Articles | Volume 18, issue 19
https://doi.org/10.5194/gmd-18-7035-2025
© Author(s) 2025. This work is distributed under
the Creative Commons Attribution 4.0 License.
the Creative Commons Attribution 4.0 License.
Special issue:
https://doi.org/10.5194/gmd-18-7035-2025
© Author(s) 2025. This work is distributed under
the Creative Commons Attribution 4.0 License.
the Creative Commons Attribution 4.0 License.
Development and technical paper
| Highlight paper
| 
10 Oct 2025
Development and technical paper | Highlight paper |
| 10 Oct 2025
| 10 Oct 2025
A dilatant visco-elasto-viscoplasticity model with globally continuous tensile cap: stable two-field mixed formulation
Institute of Geosciences, Johannes Gutenberg University, Mainz, Germany
Nicolas Berlie
Institute of Geosciences, Johannes Gutenberg University, Mainz, Germany
Boris J. P. Kaus
Institute of Geosciences, Johannes Gutenberg University, Mainz, Germany
Related authors
Maximilian O. Kottwitz, Anton A. Popov, Steffen Abe, and Boris J. P. Kaus
Solid Earth, 12, 2235–2254, https://doi.org/10.5194/se-12-2235-2021, https://doi.org/10.5194/se-12-2235-2021, 2021
Short summary
Short summary
Upscaling fluid flow in fractured reservoirs is an important practice in subsurface resource utilization. In this study, we first conduct numerical simulations of direct fluid flow at locations where fractures intersect to analyze the arising hydraulic complexities. Next, we develop a model that integrates these effects into larger-scale continuum models of fracture networks to investigate their impact on the upscaling. For intensively fractured systems, these effects become important.
Nicolas Riel, Boris J. P. Kaus, Albert de Montserrat, Evangelos Moulas, Eleanor C. R. Green, and Hugo Dominguez
Geosci. Model Dev., 18, 6951–6962, https://doi.org/10.5194/gmd-18-6951-2025, https://doi.org/10.5194/gmd-18-6951-2025, 2025
Short summary
Short summary
Our research focuses on improving the way we predict mineral assemblage. Current methods, while accurate, are slowed by complex calculations. We developed a new approach that simplifies these calculations and speeds them up significantly using a technique called the Broyden-Fletcher-Goldfarb-Shanno (BFGS) algorithm. This breakthrough reduces computation time by more than five times, potentially unlocking new horizons in modeling reactive magmatic systems.
Maximilian O. Kottwitz, Anton A. Popov, Steffen Abe, and Boris J. P. Kaus
Solid Earth, 12, 2235–2254, https://doi.org/10.5194/se-12-2235-2021, https://doi.org/10.5194/se-12-2235-2021, 2021
Short summary
Short summary
Upscaling fluid flow in fractured reservoirs is an important practice in subsurface resource utilization. In this study, we first conduct numerical simulations of direct fluid flow at locations where fractures intersect to analyze the arising hydraulic complexities. Next, we develop a model that integrates these effects into larger-scale continuum models of fracture networks to investigate their impact on the upscaling. For intensively fractured systems, these effects become important.
Cited articles
Abbo, A. and Sloan, S.: A smooth hyperbolic approximation to the Mohr-Coulomb yield criterion, Computers & Structures, 54, 427–441, https://doi.org/10.1016/0045-7949(94)00339-5, 1995. a
Ahrens, J. P., Geveci, B., and Law, C. C.: ParaView: An End-User Tool for Large-Data Visualization, in: The Visualization Handbook, edited by: Hansen, C. D. and Johnson, C. R., Elsevier Inc., Burlington, MA, USA, 717–731, https://doi.org/10.1016/B978-012387582-2/50038-1, 2005. a
Armijo, L.: Minimization of functions having Lipschitz continuous first partial derivatives, Pacific Journal of Mathematics, 16, 1–3, https://doi.org/10.2140/pjm.1966.16.1, 1966. a
Arthur, J. R., Dunstan, T., Al-Ani, Q. A., and Assadi, A.: Plastic Deformation and Failure in Granular Media, Geotechnique, 28, 125–128, https://doi.org/10.1680/geot.1978.28.1.125, 1978. a
Benedetti, L., Cervera, M., and Chiumenti, M.: Stress-accurate Mixed FEM for soil failure under shallow foundations involving strain localization in plasticity, Computers and Geotechnics, 64, https://doi.org/10.1016/j.compgeo.2014.10.004, 2014. a
Berlie, N.: Numerical modeling of magma ascent through the lithosphere, PhD thesis, Johannes Gutenberg University, Mainz, https://doi.org/10.25358/openscience-9671, 2023. a
Biot, M. A.: General Theory of Three‐Dimensional Consolidation, Journal of Applied Physics, 12, 155–164, https://doi.org/10.1063/1.1712886, 1941. a
Biot, M. A. and Willis, D. G.: The Elastic Coefficients of the Theory of Consolidation, Journal of Applied Mechanics, 24, 594–601, https://doi.org/10.1115/1.4011606, 1957. a
Carol, I., Prat, P. C., and López, C. M.: Normal/Shear Cracking Model: Application to Discrete Crack Analysis, Journal of Engineering Mechanics, 123, 765–773, https://doi.org/10.1061/(ASCE)0733-9399(1997)123:8(765), 1997. a
Cioncolini, A. and Boffi, D.: The MINI mixed finite element for the Stokes problem: An experimental investigation, Computers & Mathematics with Applications, 77, 2432–2446, https://doi.org/10.1016/j.camwa.2018.12.028, 2019. a
Commend, S.: Stabilized finite elements in geomechanics, PhD thesis, EPFL, Lausanne, https://doi.org/10.5075/epfl-thesis-2391, 2001. a
Commend, S., Truty, A., and Zimmermann, T.: Stabilized finite elements applied to elastoplasticity: I. Mixed displacement – pressure formulation, Computer Methods in Applied Mechanics and Engineering, 193, 3559–3586, https://doi.org/10.1016/j.cma.2004.01.007, 2004. a, b
Crameri, F., Shephard, G. E., and Heron, P. J.: The misuse of colour in science communication, Nature Communications, 11, 5444, https://doi.org/10.1038/s41467-020-19160-7, 2020. a
Crouzeix, M. and Raviart, P.-A.: Conforming and nonconforming finite element methods for solving the stationary Stokes equations I, Revue française d'automatique, informatique, recherche opérationnelle, Analyse numérique, 7, 33–75, https://doi.org/10.1051/m2an/197307R300331, 1973. a, b, c
Dabrowski, M., Krotkiewski, M., and Schmid, D. W.: MILAMIN: MATLAB-based finite element method solver for large problems, Geochemistry, Geophysics, Geosystems, 9, https://doi.org/10.1029/2007GC001719, 2008. a, b, c
de Borst, R. and Duretz, T.: On viscoplastic regularisation of strain‐softening rocks and soils, International Journal for Numerical and Analytical Methods in Geomechanics, I, 3046, https://doi.org/10.1002/nag.3046, 2020. a, b
de Borst, R. and Groen, A. E.: Some observations on element performance in isochoric and dilatant plastic flow, International Journal for Numerical Methods in Engineering, 38, 2887–2906, https://doi.org/10.1002/nme.1620381704, 1995. a
de Borst, R. and Mühlhaus, H.-B.: Gradient-dependent plasticity: Formulation and algorithmic aspects, International Journal for Numerical Methods in Engineering, 35, 521–539, https://doi.org/10.1002/nme.1620350307, 1992. a
de Borst, R., Crisfield, M. A., Remmers, J. J. C., and Verhoosel, C. V.: Non‐Linear Finite Element Analysis of Solids and Structures, Chap. 7, John Wiley & Sons, Ltd, 219–280, ISBN 9781118375938, https://doi.org/10.1002/9781118375938.ch7, 2012. a
Deubelbeiss, Y. and Kaus, B. J. P.: Comparison of Eulerian and Lagrangian numerical techniques for the Stokes equations in the presence of strongly varying viscosity, Physics of the Earth and Planetary Interiors, 171, 92–111, https://doi.org/10.1016/j.pepi.2008.06.023, 2008. a, b
Dohrmann, C. R. and Bochev, P. B.: A stabilized finite element method for the Stokes problem based on polynomial pressure projections, International Journal for Numerical Methods in Fluids, 46, 183–201, https://doi.org/10.1002/fld.752, 2004. a
Dolarevic, S. and Ibrahimbegovic, A.: A modified three-surface elasto-plastic cap model and its numerical implementation, Comput. Struct., 85, 419–430, https://doi.org/10.1016/j.compstruc.2006.10.001, 2007. a, b
Dunavant, D. A.: High degree efficient symmetrical Gaussian quadrature rules for the triangle, International Journal for Numerical Methods in Engineering, 21, 1129–1148, https://doi.org/10.1002/nme.1620210612, 1985. a
Duretz, T., Räss, L., de Borst, R., and Hageman, T.: A Comparison of Plasticity Regularization Approaches for Geodynamic Modeling, Geochemistry, Geophysics, Geosystems, 24, https://doi.org/10.1029/2022GC010675, 2023. a, b, c
Duvaut, G. and Lions, J. L.: Inéquations en thermoélasticité et magnétohydrodynamique, Archive for Rational Mechanics and Analysis, 46, 241–279, https://doi.org/10.1007/BF00250512, 1972. a
Eymard, R., Gallouët, T., Herbin, R., and Latché, J.-C.: Convergence of the MAC Scheme for the Compressible Stokes Equations, SIAM Journal on Numerical Analysis, 48, 2218–2246, https://doi.org/10.1137/090779863, 2010. a
Feenstra, P. H. and de Borst, R.: A composite plasticity model for concrete, International Journal of Solids and Structures, 33, 707–730, https://doi.org/10.1016/0020-7683(95)00060-N, 1996. a
Gatica, G.: A Simple Introduction to the Mixed Finite Element Method: Theory and Applications, SpringerBriefs in Mathematics, Springer International Publishing, ISBN 9783319036953, 2014. a
Gerya, T.: Introduction to Numerical Geodynamic Modelling, Cambridge University Press, 2 edn., https://doi.org/10.1017/9781316534243, 2019. a
Gerya, T. V. and Yuen, D. A.: Robust characteristics method for modelling multiphase visco-elasto-plastic thermo-mechanical problems, Physics of the Earth and Planetary Interiors, 163, 83–105, https://doi.org/10.1016/j.pepi.2007.04.015, 2007. a, b
Gerya, T. V., May, D. A., and Duretz, T.: An adaptive staggered grid finite difference method for modeling geodynamic Stokes flows with strongly variable viscosity., Geochemistry, Geophysics, Geosystems, 14, 1200–1225, https://doi.org/10.1002/ggge.20078, 2013. a
Glerum, A., Thieulot, C., Fraters, M., Blom, C., and Spakman, W.: Nonlinear viscoplasticity in ASPECT: benchmarking and applications to subduction, Solid Earth, 9, 267–294, https://doi.org/10.5194/se-9-267-2018, 2018. a
Golchin, A., Vardon, P. J., Coombs, W. M., and Hicks, M. A.: A flexible and robust yield function for geomaterials, Computer Methods in Applied Mechanics and Engineering, 387, 114162, https://doi.org/10.1016/j.cma.2021.114162, 2021. a
Harlow, F. H. and Welch, J. E.: Numerical Calculation of Time‐Dependent Viscous Incompressible Flow of Fluid with Free Surface, The Physics of Fluids, 8, 2182–2189, https://doi.org/10.1063/1.1761178, 1965. a
Harris, C. R., Millman, K. J., van der Walt, S. J., Gommers, R., Virtanen, P., Cournapeau, D., Wieser, E., Taylor, J., Berg, S., Smith, N. J., Kern, R., Picus, M., Hoyer, S., van Kerkwijk, M. H., Brett, M., Haldane, A., Fernández del Río, J., Wiebe, M., Peterson, P., Gérard-Marchant, P., Sheppard, K., Reddy, T., Weckesser, W., Abbasi, H., Gohlke, C., and Oliphant, T. E.: Array programming with NumPy, Nature, 585, 357–362, https://doi.org/10.1038/s41586-020-2649-2, 2020. a
Heeres, O. M., Suiker, A. S., and de Borst, R.: A comparison between the Perzyna viscoplastic model and the consistency viscoplastic model, European Journal of Mechanics, A/Solids, 21, 1–12, https://doi.org/10.1016/S0997-7538(01)01188-3, 2002. a
Herrera, P.: PyEVTK, GitHub [code], https://github.com/paulo-herrera/PyEVTK (last access: 25 September 2025), 2021. a
Jacquey, A. B. and Cacace, M.: Multiphysics Modeling of a Brittle-Ductile Lithosphere: 1. Explicit Visco-Elasto-Plastic Formulation and Its Numerical Implementation, Journal of Geophysical Research: Solid Earth, 125, https://doi.org/10.1029/2019JB018474, 2020a. a, b, c
Jacquey, A. B. and Cacace, M.: Multiphysics Modeling of a Brittle-Ductile Lithosphere: 2. Semi-brittle, Semi-ductile Deformation and Damage Rheology, Journal of Geophysical Research: Solid Earth, 125, https://doi.org/10.1029/2019JB018475, 2020b. a, b, c
Jacquey, A. B., Rattez, H., and Veveakis, M.: Strain localization regularization and patterns formation in rate-dependent plastic materials with multiphysics coupling, Journal of the Mechanics and Physics of Solids, 152, 104422, https://doi.org/10.1016/j.jmps.2021.104422, 2021. a
Jiang, H. and Xie, Y.: A note on the Mohr-Coulomb and Drucker-Prager strength criteria, Mechanics Research Communications, 38, 309–314, https://doi.org/10.1016/j.mechrescom.2011.04.001, 2011. a
Kaus, B. J. P.: Factors that control the angle of shear bands in geodynamic numerical models of brittle deformation, Tectonophysics, 484, 36–47, https://doi.org/10.1016/j.tecto.2009.08.042, 2010. a, b, c, d
Keller, T., May, D. A., and Kaus, B. J. P.: Numerical modelling of magma dynamics coupled to tectonic deformation of lithosphere and crust, Geophysical Journal International, 195, 1406–1442, https://doi.org/10.1093/gji/ggt306, 2013. a, b
Kiss, D., Moulas, E., Kaus, B. J. P., and Spang, A.: Decompression and Fracturing Caused by Magmatically Induced Thermal Stresses, Journal of Geophysical Research: Solid Earth, 128, 1–25, https://doi.org/10.1029/2022JB025341, 2023. a, b
Kloeckner, A., Brun, L., Liu, B., Klemenc, S., Fkikl, A., Gohlke, C., Coon, E., Oxberry, G., Veselý, J., Wala, M., Smith, M., Potrowl, P., and Kurtz, A.: MeshPy, Zenodo [code], https://doi.org/10.5281/zenodo.7296572, 2022. a
Kronbichler, M., Heister, T., and Bangerth, W.: High Accuracy Mantle Convection Simulation through Modern Numerical Methods, Geophysical Journal International, 191, 12–29, https://doi.org/10.1111/j.1365-246X.2012.05609.x, 2012. a
Lee, N.-S. and Bathe, K.-J.: Effects of element distortions on the performance of isoparametric elements, International Journal for Numerical Methods in Engineering, 36, 3553–3576, https://doi.org/10.1002/nme.1620362009, 1993. a
Li, Y., Pusok, A. E., Davis, T., May, D. A., and Katz, R. F.: Continuum approximation of dyking with a theory for poro-viscoelastic–viscoplastic deformation, Geophysical Journal International, 234, 2007–2031, https://doi.org/10.1093/gji/ggad173, 2023. a, b, c
May, D. A., Brown, J., and Le Pourhiet, L.: pTatin3D: High-performance Methods for Long-term Lithospheric Dynamics, in: Proceedings of the International Conference for High Performance Computing, Networking, Storage and Analysis, SC '14, IEEE Press, Piscataway, NJ, USA, 274–284, ISBN 978-1-4799-5500-8, https://doi.org/10.1109/SC.2014.28, 2014. a, b
Motamedi, M. H. and Foster, C. D.: An improved implicit numerical integration of a non-associated, three-invariant cap plasticity model with mixed isotropic–kinematic hardening for geomaterials, International Journal for Numerical and Analytical Methods in Geomechanics, 39, 1853–1883, https://doi.org/10.1002/nag.2372, 2015. a
Pastor, M., Quecedo, M., and Zienkiewicz, O. C.: A mixed displacement-pressure formulation for numerical analysis of plastic failure, Computers & Structures, 62, 13–23, https://doi.org/10.1016/S0045-7949(96)00208-8, 1997. a
Pelletier, D., Fortin, A., and Camarero, R.: Are fem solutions of incompressible flows really incompressible? (or how simple flows can cause headaches!), International Journal for Numerical Methods in Fluids, 9, 99–112, https://doi.org/10.1002/fld.1650090108, 1989. a
Perzyna, P.: Fundamental Problems in Viscoplasticity, Advances in Applied Mechanics, 9, 243–377, https://doi.org/10.1016/S0065-2156(08)70009-7, 1966. a, b
Popov, A. A.: Globally continuous tensile cap yield function plot, Zenodo [code], https://doi.org/10.5281/zenodo.16877978, 2025. a
Popov, A. A. and Kaus, B. J. P.: GeoTech2D: Initial release, v1.0.0, Zenodo [code], https://doi.org/10.5281/zenodo.15496843, 2025. a, b
Popov, A. A. and Sobolev, S. V.: SLIM3D: A tool for three-dimensional thermomechanical modeling of lithospheric deformation with elasto-visco-plastic rheology, Physics of the Earth and Planetary Interiors, 171, 55–75, https://doi.org/10.1016/j.pepi.2008.03.007, 2008. a
Räss, L., Duretz, T., Podladchikov, Y. Y., and Schmalholz, S. M.: M2Di: Concise and efficient MATLAB 2-D Stokes solvers using the Finite Difference Method, Geochemistry, Geophysics, Geosystems, 18, 755–768, https://doi.org/10.1002/2016GC006727, 2017. a
Read, H. and Hegemier, G.: Strain softening of rock, soil and concrete – a review article, Mechanics of Materials, 3, 271–294, https://doi.org/10.1016/0167-6636(84)90028-0, 1984. a, b
Rivalta, E., Taisne, B., Bunger, A. P., and Katz, R. F.: A review of mechanical models of dike propagation: Schools of thought, results and future directions, Tectonophysics, 638, 1–42, https://doi.org/10.1016/j.tecto.2014.10.003, 2015. a
Rozhko, A. Y., Podladchikov, Y. Y., and Renard, F.: Failure patterns caused by localized rise in pore-fluid overpressure and effective strength of rocks, Geophysical Research Letters, 34, 1–5, https://doi.org/10.1029/2007GL031696, 2007. a
Rubinstein, R. and Atluri, S.: Objectivity of incremental constitutive relations over finite time steps in computational finite deformation analyses, Computer Methods in Applied Mechanics and Engineering, 36, 277–290, https://doi.org/10.1016/0045-7825(83)90125-1, 1983. a
Sandler, I. S. and Rubin, D.: An algorithm and a modular subroutine for the CAP model, International Journal for Numerical and Analytical Methods in Geomechanics, 3, 173–186, https://doi.org/10.1002/nag.1610030206, 1979. a
Schmalholz, S., Kaus, B. J. P., and Burg, J.-P.: Stress-strength relationship in the lithosphere during continental collision, Geology, 37, 775–778, https://doi.org/10.1130/G25678A.1, 2009. a
Schroeder, W., Martin, K., and Lorensen, B.: The Visualization Toolkit, 4th edn., Kitware, ISBN 978-1-930934-19-1, 2006. a
Shewchuk, J. R.: Triangle: Engineering a 2D Quality Mesh Generator and Delaunay Triangulator, in: Applied Computational Geometry: Towards Geometric Engineering, edited by: Lin, M. C. and Manocha, D., Vol. 1148, Lecture Notes in Computer Science, Springer-Verlag, 203–222, https://doi.org/10.1007/BFb0014497, 1996. a
Shin, D. and Strikwerda, J. C.: Inf-sup conditions for finite-difference approximations of the stokes equations, The Journal of the Australian Mathematical Society. Series B. Applied Mathematics, 39, 121–134, https://doi.org/10.1017/S0334270000009255, 1997. a
Simo, J. C.: Strain Softening and Dissipation: A Unification of Approaches, in: Cracking and Damage: Strain Localization and Size Effects, edited by Bazant, Z. and Mazars, J., CRC Press, 440–461, ISBN 978-0-4290-8051-7, https://doi.org/10.1201/9781482296525, 1989. a
Simo, J. C., Kennedy, J. G., and Govindjee, S.: Non-smooth multisurface plasticity and viscoplasticity. Loading/unloading conditions and numerical algorithms, International Journal for Numerical Methods in Engineering, 26, 2161–2185, https://doi.org/10.1002/nme.1620261003, 1988. a
Spiegelman, M., May, D. A., and Wilson, C. R.: On the solvability of incompressible Stokes with viscoplastic rheologies in geodynamic, Geochemistry Geophysics Geosystems, 17, 2213–2238, https://doi.org/10.1002/2015GC006228, 2016. a, b
Stathas, A. and Stefanou, I.: The role of viscous regularization in dynamical problems, strain localization and mesh dependency, Computer Methods in Applied Mechanics and Engineering, 388, 114185, https://doi.org/10.1016/j.cma.2021.114185, 2022. a
Stupkiewicz, S., Denzer, R., Piccolroaz, A., and Bigoni, D.: Implicit yield function formulation for granular and rock-like materials, Computational Mechanics, 54, 1163–1173, https://doi.org/10.1007/s00466-014-1047-8, 2014. a, b
Tackley, P. J.: Modelling compressible mantle convection with large viscosity contrasts in a three-dimensional spherical shell using the yin-yang grid, Physics of the Earth and Planetary Interiors, 171, 7–18, https://doi.org/10.1016/j.pepi.2008.08.005, 2008. a
Terzaghi, K.: Erdbaumechanik auf bodenphysikalischer grundlage, F. Deuticke, 1925. a
Thielmann, M., Kaus, B. J. P., and Popov, A. A.: Lithospheric stresses in Rayleigh-Bénard convection: Effects of a free surface and a viscoelastic Maxwell rheology, Geophysical Journal International, 203, 2200–2219, https://doi.org/10.1093/gji/ggv436, 2015. a, b, c
Thieulot, C. and Bangerth, W.: On the choice of finite element for applications in geodynamics, Solid Earth, 13, 229–249, https://doi.org/10.5194/se-13-229-2022, 2022. a, b
Virtanen, P., Gommers, R., Oliphant, T. E., Haberland, M., Reddy, T., Cournapeau, D., Burovski, E., Peterson, P., Weckesser, W., Bright, J., van der Walt, S. J., Brett, M., Wilson, J., Millman, K. J., Mayorov, N., Nelson, A. R. J., Jones, E., Kern, R., Larson, E., Carey, C. J., Polat, İ., Feng, Y., Moore, E. W., VanderPlas, J., Laxalde, D., Perktold, J., Cimrman, R., Henriksen, I., Quintero, E. A., Harris, C. R., Archibald, A. M., Ribeiro, A. H., Pedregosa, F., van Mulbregt, P., and SciPy 1.0 Contributors: SciPy 1.0: Fundamental Algorithms for Scientific Computing in Python, Nature Methods, 17, 261–272, https://doi.org/10.1038/s41592-019-0686-2, 2020. a
Wang, W. M., Sluys, L. J., and de Borst, R.: Viscoplasticity for instabilities due to strain softening and strain-rate softening, International Journal for Numerical Methods in Engineering, 40, 3839–3864, https://doi.org/10.1002/(SICI)1097-0207(19971030)40:20<3839::AID-NME245>3.0.CO;2-6, 1997. a
Zienkiewicz, O. and Cormeau, I.: Visco-plasticity solution by finite element process, Archives of Mechanics, 24, 873–889, 1972. a
Zienkiewicz, O. C. and Cormeau, I. C.: Visco-plasticity-plasticity and creep in elastic solids – a unified numerical solution approach, International Journal for Numerical Methods in Engineering, 8, 821–845, https://doi.org/10.1002/nme.1620080411, 1974. a
Executive editor
The paper by Popov et al. provides a comprehensive insight into overcoming challenges related to modelling the brittle-ductile transition in materials. This study offers a detailed description and bridges to similar concepts used beyond the geosciences, such as in engineering.
The paper by Popov et al. provides a comprehensive insight into overcoming challenges related to...
Short summary
We present a simple plasticity model that can be used for robust modeling of strain localization in both shear and tensile failure regimes. The new model overcomes the difficulty related to combining these regimes and enables for particularly simple and reliable numerical implementation, which delivers regularized solutions that are insensitive to mesh resolution. We describe algorithmic details and demonstrate the applications to a number of relevant strain localization problems.
We present a simple plasticity model that can be used for robust modeling of strain localization...
Special issue