Investigations into the applicability of adaptive finite element methods to two‐dimensional infinite Prandtl number thermal and thermochemical convection

Davies, David R.; Davies, John H.; Hassan, O.; Morgan, K.; Nithiarasu, Perumal

doi:10.1029/2006gc001470

Cited by 35 publications

(32 citation statements)

References 36 publications

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“…Such techniques (and similar adaptive techniques that are based upon hierarchical mesh refinement) have recently been applied within the mantle dynamics community (e.g. Davies et al, 2007Davies et al, , 2008Stadler et al, 2010;Leng and Zhong, 2011), leading to the development of several state-of-the-art computational frameworks for simulating global mantle convection. The most notable examples are: (i) Fluidity Kramer et al, 2012); (ii) ASPECT (Kronbichler et al, 2012); and (iii) RHEA (Stadler et al, 2010;Burstedde et al, 2013).…”

Section: R Davies Et Al: Geometric Multigrid Refinement Techniqumentioning

confidence: 99%

A hierarchical mesh refinement technique for global 3-D spherical mantle convection modelling

Davies

Bollada

et al. 2013

Geosci. Model Dev.

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Abstract.A method for incorporating multi-resolution capabilities within pre-existing global 3-D spherical mantle convection codes is presented. The method, which we term "geometric multigrid refinement", is based upon the application of a multigrid solver on non-uniform, structured grids and allows for the incorporation of local high-resolution grids within global models. Validation tests demonstrate that the method is accurate and robust, with highly efficient solutions to large-scale non-uniform problems obtained. Significantly, the scheme is conceptually simple and straightforward to implement, negating the need to reformulate and restructure large sections of code. Consequently, although more advanced techniques are under development at the frontiers of mesh refinement and solver technology research, the technique presented is capable of extending the lifetime and applicability of pre-existing global mantle convection codes.

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Section: R Davies Et Al: Geometric Multigrid Refinement Techniqumentioning

confidence: 99%

A hierarchical mesh refinement technique for global 3-D spherical mantle convection modelling

Davies

Bollada

et al. 2013

Geosci. Model Dev.

Self Cite

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“…This results in an unstructured, Lagrangian mantle convection solver capable of efficiently dealing with hundreds of thousands of nodes [cf. Davies et al, 2007].…”

Section: Comparison To Other Softwarementioning

confidence: 99%

MILAMIN: MATLAB‐based finite element method solver for large problems

Dąbrowski

Krotkiewski

Schmid

2008

Geochem Geophys Geosyst

228

145

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[1] The finite element method (FEM) combined with unstructured meshes forms an elegant and versatile approach capable of dealing with the complexities of problems in Earth science. Practical applications often require high-resolution models that necessitate advanced computational strategies. We therefore developed ''Million a Minute'' (MILAMIN), an efficient MATLAB implementation of FEM that is capable of setting up, solving, and postprocessing two-dimensional problems with one million unknowns in one minute on a modern desktop computer. MILAMIN allows the user to achieve numerical resolutions that are necessary to resolve the heterogeneous nature of geological materials. In this paper we provide the technical knowledge required to develop such models without the need to buy a commercial FEM package, programming compiler-language code, or hiring a computer specialist. It has been our special aim that all the components of MILAMIN perform efficiently individually and as a package. While some of the components rely on readily available routines, we develop others from scratch and make sure that all of them work together efficiently. One of the main technical focuses of this paper is the optimization of the global matrix computations. The performance bottlenecks of the standard FEM algorithm are analyzed. An alternative approach is developed that sustains high performance for any system size. Applied optimizations eliminate Basic Linear Algebra Subprograms (BLAS) drawbacks when multiplying small matrices, reduce operation count and memory requirements when dealing with symmetric matrices, and increase data transfer efficiency by maximizing cache reuse. Applying loop interchange allows us to use BLAS on large matrices. In order to avoid unnecessary data transfers between RAM and CPU cache we introduce loop blocking. The optimization techniques are useful in many areas as demonstrated with our MILAMIN applications for thermal and incompressible flow (Stokes) problems. We use these to provide performance comparisons to other open source as well as commercial packages and find that MILAMIN is among the best performing solutions, in terms of both speed and memory usage. The corresponding MATLAB source code for the entire MILAMIN, including input generation, FEM solver, and postprocessing, is available from the authors (http://www.milamin.org) and can be downloaded as auxiliary material.

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“…Although different techniques each have advantages (level set -Samuel and Evonuk, 2010; field tracking - Davies et al, 2007; and marker-net based method - Oldham and Davies, 2004), particle-based methods have proven to be most useful for systems that involve strong mixing, e.g. the Earth's mantle evolving over billions of years.…”

Section: Introductionmentioning

confidence: 99%

Global-scale modelling of melting and isotopic evolution of Earth's mantle: melting modules for TERRA

et al. 2016

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Abstract. Many outstanding problems in solid-Earth science relate to the geodynamical explanation of geochemical observations. Currently, extensive geochemical databases of surface observations exist, but satisfying explanations of underlying mantle processes are lacking. One way to address these problems is through numerical modelling of mantle convection while tracking chemical information throughout the convective mantle.We have implemented a new way to track both bulk compositions and concentrations of trace elements in a finiteelement mantle convection code. Our approach is to track bulk compositions and trace element abundances via particles. One value on each particle represents bulk composition and can be interpreted as the basalt component. In our model, chemical fractionation of bulk composition and trace elements happens at self-consistent, evolving melting zones. Melting is defined via a composition-dependent solidus, such that the amount of melt generated depends on pressure, temperature and bulk composition of each particle. A novel aspect is that we do not move particles that undergo melting; instead we transfer the chemical information carried by the particle to other particles. Molten material is instantaneously transported to the surface layer, thereby increasing the basalt component carried by the particles close to the surface and decreasing the basalt component in the residue.The model is set to explore a number of radiogenic isotopic systems, but as an example here the trace elements we choose to follow are the Pb isotopes and their radioactive parents. For these calculations we will show (1) the evolution of the distribution of bulk compositions over time, showing the buildup of oceanic crust (via melting-induced chemical separation in bulk composition), i.e. a basalt-rich layer at the surface, and the transportation of these chemical heterogeneities through the deep mantle; (2) the amount of melt generated over time; (3) the evolution of the concentrations and abundances of different isotopes of the trace elements (U, Th, K and Pb) throughout the mantle; and (4) a comparison to a semi-analytical theory relating observed arrays of correlated Pb isotope compositions to melting age distributions (Rudge, 2006).

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Investigations into the applicability of adaptive finite element methods to two‐dimensional infinite Prandtl number thermal and thermochemical convection

Cited by 35 publications

References 36 publications

A hierarchical mesh refinement technique for global 3-D spherical mantle convection modelling

A hierarchical mesh refinement technique for global 3-D spherical mantle convection modelling

MILAMIN: MATLAB‐based finite element method solver for large problems

Global-scale modelling of melting and isotopic evolution of Earth's mantle: melting modules for TERRA

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