-10-1785-2019, https://doi.org/10.5194/se-10-1785-2019, 2019
Short summary
Short summary
Three-dimensional numerical modelling of geodynamic processes may benefit strongly from using realistic 3-D starting models that approximate, e.g. natural subduction settings in the geological past or at present. To this end, we developed the Geodynamic World Builder (GWB), which enables relatively straightforward parameterization of complex 3-D geometric structures associated with geodynamic processes. The GWB is an open-source community code designed to easily interface with geodynamic codes.
Cedric Thieulot
Solid Earth, 9, 1169–1177, https://doi.org/10.5194/se-9-1169-2018, https://doi.org/10.5194/se-9-1169-2018, 2018
Short summary
Short summary
I present the GHOST (Geoscientific Hollow Sphere Tessellation) software which allows for the fast generation of computational meshes in hollow sphere geometries counting up to a hundred million cells. Each mesh is composed of concentric spherical shells made of quadrilaterals or triangles. I focus here on three commonly used meshes used in the geodynamics/geophysics community and further benchmark the gravity and gravitational potential procedures in the simple case of a constant density.
Alexis Plunder, Cédric Thieulot, and Douwe J. J. van Hinsbergen
Solid Earth, 9, 759–776, https://doi.org/10.5194/se-9-759-2018, https://doi.org/10.5194/se-9-759-2018, 2018
Short summary
Short summary
The thermal state of the Earth's crust determines how it reacts to tectonic forces and to fluid flow responsible for ore formation. We hypothesize that the angle between plate motion and convergent boundaries determines the thermal regime of subduction zones (where a plate goes under another one). Computer models and a geological reconstruction of Turkey were used to validate this hypothesis.
This research was done to validate a hypothesis made on the basis of nonquantitative field data.
Anne Glerum, Cedric Thieulot, Menno Fraters, Constantijn Blom, and Wim Spakman
Solid Earth, 9, 267–294, https://doi.org/10.5194/se-9-267-2018, https://doi.org/10.5194/se-9-267-2018, 2018
Short summary
Short summary
A nonlinear viscoplastic rheology is implemented and benchmarked in the ASPECT software, allowing for the modeling of lithospheric deformation. We showcase the new functionality with a four-dimensional model of thermomechanically coupled subduction.
Cedric Thieulot
Solid Earth, 8, 1181–1191, https://doi.org/10.5194/se-8-1181-2017, https://doi.org/10.5194/se-8-1181-2017, 2017
Short summary
Short summary
I present a new family of analytical flow solutions to the incompressible Stokes equation in a spherical shell. The velocity is tangential to both inner and outer boundaries, the viscosity is radial, and the solution has been designed so that the expressions for velocity, pressure, and body force are simple to implement in (geodynamics) codes. This forms the basis of a numerical benchmark for convection codes, and I have implemented it in two finite-element codes.
B. Hillebrand, C. Thieulot, T. Geenen, A. P. van den Berg, and W. Spakman
Solid Earth, 5, 1087–1098, https://doi.org/10.5194/se-5-1087-2014, https://doi.org/10.5194/se-5-1087-2014, 2014
Short summary
Short summary
Our paper demonstrates that the level set method is a viable method for material tracking in multi-material flow models. The different benchmarks illustate several advantages that the level set method provides over tracer-based methods. We therefore conclude that the level set method is well suited for geodynamical modeling.
C. Thieulot
Solid Earth Discuss., https://doi.org/10.5194/sed-6-1949-2014, https://doi.org/10.5194/sed-6-1949-2014, 2014
Revised manuscript has not been submitted
Erik van der Wiel, Cedric Thieulot, and Douwe J. J. van Hinsbergen
Solid Earth, 15, 861–875, https://doi.org/10.5194/se-15-861-2024, https://doi.org/10.5194/se-15-861-2024, 2024
Short summary
Short summary
Geodynamic models of mantle convection provide a powerful tool to study the structure and composition of the Earth's mantle. Comparing such models with other datasets is difficult. We explore the use of
configurational entropy, which allows us to quantify mixing in models. The entropy may be used to analyse the mixed state of the mantle as a whole and may also be useful to validate numerical models against anomalies in the mantle that are obtained from seismology and geochemistry.
Cedric Thieulot and Wolfgang Bangerth
EGUsphere, https://doi.org/10.5194/egusphere-2024-1668, https://doi.org/10.5194/egusphere-2024-1668, 2024
Short summary
Short summary
One of the main numerical methods in geodynamics is the finite-element method. Many types of elements have been used in the past decades in hundreds of publications. They usually fall under two categories: quadrilaterals and triangles. For the first time we compare results obtained with the most used elements of each type on a series of geodynamical benchmarks and draw conclusions as to which are the best ones and which are to be preferably avoided.
Rene Gassmöller, Juliane Dannberg, Wolfgang Bangerth, Elbridge Gerry Puckett, and Cedric Thieulot
Geosci. Model Dev., 17, 4115–4134, https://doi.org/10.5194/gmd-17-4115-2024, https://doi.org/10.5194/gmd-17-4115-2024, 2024
Short summary
Short summary
Numerical models that use simulated particles are a powerful tool for investigating flow in the interior of the Earth, but the accuracy of these models is not fully understood. Here we present two new benchmarks that allow measurement of model accuracy. We then document that better accuracy matters for applications like convection beneath an oceanic plate. Our benchmarks and methods are freely available to help the community develop better models.
Iris van Zelst, Cedric Thieulot, and Timothy J. Craig
Solid Earth, 14, 683–707, https://doi.org/10.5194/se-14-683-2023, https://doi.org/10.5194/se-14-683-2023, 2023
Short summary
Short summary
A common simplification in subduction zone models is the use of constant thermal parameters, while experiments have shown that they vary with temperature. We test various formulations of temperature-dependent thermal parameters and show that they change the thermal structure of the subducting slab. We recommend that modelling studies of the thermal structure of subduction zones take the temperature dependence of thermal parameters into account, especially when providing insights into seismicity.
Barend Cornelis Root, Josef Sebera, Wolfgang Szwillus, Cedric Thieulot, Zdeněk Martinec, and Javier Fullea
Solid Earth, 13, 849–873, https://doi.org/10.5194/se-13-849-2022, https://doi.org/10.5194/se-13-849-2022, 2022
Short summary
Short summary
Several alternative gravity modelling techniques and associated numerical codes with their own advantages and limitations are available for the solid Earth community. With upcoming state-of-the-art lithosphere density models and accurate global gravity field data sets, it is vital to understand the differences of the various approaches. In this paper, we discuss the four widely used techniques: spherical harmonics, tesseroid integration, triangle integration, and hexahedral integration.
Iris van Zelst, Fabio Crameri, Adina E. Pusok, Anne Glerum, Juliane Dannberg, and Cedric Thieulot
Solid Earth, 13, 583–637, https://doi.org/10.5194/se-13-583-2022, https://doi.org/10.5194/se-13-583-2022, 2022
Short summary
Short summary
Geodynamic modelling provides a powerful tool to investigate processes in the Earth’s crust, mantle, and core that are not directly observable. In this review, we present a comprehensive yet concise overview of the modelling process with an emphasis on best practices. We also highlight synergies with related fields, such as seismology and geology. Hence, this review is the perfect starting point for anyone wishing to (re)gain a solid understanding of geodynamic modelling as a whole.
Melchior Schuh-Senlis, Cedric Thieulot, Paul Cupillard, and Guillaume Caumon
Solid Earth, 11, 1909–1930, https://doi.org/10.5194/se-11-1909-2020, https://doi.org/10.5194/se-11-1909-2020, 2020
Short summary
Short summary
This paper presents a numerical method for restoring models of the subsurface to a previous state in their deformation history, acting as a numerical time machine for geological structures. The method relies on the assumption that rock layers can be modeled as highly viscous fluids. It shows promising results on simple setups, including models with faults and non-flat topography. While issues still remain, this could open a way to add more physics to reverse time structural modeling.
Menno Fraters, Cedric Thieulot, Arie van den Berg, and Wim Spakman
Solid Earth, 10, 1785–1807, https://doi.org/10.5194/se-10-1785-2019, https://doi.org/10.5194/se-10-1785-2019, 2019
Short summary
Short summary
Three-dimensional numerical modelling of geodynamic processes may benefit strongly from using realistic 3-D starting models that approximate, e.g. natural subduction settings in the geological past or at present. To this end, we developed the Geodynamic World Builder (GWB), which enables relatively straightforward parameterization of complex 3-D geometric structures associated with geodynamic processes. The GWB is an open-source community code designed to easily interface with geodynamic codes.
Cedric Thieulot
Solid Earth, 9, 1169–1177, https://doi.org/10.5194/se-9-1169-2018, https://doi.org/10.5194/se-9-1169-2018, 2018
Short summary
Short summary
I present the GHOST (Geoscientific Hollow Sphere Tessellation) software which allows for the fast generation of computational meshes in hollow sphere geometries counting up to a hundred million cells. Each mesh is composed of concentric spherical shells made of quadrilaterals or triangles. I focus here on three commonly used meshes used in the geodynamics/geophysics community and further benchmark the gravity and gravitational potential procedures in the simple case of a constant density.
Alexis Plunder, Cédric Thieulot, and Douwe J. J. van Hinsbergen
Solid Earth, 9, 759–776, https://doi.org/10.5194/se-9-759-2018, https://doi.org/10.5194/se-9-759-2018, 2018
Short summary
Short summary
The thermal state of the Earth's crust determines how it reacts to tectonic forces and to fluid flow responsible for ore formation. We hypothesize that the angle between plate motion and convergent boundaries determines the thermal regime of subduction zones (where a plate goes under another one). Computer models and a geological reconstruction of Turkey were used to validate this hypothesis.
This research was done to validate a hypothesis made on the basis of nonquantitative field data.
Anne Glerum, Cedric Thieulot, Menno Fraters, Constantijn Blom, and Wim Spakman
Solid Earth, 9, 267–294, https://doi.org/10.5194/se-9-267-2018, https://doi.org/10.5194/se-9-267-2018, 2018
Short summary
Short summary
A nonlinear viscoplastic rheology is implemented and benchmarked in the ASPECT software, allowing for the modeling of lithospheric deformation. We showcase the new functionality with a four-dimensional model of thermomechanically coupled subduction.
Cedric Thieulot
Solid Earth, 8, 1181–1191, https://doi.org/10.5194/se-8-1181-2017, https://doi.org/10.5194/se-8-1181-2017, 2017
Short summary
Short summary
I present a new family of analytical flow solutions to the incompressible Stokes equation in a spherical shell. The velocity is tangential to both inner and outer boundaries, the viscosity is radial, and the solution has been designed so that the expressions for velocity, pressure, and body force are simple to implement in (geodynamics) codes. This forms the basis of a numerical benchmark for convection codes, and I have implemented it in two finite-element codes.
B. Hillebrand, C. Thieulot, T. Geenen, A. P. van den Berg, and W. Spakman
Solid Earth, 5, 1087–1098, https://doi.org/10.5194/se-5-1087-2014, https://doi.org/10.5194/se-5-1087-2014, 2014
Short summary
Short summary
Our paper demonstrates that the level set method is a viable method for material tracking in multi-material flow models. The different benchmarks illustate several advantages that the level set method provides over tracer-based methods. We therefore conclude that the level set method is well suited for geodynamical modeling.
C. Thieulot
Solid Earth Discuss., https://doi.org/10.5194/sed-6-1949-2014, https://doi.org/10.5194/sed-6-1949-2014, 2014
Revised manuscript has not been submitted
Related subject area
Subject area: Core and mantle structure and dynamics | Editorial team: Geodynamics and quantitative modelling | Discipline: Geodynamics
ECOMAN: an open-source package for geodynamic and seismological modelling of mechanical anisotropy
Quantifying mantle mixing through configurational entropy
On the impact of true polar wander on heat flux patterns at the core–mantle boundary
Modeling liquid transport in the Earth's mantle as two-phase flow: effect of an enforced positive porosity on liquid flow and mass conservation
Transport mechanisms of hydrothermal convection in faulted tight sandstones
Influence of heterogeneous thermal conductivity on the long-term evolution of the lower-mantle thermochemical structure: implications for primordial reservoirs
Coupled dynamics and evolution of primordial and recycled heterogeneity in Earth's lower mantle
Comparing global seismic tomography models using varimax principal component analysis
Manuele Faccenda, Brandon P. VanderBeek, Albert de Montserrat, Jianfeng Yang, Francesco Rappisi, and Neil Ribe
Solid Earth, 15, 1241–1264, https://doi.org/10.5194/se-15-1241-2024, https://doi.org/10.5194/se-15-1241-2024, 2024
Short summary
Short summary
The Earth's internal dynamics and structure can be well understood by combining seismological and geodynamic modelling with mineral physics, an approach that has been poorly adopted in the past. To this end we have developed ECOMAN, an open-source software package that is intended to overcome the computationally intensive nature of this multidisciplinary methodology and the lack of a dedicated and comprehensive computational framework.
Erik van der Wiel, Cedric Thieulot, and Douwe J. J. van Hinsbergen
Solid Earth, 15, 861–875, https://doi.org/10.5194/se-15-861-2024, https://doi.org/10.5194/se-15-861-2024, 2024
Short summary
Short summary
Geodynamic models of mantle convection provide a powerful tool to study the structure and composition of the Earth's mantle. Comparing such models with other datasets is difficult. We explore the use of
configurational entropy, which allows us to quantify mixing in models. The entropy may be used to analyse the mixed state of the mantle as a whole and may also be useful to validate numerical models against anomalies in the mantle that are obtained from seismology and geochemistry.
Thomas Frasson, Stéphane Labrosse, Henri-Claude Nataf, Nicolas Coltice, and Nicolas Flament
Solid Earth, 15, 617–637, https://doi.org/10.5194/se-15-617-2024, https://doi.org/10.5194/se-15-617-2024, 2024
Short summary
Short summary
Heat flux heterogeneities at the bottom of Earth's mantle play an important role in the dynamic of the underlying core. Here, we study how these heterogeneities are affected by the global rotation of the Earth, called true polar wander (TPW), which has to be considered to relate mantle dynamics with core dynamics. We find that TPW can greatly modify the large scales of heat flux heterogeneities, notably at short timescales. We provide representative maps of these heterogeneities.
Changyeol Lee, Nestor G. Cerpa, Dongwoo Han, and Ikuko Wada
Solid Earth, 15, 23–38, https://doi.org/10.5194/se-15-23-2024, https://doi.org/10.5194/se-15-23-2024, 2024
Short summary
Short summary
Fluids and melts in the mantle are key to the Earth’s evolution. The main driving force for their transport is the compaction of the porous mantle. Numerically, the compaction equations can yield unphysical negative liquid fractions (porosity), and it is necessary to enforce positive porosity. However, the effect of such a treatment on liquid flow and mass conservation has not been quantified. We found that although mass conservation is affected, the liquid pathways are well resolved.
Guoqiang Yan, Benjamin Busch, Robert Egert, Morteza Esmaeilpour, Kai Stricker, and Thomas Kohl
Solid Earth, 14, 293–310, https://doi.org/10.5194/se-14-293-2023, https://doi.org/10.5194/se-14-293-2023, 2023
Short summary
Short summary
The physical processes leading to the kilometre-scale thermal anomaly in faulted tight sandstones are numerically investigated. The fluid-flow pathways, heat-transfer types and interactions among different convective and advective flow modes are systematically identified. The methodologies and results can be applied to interpret hydrothermal convection-related geological phenomena and to draw implications for future petroleum and geothermal exploration and exploitation in analogous settings.
Joshua Martin Guerrero, Frédéric Deschamps, Yang Li, Wen-Pin Hsieh, and Paul James Tackley
Solid Earth, 14, 119–135, https://doi.org/10.5194/se-14-119-2023, https://doi.org/10.5194/se-14-119-2023, 2023
Short summary
Short summary
The mantle thermal conductivity's dependencies on temperature, pressure, and composition are often suppressed in numerical models. We examine the effect of these dependencies on the long-term evolution of lower-mantle thermochemical structure. We propose that depth-dependent conductivities derived from mantle minerals, along with moderate temperature and compositional correction, emulate the Earth's mean lowermost-mantle conductivity values and produce a stable two-pile configuration.
Anna Johanna Pia Gülcher, Maxim Dionys Ballmer, and Paul James Tackley
Solid Earth, 12, 2087–2107, https://doi.org/10.5194/se-12-2087-2021, https://doi.org/10.5194/se-12-2087-2021, 2021
Short summary
Short summary
The lower mantle extends from 660–2890 km depth, making up > 50 % of the Earth’s volume. Its composition and structure, however, remain poorly understood. In this study, we investigate several hypotheses with computer simulations of mantle convection that include different materials: recycled, dense rocks and ancient, strong rocks. We propose a new integrated style of mantle convection including
piles,
blobs, and
streaksthat agrees with various observations of the deep Earth.
Olivier de Viron, Michel Van Camp, Alexia Grabkowiak, and Ana M. G. Ferreira
Solid Earth, 12, 1601–1634, https://doi.org/10.5194/se-12-1601-2021, https://doi.org/10.5194/se-12-1601-2021, 2021
Short summary
Short summary
As the travel time of seismic waves depends on the Earth's interior properties, seismic tomography uses it to infer the distribution of velocity anomalies, similarly to what is done in medical tomography. We propose analysing the outputs of those models using varimax principal component analysis, which results in a compressed objective representation of the model, helping analysis and comparison.
Cited articles
Alisic, L., Gurnis, M., Stadler, G., Burstedde, C., and Ghattas, O.:
Multi-scale dynamics and rheology of mantle flow with plates,
J. Geophys. Res., 117, B10402, https://doi.org/10.1029/2012JB009234, 2012. a
Arndt, D., Bangerth, W., Davydov, D., Heister, T., Heltai, L., Kronbichler, M.,
Maier, M., Pelteret, J.-P., Turcksin, B., and Wells, D.: The deal. II finite
element library: Design, features, and insights, Comput. Math.
Appl., 81, 407–422, https://doi.org/10.1016/j.camwa.2020.02.022, 2020. a
Arrial, P.-A. and Billen, M.: Influence of geometry and eclogitization on
oceanic plateau subduction, Earth Planet. Sc. Lett., 363, 34–43,
https://doi.org/10.1016/j.epsl.2012.12.011, 2013. a, b, c, d
ASPECT developers: ASPECT: Advanced Solver for Problems in Earth's ConvecTion, available at: https://aspect.geodynamics.org/, last access: 17 January 2022. a
Bangerth, W., Hartmann, R., and Kanschat, G.: A General-Purpose Object-Oriented Finite Element Library, ACM T. Math.
Software, 33, 24–51, https://doi.org/10.1145/1268776.1268779, 2007. a
Bangerth, W., Dannberg, J., Fraters, M., Gassmöller, R., Glerum, A., Heister, T., Myhill, B., and Naliboff, J.: ASPECT: Advanced Solver for
Problems in Earth’s ConvecTion, Computational Infrastructure for
Geodynamics, available at: https://www.math.clemson.edu/~heister/manual.pdf, last access: 17 January 2022. a
Bochev, P. B., Dohrmann, C. R., and Gunzburger, M. D.: Stabilization of
Low-order Mixed Finite Elements for the Stokes Equations, SIAM Journal on
Numerical Analysis, 44, 82–101, https://doi.org/10.1137/s0036142905444482, 2006. a, b
Boffi, D. and Gastaldi, L.: On the quadrilateral Q2–P1 element for the Stokes
problem, Int. J. Numer. Meth. Fl., 39,
1001–1011, 2002. a
Boffi, D., Cavallini, N., Gardini, F., and Gastaldi, L.: Local Mass
Conservation of Stokes Finite Elements, J. Sci. Comput., 52,
383–400, https://doi.org/10.1007/s10915-011-9549-4, 2011. a
Braun, J., Thieulot, C., Fullsack, P., DeKool, M., and Huismans, R.: DOUAR: a
new three-dimensional creeping flow model for the solution of geological
problems, Phys. Earth. Planet. Int., 171, 76–91, 2008. a
Brezzi, F. and Douglas, J.: Stabilised mixed methods for the Stokes problem,
Numer. Math., 53, 225–235, 1988. a
Brezzi, F. and Pitkäranta, J.: On the stabilization of finite element
approximations of the Stokes equations, Vol. 10, Notes on numerical
fluid mechanics, edited by: Hackbusch, W., 11–19, Vieweg+Teubner Verlag,
https://doi.org/10.1007/978-3-663-14169-3_2, 1984. a
Brune, S., Heine, C., Clift, P. D., and Pérez-Gussinyé, M.: Rifted
margin architecture and crustal rheology: reviewing Iberia-Newfoundland,
central South Atlantic, and South China sea, Mar. Petrol. Geol.,
79, 257–281, 2017. a
Burstedde, C., Ghattas, O., Stadler, G., Tu, T., and Wilcox, L.: Parallel
scalable adjoint-based adaptive solution of variable-viscosity Stokes flow
problems, Comput. Method. Appl. M., 198,
1691–1700, https://doi.org/10.1016/j.cma.2008.12.015, 2009. a, b
Burstedde, C., Stadler, G., Alisic, L., Wilcox, L., Tan, E., Gurnis, M., and
Ghattas, O.: Large-scale adaptive mantle convection simulation,
Geophy. J. Int., 192, 889–906, https://doi.org/10.1093/gji/ggs070, 2013. a, b
Cerpa, N., Hassani, R., Gerbault, M., and Prévost, J.-H.: A fictitious
domain method for lithosphere-asthenosphere interaction: Application to
periodic slab folding in the upper mantle, Geochem. Geophy. Geosy., 15, 1852–1877, https://doi.org/10.1002/2014GC005241,
2014. a
Cerpa, N., Araya, R., Gerbault, M., and Hassani, R.: Relationship between slab
dip and topography segmentation in an oblique subduction zone: Insights from
numerical modeling, Geophys. Res. Lett., 42, 5786–5795, https://doi.org/10.1002/2015GL064047, 2015. a
Cerpa, N., Guillaume, B., and Martinod, J.: The interplay between overriding
plate kinematics, slab dip and tectonics, Geophys. J. Int., 215, 1789–1802,
2018. a
Clevenger, T., Heister, T., Kanschat, G., and Kronbichler, M.: A flexible,
parallel, adaptive geometric multigrid method for FEM, ACM T. Math.
Software, 47, 7, https://doi.org/10.1145/3425193, 2020. a
Crameri, F., Schmeling, H., Golabek, G., Duretz, T., Orendt, R., Buiter, S.,
May, D., Kaus, B., Gerya, T., and Tackley, P.: A comparison of numerical
surface topography calculations in geodynamic modelling: an evaluation of the
“sticky air” method, Geophys. J. Int., 189, 38–54, 2012. a
Crouzeix, M. and Raviart, P.-A.: Conforming and nonconforming finite element
methods for solving the stationary Stokes equations I, RAIRO, 7,
33–75, 1973. a
Dabrowski, M., Krotkiewski, M., and Schmid, D.: MILAMIN: Matlab based finite
element solver for large problems, Geochem. Geophy. Geosy., 9, Q04030,
https://doi.org/10.1029/2007GC001719, 2008. a, b
Davies, D., Wilson, C., and Kramer, S.: Fluidity: A fully unstructured
anisotropic adaptive mesh computational modeling framework for geodynamics,
Geochem. Geophy. Geosy., 12, Q06001, https://doi.org/10.1029/2011GC003551, 2011. a
Dawson, C., Sun, S., and Wheeler, M. F.: Compatible algorithms for coupled flow
and transport, Comput. Method. Appl. M., 193,
2565–2580, 2004. a
de Montserrat, A., Morgan, J. P., and Hasenclever, J.: LaCoDe: a Lagrangian
two-dimensional thermo-mechanical code for large-strain compressible
visco-elastic geodynamical modeling, Tectonophysics, 767, 228173, https://doi.org/10.1016/j.tecto.2019.228173, 2019. a
Deubelbeiss, Y. and Kaus, B.: Comparison of Eulerian and Lagrangian numerical
techniques for the Stokes equations in the presence of strongly varying
viscosity, Phys. Earth Planet. In., 171, 92–111,
https://doi.org/10.1016/j.pepi.2008.06.023, 2008. a, b
Douglas, J., Santos, J. E., Sheen, D., and Ye, X.: Nonconforming Galerkin
methods based on quadrilateral elements for second order elliptic problems,
ESAIM: Math. Model. Num., 33, 747–770, 1999. a
Duretz, T., May, D., Gerya, T., and Tackley, P.: Discretization errors and
free surface stabilisation in the finite difference and marker-in-cell method
for applied geodynamics: A numerical study, Geochem. Geophy. Geosy., 12, Q07004,
https://doi.org/10.1029/2011GC003567, 2011. a, b
Duretz, T., de Borst, R., Yamato, P., and Le Pourhiet, L.: Towards robust and
predictive geodynamic modelling: the way forward in frictional plasticity,
Geophys. Res. Lett., 47, e2019GL086027,
https://doi.org/10.1029/2019GL086027, 2020. a
Fraters, M., Bangerth, W., Thieulot, C., Glerum, A., and Spakman, W.:
Efficient and Practical Newton Solvers for Nonlinear Stokes Systems in
Geodynamic Problems, Geophys. J. Int., 218, 873–894, https://doi.org/10.1093/gji/ggz183, 2019. a
Fullsack, P.: An arbitrary Lagrangian-Eulerian formulation for creeping
flows and its application in tectonic models, Geophys. J. Int., 120, 1–23,
https://doi.org/10.1111/j.1365-246X.1995.tb05908.x, 1995. a
Gerya, T.: Numerical Geodynamic Modelling, 2nd Edn., Cambridge University Press, ISBN 978-1-107-14314-2, 2019. a
Gerya, T., May, D., and Duretz, T.: An adaptive staggered grid finite
difference method for modeling geodynamic Stokes flows with strongly variable
viscosity, Geochem. Geophy. Geosy., 14, 1200–1225, https://doi.org/10.1002/ggge.20078, 2013. a
Gmeiner, B., Rüde, U., Stengel, H., Waluga, C., and Wohlmuth, B.: Performance
and scalability of hierarchical hybrid multigrid solvers for Stokes systems,
SIAM J. Sci. Comput., 37, C143–C168,
https://doi.org/10.1137/130941353, 2015. a
Gresho, P. and Sani, R.: Incompressible flow and the Finite Element Method, vol
II, John Wiley and Sons, Ltd, ISBN 978-0471492504, 2000. a
Gresho, P., Chan, S., Christon, M., and Hindmarsch, A.: A little more on
stabilised Q1Q1 for transient viscous incompressible flow,
Int. J. Numer. Meth. Fl., 21, 837–856, 1995. a
Heister, T., Dannberg, J., Gassmöller, R., and Bangerth, W.: High Accuracy
Mantle Convection Simulation through Modern Numerical Methods. II: Realistic
Models and Problems, Geophys. J. Int., 210, 833–851,
https://doi.org/10.1093/gji/ggx195, 2017. a, b, c
Huismans, R. and Beaumont, C.: Complex rifted continental margins explained by
dynamical models of depth-dependent lithospheric extension, Geology, 30,
211–214, 2002. a
Jacquey, A. B., Rattez, H., and Veveakis, M.: Strain localization
regularization and patterns formation in rate-dependent plastic materials
with multiphysics coupling, J. Mech. Phys. Solids,
152, 104422, https://doi.org/10.1016/j.jmps.2021.104422, 2021. a
Jadamec, M. and Billen, M.: The role of rheology and slab shape on rapid
mantle flow: Three-dimensional numerical models of the Alaska slab edge,
J. Geophys. Res.-Sol. Ea., 117, B02304, https://doi.org/10.1029/2011JB008563, 2012. a
Jammes, S. and Huismans, R.: Structural styles of mountain building: Controls
of lithospheric rheologic stratification and extensional inheritance,
J. Geophys. Res., 117, B10403, https://doi.org/10.1029/2012JB009376, 2012. a
John, V.: Finite Element Methods for Incompressible Flow Problems, Springer, https://doi.org/10.1007/978-3-319-45750-5,
2016. a, b
Kaus, B., Mühlhaus, H., and May, D.: A stabilization algorithm for
geodynamic numerical simulations with a free surface,
Phys. Earth Planet. In., 181, 12–20, https://doi.org/10.1016/j.pepi.2010.04.007,
2010. a
Kaus, B., Popov, A., Baumann, T., Pusok, A., Bauville, A., Fernandez, N., and
Collignon, M.: Forward and Inverse Modelling of Lithospheric Deformation on
Geological Timescales, 11–12 February 2016, NIC Symposium 2016, Forschungszentrum Jülich GmbH, John von Neumann Institute for Computing (NIC), 299–307, 2016. a
King, S., Raefsky, A., and Hager, B.: ConMan: Vectorizing a finite element
code for incompressible two-dimensional convection in the Earth’s mantle,
Phys. Earth Planet. In., 59, 195–208,
https://doi.org/10.1016/0031-9201(90)90225-M, 1990. a
Le Pourhiet, L., Huet, B., May, D., Labrousse, L., and Jolivet, L.:
Kinematic interpretation of the 3D shapes of metamorphic core complexes,
Geochem. Geophy. Geosy., 13, Q09002, https://doi.org/10.1029/2012GC004271, 2012. a
Lechmann, S., May, D., Kaus, B., and Schmalholz, S.: Comparing thin-sheet
models with 3-D multilayer models for continental collision,
Geophys. J. Int., 187, 10–33, 2011. a
Lehmann, R., Lukacova-Medvidova, M., Kaus, B., and Popov, A.: Comparison of
continuous and discontinuous Galerkin approaches for variable-viscosity
Stokes flow, Z. Angew. Math. Mech., 96, 733–746, https://doi.org/10.1002/zamm.201400274,
2015. a
Leng, W. and Zhong, S.: Implementation and application of adaptive mesh
refinement for thermochemical mantle convection studies,
Geochem. Geophy. Geosy., 12, Q04006, https://doi.org/10.1029/2010GC003425, 2011. a
Matthies, G. and Tobiska, L.: The Inf-Sup Condition for the Mapped
Element in Arbitrary Space Dimensions, Computing,
69, 119–139, https://doi.org/10.1007/s00607-002-1451-3, 2002. a
May, D. and Moresi, L.: Preconditioned iterative methods for Stokes flow
problems arising in computational geodynamics, Phys. Earth Planet. In.,
171, 33–47, https://doi.org/10.1016/j.pepi.2008.07.036, 2008. a, b
May, D., Brown, J., and Le Pourhiet, L.: A scalable, matrix-free multigrid
preconditioner for finite element discretizations of heterogeneous Stokes
flow, Comput. Method. Appl. M., 290, 496–523,
https://doi.org/10.1016/j.cma.2015.03.014, 2015. a, b, c, d
McNamara, A. K. and Zhong, S.: Thermochemical structures within a spherical
mantle: Superplumes or piles?, J. Geophys. Res.-Sol. Ea.,
109, B07402, https://doi.org/10.1029/2003JB002847, 2004. a
Mishin, Y.: Adaptive multiresolution methods for problems of computational
geodynamics, PhD thesis, ETH Zurich, https://doi.org/10.3929/ethz-a-007347901, 2011. a
Moresi, L. and Gurnis, M.: Constraints on the lateral strength of slabs from
three-dimensional dynamic flow models, Earth Planet. Sc. Lett.,
138, 15–28, https://doi.org/10.1016/0012-821X(95)00221-W, 1996. a
Moresi, L., Dufour, F., and Mühlhaus, H.: A Lagrangian integration point
finite element method for large deformation modeling of visco-elastic
geomaterials, J. Comput. Phys., 184, 476–497,
https://doi.org/10.1016/S0021-9991(02)00031-1, 2003. a, b
Moresi, L., Landry, W., Hodkison, L., Turnbull, R., Lemiale, V., May, D.,
Stegman, D., Velic, M., Sunter, P., and Giordani, J.: Gale v2.0.1, Computational Infrastructure for Geodynamics [software], available at: https://geodynamics.org/cig/software/gale/ (last access: 17 January 2022), 2012. a
Norburn, S. and Silvester, D.: Fourier analysis of stabilized Q1-Q1 mixed
finite element approximation, SIAM J. Numer. Anal., 39, 817–833,
https://doi.org/10.1137/S0036142999362274, 2001. a
O'Neill, C., Moresi, L., Müller, D., Albert, R., and Dufour, F.: Ellipsis
3D: a particle-in-cell finite element hybrid code for modelling mantle
convection and lithospheric deformation, Comput. Geosci., 32,
1769–1779, 2006. a
Plunder, A., Thieulot, C., and van Hinsbergen, D. J. J.: The effect of obliquity on temperature in subduction zones: insights from 3-D numerical modeling, Solid Earth, 9, 759–776, https://doi.org/10.5194/se-9-759-2018, 2018. a
Popov, A. and Sobolev, S.: SLIM3D: a tool for three-dimensional
thermomechanical modelling of lithospheric deformation with
elasto-visco-plastic rheology, Phys. Earth Planet. In., 171, 55–75,
https://doi.org/10.1016/j.pepi.2008.03.007, 2008. a, b
Rannacher, R. and Turek, S.: Simple Nonconforming Quadrilateral Stokes
Element, Numer. Meth. Part. D. E., 8, 97–111,
1992. a
Rose, I., Buffet, B., and Heister, T.: Stability and accuracy of free surface
time integration in viscous flows, Phys. Earth Planet. In., 262,
90–100, 2017. a
Sani, R., Gresho, P., Lee, R., and Griffiths, D.: The cause and cure (?) of the
spurious pressures generated by certain FEM solutions of the incompressible
Navier-Stokes equations: part 1, Int. J. Numer. Meth. Fl., 1, 17–43,
https://doi.org/10.1002/fld.1650010104, 1981a. a
Sani, R., Gresho, P., Lee, R., Griffiths, D., and Engelman, M.: The cause and
cure (?) of the spurious pressures generated by certain FEM solutions of the
incompressible Navier-Stokes equations: part 2, Int. J. Numer. Meth. Fl., 1,
171–204, https://doi.org/10.1002/fld.1650010206, 1981b. a
Schmid, D. and Podlachikov, Y.: Analytical solutions for deformable elliptical
inclusions in general shear, Geophys. J. Int., 155, 269–288,
https://doi.org/10.1046/j.1365-246X.2003.02042.x, 2003. a
Schubert, G., Turcotte, D. L., and Olson, P.: Mantle Convection in the Earth
and Planets, Cambridge University Press, https://doi.org/10.1017/CBO9780511612879, 2001. a, b
Schuh-Senlis, M., Thieulot, C., Cupillard, P., and Caumon, G.: Towards the application of Stokes flow equations to structural restoration simulations, Solid Earth, 11, 1909–1930, https://doi.org/10.5194/se-11-1909-2020, 2020. a
Silvester, D. and Kechkar, N.: Stabilised bilinear-constant velocity-pressure
finite elements for the conjugate gradient solution of the stokes problem,
Comput. Method. Appl. M., 79, 71–86,
https://doi.org/10.1016/0045-7825(90)90095-4, 1990. a
Spiegelman, M., May, D., and Wilson, C.: On the solvability of incompressible
Stokes with viscoplastic rheologies in geodynamics,
Geochem. Geophy. Geosy., 17, 2213–2238, 2016. a
Stadler, G., Gurnis, M., Burstedde, C., Wilcox, L., Alisic, L., and Ghattas,
O.: The dynamics of plate tectonics and mantle flow: from local to global
scales, Science, 329, 1033–1038, https://doi.org/10.1126/science.1191223, 2010. a
Suckale, J., Nave, J.-C., and Hager, B.: It takes three to tango: 1.
Simulating buoyancy-driven flow in the presence of large viscosity
contrasts, J. Geophys. Res., 115, B07409, https://doi.org/10.1029/2009JB006916, 2010. a
Thielmann, M. and Kaus, B.: Shear heating induced lithospheric-scale
localization: Does it result in subduction?, Earth Planet. Sc. Lett.,
359–360, 1–13, 2012. a
Thielmann, M., May, D., and Kaus, B.: Discretization errors in the Hybrid
Finite Element Particle-In-Cell Method, Pure Appl. Geophys., 171,
2164–2184, https://doi.org/10.1007/s00024-014-0808-9, 2014. a, b
Thieulot, C.: FANTOM: two- and three-dimensional numerical modelling of
creeping flows for the solution of geological problems,
Phys. Earth. Planet. In., 188, 47–68, https://doi.org/10.1016/j.pepi.2011.06.011,
2011. a, b, c
Thieulot, C.: ELEFANT: a user-friendly multipurpose geodynamics code, Solid Earth Discuss., 6, 1949–2096, https://doi.org/10.5194/sed-6-1949-2014, 2014. a
Thieulot, C., Fullsack, P., and Braun, J.: Adaptive octree-based finite element
analysis of two- and three-dimensional indentation problems,
J. Geophys. Res., 113, B12207, https://doi.org/10.1029/2008JB005591, 2008. a
Turcotte, D. and Schubert, G.: Geodynamics, 2nd Edn., Cambridge, https://doi.org/10.1017/CBO9780511807442, 2012. a
Turek, S.: Efficient Solvers for Incompressible Flow Problems, Springer, https://doi.org/10.1007/978-3-642-58393-3,
1999. a
Weismüller, J., Gmeiner, B., Ghelichkhan, S., Huber, M., John, L.,
Wohlmuth, B., Rüde, U., and Bunge, H.-P.: Fast asthenosphere motion in
high-resolution global mantle flow models, Geophys. Res. Lett., 42,
7429–7435, https://doi.org/10.1002/2015GL063727, 2015.
a
Zhong, S.: Analytic solutions for Stokes’ flow with lateral variations in
viscosity, Geophys. J. Int., 124, 18–28,
https://doi.org/10.1111/j.1365-246X.1996.tb06349.x, 1996. a, b
Zhong, S.: Constraints on thermochemical convection of the mantle from plume
heat flux, plume excess temperature, and upper mantle temperature, J.
Geophys. Res., 111, B04409, https://doi.org/10.1029/2005JB003972, 2006. a
Zhong, S., Zuber, M., Moresi, L., and Gurnis, M.: The role of
temperature-dependent viscosity and surface plates in spherical shell models
of mantle convection, J. Geophys. Res., 105, 11063–11082,
https://doi.org/10.1029/2000JB900003, 2000. a
Zhong, S., McNamara, A., Tan, E., Moresi, L., and Gurnis, M.: A benchmark
study on mantle convection in a 3-D spherical shell using CitcomS,
Geochem. Geophy. Geosy., 9, Q10017, https://doi.org/10.1029/2008GC002048, 2008. a
Zienkiewicz, O. and Taylor, R.: The Finite Element Method. Vol. 1: The basis,
Butterworth and Heinemann, ISBN 0-7506-5049-4, 2002. a
Short summary
One of the main numerical methods to solve the mass, momentum, and energy conservation equations in geodynamics is the finite-element method. Four main types of elements have been used in the past decades in hundreds of publications. For the first time we compare results obtained with these four elements on a series of geodynamical benchmarks and applications and draw conclusions as to which are the best ones and which are to be preferably avoided.
One of the main numerical methods to solve the mass, momentum, and energy conservation equations...