Investigations into the applicability of adaptive finite element methods to two-dimensional infinite...

Investigations into the applicability of adaptive finite elementmethods to two-dimensional infinite Prandtl number thermaland thermochemical convection

D. R. DaviesSchool of Earth, Ocean and Planetary Sciences, Cardiff University, Park Place, Cardiff CF10 3YE, Wales, UK([email protected])

Civil and Computational Engineering Center, School of Engineering, Swansea University, Singleton Park, SwanseaSA2 8PP,Wales, UK ([email protected])

J. H. DaviesSchool of Earth, Ocean and Planetary Sciences, Cardiff University, Park Place, Cardiff CF10 3YE, Wales, UK([email protected])

O. Hassan, K. Morgan, and P. NithiarasuCivil and Computational Engineering Center, School of Engineering, Swansea University, Singleton Park, SwanseaSA2 8PP, Wales, UK ([email protected]; [email protected]; [email protected])

[1] An adaptive finite element procedure is presented for improving the quality of solutions to convection-dominated problems in geodynamics. The method adapts the mesh automatically around regions of highsolution gradient, yielding enhanced resolution of the associated flow features. The approach requires thecoupling of an automatic mesh generator, a finite element flow solver, and an error estimator. In this study,the procedure is implemented in conjunction with the well-known geodynamical finite element codeConMan. An unstructured quadrilateral mesh generator is utilized, with mesh adaptation accomplishedthrough regeneration. This regeneration employs information provided by an interpolation-based localerror estimator, obtained from the computed solution on an existing mesh. The technique is validated bysolving thermal and thermochemical problems with well-established benchmark solutions. In a purelythermal context, results illustrate that the method is highly successful, improving solution accuracy whileincreasing computational efficiency. For thermochemical simulations the same conclusions can be drawn.However, results also demonstrate that the grid-based methods employed for simulating the compositionalfield are not competitive with the other methods (tracer particle and marker chain) currently employed inthis field, even at the higher spatial resolutions allowed by the adaptive grid strategies.

Components: 10,911 words, 17 figures, 11 tables.

Keywords: convection; adaptivity; finite element methods; geodynamics; error estimation; mantle.

Index Terms: 0545 Computational Geophysics: Modeling (4255); 0560 Computational Geophysics: Numerical solutions

(4255); 1213 Geodesy and Gravity: Earth’s interior: dynamics (1507, 7207, 7208, 8115, 8120).

Received 5 September 2006; Revised 17 January 2007; Accepted 31 January 2007; Published 10 May 2007.

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AN ELECTRONIC JOURNAL OF THE EARTH SCIENCES

GeochemistryGeophysics

Geosystems

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Volume 8, Number 510 May 2007

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Davies, D. R., J. H. Davies, O. Hassan, K. Morgan, and P. Nithiarasu (2007), Investigations into the applicability of adaptive

finite element methods to two-dimensional infinite Prandtl number thermal and thermochemical convection, Geochem.

Geophys. Geosyst., 8, Q05010, doi:10.1029/2006GC001470.

1. Introduction

[2] Over recent decades, numerical modeling hasstirred significant interest in the geodynamicalcommunity, leading the way in studies of numer-ous geological processes. This interest is due to thefact that analytical solutions to the various phe-nomena are normally unavailable, while experi-mental methods are sometimes time-consumingand often have limitations. The numerical methodsthat have been employed have generally beenbased upon finite-difference [e.g., McKenzie etal., 1974; Bodri and Bodri, 1978; Matyska andYuen, 2001] and, occasionally, finite-volume tech-niques [Tackley, 1996, 1998; Ratcliff et al., 1998;Albers and Christensen, 2001], although recently,the finite element method has become more estab-lished [e.g., Baumgardner, 1985; Farnetani andRichards, 1995; Sidorin et al., 1998; Zhong, 2000,2006]. Methods based upon the finite-elementapproach are attractive since they lead to general-purpose computer codes. However, it is not thisfeature that has been largely responsible for therecent interest shown in the method, but the factthat it, as indeed is the finite volume method, isbased upon an integral formulation and hence isreadily implemented on arbitrary discretizations,i.e., unstructured grids. This final point is centralto our study.

[3] It is a well known fact that even the use ofsophisticated computational models can give inac-curate results, if the numerical grid upon which themodel is based is unable to capture the significantfeatures of the problem. Indeed, for the large-scaleproblems encountered in geodynamics, inadequategrid resolution has become a major concern. Themajority of phenomena studied (e.g., subductionzone and mid-ocean ridge magmatism) are charac-terized by the interaction of complex geometries,complex material properties and complex boundaryconditions. Such a combination often yields unpre-dictable and intricate solutions, with narrowregions of high solution gradient frequently foundembedded in large areas where the solution variesslowly. It is these high gradient regions that presenta serious challenge for computational methods:their location and extent is very difficult to deter-

mine a priori and, even if their location is identi-fied, with current methods it is often impossible toresolve localized features. The net result is thatachieving accurate solutions is a very demandingtask for the analyst. Indeed, one of the mostchallenging problems currently facing geodynam-icists is the accurate solution of such ‘‘multi-scale’’flows.

[4] This issue of accuracy must be balanced withcomputational considerations. An accurate solutionrequires that one properly resolves the activefeatures in the simulation. Resolution, in turn, isrelated to the number of nodes employed. Obvi-ously, one could generate a solution of high accu-racy by employing an extremely fine meshthroughout the computational domain. However,the larger the number of nodes, or degrees offreedom, the greater the demands on computationalmemory and processing power. Finding the rightbalance between accuracy and computational effi-ciency is therefore a difficult task. Ideally, what isneeded is a method capable of yielding an accuratesolution, while employing as few degrees of free-dom as possible.

[5] Standard methods have attempted to achievethis by utilizing nonuniform grids generated apriori, with the user exploiting previous experienceto define the grid [e.g., Davies and Stevenson,1992; Scott, 1992]. Such methods, however, arenot applicable to unsteady problems, since theactive regions within the solution domain areconstantly mobile and predicting their location atany given time is an impossible task. The major-ity of previous studies have overcome this issueby employing a uniformly fine grid throughoutthe computational domain, as described above[e.g., Oldham and Davies, 2004; Bunge et al.,2003]. This ensures that, as the simulationevolves, active regions are continually in zonesof fine resolution and solution accuracy is main-tained. Other methods have also been utilized,although more rarely, including both Lagrangianand Arbitrary Lagrangian-Eulerian (ALE) formu-lations [Fullsack, 1995]. However, over time, ithas become apparent that these methods havetheir own restrictions and, consequently, a majorarea of research in the field of geodynamics is the

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https://www.researchgate.net/publication/229574626_Arbitrary_Lagrangian-Eulerian_formulation_for_creeping_flows_and_its_application_in_tectonic_models?el=1_x_8&enrichId=rgreq-c38b0b7e-02cb-4ea7-b767-a4f78e3c2040&enrichSource=Y292ZXJQYWdlOzI1NjU0MjY4NztBUzo5OTU2Nzk1NzcwODgxMEAxNDAwNzUwMjU5NjA3

generation of numerical models that accuratelyportray the nature of the problem, while main-taining computational efficiency.

[6] Here, we introduce grid adaptivity, which is amethod commonly employed within the field ofengineering [see Babuska and Rheinbolt, 1978;Lohner et al., 1985; Peraire et al., 1987; Pelletierand Ilinca, 1995; Nithiarasu and Zienkiewicz,2000]. Since we demonstrate the method in thecontext of finite elements it is termed the AdaptiveFinite Element Method (AFEM). The method pro-vides a powerful approach for the accurate andefficient solution of the complex problems encoun-tered in geodynamics. Grid points are automaticallyclustered in regions of rapid solution variation toimprove accuracy, leading to a ‘‘multi-resolution’’solution, with the highest resolutions being analo-gous to zones of high solution gradient. In simpleterms, the method automatically increases ordecreases grid resolution where required, leadingto more accurate solutions, while employing fewerdegrees of freedom.

[7] In this study, the method is applied to infinitePrandtl number thermal and thermochemical con-vection in 2-D Cartesian geometry, to investigateits potential benefits within the field of geodynam-ics. We begin by investigating a benchmark ther-mal convection problem, with results illustratingthat the method is highly successful, improvingsolution accuracy while increasing computationalefficiency.

[8] The method is then applied to a more challeng-ing thermochemical benchmark problem, employ-ing a grid-based method to solve for thecompositional field. Our reasons for selecting agrid-based method are simple. Recent work [e.g.,van Keken et al., 1997; Tackley and King, 2003]has highlighted the fact that such methods sufferfrom numerical diffusion, leading to greater entrain-ment rates in numerical simulations when comparedto marker chain and tracer particle methods. Thisnumerical diffusion is predominantly caused byinsufficient resolution, a factor that is naturallyaddressed by the AFEM. Consequently, two hy-potheses are tested:

[9] 1. The greater resolution endorsed by theAFEM will reduce artificial diffusion.

[10] 2. This reduced diffusion, in turn, will seegrid-based methods yielding results that are con-sistent with those achieved using tracer particle andmarker chain methods.

[11] As expected, results show that adapted gridsyield large improvements over regular, uniformgrids, generating less diffusive results while reduc-ing the number of degrees of freedom. However,results also demonstrate that grid-based methods,even when coupled with the AFEM, are notcompetitive with the other methods currentlyemployed for the tracking of compositional heter-ogeneities.

[12] The remainder of this paper will provide anoverview of the test problems that are studied, asummary of the numerical techniques employed(an appendix is included with a more detailedanalysis) and a comparison of the results obtained.These demonstrate the applicability of grid adap-tivity to the modeling of mantle dynamics.

2. Problem Description

[13] We consider two dimensional thermal andthermochemical convection in an isoviscous, infi-nite Prandtl number Cartesian box. The equations(in dimensionless, vector form) describing suchincompressible convection are the equations of

Momentum

r2u�rp ¼ RaTk̂ ð1Þ

Continuity (mass)

r � u ¼ 0 ð2Þ

Energy

@T

@tþ u � rT ¼ r2T ð3Þ

where u is the dimensionless velocity, T is thedimensionless temperature, p is the dimensionlessnonlithostatic pressure, k̂ is the unit vector in thevertical direction and t is the dimensionless time.In this form, all material properties are combinedinto one dimensionless parameter, the Rayleighnumber:

Ra ¼ brgDTd3

kmð4Þ

where g is the acceleration due to gravity, r isdensity, b is the coefficient of thermal expansion,DT is the temperature drop across the domain, d isthe domain length, k is the thermal diffusivity andm is the dynamic viscosity.


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[14] In thermochemical simulations, a second ad-vection-diffusion equation is solved for composi-tion:

@C

@tþ u � r C ¼ 1

Ler2 C ð5Þ

where C represents composition and Le is theLewis number, the ratio of thermal diffusivity tochemical diffusivity. The desired limit approachesinfinite Le; however, for numerical reasons, a finiteLe is often assumed. The momentum equation alsodiffers from equation (1) and is now taken in theform

r2u�rp ¼ RaT � RbCð Þk̂ ð6Þ

Here, Rb is the compositional Rayleigh number:

Rb ¼ Drgd3

kmð7Þ

where Dr is the density difference between thedense (C = 1) and light (C = 0) material.

[15] These equations are solved using a modifiedversion of the widely used 2D geodynamics finiteelement program ConMan. A brief overview of thecode is provided here; however, a more detaileddescription is given by King et al. [1990].

[16] The momentum and energy equations form acoupled set of differential equations, although thecoupling is not strong since the density, r, isconstant, other than in the buoyancy term of themomentum equation (Boussinesq approximation).The incompressibility (continuity) equation is trea-ted as a constraint on the momentum equation,with incompressibility enforced using a penalty

formulation. The energy equation is solved usinga streamline upwind Petrov Galerkin (SUPG)method [Hughes and Brooks, 1979], with timestepping accomplished by means of a second orderexplicit predictor-corrector algorithm.

[17] A grid-based method, identical to that used forsolving the energy equation, is utilized for model-ing the compositional field. A small chemicaldiffusivity is assumed and a filtering scheme isemployed to remove spurious numerical overshootand undershoot features, common with advectiondiffusion problems of this nature. This filter con-serves mass by design and has been shown to workremarkably well at limiting the aforementionednumerical errors (see Lenardic and Kaula [1993]for further details).

2.1. Case 1: Thermal Convection in aSquare Cavity

[18] The first example considered is buoyancy-driven flow in an iso-chemical square cavity. Thecavity is filled with a material of constant viscosityand there are no internal heat sources. Boundaryconditions are summarized in Figure 1. This prob-lem is solved at Ra = 104, 105, and 106, initially onuniform, structured meshes, and subsequently, onadapted, unstructured meshes.

[19] The following data or sets of data are calcu-lated during the simulations:

[20] 1. The Nusselt Number, i.e., the mean surfacetemperature gradient:

Nu ¼Z l

0

� @T

@yx; y ¼ 1ð Þdx ð8Þ

where l is the length (= 1).

[21] The nondimensional Root-Mean-Square(RMS) velocity:

VRMS ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1

V

ZV

jjujj2vuut ð9Þ

where V is the area of the computational domain.

[22] 3. Nondimensional temperature gradients atdomain corners:

q ¼ � @T

@yð10Þ

with q1 at x = 0, y = 1; and q2 at x = y = 1.

Figure 1. The boundary conditions utilized whenstudying thermal convection in a square cavity. T,V(x), and V(y) represent temperature and the velocitiesin the x and y directions, respectively.



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2.2. Case 2: Entrainment of a Thin DenseLayer Through ThermochemicalConvection

[23] The second example considered is a well-established thermochemical benchmark problemfrom van Keken et al. [1997]. We model theentrainment of a deep-seated, thin (0.025), denselayer in an aspect ratio 2 box of unit height. Thislayer is prescribed a compositional value C = 1,while the overlying, lighter material has a value ofC = 0. This problem is analogous to the entrainmentof a compositionally dense layer in the D00 region atthe Core-Mantle boundary. Results are computedfor 1/Le = 10�6. The thermal Rayleigh Number,Ra, is set to 3 105, the compositional RayleighNumber, Rb, is set to 4.5 105, while the viscosityis assumed to be constant. Boundary conditions aresummarized in Figure 2. As an initial condition, ananalytical expression of the temperature based onboundary layer theory is taken [see van Keken etal., 1997, Appendix A]. Once again, this problemis solved initially on uniform, structured meshes,and subsequently, on adapted, unstructured meshes.

[24] For this case, the relative entrainment (e) iscalculated as a function of time, from

e ¼ 1

ldb

Z l

de

CdV ð11Þ

where l is the aspect ratio of the box (= 2), db isthe thickness of the dense layer and de is anarbitrarily chosen height (= 0.2 here). We focus ourattention on the relative entrainment as opposed toother parameters, such as RMS velocity, since therate of entrainment provides an excellent means totrack the evolution of a thermochemical simulation.

Additionally, entrainment rates have been calcu-lated in previous studies [e.g., van Keken et al.,1997; Tackley and King, 2003], and hence directcomparisons can be made with ease.

[25] Before providing a summary of the adaptivestrategies employed, we should point out that thetwo cases presented above not only allow a meansto test the applicability of the AFEM to thermaland thermochemical convection problems, but theyalso allow us to test the AFEM in both steady state(Case 1) and unsteady (Case 2) situations. Both arecommon within the field of geodynamics.

3. Adaptive Methodology

[26] As is clear from the introduction, the methodsemployed in modeling thermal and thermochemi-cal convection must provide an adequate definitionof the problem, in a computationally efficientmanner. In other words, the methods must be adeptat resolving narrow regions of high gradients thatfrequently occur and are normally found embeddedin large areas where the flow variables vary slowly.Since the exact location of these high gradientregions are not always known to the analyst apriori, particularly with unsteady problems, it isapparent that adaptive mesh methods, with a pos-teriori error estimators, could have an importantrole to play in the development of efficient solutiontechniques for such problems. At present, a widevariety of adaptive procedures are being utilizedwithin the engineering community. Broadly speak-ing, these fall into two categories:

[27] 1. h-refinement, in which the same class ofelements continue to be used but are changed insize, in some locations made larger, and in othersmade smaller, to provide the maximum economy inreaching the desired solution.

[28] 2. p-refinement, in which the same elementsize is utilized, but the order of the polynomial isincreased or decreased as required (e.g., linearshape functions are ‘‘adapted’’ to quadratic orhigher order).

[29] A variant of the h-method, known as ‘‘adap-tive remeshing,’’ is employed in this study. Itprovides the greatest control of mesh size andgrading to better resolve the flow features. In thismethod, for steady state simulations, the problem issolved initially on a grid fine enough to roughlycapture the physics of the flow. Remeshing theninvolves the following steps:

Figure 2. The boundary conditions utilized whenstudying the entrainment of a thin dense layer, in anaspect ratio 2 box, through thermochemical simulations.T, C, V(x), and V(y) represent temperature, composition,and the velocities in the x and y directions, respectively.



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[30] 1. The solution is analyzed through some kindof error estimation procedure, to determine loca-tions where the mesh fails to provide an adequatedefinition of the problem. This can mean that eitheradditional grid points are needed, or indeed, thatthere are too many grid points at certain locationswithin the domain. An interpolation-based localerror estimator is employed in this study, basedupon nodal solution gradients and curvatures.

[31] 2. The information yielded by this error esti-mation process is utilized to generate an improvedmesh through an automatic mesh generator. Avariant of the so-called advancing front techniqueis utilized here, being capable of generating meshesthat conform to a user prescribed spatial distribu-tion of element size. Elements can locally increaseor decrease in size as required, leading to what weterm an ‘‘optimal’’ mesh.

[32] 3. The original solution is interpolated betweenmeshes using higher-order cubic interpolation[Nielson, 1979; El Hachemi et al., 2003].

[33] 4. The solution procedure continues on thenew mesh.

[34] The remeshing process is repeated until adesired level of accuracy has been achieved.

[35] For unsteady problems, the process is almostidentical; however, it is fundamental that the initialmesh is suitably defined. If the mesh is inadequate,the errors generated during the calculation’s earlystages propagate through the computational domain,generating misleading results. To ensure such errorsdo not arise in our simulations, we generate anoptimal initial mesh. This process is straightforward.The initial condition is set up on a structured grid. Inthe same way as is described in points 1-4 above, thedata from this structured grid is analyzed to determinewhere themesh needsmodification. This informationis then used to regenerate the mesh, and the solution(i.e., the initial condition) is transferred onto the newmesh using cubic interpolation. Consequently, theinitial grid naturally provides an optimal definition ofthe problem. With unsteady problems, the remeshingprocedure can be continued indefinitely as the sim-ulation evolves. The remeshing ‘‘loop’’ is activatedafter a user-defined time interval, or dynamically,upon the basis of an approximation to the error.

[36] It is important to point out that although theprocess seems similar to Lagrangian formulations(the computational mesh appears to follow thesolution), it is indeed an Eulerian formulation,

perhaps with some of the advantages commonlyassociated with Lagrangian schemes.

[37] For a more detailed description of the meshgeneration process, the error estimation procedureand the adaptive strategy, please refer toAppendixA.A flow chart summarizing the essential stagesinvolved is also included in Figure 3.

3.1. Remeshing Procedure for Case 1:Steady-State Adaptivity

[38] A steady-state solution is achieved here.Remeshing is therefore a simple task, being per-formed when the solution converges to a steadystate on a given grid. The process is terminatedwhen an optimal mesh has been produced, i.e., thesolution does not improve with the remeshingprocedure. The error estimator employed in thiscase is based upon nodal temperature gradients andcurvatures.

3.2. Remeshing Procedure for Case 2:Temporal Adaptivity

[39] The results presented are based upon simula-tions with a remeshing frequency of 2000 timesteps. This value was selected after a series of tests,both visual and analytical, tracing the temporalevolution of the model. Ideally, the remeshingprocedure would be linked to the dynamics (i.e.,it should be tied to some measure of how much thesolution has changed or whether derivatives in themesh exceed a certain tolerance). Nonetheless, wehave verified, through tests at various remeshingfrequencies, that for this simulation, the remeshingfrequency selected (i.e., 2000 time steps) does notdegenerate the results.

[40] The error estimator employed is similar to thatin Case 1; however, it is based upon a combinationof temperature and composition, as opposed totemperature alone. Nodal solution derivatives arecalculated for both the temperature and composi-tional fields. The highest values yielded are thenselected as derivatives for that particular node.Such a scheme engenders high resolving powerat the density interface, as well as sufficient reso-lution to accurately solve the thermal field. Wehave verified that this combination yields superiorresults to simulations employing a combination ofthe composition and velocity variables.

[41] The remeshing strategy is slightly different tothat in Case 1. Rather than simply refining zones ofhigh solution gradient, we also allocate fine reso-



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Figure

3.

Aflow

diagram

summarizingthemajorstepsinvolved

intheadaptiveremeshingprocedure.



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lution to neighboring regions. This is done byensuring that the transition from fine to coarseelements is extremely gradational. In the steady-state cases, we specify that a minimum of 10elements is required to make this transition. How-ever, in our thermochemical, unsteady simulations,this value is set to 30. In this way, fine regions ofthe mesh are always surrounded by relatively finezones. Accordingly, as the simulation evolves,accuracy is maintained, since zones of high solu-tion gradient have not departed the fine gridregions before the next remeshing. Such a schemeallows a greater time interval between remeshingsand was a key consideration when selecting theremeshing frequency.

[42] It is important to note that the minimumelement size (dmin) permitted in our simulations is�0.002 which yields localized resolution equiva-lent to that achieved during a uniform mesh sim-ulation of 1000 500 elements, in a box of aspectratio 2.

4. Results and Discussion

4.1. Thermal Convection in a SquareCavity

4.1.1. Uniform Structured Meshes

[43] The results obtained for uniform mesh calcu-lations at Ra = 105 are displayed in Table 1. Theresults at both Ra = 104 (Table 2) and Ra = 106

(Table 3) demonstrate a comparable relationship,and consequently only one set of results are pre-sented fully.

[44] Figure 4 displays the relationship between thenumber of elements in a mesh and the RMSvelocity and mean Nusselt Number, respectively.The figures illustrate that a large number of ele-ments (i.e., >4000) is required before results beginto show some sort of consistency. Even moreelements are required before results begin to con-

verge to the benchmark solutions of Blankenbachet al. [1989].

[45] By calculating the percentage error for eachoutput analyzed (i.e., mean Nu, RMS velocity, q1and q2) and subsequently taking the mean of thesefour percentages, we have determined the discrep-ancy, i.e., the solution error, between uniform meshsolutions and benchmark solutions at various Ray-leigh numbers. Our results are summarized inTable 4.

[46] Clearly, as convection intensifies, the solutionerror becomes more prominent, as would beexpected. At higher Rayleigh numbers (105, 106),RMS velocities, mean Nusselt numbers, and cornertemperature gradients lie far beyond the realms ofuncertainty of the benchmark solution. Even atRa = 104 solutions fail to achieve a suitable levelof accuracy. However, as we show next, with theuse of adaptive, unstructured meshes, solutionaccuracy is greatly enhanced.

4.1.2. Adapted Meshes

[47] The results obtained at Ra = 104, 105 and 106

alongside their final adapted meshes are displayedin Figure 5. They are also summarized in Tables 5,6, and 7. For completeness, the sequence of meshesemployed at Ra = 105 are displayed in Figure 6,together with the corresponding temperature con-tours. Once again, results for both Ra = 104 and

Table 1. Results Obtained on Various Uniform, Structured Meshes for Simulations of Thermal Convection in aSquare Cavity at Ra = 105 a

Mesh 1 Mesh 2 Mesh 3 Mesh 4 Mesh 5 Mesh 6 Benchmark Uncertainty

Elements 256 1024 4096 7225 10000 16384 **** ****Nu 7.8757 9.6838 10.3361 10.4453 10.4712 10.5058 10.5341 0.00001VRMS 197.4998 194.2366 193.4223 193.3222 193.2968 193.2787 193.2145 0.0001q1 11.8004 16.5079 18.3362 18.6467 18.7641 18.8861 19.0794 0.00004q2 1.4204 0.9397 0.7754 0.7529 0.7445 0.7358 0.7228 0.00002

aThe benchmark results of Blankenbach et al. [1989] are included for comparison, together with their uncertainties.

Table 2. 1Results Obtained on Uniform Meshes forSimulations of Thermal Convection in a Square Cavityat Ra = 104

UMa

Benchmark Uncertainty

Elements 16384 **** ****Nu 4.8952 4.8844 0.00001VRMS 42.8713 42.8649 0.00002q1 8.0457 8.0594 0.000003q2 0.5905 0.5888 0.000003

aUM, uniform mesh.



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106 display a comparable relationship, and, conse-quently, only one set of results is presented fully.The characteristics of each mesh at Ra = 105 aresummarized in Table 8, which shows the numberof quadrilateral elements and nodal points, and the

values of the generation parameters (dMin, dMax,sMax and C) defined in Appendix A. It is importantto note that for consistency, the generation param-eters displayed in Table 8 are also utilized in oursimulations at Ra = 104 and Ra = 106. It shouldalso be noted that we have intentionally restrictedthe number of elements to less than 16,500, toallow a simple comparison between adapted anduniform structured meshes.

[48] As seen, the proposed procedure has refinedlocations of thermal boundary layers, whereverthey are strong. Consequently, when compared touniform structured meshes, the majority of resultsshow far superior concurrence with the benchmarksolution, in spite of a reduction in the number ofelements (Table 9). As noted previously, at Ra =105, a uniform mesh of more than 16,000 elementsis required for a reasonably well-resolved solution,i.e., within 1% of the benchmark results. At Ra =106, extrapolating from a series of uniform meshsimulations, we expect that more than 50,000elements would be required before results convergeto within �4% of the benchmark solution. Moreaccurate results are inaccessible with the lower-order, serial configuration of ConMan. However,with the proposed adaptive procedure this is not thecase. At Ra = 105, solutions within 0.1% of thebenchmark are achieved on a mesh of �15,700elements and at Ra = 106, results converge towithin 1% on a mesh of �16,200 elements. Usingthe AFEM, the number of elements required foradequate solution varies with Rayleigh number, butis always significantly less than that of a uniformmesh for a specified precision.

[49] It is important to point out that the benefits ofthis technique become more noticeable when con-vection is intense, and temperature gradients aregreater. At Ra = 104, only a moderate increase inaccuracy is observed between adapted and uniformmeshes, for approximately the same number ofelements (solution error decreases by a factor of3, from �0.2% to �0.06%). However, at Ra = 106,

Table 3. Results Obtained on Uniform Meshes forSimulations of Thermal Convection in a Square Cavityat Ra = 106

UM Benchmark Uncertainty

Elements 16384 **** ****Nu 21.3773 21.9725 0.00002VRMS 834.9486 833.9898 0.0002q1 43.3217 45.9673 0.0003q2 0.9737 0.8772 0.00001

Table 4. Solution Errors Obtained on a Uniform Meshof 16384 Elements (128 128 Elements), at VariousRayleigh Numbers, for Thermal Convection in a SquareCavity

Rayleigh Number Solution Error, %

104 0.2105 0.9106 4.9

Figure 4. The relationship between the number ofelements in a mesh and (a) RMS velocity and (b) meanNusselt Number. The results represent thermal convec-tion in a square cavity, on uniform, structured meshes atRa = 105. Benchmark values are represented byhorizontal dashed lines.



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results display a superior concordance with thebenchmark solution on adapted meshes (the solu-tion error decreases by a factor of 5, from �5% to�1%). Indeed, on fully uniform meshes with linearshape functions, obtaining an accurate solution on

a single processor would be highly impractical, atRa = 106.

[50] Other conclusions can be drawn by analyzingthe 4 outputs individually. The remeshing processclearly has a positive effect on the global measures(i.e., VRMS and Nu). However, the procedure seems

Figure 5. Final adapted meshes and corresponding temperature contours for purely thermal convection in a squarecavity at Ra = 104, 105, and 106. Red is hot (T = 1), blue is cold (T = 0), and the contour spacing is 0.02.

Table 5. Results Obtained After Each Remeshing forSimulations at Ra = 104 on Nonuniform, AdaptedMeshes

Mesh 1 Mesh 2 Mesh 3 Mesh 4 Benchmark

Elements 412 1977 6474 14972 ****Nu 4.2687 4.8565 4.8683 4.8790 4.8844VRMS 40.8536 42.9019 42.8792 42.8679 42.8649q1 7.2917 8.0345 8.0479 8.0538 8.0594q2 0.6366 0.5887 0.5886 0.5885 0.5888

Table 6. Results Obtained After Each Remeshing atRa = 105





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to have a more dramatic effect on the accuracy ofthe heat flux at domain corners (q1 and q2),particularly at higher Rayleigh numbers. This iseasy to understand; the heat flux at these corners isstrongly influenced by the resolution achieved inthe upper thermal boundary layer. Since thisboundary layer is characterized by high tempera-ture gradients, the remeshing procedure refines thegrid significantly in these regions. Accordingly, thecorner solutions yielded by adapted meshes are farsuperior to those yielded on uniform structuredmeshes. Global measures on the other hand, arenot influenced by these boundary layers to such anextent. Consequently, the improvement observed inglobal measures between adapted and uniformgrids is less dramatic.

4.1.3. AFEM and Processing Efficiency

[51] We have demonstrated that the number ofnodes and elements required for accuracy is lesswith the AFEM. Consequently, the AFEM is moreefficient in terms of memory requirements. How-ever, is the AFEM economical in terms of compu-tational processing time?

[52] Figure 7 illustrates the relationship betweensolution error and the time taken in obtaining thesesolutions on both uniform and adapted meshes (atvarious Rayleigh numbers). It should be noted thatthe timings displayed for the adaptive cases includethe time allocated for remeshing. The main pointsarising are summarized below:

[53] 1. In general, at Ra = 104, the AFEM is lessefficient than uniform meshes for a prescribed levelof accuracy. However, when a solution error of lessthan �0.2% is required, the AFEM becomes moreeconomical.

[54] 2. At Ra = 105 and Ra = 106 the AFEM ismore efficient than uniform structured meshes,decreasing computational processing time whileincreasing solution accuracy. Indeed, the lower

Table 7. Results Obtained After Each Remeshing atRa = 106



Figure 6. The series of meshes employed at Ra = 105 along with corresponding temperature contours. Red is hot(T = 1), blue is cold (T = 0), and the contour spacing is 0.02.



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graph in Figure 7 (Ra = 106) suggests that solutionerrors of less than �4% cannot be obtained onuniform meshes. With the AFEM this is not thecase; far superior results are attained, although dueto the lower order nature of ConMan, minor errorspersist.

[55] Table 10 summarizes the percentage of calcu-lation time taken by the remeshing procedure,compared to the time spent solving the governingequations. It is important to note that althoughremeshing appears more efficient at higher Ray-leigh numbers, this is not strictly true. At higherconvective vigors, simulations take longer to con-verge toward a steady state solution. Since the

number of remeshing loops employed during eachsimulation is constant (i.e., 3), it is apparent thatremeshing will take a smaller fraction of thecalculation time at higher Rayleigh numbers (sincethe calculation time as a whole is greater). Regard-less of this point, the main conclusion to be drawnfrom the data is that the remeshing procedure iscomputationally inexpensive, expending only asmall percentage of the calculation time.

[56] In summary, in the context of purely thermalconvection, the number of degrees of freedomrequired for accuracy on uniform structuredmeshes is greater than that required for adaptedmeshes. Thus, for the same precision, the numberof nodes and elements is reduced when the adap-tive procedure is used. Additionally, the remeshing

Figure 6. (continued)

Table 8. Sequence of Meshes Employed for theProblem of Buoyancy-Driven Flow in a Square Cavityat Ra = 105

Mesh Elements Nodes d Min d Max s Max C

1 412 453 0.05 0.05 1.0 -2 2390 2493 0.009 0.05 5.0 0.63 7321 7469 0.0045 0.030 5.0 0.34 15722 15961 0.0025 0.025 5.0 0.2

Table 9. Solution Errors Yielded by NonuniformAdapted Meshes at Various Rayleigh Numbers

Rayleigh Number Elements Solution Error, %

104 14972 0.06105 15722 0.09106 16195 1



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procedure is computationally inexpensive and,consequently, particularly at higher Ra (>105), theAFEM allows one to attain a desired solution inless processing time.

4.2. Thermochemical Convection

[57] Having demonstrated the applicability andbenefits of the AFEM for thermal convection, wemove on to thermochemical simulations.

4.2.1. Uniform Structured Meshes

[58] Curves illustrating the entrainment (e) yieldedby uniform mesh simulations at various grid reso-lutions are displayed in Figure 8. The results areconsistent with previous work in that relative

entrainment decreases with increased resolution.As a quantitative example, entrainment at t =0.02 decreases from a value of 0.4 on a grid of64 64 elements, to a value of 0.08 on a grid of256 256 elements. This large reduction can beeasily understood; grid-based methods suffer fromnumerical diffusion, which smears density interfa-ces over several grid spacings. Consequently, withcoarser grids, material at the density interface itselfbecomes easier to entrain. As one increases gridresolution, grid spacings decrease, leading to lessartificial diffusion and hence reduced entrainment.

[59] Perhaps a more fundamental point to notefrom Figure 8, however, is that even after thissignificant increase in resolution, when t = 0.01,and indeed for a high percentage of the calculation,the grid-based method displays almost an order ofmagnitude greater entrainment than the other meth-ods currently employed in this field. The markerchain results produced by Christensen, Neumeisterand Dion from van Keken et al. [1997] are dis-played for comparison. The reader should note thatthe tracer particle studies of Tackley and King[2003] are consistent with these results. Clearly,there is no resemblance between the grid-basedmethods and these results, particularly during theearly stages of the simulation. At first glance, itdoes appear that when t > 0.02, the results yieldedby grid-based simulations are reasonably consistentwith those of previous particle studies. Closeranalysis, however, reveals that this is misleading.The drastic difference in entrainment during theearly development of these models means that, bythis time, both simulations have evolved intocompletely different problems. Consequently, theapparent consistent relationship between bothmethods is purely coincidental. This point is rein-forced when studying the visual patterns; there isonly a poor resemblance between Figure 9 here,which shows the temporal evolution of the com-positional field, and equivalent Figures 8 and 10 ofvan Keken et al. [1997] and Figure 3 of Tackley andKing [2003]. The similarities also diminish as thesimulation evolves. The main differences includethe position of the dense pile after reorganization

Figure 7. The time taken to converge on varioussolution errors with both uniform (continuous line) andadapted (dashed line) meshes at (a) Ra = 104, (b) Ra =105, and (c) Ra = 106.

Table 10. Percentage of Computational Time Taken bythe Remeshing Process at Various Rayleigh Numbers

Rayleigh Number% Time

Remeshing

104 4105 1.7106 0.8



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into a two cell pattern and the amount of materialtrapped at the stagnation point. An importantobservation to make here is that, during the laterstages of the calculation, diffusion dominates and,by t = 0.07, the dense layer has virtually disap-peared. This is not the case when tracer particle andmarker chain methods are employed, as has beenpointed out by van Keken et al. [1997].

[60] It is clear from the simple tests performedhere, on uniform meshes, that increased grid reso-lution reduces entrainment. Results move toward

those yielded with tracer particle and marker chainmethods; however, entrainment rates remain sig-nificantly higher. Next, we will attempt to answerwhether this is the case when the AFEM isemployed or does the method provide sufficientresolution to resolve these discrepancies?

4.2.2. Adapted Unstructured Meshes

[61] The generation parameters (dMin, dMax, sMax

and C) utilized within these models are summa-rized in Table 11. Unlike those previously defined

Figure 8. (a) Relative entrainment (e) against time on a series of uniform meshes. The ‘‘best’’ results from vanKeken et al. [1997] (CND Markerchain) are also displayed for ease of comparison. (b) An enlargement of the resultsin Figure 8a for time �0.02.



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for the purely thermal case (Table 8), the valuesremain constant throughout the simulation. Thereis a simple explanation for this. Since the problemunder study does not converge to a steady-statesolution, one must ensure that not only is the errorequally distributed spatially, but also temporally.By fixing the generation parameters over time, oneguarantees that this is the case.

[62] Entrainment curves from these simulations aredisplayed in Figure 10. The result obtained from auniform mesh simulation on a grid of 256 256elements is also displayed for ease of comparison,along with the ‘‘best’’ results of van Keken et al.[1997].

[63] Plots are displayed from simulations employ-ing linear (AM-Linear) and cubic (AM-Cubic)interpolation between grids. The accuracy of thisinterpolation process is of fundamental importanceduring unsteady problems of this nature. Withsteady state problems, the solution always con-verges toward a certain end member. Consequently,minor errors arising due to inaccurate interpolationcan be overcome. However, with unsteady prob-lems, errors arising during the interpolation processpropagate through the computational domain, lead-ing to a solution that is unrepresentative of the trueproblem. In essence, the solution emerging fromthe remeshing procedure must be exactly thatwhich enters. Otherwise, the simulation evolvesfalsely.

[64] The results involving linear interpolation aretherefore only presented for completeness; themethod fails to accurately interpolate features atthe density interface. At certain locations, thedensity jumps from a value of 0 to 1 within asingle element. Linear interpolation is not capableof resolving such a feature and, consequently, theremeshing process employing linear interpolationgenerates diffusive and nonconservative results.The cubic interpolation strategy employed, however

Figure 9. Seven figures, at regular time intervals of0.01, illustrating the evolution of the compositionalfield, modeled on a uniform mesh of 256 256elements. Red represents dense material (C = 1), whileblue represents lighter material (C = 0).

Table 11. Mesh Parameters Employed forThermochemical Entrainment Simulations

dMin dMax sMax C

0.002 0.02 5 0.1



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[Nielson, 1979; El Hachemi et al., 2003], accuratelycaptures such features, being both locally and glob-ally conservative.

[65] The results demonstrate that in general, theAFEM leads to a significant reduction in artificialdiffusion and hence entrainment rates, when com-pared to uniform mesh simulations. This point isreinforced by making a comparison of the longev-ity of the dense pile in the visual output (Figures 9

and 11). As was noted earlier, with a uniform meshsimulation on a grid of 256 256 elements, byt = 0.07, the dense pile has almost completelydiffused. This is not the case with simulationsemploying the AFEM; the dense pile remainsextremely coherent until this time. It is cleartherefore that the higher resolution permitted bythe AFEM leads to decreased artificial diffusionand, consequently, results that provide a moreprecise representation of the problem.

Figure 10. (a) Relative entrainment (e) against time on adapted, unstructured meshes. The results obtained on a256 256 element uniform mesh are also displayed, as well as the ‘‘best’’ results from van Keken et al. [1997].(b) An enlargement of the results in Figure 10a for time �0.02.



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Figure 11. Evolution of (left) the compositional field and (right) the temperature field on (middle) a series ofadapted grids. The grids are adapted around temperature and compositional solution gradients. Additionally, a regionof fine resolution is generated adjacent to zones of high solution gradient. Consequently, as the simulation evolves,high gradient zones remain in regions of high resolution, leading to less numerical error.



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[66] It is important to point out that, in addition toyielding superior results, our adaptive grid simu-lations are computationally more efficient, both interms of memory and processing time, when com-pared to uniform mesh simulations. The number ofelements utilized in this simulation varies with timebetween �40,000 and �58,000, depending uponthe configuration of the calculation. Additionally,the generation of a new optimal mesh is aninexpensive procedure, typically taking between15 and 20 time steps, compared to the time takenfor one time step with a nonchanging mesh(although the time expended in remeshing can bedecreased significantly by specifying a largererr_max - the remeshing tolerance; see Appendix A,section A2.2). Obviously, this gain in computa-tional efficiency is only valid provided one doesnot set the minimum element size, dmin, to anunreasonably small value. Our experience suggeststhat simulations with dmin < 0.002 are highlyimpractical, due to the tradeoff between minimumelement size and time stepping, dictated by theCourant-Friedrichs-Levy condition. Of course, thissituation could be remedied by employing a localtime stepping algorithm; however, this was beyondthe scope of our study. It should be noted that theefficiency of the method as a whole depends onhow often it is necessary to remesh, which dependson how time-dependent the simulation is. Meshadaptivity therefore becomes less efficient forhighly time-dependent cases needing frequentremeshing.

[67] Although the results presented demonstratethe benefits of the AFEM, grid-based thermochem-ical methods, even when coupled with the AFEM,yield results that are beyond the realms of uncer-tainty of those achieved via tracer particle andmarker chain methods. The method remains par-ticularly diffusive, yielding erroneous entrainmentrates, even at the extremely fine resolutions per-mitted in our simulations. Once again, these find-ings are reinforced by the visual patterns, withFigure 11 displaying only a marginal resemblanceto Figures 8 and 10 of van Keken et al. [1997] andFigure 3 of Tackley and King [2003]. This resem-blance, however, is stronger than that observedwith our uniform mesh simulations (Figure 9).

5. Conclusions

[68] An adaptive finite element procedure has beenpresented for solving convective heat transferproblems within the field of geodynamics. Themethod adapts the mesh automatically around

regions of high solution gradient, yielding en-hanced resolution of the associated flow features.

5.1. Applicability to Thermal Convection

[69] The results obtained from thermal convectivesimulations are extremely positive. The error esti-mator presented has proven reliable and the adap-tive procedure is shown to be robust. Predictionsfor heat transfer agree well with benchmark sol-utions, suggesting that the technique is valid andaccurate.

5.2. Applicability to ThermochemicalConvection

[70] The results obtained from thermochemical sim-ulations are somewhat less conclusive. However,two major conclusions can be drawn:

[71] 1. The AFEM provides a suitable means forincreasing grid resolution in localized regions. Thisleads to a reduction in numerical diffusion andhence entrainment rates, provided that the interpo-lation employed during the remeshing procedureaccurately captures all underlying features. Ourresults suggest that an extension of this work toboth tracer particle and marker chain methodswould be a worthwhile exercise, with the higherspatial resolution yielded leading to the moreaccurate tracking of particles (or the marker chain),generating results of greater accuracy.

[72] 2. Even using the AFEM, grid-based meth-ods fail to achieve results that are consistentwith other methods. Consequently, we concludethat the method, at least in its current format,requires unrealistically high resolution to limitartificial diffusion and accurately track chemicalheterogeneities.

[73] In summary, the number of degrees of freedomrequired for accuracy on uniform structuredmeshes is greater than that on adapted unstructuredmeshes. Thus for the same, or often superiorprecision, the number of degrees of freedom isreduced when the present adaptive procedure isused. However, perhaps the most important advan-tages of the AFEM are the following:

[74] 1. The unstructured nature of the techniqueallows its use when modeling many of the complexgeometries encountered on Earth.

[75] 2. Nodes automatically cluster around zones ofhighest solution gradient, without the need forcomplicated a priori mesh generation.



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[76] 3. The reduction in the number of degrees offreedom leads to a decrease in computationalprocessing time and memory use (both in termsof disk space and RAM), meaning that complexproblems can be solved efficiently.

[77] To date, successful goal-orientated/error-guidedgrid adaptation techniques have, to our knowledge,not been utilized in the field of geodynamics.Potential applications of the method are wide-ranging and specific elements of the method couldeven be applied alone in certain situations. Forexample, the error-guided remeshing procedurewould prove extremely useful in Lagrangian simu-lations, when large distortions of the computationaldomain necessitate a total regridding.Within the fieldof geodynamics applications of the AFEM wouldinclude studies into subduction zone dynamics, D00

and its interaction with the post-perovskite phasetransition, upper mantle phase transitions, mid-oceanridge magmatism and plume dynamics. These phe-nomena have one thing in common: ‘‘active’’ regions

of high solution gradient, be it in temperature,composition, pressure or velocity, that are foundembedded in large, ‘‘passive’’ regions, whose loca-tion is difficult to determine a priori. It is cleartherefore that adaptive grid methods, with a poste-riori error estimators, should have an important roleto play in the development of efficient solutiontechniques for such problems. This developmentshould not be restricted to the method describedhere (i.e., adaptive remeshing), but to adaptive pro-cedures as a whole. Due to its flexibility, numericalmodeling will undoubtedly continue as a primarytool in helping us to understand various geologicalprocesses. The AFEM and the ‘‘multi-resolution’’solutions yielded, should, for the moment at least,ensure that progress is not unnecessarily restricted bycomputer power.

Appendix A: Mesh Generation andAdaptive Remeshing Procedure

A1. Mesh Generation

[78] The algorithmic procedure to be described forthe mesh generation process is based upon themethod originally proposed by Peraire et al.[1987]. The advocated approach is regarded as avariant of the so-called ‘‘advancing front’’ tech-nique [George, 1971; Lo, 1985] with the distinctivefeature that elements and nodes are generatedsimultaneously. This technique is capable of gener-ating meshes that conform to an externally pre-scribed spatial distribution of element size. Theability to generate meshes that are locally stretchedalong prescribed directions is also included, leadingto highly efficient definition of one-dimensionalflow features. For simplicity, triangular elementsare generated initially. These are subsequentlycombined or subdivided to form quadrilaterals,the elements utilized by ConMan.

A1.1. Generation of the Initial Mesh

[79] The underlying process in the advancing fronttechnique is illustrated in Figure A1 (see Peraire etal. [1987, 1990] for further discussion). Theboundary of the domain is discretized first. Nodalpoints are placed on the boundary curves in such away that the distance between them is as close aspossible to the desired mesh spacing. Contiguousnodes on the boundary curves are joined bystraight-line segments and assembled to form theinitial generation front. At this stage, the triangu-lation loop begins. A side from the front is chosen

Figure A1. An illustration of the advancing fronttechnique. The figure shows different stages during thetriangulation process.



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and a triangle is generated that will have thisselected side as one edge. In generating thisnew triangle an interior node may be created oran existing node in the front may be chosen. Aftergenerating the new element the front is conve-niently updated in such a way that it alwayscontains the sides that are available to form anew triangle. The generation is complete when nosides are left in the front.

A1.2. Mesh Control: The BackgroundMesh

[80] The inclusion of adequate mesh control is akey ingredient in ensuring the generation of a meshof the desired form. Control over the characteristicsis obtained by the specification of a spatial distri-bution of mesh parameters by means of a back-ground mesh. The background mesh is used forinterpolation purposes only and is made up oftriangles.

[81] To control the elements generated, the userdefines the node spacing, d, the value of a stretchingparameter, s, and a direction of stretching a. Thegenerated elements will then have a typical lengthsd in the direction parallel to a, and a typical lengthd in the direction normal to a (Figure A2). Thus, ateach node on the background grid, the nodal valuesof d, s, and a must be specified. These values aredependent upon the solution gradients and curva-tures yielded by the error estimation process(described further in sections A2.1 and A2.2). Localvalues of these quantities are then obtained duringthe generation process by cubic interpolation, overthe triangles of the background grid, between thespecified nodal values (see Nielson [1979] and El

Hachemi et al. [2003] for a detailed description ofthis interpolation process).

[82] In general, for the initial mesh, the location ofone-dimensional features is not known. Conse-quently a value of s = 1 (i.e., no stretching) isspecified. The node spacing d is also defined to beuniform, although a variation of d can be achieved(by suitable construction of the background grid) ifit is apparent that increased mesh resolution isrequired in certain regions of the flow domain.Note that if d is required to be uniform initially andno stretching is to be specified, then the back-ground grid need only consist of a single element,which completely covers the solution domain.

A1.3. Boundary Representation

[83] The boundary of the solution domain is rep-resented by closed loops of oriented piecewisecubic spline curves. For simply connecteddomains, these boundary curves are orientated ina counter-clockwise sense, while for multi-connected regions the exterior boundary curvesare given a counter-clockwise orientation and allinterior boundary curves are orientated in a clock-wise sense (Figure A3). This means that, as theboundary curve is traversed, the region to bemeshed always lies to the left.

[84] When these boundary curves are discretized,the boundary edges forming the initial front areorientated in the same fashion. In this study, theorientation of a boundary edge is defined by theorder in which the two nodes of the edge are listedin the front. The orientation of the edge is impor-tant as it identifies the area of the plane in which a

Figure A2. The definition of the mesh parameters a,s, and d.

Figure A3. Orientation of the boundary. The domainof interest is shaded.



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valid triangle can be created using that edge as abase.

A1.4. Triangle Element Generation

[85] The generation of a regular triangular elementof size d is illustrated in Figure A4. The processinvolves the following steps:

[86] 1. Select an edge AB from the generationfront.

[87] 2. Using the orientation of the edge, determinethe position of a point C1, which lies at a distance dfrom A and B.

[88] 3. Determine all points in the front that lieinside a circle of radius d and center at C1. Letthese points be denoted by Qi, where the subscript ivaries between 0 and the number of ‘‘front’’ pointsinside the circle (i = 2 in Figure A4).

[89] 4. Determine the positions of the equallyspaced points C2, C3, C4 and C5 on the line joiningC1 and the mid-side (M) point of AB.

[90] 5. Form a list containing all the points deter-mined in step 3 as well as points C1, C2, C3, C4 andC5. The points in this list will then be orderedaccording to their distance from the point C1.

[91] 6. Create an element with nodes A, B and thefirst point in the list which satisfies the mesh

consistency requirement, i.e., the two newly creatededges do not cross any of the existing edges in thefront.

[92] 7. Update the front by removing the edge AB,and adding the appropriate number of new edgeswith the correct orientation.

A1.5. Quadrilateral Element GenerationUsing an Existing Triangular Mesh

[93] Since ConMan can only handle quadrilateralgrids, the triangular meshes generated by the pro-cedures described above must be altered intoquadrilaterals. This can be done by doing thefollowing:

[94] 1. Combine two triangles together to form aquadrilateral. However, this method comes unstuckwhen an odd number of triangular elements exist inthe original mesh.

[95] 2. Generate mid-side nodes on triangles andone in the center, and interconnect them, generat-ing three quadrilaterals.

[96] In this study, a combination of the two meth-ods is employed (Figure A5):

[97] 1. The triangles of the mesh are combined inpairs to give quadrilaterals.

[98] 2. Since there may be several triangles remain-ing, all elements, triangular and quadrilateral, aresplit into quadrilaterals by placing new nodes at themid-sides, and one in the middle of each element.This ensures the generation of an all-quadrilateralmesh.

[99] It should be pointed out that in order toproduce a mesh with desired element size, d, the

Figure A4. Generation of a new triangle.

Figure A5. Agglomeration of triangles.



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spacing of the triangles generated originally mustbe 2d.

A1.6. Mesh Quality Enhancement

[100] Three post-processing procedures are appliedto enhance mesh quality. These procedures do notalter the total number of points or elements in themesh:

[101] 1. Diagonal swapping: This techniquechanges the connectivities among nodes in themesh without altering their position. This processrequires a loop over all the element sides excludingthose sides on the boundary. For each side commonto the triangles ACD and BCD (see Figure A6),one considers the possibility of swapping CD byAB, thus replacing the two triangles ACD andBCD by the triangles ABC and ABD. The swap-ping is performed if a prescribed regularity criteri-on is better satisfied by the new configuration thanby the existing one. In our implementation, theswapping operation is performed if the minimumangle (formed by element edges at each elementnode) occurring in the new configuration is largerthan in the original one.

[102] 2. Mesh smoothing: This alters the position ofthe interior nodes without changing the topology ofthe mesh. The element sides are considered assprings of stiffness proportional to the length ofthe side. The nodes are moved until the springsystem is in equilibrium. The equilibrium positionsare found by iteration. Each iteration amounts toperforming a loop over the interior points andmoving their coordinates to coincide with thoseof the centroid of the neighboring points. Usuallythree to five iterations are performed [Hassan andProbert, 1999].

[103] 3. Bandwidth reduction: A variant of the‘‘Cuthill-McKee’’ algorithm (an algorithm to re-

duce the bandwidth of sparse symmetric matrices)is utilized to renumber elements and nodes [Cuthilland McKee, 1969]. This process speeds up thecalculation and reduces memory requirements,since the resulting solution matrices are far morecompact, with reduced bandwidth.

A2. Error Indicator and Adaptive Strategy

[104] Having obtained an approximation to a solu-tion for a given problem, one can improve theaccuracy of this solution by adaptively refining themesh. In this study, mesh adaptation is achieved byusing the computed solution to determine ‘‘opti-mum’’ nodal values for d, s, and a. The mesh isthen regenerated with the initial computationalmesh acting as a background grid.

[105] To determine the values for the mesh param-eters, it is necessary to use the current solution togive some indication of the error magnitude anddirection. A certain ‘‘key’’ variable must be iden-tified and then the error indication process can beperformed in terms of this variable. In this study,the error indicator is based on the temperaturevariable, T, in purely thermal convective simula-tions, and a combination of T and C, the compo-sition, in thermochemical simulations. Of course,other variables (e.g., pressure) or any combinationof variables (e.g., temperature and velocity[Nithiarasu, 2000]) can be chosen, depending uponthe nature of the problem under investigation.

[106] The construction of the error estimator cantake various forms depending upon the nature of theproblem. It is obvious from the large number ofpublications available on error estimation and ad-aptivity [Lohner et al., 1986; Zienkiewicz and Zhu,1987; Lohner, 1995; Hassan et al., 1995; Fortin etal., 1996] that research in this area of computationalmechanics remains very active. However, as hasbeen pointed out by Nithiarasu and Zienkiewicz[2000], most of the well-known literature on errorestimates deals with self-adjoint problems [e.g.,Zienkiewicz and Zhu, 1987]. Fluid mechanics prob-lems, which involve non self-adjoint operators, aremore difficult concepts, and traditional methodssuch as the ‘‘energy norm’’ are not always suitablefor measuring the error. For this reason, most of thework in fluids utilizes local indicators, such as thelocal interpolation error, to refine the grid withoutspecifying a total error.

[107] Error indicators based upon the interpolationtheory make the following assumptions:

Figure A6. Diagonal swapping procedure.



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[108] 1. The nodal error is zero.

[109] 2. The solution is smooth.

[110] This allows one to approximate the elementalerror by a derivative one order higher than theelement shape function. We make use of thisapproach to refine the grid, by considering thesecond derivatives (curvatures) of T and C. Notethat for the remainder of this appendix we willrestrict our discussion to the solution variable f,rather than refer to T and C explicitly.

A2.1. Error Indicator

[111] Consider a one-dimensional situation inwhich the exact values of the key variable f areapproximated by a piecewise linear function f̂. Theerror E is then defined as

E ¼ f xð Þ � f̂ xð Þ ðA1Þ

If the exact solution is a linear function of x, thiserror will vanish, as the approximation has beenobtained using piecewise linear finite elementshape functions. To a first order of approximation,the error E can be evaluated as the differencebetween a quadratic finite element solution f̂ andthe linear computed solution. To obtain a piecewisequadratic approximation, one could obviouslysolve a new problem using quadratic shapefunctions. This, however, would be costly and analternative approach for estimating a quadraticapproximation from the linear finite elementsolution can be employed. Assuming that the nodalvalues of the quadratic and the linear approxima-tions coincide, i.e., that the nodal values of E arezero, a quadratic solution can be constructed oneach element, once the value of the secondderivative is known (assuming that second deriva-tives are constant over each element).

[112] The variation of the error E within an elemente is then expressed as

Ee ¼1

2z he � zð Þ @

2f̂@x2

ðA2Þ

where z denotes a local element coordinate and hedenotes the element length [Peraire et al., 1987].The root mean square value Ee

RMS of this error overthe element is computed as

ERMSe ¼

Zhe0

E2e

hedz

8<:

9=;

1=2

¼ 1ffiffiffiffiffiffiffiffi120

p h2e@2f̂@x2

��e

ðA3Þ

where k denotes absolute value. Several previousstudies [Demkowicz et al., 1984; Peraire et al.,1987; Nithiarasu, 2000] have demonstrated thatequi-distribution of the element error leads to anoptimal mesh and in what follows we employ thesame criterion. This requirement implies that

h2@2f̂@x2

�� ¼ C ðA4Þ

where C denotes a positive constant. Finally, therequirement of equation (A4) suggests that theoptimal spacing d on the new adapted mesh shouldbe computed according to

d2@2f̂@x2

�� ¼ C ðA5Þ

Equation (A5) can be directly extended to the two-dimensional case by writing the quadratic form:

d2b mijbibj

�¼ C ðA6Þ

where b is an arbitrary unit vector, db is the spacingalong the direction of b, and mij are thecomponents of a 2 2 symmetric matrix, m, ofsecond derivatives defined by

mij ¼@2f̂@xi@xj

ðA7Þ

These derivatives are computed at each node of thecurrent mesh by using the two-dimensionalequivalent of the variational recovery procedure.This procedure allows one to recover the nodalvalues of second derivatives from the elementalvalues of the first derivatives of f̂; refer toAppendix B for a detailed analysis of thisprocedure.

A2.2. Adaptive Remeshing

[113] The basic concept behind the adaptiveremeshing technique is to use the computed solu-tion to provide information on the spatial distribu-tion of the mesh parameters. This information willbe used by the mesh generator to generate a newadapted mesh in those areas where the values of theoptimal mesh parameters differ from the values ofthe current mesh parameters by greater than a userprescribed amount, err_max (set as 0.5% in thisstudy).

[114] The optimal values for the mesh parametersare calculated at each node of the current mesh.The directions ai; i = 1,2 are taken to be theprincipal directions of the matrix m. The corre-



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sponding mesh spacings are computed from theeigenvalues li of m, as

di ¼ffiffiffiffiC

li

r; i ¼ 1; 2 ðA8Þ

The spatial distribution of the mesh parameters isdefined when a value is specified for the constant C.The total number of elements in the adapted meshwill depend upon the choice of this constant. Themagnitude of the stretching parameter, s, at node n,is simply defined as the ratio between the twospacings:

sn ¼

ffiffiffiffiffiffiffiffiffid1nj jd2nj j

sðA9Þ

where d1n is the spacing in principal direction 1, andd2n is the spacing in principal direction 2.

[115] In the practical implementation of this method,two threshold values are used: a minimum spacingdmin, and a maximum spacing dmax, with

dmin � di � dmax; i ¼ 1; 2 ðA10Þ

It is apparent that in regions of uniform flow, thecomputed values of dn will be very large.Consequently, the user must specify a maximumallowable value, dmax, for the local spacing on thenew mesh. Then, if dn is such that dn � dmax, thevalue of dn is set to dmax. Similarly, the userprescribes a maximum allowable stretching ratioon the new mesh.

[116] The new mesh is generated according to thecomputed distribution of mesh parameters. Theoriginal solution is then transferred onto thenew mesh using cubic interpolation [Nielson,1979; El Hachemi et al., 2003] and the solutionprocedure continues on the new mesh. It should benoted that the increase in definition of flow featuresis achieved by decreasing the value of dmin. Thevalue of dmin is therefore the major parametergoverning the number of elements in the newmesh.

Appendix B: Calculation of NodalGradients and Curvatures

[117] The discrete, finite element solution providesthe values of fh in terms of the nodal values �q as

fh ¼ Nf ðB1Þ

where N are the appropriate shape functions used.

The nodal values of the second derivatives, @2fh

@x2 ,

can be obtained by using a similar approximation:

@2fh

@x2¼ N

@2fh

@x2

( )ðB2Þ

with similar expressions for @2fh

@y2 . The projection,

ZWNT N

@2f@x2

� �� @2N

@x2f

� �dW ¼ 0 ðB3Þ

can be used to determine the nodal curvatures.

Thus

@2f@x2

¼ M�1

ZW

@NT

@x

@N

@x

� �dWf ðB4Þ

where

M ¼ZWNTNdW ðB5Þ

is the well-known mass matrix, which is lumped

for convenience. The contribution at any node thus

involves only the elements surrounding it. Similar

expressions can be written for @2f@y2 and

@2f@x@y.

Acknowledgments

[118] D.R.D. would like to acknowledge support from both

NERC and EPSRC as part of the Environmental Mathematics

and Statistics (EMS) studentship programme (NER/S/E/2004/

12725). The authors also thank Scott King for help and support

with ConMan, as well as a number of colleagues for helpful and

informative discussions. Additionally, Paul Tackley, Magali

Billen, and Peter van Keken are acknowledged for detailed

comments and suggestions during review.

References

Albers, M., and U. R. Christensen (2001), Channeling ofplume flow beneath mid-ocean ridges, Earth Planet. Sci.Lett., 187, 207–220.

Babuska, I., and W. C. Rheinbolt (1978), A posteriori errorestimates for the finite element method, Int. J. Numer. Meth-ods, 12, 1597–1615.

Baumgardner, J. R. (1985), Three-dimensional treatment ofconvective flow in the Earth’s mantle, J. Stat. Phys., 38,501–511.

Blankenbach, B., et al. (1989), A benchmark comparison formantle convection codes, Geophys. J. Int., 98, 23–38.

Bodri, L., and B. Bodri (1978), Numerical investigation of tec-tonic flow in island-arc areas, Tectonophysics, 50, 163–175.

Bunge, H. P., C. R. Hagelberg, and B. J. Travis (2003), Mantlecirculation models with variational data assimilation: Infer-ring past mantle flow and structure from plate motion historiesand seismic tomography, Geophys. J. Int., 152, 280–301.



24 of 25

Cuthill, E. H., and J. McKee (1969), Reducing the bandwidthof sparse symmetric matrices, paper presented at ACM 24thNational Conference, Assoc. for Comput. Mach., New York.

Davies, J. H., and D. J. Stevenson (1992), Physical model ofsource region of subduction zone volcanics, J. Geophys.Res., 97, 2037–2070.

Demkowicz, L., J. T. Oden, and T. Strouboulis (1984), Adap-tive finite-elements for flow problems with moving bound-aries— Variational principles and a posteriori estimates,Comput. Methods Appl. Mech., 46, 217–251.

El Hachemi, M., O. Hassan, K. Morgan, D. P. Rowse, and N. P.Weatherill (2003), Hybrid methods for electromagnetic scat-tering simulations on overlapping grids, Commun. Numer.Methods Eng., 19, 749–760.

Farnetani, C. G., and M. A. Richards (1995), Thermal entrain-ment and melting in mantle plumes, Earth Planet. Sci. Lett.,136, 251–267.

Fortin, M., M.-G. Vallet, J. Dompierre, J. Bourgault, and W. G.Habashi (1996), Anisotropic mesh adaptation: Theory, va-lidation and applications, Comput. Fluid. Dyn., 96, 174–180.

Fullsack, P. (1995), An arbitrary Lagrangian-Eulerian formula-tion for creeping flows and its application in tectonic models,Geohpys. J. Int., 120, 1–23.

George, A. J. (1971), Computer implementation of the finiteelement method, Ph.D. thesis, STAN-CS-71-208, StanfordUniv., Stanford, Calif.

Hassan, O., and E. J. Probert (1999), Grid control and adapta-tion, in Handbook of Grid Generation, pp. (35)1–(35)29,CRC Press, Boca Raton, Fla.

Hassan, O., E. J. Probert, K. Morgan, and J. Peraire (1995),Mesh generation and adaptivity for the solution of compres-sible viscous high speed flows, Int. J. Numer. Methods Eng.,38, 1123–1148.

Hughes, T. J. R., and A. Brooks (1979), A multi-dimensionalupwind scheme with no crosswind diffusion, in FiniteElement Methods for Convection Dominated Flows, AMD,vol. 34, pp. 19–35, Am. Soc. of Mech. Eng., New York.

King, S. D., A. Raefsky, and B. H. Hager (1990), ConMan:Vectorizing a finite element code for incompressible two-dimensional convection in the Earth’s mantle, Phys. EarthPlanet. Inter., 59, 195–207.

Lenardic, A., and W. M. Kaula (1993), A numerical treatmentof geodynamic viscous-flow problems involving the advec-tion of material interfaces, J. Geophys. Res., 98, 8243–8260.

Lo, S. H. (1985), A new mesh generation scheme for arbitraryplanar domains, Int. J. Numer. Methods Eng., 21, 1403–1426.

Lohner, R. (1995), Mesh adaptation in fluid mechanics, Eng.Frac. Mech., 50, 819–847.

Lohner, R., K. Morgan, and O. C. Zienkiewicz (1985), Anadaptive finite-element method for high-speed compressibleflow, Lect. Notes Phys., 218, 388–392.

Lohner, R., K.Morgan, and O. C. Zienkiewicz (1986), Adaptivegrid refinement for the Euler and compressible Navier-Stokesequations, in Accuracy Estimates and Adaptive Refinementin Finite Element Computations, edited by I. Babuska et al.,pp. 281-296, John Wiley, Hoboken, N. J.

Matyska, C., and D. A. Yuen (2001), Are mantle plumes adaia-batic?, Earth Planet. Sci. Lett., 189, 165–176.

McKenzie, D. P., J. M. Roberts, and N. O. Weiss (1974),Convection in the Earth’s mantle: Towards a numerical si-mulation, J. Fluid Mech., 62, 465–538.

Nielson, G. M. (1979), The side-vertex method for interpola-tion in triangles, J. Approx. Theory, 25, 318–336.

Nithiarasu, P. (2000), An adaptive finite element procedure forsolidification problems, Heat Mass Transfer, 36, 223–229.

Nithiarasu, P., and O. C. Zienkiewicz (2000), Adaptive meshgeneration for fluid mechanics problems, Int. J. Numer.Methods Eng., 47, 629–662.

Oldham, D., and J. H. Davies (2004), Numerical investigationof layered convection in a three-dimensional shell with ap-plication to planetary mantles, Geochem. Geophys. Geosyst.,5, Q12C04, doi:10.1029/2003GC000603.

Pelletier, D., and F. Ilinca (1995), Adaptive finite elementmethod for mixed convection, J. Thermophys. Heat Transfer,9, 708–714.

Peraire, J., M. Vahdati, K. Morgan, and O. C. Zienkiewicz(1987), Adaptive remeshing for compressible flow computa-tions, J. Comput. Phys., 72, 449–466.

Peraire, J., K. Morgan, and J. Peiro (1990), Unstructured finiteelement mesh generation and adaptive procedures for CFD,AGARD Conf. Proc., 464, 18.1–18.12.

Ratcliff, J. T., D. Bercovici, G. Schubert, and L. W. Kroenke(1998), Mantle plume heads and the initiation of plate tec-tonic reorganizations, Earth Planet. Sci. Lett., 156, 195–207.

Scott, D. R. (1992), Small-scale convection and mantle melt-ing beneath mid-ocean ridges, in Mantle Flow and MeltGeneration at Mid-ocean Ridges, Geophys. Monogr. Ser.,vol. 71, pp. 327–352, AGU, Washington, D. C.

Sidorin, I., M. Gurnis, and D. V. Helmberger (1998), Dynamicsof a phase change at the base of the mantle consistent withseismological observations, J. Geophys. Res., 104, 15,005–15,023.

Tackley, P. J. (1996), Effects of strongly variable viscosity onthree dimensional compressible convection in planetarymantles, J. Geophys. Res., 101, 3311–3332.

Tackley, P. J. (1998), Three-dimensional simulations of mantleconvection with a thermochemical CMB boundary layer:D00?, in The Core-Mantle Boundary Region, Geodyn. Ser.,vol. 28, edited by M. Gurnis et al., pp. 231–253, AGU,Washington, D. C.

Tackley, P. J., and S. D. King (2003), Testing the tracer ratiomethod for modeling active compositional fields in mantleconvection simulations, Geochem. Geophys. Geosyst., 4(4),8302, doi:10.1029/2001GC000214.

van Keken, P. E., S. D. King, H. Schmeling, U. R. Christensen,D. Neumeister, and M. P. Doin (1997), A comparison ofmethods for the modeling of thermochemical convection,J. Geophys. Res., 102, 22,477–22,495.

Zhong, S. (2000), Role of temperature-dependent viscosity andsurface plates in spherical shell models of mantle convection,J. Geophys. Res., 105, 11,063–11,082.

Zhong, S. (2006), Constraints on thermochemical convectionof the mantle from plume heat flux, plume excess tempera-ture, and upper mantle temperature, J. Geophys. Res., 111,B04409, doi:10.1029/2005JB003972.

Zienkiewicz, O. C., and J. Z. Zhu (1987), A simple errorestimator and adaptive procedure for practical engineeringanalysis, Int. J. Numer. Methods Eng., 24, 337–357.



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