Articles | Volume 18, issue 19
https://doi.org/10.5194/gmd-18-6951-2025
© 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-18-6951-2025
© 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
| 
09 Oct 2025
Development and technical paper |
| 09 Oct 2025
| 09 Oct 2025
A bound-constrained formulation for complex solution phase minimization
Nicolas Riel
CORRESPONDING AUTHOR
Institute of Geosciences, Johannes Gutenberg-University, Mainz, Germany
Terrestrial Magmatic Systems (TeMaS) research center, Johannes Gutenberg-University, Mainz, Germany
Boris J. P. Kaus
Institute of Geosciences, Johannes Gutenberg-University, Mainz, Germany
Terrestrial Magmatic Systems (TeMaS) research center, Johannes Gutenberg-University, Mainz, Germany
Mainz Institute of Multiscale Modelling (M³ODEL), Johannes Gutenberg-University, Mainz, Germany
Albert de Montserrat
Department of Earth Sciences, Institut für Geophysik, ETH Zürich, 8092 Zürich, Switzerland
Evangelos Moulas
Institute of Geosciences, Johannes Gutenberg-University, Mainz, Germany
Terrestrial Magmatic Systems (TeMaS) research center, Johannes Gutenberg-University, Mainz, Germany
Mainz Institute of Multiscale Modelling (M³ODEL), Johannes Gutenberg-University, Mainz, Germany
Eleanor C. R. Green
School of Geography, Earth and Atmospheric Sciences, The University of Melbourne, Victoria 3010, Australia
Hugo Dominguez
Institute of Geosciences, Johannes Gutenberg-University, Mainz, Germany
Related authors
Hugo Dominguez, Nicolas Riel, and Pierre Lanari
Geosci. Model Dev., 17, 6105–6122, https://doi.org/10.5194/gmd-17-6105-2024, https://doi.org/10.5194/gmd-17-6105-2024, 2024
Short summary
Short summary
Predicting the behaviour of magmatic systems is important for understanding Earth's matter and heat transport. Numerical modelling is a technique that can predict complex systems at different scales of space and time by solving equations using various techniques. This study tests four algorithms to find the best way to transport the melt composition. The "weighted essentially non-oscillatory" algorithm emerges as the best choice, minimising errors and preserving system mass well.
Anton A. Popov, Nicolas Berlie, and Boris J. P. Kaus
Geosci. Model Dev., 18, 7035–7058, https://doi.org/10.5194/gmd-18-7035-2025, https://doi.org/10.5194/gmd-18-7035-2025, 2025
Short summary
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.
Arne Spang, Marcel Thielmann, Casper Pranger, Albert de Montserrat, and Ludovic Räss
EGUsphere, https://doi.org/10.5194/egusphere-2025-2417, https://doi.org/10.5194/egusphere-2025-2417, 2025
Short summary
Short summary
Concentration of deformation is difficult to capture accurately in computer simulations. We present a number of challenges associated with concentrated viscous deformation and demonstrate strategies to overcome them. These strategies include automatic selection of appropriate time steps to react to rapid changes in model behavior, automatic rescaling to avoid rounding errors, and two methods to prevent model instability. This way, we are able to accurately capture very fast viscous deformation.
Annalena Stroh, Pascal S. Aellig, and Evangelos Moulas
EGUsphere, https://doi.org/10.5194/egusphere-2025-2511, https://doi.org/10.5194/egusphere-2025-2511, 2025
This preprint is open for discussion and under review for Geoscientific Model Development (GMD).
Short summary
Short summary
Crystal growth and diffusion are common processes in geology. Our software MovingBoundaryMinerals.jl calculates compositional profiles in diffusion couples by simulating diffusion-growth processes for geometries with planar/cylindrical/spherical symmetries. Our software has been tested versus various benchmark cases and is provided as an open access software package. This package allows the further use of diffusion/growth phenomena in the calculation of the thermal histories of rocks.
Simon Boisserée, Evangelos Moulas, and Markus Bachmayr
EGUsphere, https://doi.org/10.48550/arXiv.2411.14211, https://doi.org/10.48550/arXiv.2411.14211, 2025
Short summary
Short summary
Understanding porous fluid flow is key for many geology applications. Traditional methods cannot resolve cases with sharp discontinuities in hydraulic/mechanical properties across those layers. Here we present a new space-time method that can handle such discontinuities. This approach is coupled with trace element transport. Our study reveals that the layering of rocks significantly influences the formation of fluid-rich channels and the material distribution adjacent to discontinuities.
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.
Hugo Dominguez, Nicolas Riel, and Pierre Lanari
Geosci. Model Dev., 17, 6105–6122, https://doi.org/10.5194/gmd-17-6105-2024, https://doi.org/10.5194/gmd-17-6105-2024, 2024
Short summary
Short summary
Predicting the behaviour of magmatic systems is important for understanding Earth's matter and heat transport. Numerical modelling is a technique that can predict complex systems at different scales of space and time by solving equations using various techniques. This study tests four algorithms to find the best way to transport the melt composition. The "weighted essentially non-oscillatory" algorithm emerges as the best choice, minimising errors and preserving system mass well.
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.
Lorenzo G. Candioti, Thibault Duretz, Evangelos Moulas, and Stefan M. Schmalholz
Solid Earth, 12, 1749–1775, https://doi.org/10.5194/se-12-1749-2021, https://doi.org/10.5194/se-12-1749-2021, 2021
Short summary
Short summary
We quantify the relative importance of forces driving the dynamics of mountain building using two-dimensional computer simulations of long-term coupled lithosphere–upper-mantle deformation. Buoyancy forces can be as high as shear forces induced by far-field plate motion and should be considered when studying the formation of mountain ranges. The strength of rocks flooring the oceans and the density structure of the crust control deep rock cycling and the topographic elevation of orogens.
Roger Powell, Eleanor C. R. Green, Estephany Marillo Sialer, and Jon Woodhead
Geochronology, 2, 325–342, https://doi.org/10.5194/gchron-2-325-2020, https://doi.org/10.5194/gchron-2-325-2020, 2020
Short summary
Short summary
The standard approach to isochron calculation assumes that the distribution of uncertainties on the data arising from isotopic analysis is strictly Gaussian. This excludes datasets that have more scatter, even though many appear to have age significance. Our new approach requires only that the central part of the uncertainty distribution of the data defines a "spine" in the trend of the data. A robust statistics approach is used to locate the spine, and an implementation in Python is given.
Cited articles
Bouilhol, P., Magni, V., van Hunen, J., and Kaislaniemi, L.: A numerical approach to melting in warm subduction zones, Earth and Planetary Science Letters, 411, 37–44, 2015. a
Candioti, L. G., Nathwani, C. L., and Chelle-Michou, C.: Towards fully-coupled thermodynamic-thermomechanical two-phase flow models of transcrustal magmatic systems, in: EGU General Assembly Conference Abstracts, p. 16936, https://doi.org/10.5194/egusphere-egu25-8644, 2024. a
de Capitani, C. and Brown, T. H.: The computation of chemical equilibrium in complex systems containing non-ideal solutions, Geochimica et Cosmochimica Acta, 51, 2639–2652, https://doi.org/10.1016/0016-7037(87)90145-1, 1987. a, b
Dennis, J. E. and Schnabel, R. B.: Numerical methods for unconstrained optimization and nonlinear equations, SIAM, https://doi.org/10.1137/1.9781611971200, 1996. a
Fletcher, R.: Practical methods of optimization, John and Sons, Chichester, https://doi.org/10.1002/9781118723203, 1987. a
Ganguly, J.: Thermodynamic modelling of solid solutions, EMU Notes in Mineralogy, 3, 37–69, 2001. a
Ghiorso, M. S. and Sack, R. O.: Chemical mass transfer in magmatic processes IV. A revised and internally consistent thermodynamic model for the interpolation and extrapolation of liquid-solid equilibria in magmatic systems at elevated temperatures and pressures, Contributions to Mineralogy and Petrology, 119, 197–212, 1995. a, b
Helgeson, H. C.: Summary and critique of the thermodynamic properties of rock-forming minerals, American Journal of Science, 278, 1–229, 1978. a
Hestenes, M. R. and Stiefel, E.: Methods of conjugate gradients for solving, Journal of research of the National Bureau of Standards, 49, 409–436, https://doi.org/10.6028/jres.049.044, 1952. a
Holland, T. and Powell, R.: Activity–composition relations for phases in petrological calculations: an asymmetric multicomponent formulation, Contributions to Mineralogy and Petrology, 145, 492–501, 2003. a
Holland, T. J. B. and Powell, R.: An internally consistent thermodynamic data set for phases of petrological interest, Journal of Metamorphic Geology, 16, 309–343, https://doi.org/10.1111/j.1525-1314.1998.00140.x, 1998. a
Hu, H., Jackson, M. D., and Blundy, J.: Melting, compaction and reactive flow: Controls on melt fraction and composition change in crustal mush reservoirs, Journal of Petrology, 63, egac097, https://doi.org/10.1093/petrology/egac097, 2022. a
Jackson, M. D., Cheadle, M. J., and Atherton, M. P.: Quantitative modeling of granitic melt generation and segregation in the continental crust, Journal of Geophysical Research: Solid Earth, 108, https://doi.org/10.1029/2001JB001050, 2003. a
Johnson, S. G.: The NLopt nonlinear-optimization package, https://doi.org/10.7283/633E-1497, 2021. a, b
Katz, R. F., Jones, D. W. R., Rudge, J. F., and Keller, T.: Physics of Melt Extraction from the Mantle: Speed and Style, Annual Review of Earth and Planetary Sciences, 50, 507–540, https://doi.org/10.1146/annurev-earth-032320-083704, 2022. a
Keller, T. and Katz, R. F.: The Role of Volatiles in Reactive Melt Transport in the Asthenosphere, Journal of Petrology, 57, 1073–1108, https://doi.org/10.1093/petrology/egw030, 2016. a
Keller, T. and Suckale, J.: A continuum model of multi-phase reactive transport in igneous systems, Geophysical Journal International, 219, 185–222, https://doi.org/10.1093/gji/ggz287, 2019. a
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
Keller, T., Katz, R. F., and Hirschmann, M. M.: Volatiles beneath mid-ocean ridges: Deep melting, channelised transport, focusing, and metasomatism, Earth and Planetary Science Letters, 464, 55–68, 2017. a
Kraft, D.: A software package for sequential quadratic programming, Forschungsbericht- Deutsche Forschungs- und Versuchsanstalt fur Luft- und Raumfahrt, https://primo.uni-due.de/permalink/49HBZ_UDE/1gpd7fl/alma990012356640206446 (last access: 7 October 2025), 1988. a
Kraft, D.: Algorithm 733: TOMP–Fortran modules for optimal control calculations, ACM Transactions on Mathematical Software (TOMS), 20, 262–281, 1994. a
Leal, A. M., Kyas, S., Kulik, D. A., and Saar, M. O.: Accelerating reactive transport modeling: on-demand machine learning algorithm for chemical equilibrium calculations, Transport in Porous Media, 133, 161–204, 2020. a
Magni, V., Bouilhol, P., and Van Hunen, J.: Deep water recycling through time, Geochemistry, Geophysics, Geosystems, 15, 4203–4216, 2014. a
Nocedal, J., Sartenaer, A., and Zhu, C.: On the behavior of the gradient norm in the steepest descent method, Computational Optimization and Applications, 22, 5–35, 2002. a
Pincus, M.: A Monte Carlo method for the approximate solution of certain types of constrained optimization problems, Operations research, 18, 1225–1228, 1970. a
Piro, M. H. A.: Computation of Thermodynamic Equilibria Pertinent to Nuclear Materials in Multi-Physics Codes, PhD thesis, Royal Military College of Canada (Canada), ISBN 9780494822272, 2011. a
Powell, R. and Holland, T.: An internally consistent dataset with uncertainties and correlations: 3. Applications to geobarometry, worked examples and a computer program, Journal of metamorphic Geology, 6, 173–204, 1988. a
Powell, R. and Holland, T.: On the formulation of simple mixing models for complex phases, American Mineralogist, 78, 1174–1180, 1993. a
Riel, N., Bouilhol, P., van Hunen, J., Cornet, J., Magni, V., Grigorova, V., and Velic, M.: Interaction between mantle-derived magma and lower arc crust: quantitative reactive melt flow modelling using STyx, Geological Society, London, Special Publications, 478, 65–87, 2019. a
Riel, N., Kaus, B., de Montserrat, A., Moulas, E., Green, E., and Dominguez, H.: An unconstrained formulation for complex solution phase minimization (Codes) (1.0), Zenodo [code, data set], https://doi.org/10.5281/zenodo.13982544, 2024. a
Rivaie, M., Mamat, M., June, L. W., and Mohd, I.: A new class of nonlinear conjugate gradient coefficients with global convergence properties, Applied Mathematics and Computation, 218, 11323–11332, 2012. a
Rummel, L., Kaus, B. J. P., Baumann, T. S., White, R. W., and Riel, N.: Insights into the Compositional Evolution of Crustal Magmatic Systems from Coupled Petrological-Geodynamical Models, Journal of Petrology, 61, https://doi.org/10.1093/petrology/egaa029, 2020. a, b, c
Sherman, J. and Morrison, W. J.: Adjustment of an inverse matrix corresponding to a change in one element of a given matrix, The Annals of Mathematical Statistics, 21, 124–127, 1950. a
Stixrude, L. and Lithgow-Bertelloni, C.: Thermodynamics of mantle minerals – II. Phase equilibria, Geophysical Journal International, 184, 1180–1213, https://doi.org/10.1111/j.1365-246X.2010.04890.x, 2011. a
Svanberg, K.: A class of globally convergent optimization methods based on conservative convex separable approximations, SIAM Journal on Optimization, 12, 555–573, https://doi.org/10.1137/S1052623499362822, 2002. a, b, c
Taylor-West, J. and Katz, R. F.: Melt-preferred orientation, anisotropic permeability and melt-band formation in a deforming, partially molten aggregate, Geophysical Journal International, 203, 1253–1262, 2015. a
Turner, A. J., Katz, R. F., Behn, M. D., and Keller, T.: Magmatic focusing to mid-ocean ridges: The role of grain-size variability and non-newtonian viscosity, Geochemistry, Geophysics, Geosystems, 18, 4342–4355, 2017. a
Wolfe, P.: Convergence conditions for ascent methods, SIAM review, 11, 226–235, 1969. a
Wong, Y.-Q., and Keller, T.: A unified numerical model for two-phase porous, mush and suspension flow dynamics in magmatic systems, 233, 769–795, https://doi.org/10.1093/gji/ggac481, 2023. a
Yuan, Q., Asimow, P. D., Gurnis, M., and Antoshechkina, P. M.: Enabling large-scale geodynamic-geochemical modeling via neural network acceleration, in: AGU Fall Meeting Abstracts, vol. 2024, V41E–3161, 2024AGUFMV41E.3161Y, 2024. a
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.
Our research focuses on improving the way we predict mineral assemblage. Current methods, while...
Special issue