Towards automatic finite-element methods for  geodynamics via Firedrake

Davies, D. Rhodri; Kramer, Stephan C.; Ghelichkhan, Siavash; Gibson, A. H.

doi:10.5194/gmd-15-5127-2022

Cited by 7 publications

(10 citation statements)

References 141 publications

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“…The PT solving strategy enabled us to analyze the different regularization in a single framework. Nevertheless, the inclusion of spatial regularization techniques may also be easily achieved with other solving strategies that are designed to automatically discretize and solve coupled non‐linear systems of equations (Davies et al., 2022; Wilson et al., 2017). Alternatively, it may also be possible to include effects of spatial regularization by decoupling the equations (operator split) and using staggered schemes such as usually applied in damage modeling (Miehe et al., 2010).…”

Section: Implications For Geodynamic Simulation Toolsmentioning

confidence: 99%

A Comparison of Plasticity Regularization Approaches for Geodynamic Modeling

Duretz

Räss

Borst

et al. 2023

Geochem Geophys Geosyst

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Strain localization is ubiquitous and controls the development of shear zones and the establishment of plate boundaries. The outermost portion of the lithosphere, the crust, is relatively cold. In these regions, deformation is governed by elasto-plasticity, which allows tectonic plates to behave both as rigid solid and, locally, as a deformable solid. The rocks that make up the crust exhibit a frictional behavior, which causes their strength to depend on pressure (Byerlee, 1978). Moreover, the dilatancy angles of rocks are generally smaller than friction angles (Zhao & Cai, 2010), therefore frictional plasticity models should be non-associated. Geodynamic models are typically employed to simulate the deformations of the Earth's upper shell, such as the opening of rifts zone (e.g., Naliboff et al., 2017), the growth of orogenic wedges (e.g., Buiter et al., 2016), or the formation of transform boundaries (e.g., Gerya, 2013) heavily rely on the simulation of frictional deformation processes. Nonetheless, the inclusion of frictional plasticity in numerical models poses problems as standard model implementations may not achieve force equilibrium and may exhibit mesh dependence. The absence of characteristic spatial or temporal scales in standard frictional constitutive laws is the primary cause for these problems. To overcome these issues, models should be augmented by including additional physics (e.g., fluid pressure diffusion, viscoplastic dissipation) (e.g., Brantut et al., 2017), which from a mathematical point of view can be considered as regularization techniques (e.g., de Borst et al., 1993).Numerous regularization strategies have been proposed, which have the effect of introducing a length or a time scale in the model. Viscoplastic regularization relies on the inclusion of a time scale W. M. Wang et al., 1997) which implicitly rather than directly introduces a length scale like in (e.g., H. Wang, 2019). The yield function does not depend on the gradients of plastic strain: it is defined locally. As a consequence, return mapping can be achieved without solving an additional partial differential equation. Thus, the total number of degrees of freedom is not increased. The implementation of viscoplastic regularization in existing codes is

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Section: Implications For Geodynamic Simulation Toolsmentioning

confidence: 99%

A Comparison of Plasticity Regularization Approaches for Geodynamic Modeling

Duretz

Räss

Borst

et al. 2023

Geochem Geophys Geosyst

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“…Two prominent examples of this approach are Firedrake (www.firedrakeproject.org) and FEniCS (www.fenicsproject.org). Examples of the use of these two approaches in geodynamical applications are in Davies et al (2022) and Vynnytska et al (2013).…”

Section: Practical Approaches Software Availability and Comparisonmentioning

confidence: 99%

An introductory review of the thermal structure of subduction zones: II—numerical approach and validation

Wilson,

van Keken

2023

Prog Earth Planet Sci

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The thermal structure of subduction zones is fundamental to our understanding of the physical and chemical processes that occur at active convergent plate margins. These include magma generation and related arc volcanism, shallow and deep seismicity, and metamorphic reactions that can release fluids. Computational models can predict the thermal structure to great numerical precision when models are fully described but this does not guarantee accuracy or applicability. In a trio of companion papers, the construction of thermal subduction zone models, their use in subduction zone studies, and their link to geophysical and geochemical observations are explored. In this part II, the finite element techniques that can be used to predict thermal structure are discussed in an introductory fashion along with their verification and validation.Steady-state thermal structure for the updated subduction zone benchmark. a) Temperature predicted by TF for case 1; b) temperature difference between TF and Sepran using the penalty function (PF) method for case 1 at fm=1 where fm represents the smallest element sizes in the finite element grids near the coupling point; c) slab top temperature comparison for case 1; and d)–f) as a)–c) but now for case 2. The star indicates the position or temperature conditions at the coupling point.

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“…Key among these features is support for a variety of Finite Element discretisations, including a highly efficient implementation of discretisations based on extruded meshes, programmable non-linear solvers and operator-aware solver preconditioners that can be combined in a flexible manner to create linear or non-linear systems, which are solved by PETSc (e.g., Balay et al, 1997;Dalcin et al, 2011;Balay et al, 2023). The suitability of Firedrake for geodynamics has been demonstrated via comparison with a comprehensive set of analytical solutions and community benchmarks in Davies et al (2022). We refer the reader to this study, alongside Rathgeber et al (2016), for a more in-depth discussion on Firedrake and its dependencies, alongside an outline of the solution strategy employed herein.…”

Section: Solution Strategy: Leveraging Firedrake Through G-adoptmentioning

confidence: 99%

“…As explained in more detail in Davies et al (2022), the forward geodynamical model can be described as a series of linear and nonlinear solves. Although we solve for temperature using Q 2 elements, we choose the control T IC to be in the Q 1 function space as a means to regularise the inversion problem.…”

Section: Discrete Forward Modelmentioning

confidence: 99%

“…In this paper, we introduce the Geoscientific Adjoint Optimisation Platform (G-ADOPT), research software infrastructure that allows us to overcome these limitations. G-ADOPT is built around three state-of-the-art software libraries -(i) Firedrake, an automated system for solving a range of partial differential equations using the finite element method (Rathgeber et al, 2016), recently validated for geodynamics (Davies et al, 2022); (ii) Dolfin-Adjoint, which automatically derives the discrete adjoint equations in a form compatible with Firedrake (Farrell et al, 2013a;Mitusch et al, 2019); and (iii) the Rapid Optimisation Library (ROL), a Trillions package for performing highly-efficient large-scale optimisation (The ROL Project Team, 2022).…”

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confidence: 99%

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Automatic adjoint-based inversion schemes for geodynamics: Reconstructing the evolution of Earth’s mantle in space and time

Ghelichkhan,

Gibson,

Davies

et al. 2023

Preprint

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Abstract. Reconstructing the thermo-chemical evolution of Earth's mantle and its diverse surface manifestations is a widely-recognised grand challenge for the geosciences. It requires the creation of a digital twin: a digital representation of Earth's mantle across space and time that is compatible with available observational constraints on the mantle's structure, dynamics and evolution. This has led geodynamicists to explore adjoint-based approaches that reformulate mantle convection modelling as an inverse problem, in which unknown model parameters can be optimised to fit available observational data. Whilst recent years have seen a notable increase in the use of adjoint-based methods in geodynamics, the theoretical and practical challenges of deriving, implementing and validating adjoint systems for large-scale, non-linear, time-dependent problems, such as global mantle flow, has hindered their broader use. Here, we present the Geoscientific Adjoint Optimisation Platform (G-ADOPT), an advanced computational modelling framework that overcomes these challenges for coupled, non-linear, time-dependent systems. By integrating three main components: (i) Firedrake, an automated system for the solution of partial differential equations using the finite element method; (ii) Dolfin-Adjoint, which automatically generates discrete adjoint models in a form compatible with Firedrake; and (iii) the Rapid Optimisation Library, ROL, an efficient large-scale optimisation toolkit; G-ADOPT enables the application of adjoint methods across geophysical continua, showcased herein for geodynamics. Through two sets of synthetic experiments, we demonstrate application of this framework to the initial condition problem of mantle convection, in both square and annular geometries, for both isoviscous and non-linear rheologies. We confirm the validity of the gradient computations underpinning the adjoint approach, for all cases, through second-order Taylor remainder convergence tests, and subsequently demonstrate excellent recovery of the unknown initial conditions. Moreover, we show that the framework achieves theoretical computational efficiency. Taken together, this confirms the suitability of G-ADOPT for reconstructing the evolution of Earth's mantle in space and time. The framework overcomes the significant theoretical and practical challenges of generating adjoint models, and will allow the community to move from idealised forward models to data-driven simulations that rigorously account for observational constraints and their uncertainties using an inverse approach.

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Towards automatic finite-element methods for geodynamics via Firedrake

Cited by 7 publications

References 141 publications

A Comparison of Plasticity Regularization Approaches for Geodynamic Modeling

A Comparison of Plasticity Regularization Approaches for Geodynamic Modeling

An introductory review of the thermal structure of subduction zones: II—numerical approach and validation

Automatic adjoint-based inversion schemes for geodynamics: Reconstructing the evolution of Earth’s mantle in space and time

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