From 18378ca4252cf1b01a6a7b1f0f99ab2a57a87f0c Mon Sep 17 00:00:00 2001 From: lmoresi Date: Thu, 13 Aug 2026 17:36:23 +1000 Subject: [PATCH 1/2] Convert gthyagi's annulus and spherical Stokes benchmark post UWTN 2026-012, from gthyagi/UW3_Annulus_Spherical_Benchmarks/docs/benchmarks_blog_post. His prose is carried over as written; only the markup is converted. -

wrappers dropped, HTML footnote anchors replaced by real citations, and the hand-written References list removed in favour of a bibliography MyST generates. Five entries pinned in references.bib rather than resolved from doi.org at build time, for the reason UWTN 2026-011 found the hard way. Thieulot & Puckett has no DOI, so it is an @unpublished entry with the author-hosted URL. - // markup turned into real maths: L_2, q_h, P_2 x P_1, O(h^2), sigma_rr, Gamma_inner. ```math blocks became display maths. - Nine tags became eight {figure} directives with alt text on ONE line, plus the banner. The combined-fields image is the banner, so the in-body copy of it was dropped and its caption folded into the text -- otherwise it renders twice, once from the injected banner block and once as a figure. Underworld development team with AI support from Claude Code --- ...annulus-and-spherical-stokes-benchmarks.md | 354 ++++++++++++++++++ .../figures/banner.jpg | Bin 0 -> 423995 bytes .../figures/boundary_metric_convergence.jpg | Bin 0 -> 285272 bytes .../figure_3_kramer_annulus_convergence.jpg | Bin 0 -> 474945 bytes .../figure_4_kramer_spherical_convergence.jpg | Bin 0 -> 464896 bytes .../figure_5_thieulot_annulus_convergence.jpg | Bin 0 -> 414964 bytes .../figure_7_kramer_boundary_convergence.jpg | Bin 0 -> 274649 bytes .../figure_boundary_pressure_convergence.jpg | Bin 0 -> 419205 bytes ...ure_p2p1_boundary_pressure_convergence.jpg | Bin 0 -> 218040 bytes .../figures_4_5_thieulot_convergence.jpg | Bin 0 -> 233004 bytes .../metadata.yml | 35 ++ .../references.bib | 70 ++++ classification.yml | 5 + 13 files changed, 464 insertions(+) create mode 100644 articles/annulus-and-spherical-stokes-benchmarks/annulus-and-spherical-stokes-benchmarks.md create mode 100644 articles/annulus-and-spherical-stokes-benchmarks/figures/banner.jpg create mode 100644 articles/annulus-and-spherical-stokes-benchmarks/figures/boundary_metric_convergence.jpg create mode 100644 articles/annulus-and-spherical-stokes-benchmarks/figures/figure_3_kramer_annulus_convergence.jpg create mode 100644 articles/annulus-and-spherical-stokes-benchmarks/figures/figure_4_kramer_spherical_convergence.jpg create mode 100644 articles/annulus-and-spherical-stokes-benchmarks/figures/figure_5_thieulot_annulus_convergence.jpg create mode 100644 articles/annulus-and-spherical-stokes-benchmarks/figures/figure_7_kramer_boundary_convergence.jpg create mode 100644 articles/annulus-and-spherical-stokes-benchmarks/figures/figure_boundary_pressure_convergence.jpg create mode 100644 articles/annulus-and-spherical-stokes-benchmarks/figures/figure_p2p1_boundary_pressure_convergence.jpg create mode 100644 articles/annulus-and-spherical-stokes-benchmarks/figures/figures_4_5_thieulot_convergence.jpg create mode 100644 articles/annulus-and-spherical-stokes-benchmarks/metadata.yml create mode 100644 articles/annulus-and-spherical-stokes-benchmarks/references.bib diff --git a/articles/annulus-and-spherical-stokes-benchmarks/annulus-and-spherical-stokes-benchmarks.md b/articles/annulus-and-spherical-stokes-benchmarks/annulus-and-spherical-stokes-benchmarks.md new file mode 100644 index 0000000..db98069 --- /dev/null +++ b/articles/annulus-and-spherical-stokes-benchmarks/annulus-and-spherical-stokes-benchmarks.md @@ -0,0 +1,354 @@ +--- +title: Benchmarking Stokes Flow in Annulus and Spherical-Shell Geometries +description: >- + A suite of annulus and spherical-shell Stokes benchmarks reproduced in + Underworld3. What each benchmark is designed to test, what numerical + behaviour to expect from it, and how to read the convergence results — + including where reduced rates are the correct answer rather than a defect. +date: 2026-08-13 +authors: + - name: Thyagarajulu Gollapalli + orcid: 0000-0001-9394-4104 + affiliations: + - Monash University +license: CC-BY-4.0 +bibliography: + - references.bib +banner: figures/banner.jpg +keywords: + - Underworld Code + - benchmarks + - development +exports: + - format: typst + logo: ../../static/uwtn-logo.png + series: "Underworld Technical Notes" + origin_url: https://www.underworldcode.org/annulus-and-spherical-stokes-benchmarks/ + template: ../../templates/pdf + output: annulus-and-spherical-stokes-benchmarks.pdf + article_id: UWTN 2026-012 + article_version: 1.0.0 + software_version: underworld3 0.0.0 +--- +

+ +This post introduces a suite of annulus and spherical-shell Stokes benchmarks +reproduced using Underworld3 [@Moresi_2025]. These benchmark problems have +previously been implemented in several numerical codes, but here we bring them +together within a single framework to highlight both the strengths and +practical challenges of curved-domain finite-element modelling. The post is +aimed at researchers working on geodynamics numerical modelling. Rather than +focusing on heavy mathematical derivations or low-level implementation details, +the goal is to provide an intuitive and practical guide to what each benchmark +is designed to test, what numerical behaviour should be expected, and how to +interpret the results. By the end, readers should have a clear understanding of +the purpose of these benchmarks and the key ideas behind verifying Stokes flow +in curved geometries. + +## What Is Benchmarking and Why Is It Important? + +Benchmarking is the process of testing a numerical method or software +implementation against problems with known analytical solutions or +well-established reference results. In computational geodynamics, benchmarks +are essential because they help verify that a code correctly solves the +governing equations before it is applied to complex Earth-science problems +where the true solution is unknown. A good benchmark does more than produce a +visually reasonable result; it tests numerical convergence behaviour, +boundary-condition implementation, mesh geometry, and solver robustness under +controlled conditions. Curved-domain benchmarks such as annulus and +spherical-shell Stokes problems are particularly important because they expose +numerical challenges that do not appear in simple Cartesian geometries. +Successfully reproducing benchmark results therefore builds confidence that a +numerical framework can model real-world problems with reliable and physically +meaningful approximations. + +## What Are the Stokes Equations? + +The Stokes equations describe the slow, viscous flow of fluids in situations +where inertial forces are negligible compared with viscous forces. In +geodynamics, this approximation is widely used because rocks in the Earth's +mantle deform extremely slowly over geological timescales and behave like +highly viscous fluids. The incompressible Stokes equations are written as + +$$ +\begin{aligned} +-\nabla \cdot \left(2 \eta \dot{\varepsilon}(\mathbf{u})\right) ++ \nabla p &= \rho \mathbf{g}, \\ +\nabla \cdot \mathbf{u} &= 0 . +\end{aligned} +$$ + +Here $\mathbf{u}$ is velocity, $p$ is pressure, $\eta$ is viscosity, $\rho$ is +density, and $\mathbf{g}$ is gravity. The first equation represents +conservation of momentum, balancing viscous stresses, pressure gradients, and +body forces. The second equation enforces mass conservation through +incompressibility. Solving these equations allows us to model mantle +convection, lithospheric deformation, subduction, and many other large-scale +Earth processes. Although the equations appear compact, solving them accurately +in curved geometries with complex boundary conditions and variable material +properties is computationally challenging, which is why benchmark problems are +important. + +## Why Do We Need Curved Geometries? + +Curved geometries are important in geodynamics because the Earth itself is +curved. Many large-scale Earth processes, such as mantle convection, +subduction, plume dynamics, and lithospheric deformation, occur within +spherical or shell-like domains rather than simple rectangular boxes. While +Cartesian geometries are useful for developing intuition and testing numerical +methods, they cannot fully represent radial gravity, curved boundaries, or +global-scale flow patterns. Curved-domain models also introduce additional +numerical challenges, including geometric approximation errors, coordinate +transformations, and the accurate implementation of free-slip or zero-slip +boundary conditions on non-planar surfaces. As a result, annulus and +spherical-shell benchmarks provide a more realistic and demanding test of +Stokes solvers. + +## Benchmark Suite + +In this work, we reproduce four widely used Stokes benchmark suites in curved +geometries using Underworld3 [@Moresi_2025]. The first is the +Thieulot–Puckett annulus benchmark, which provides a smooth analytical solution +in an annulus and is mainly used to test optimal finite-element convergence +behaviour [@ThieulotPuckett_2018]. The second is the Kramer annulus benchmark, +which extends the problem to include both smooth volumetric forcing and +singular delta-function forcing on an internal interface, together with +free-slip and zero-slip boundary conditions [@Kramer_2021]. We then consider +the spherical-shell counterparts of these problems: the Thieulot spherical +benchmark, which tests smooth Stokes flow in spherical geometry with both +constant and radially varying viscosity [@Thieulot_2017], and the Kramer +spherical benchmark, which again introduces internal interface forcing and +reduced solution regularity [@Kramer_2021]. Together, these four benchmark +suites test curved geometries, pressure treatment, mesh approximation, +boundary-condition implementation, smooth and singular forcing, and convergence +behaviour in both two- and three-dimensional Stokes flow. + +The figure at the head of this post collects the analytical benchmark fields +for all four cases: the Thieulot–Puckett annulus, the Kramer annulus, the +Thieulot spherical shell, and the Kramer spherical shell. + +## What Do We Measure? Error Quantification + +The benchmark comparisons use $L_2$-norm errors because they give a single +quantitative measure of the difference between the numerical solution and the +analytical solution over the whole domain. For a computed field $q_h$ and +analytical field $q^{*}$, the absolute volume error is + +$$ +E_{L_2}(q) += +\left( +\int_{\Omega} |q_h-q^*|^2\,\mathrm{d}\Omega +\right)^{1/2}. +$$ + +The corresponding relative volume error is + +$$ +E_{L_2}^{\mathrm{rel}}(q) += +\frac{E_{L_2}(q)}{\|q^*\|_{L_2}} += +\left( +\frac{\int_{\Omega} |q_h-q^*|^2\,\mathrm{d}\Omega} + {\int_{\Omega} |q^*|^2\,\mathrm{d}\Omega} +\right)^{1/2}. +$$ + +The relative form is useful when comparing velocity and pressure errors across +different benchmark cases because it normalises the error by the size of the +analytical solution. For pressure, the numerical and analytical fields are +first compared in the same pressure gauge, since incompressible Stokes pressure +is determined only up to an additive constant. + +Boundary pressure errors are measured separately on the inner and outer +surfaces. For a boundary $\Gamma \in \{\Gamma_{\mathrm{inner}}, +\Gamma_{\mathrm{outer}}\}$, the absolute pressure-trace error is + +$$ +E_{L_2,\Gamma}(p) += +\left( +\int_{\Gamma} |p_h-p^*|^2\,\mathrm{d}\Gamma +\right)^{1/2}. +$$ + +Where the analytical boundary pressure has a nonzero $L_2$ norm, the relative +boundary pressure error is + +$$ +E_{L_2,\Gamma}^{\mathrm{rel}}(p) += +\left( +\frac{\int_{\Gamma} |p_h-p^*|^2\,\mathrm{d}\Gamma} + {\int_{\Gamma} |p^*|^2\,\mathrm{d}\Gamma} +\right)^{1/2}. +$$ + +Convergence is measured by comparing errors across successively refined meshes: + +$$ +\mathrm{rate} += +\frac{ +\log\left(E_{h_1}/E_{h_2}\right) +}{ +\log\left(h_1/h_2\right) +}. +$$ + +Here $h$ is the characteristic cell size. If the mesh is uniformly refined so +that $h_2 = h_1/2$, this becomes + +$$ +\mathrm{rate} += +\log_2\left(\frac{E_{h_1}}{E_{h_2}}\right). +$$ + +The expected behaviour is simple: as $h$ decreases, the error should decrease. +On a log-log convergence plot, a method with error proportional to $C h^{r}$ +appears approximately as a straight line with slope $r$. For smooth Stokes +solutions, stable mixed finite-element pairs have well-defined optimal +convergence expectations; for example, Taylor–Hood $P_2 \times P_1$ commonly +gives third-order velocity and second-order pressure convergence in the volume +$L_2$ norm when the geometry and solution are sufficiently smooth +[@Boffi_2013]. Singular forcing cases are different: the solution is less +regular near the internal interface, so reduced convergence rates are expected +even when the solver is implemented correctly. + +## What Do We Observe? Results of Benchmarks in Underworld3 + +### Volumetric Convergence + +The benchmark results show that Underworld3 reproduces the expected convergence +behaviour for both annulus and spherical-shell Stokes problems. For smooth +analytical solutions, such as the Thieulot annulus and spherical benchmarks, +the Taylor–Hood $P_2 \times P_1$ discretisation achieves close to the +theoretically expected convergence rates, with approximately third-order +velocity convergence and second-order pressure convergence in the volume $L_2$ +norm. Higher-order element pairs further improve accuracy, while lower-order +discretisations show the expected reduction in convergence order. + +For the Kramer benchmarks with smooth forcing, the velocity convergence is +closer to second order because the curved geometry is represented using linear +meshes, consistent with the original benchmark studies [@Kramer_2021]. In the +delta-function forcing cases, the convergence rates reduce significantly +because the internal singular interface lowers the regularity of the analytical +solution. Overall, the volumetric results demonstrate that Underworld3 captures +both optimal convergence behaviour for smooth problems and the expected +degradation in accuracy for singular forcing cases. + +```{figure} figures/figure_5_thieulot_annulus_convergence.jpg +:alt: Log-log convergence plots of velocity and pressure error against cell size for the Thieulot-Puckett annulus benchmark. + +Velocity and pressure convergence for the Thieulot–Puckett annulus benchmark. +``` + +```{figure} figures/figure_3_kramer_annulus_convergence.jpg +:alt: Log-log convergence plots of velocity and pressure error against cell size for the Kramer annulus benchmark. + +Velocity and pressure convergence for the Kramer annulus benchmark. +``` + +```{figure} figures/figures_4_5_thieulot_convergence.jpg +:alt: Log-log convergence plots of velocity and pressure error against cell size for the Thieulot spherical-shell benchmark. + +Velocity and pressure convergence for the Thieulot spherical-shell benchmark. +``` + +```{figure} figures/figure_4_kramer_spherical_convergence.jpg +:alt: Log-log convergence plots of velocity and pressure error against cell size for the Kramer spherical-shell benchmark. + +Velocity and pressure convergence for the Kramer spherical-shell benchmark. +``` + +### Boundary Pressure Convergence + +Boundary diagnostics provide a more local and often more sensitive measure of +solver accuracy in curved geometries. The volume pressure norm measures an +error over the full domain, but boundary pressure errors isolate the pressure +trace on the surfaces where boundary conditions, radial normal stresses, and +traction-related diagnostics are evaluated. This is useful because geometric +approximation errors and boundary quadrature errors can be more visible on +curved inner and outer boundaries than in a volume-averaged metric. + +For the Thieulot–Puckett annulus benchmark, the $P_2 \times P_1$ boundary +pressure errors decrease close to the expected $O(h^2)$ trend. The inner and +outer boundaries do not have identical error constants, which is acceptable: +the two curves have different radii, different geometric representation errors, +and different boundary integration paths. + +```{figure} figures/figure_p2p1_boundary_pressure_convergence.jpg +:alt: Log-log plot of absolute boundary pressure error against cell size for the Thieulot-Puckett annulus benchmark, with separate curves for the inner and outer boundaries. + +Absolute boundary pressure error convergence for the Thieulot–Puckett annulus +benchmark using the $P_2 \times P_1$ discretisation. Solid curves denote the +inner boundary and dashed curves denote the outer boundary. +``` + +For the Kramer annulus benchmark, the boundary pressure errors converge close +to second order for the smooth cases. The delta-function cases also show +stronger boundary pressure convergence than the corresponding volume pressure +norm because the reduced regularity is localized at the internal interface +rather than on the annulus boundaries. + +```{figure} figures/figure_boundary_pressure_convergence.jpg +:alt: Log-log plot of relative boundary pressure error against cell size for the Kramer annulus benchmark, with separate curves for the inner and outer boundaries. + +Relative boundary pressure error convergence for the Kramer annulus benchmark. +Solid curves denote the inner boundary and dashed curves denote the outer +boundary. +``` + +The Thieulot spherical-shell benchmark gives the corresponding smooth +three-dimensional boundary test. The boundary-pressure panel shows that the +absolute pressure-trace errors for both $m = -1$ and $m = 3$ decrease +systematically with refinement. These errors are shown together with radial +normal-stress errors because the stress diagnostic combines pressure and +velocity-gradient errors at the same curved spherical boundaries. + +```{figure} figures/boundary_metric_convergence.jpg +:alt: Two panels of log-log convergence plots for the Thieulot spherical-shell benchmark: boundary radial normal-stress error on the left and absolute boundary pressure error on the right. + +Boundary radial normal-stress and absolute boundary pressure error convergence +for the Thieulot spherical-shell benchmark. The right panel shows +$E_{L_2,\Gamma}(p)$ on the inner and outer spherical boundaries. +``` + +For the Kramer spherical-shell benchmark, the current boundary figure reports +boundary velocity and radial normal-stress convergence for the free-slip +delta-function case. A separate boundary pressure-trace convergence plot is not +part of the current spherical Kramer article outputs. The radial normal-stress +diagnostic is included here because it is pressure-sensitive: $\sigma_{rr}$ +contains the pressure contribution and therefore tests pressure recovery +together with velocity-gradient recovery on the spherical boundaries. + +```{figure} figures/figure_7_kramer_boundary_convergence.jpg +:alt: Log-log convergence plots of boundary velocity error and radial normal-stress error against cell size for the Kramer spherical-shell benchmark. + +Boundary velocity and radial normal-stress convergence for the Kramer +spherical-shell benchmark. This is the available pressure-sensitive boundary +diagnostic for the spherical Kramer case. +``` + +Overall, the boundary diagnostics complement the volume $L_2$ pressure errors. +Smooth benchmark cases recover approximately second-order pressure-trace +convergence for the $P_2 \times P_1$ pair, while singular forcing primarily +affects the volume pressure norm through reduced regularity at the internal +interface. Persistent differences between inner- and outer-boundary error +constants should therefore be interpreted as geometric and boundary-evaluation +effects rather than as a failure of pressure normalisation. + +The full details are in the article PDFs, which can be viewed directly from the +[benchmark figures and articles directory on GitHub](https://github.com/gthyagi/UW3_Annulus_Spherical_Benchmarks/tree/main/docs/benchmarks_figures_and_articles): + +- [Thieulot–Puckett annulus benchmark](https://github.com/gthyagi/UW3_Annulus_Spherical_Benchmarks/blob/main/docs/benchmarks_figures_and_articles/annulus/thieulot/thieulot_annulus_benchmark_article.pdf) +- [Kramer annulus benchmark](https://github.com/gthyagi/UW3_Annulus_Spherical_Benchmarks/blob/main/docs/benchmarks_figures_and_articles/annulus/kramer/kramer_annulus_benchmark_article.pdf) +- [Thieulot spherical-shell benchmark](https://github.com/gthyagi/UW3_Annulus_Spherical_Benchmarks/blob/main/docs/benchmarks_figures_and_articles/spherical/thieulot/thieulot_spherical_benchmark_article.pdf) +- [Kramer spherical-shell benchmark](https://github.com/gthyagi/UW3_Annulus_Spherical_Benchmarks/blob/main/docs/benchmarks_figures_and_articles/spherical/kramer/kramer_spherical_benchmark_article.pdf) + +Those articles contain the analytical fields, discretisation choices, +convergence tables, boundary diagnostics, and full reference details. + +
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zwssa)zH%tp&inb8{H=b%F}~D}XM#*;V20J6oNmlm;pc>KpmmlFM^2AKtVyi-eStF~ zylxO8!uzsYx>PLiyiv@Nnm@uO+SnJ6AC>47Jcdcn61T9RoJgxJx+H7@A3g=HJ;>r=^Fy)W!EG095Q4oWYFXn2QAfydmvl9w7+{%uy(Ss&G!KP0$wX zo6%G`o+&RQ=9kG@O7YjX=Sw#w#XRS>fAp(!^N)@BZ~1Q_aDrHk6uI4C++)&q5Qvs- z$Qy?hyh!z^AoJWez!g{#w7Gkij&$i4LWKjRiFGhj1)=RllxR^J4v}G#&TxB)2nAfF z;R0SF?)1=QiFsKuf$*^!iCg&SGiHANE1ANv`ay1ZN$8^+|K7U(t!DjmuAO0D9GFwL zu6XR5dq1pjS&~L|op%VLwR>qT8Uh3}*XQ9j*KFEI(;Tx&O0fqM;|nFlQc)x^{8%A8pS+p+5WboSOCT zBA>PlA=svJQ!_VgUJ<<5p*eNA9jN$9r}B&7_<)jxqn{|SIDv>DclrR{33CC0Dp0tY z_^5%c;zg0Eh#<=mTEAlqEJ#A%grCHlW#MTBT2kn*0gY+@*K~@>X2D$a1u+k#+4(ws zqrUB_9tXf*rT+H9oI1Awq=D=F~z;0L7#YVSI zb#mhUhr(hI+0f}2h|E.md, that canonical_path +# matches the slug, and that no legacy DOI is ever paired with a new registrant. +id: UWTN 2026-012 +slug: annulus-and-spherical-stokes-benchmarks +title: Benchmarking Stokes Flow in Annulus and Spherical-Shell Geometries +article_type: technical-note +status: review +authors: + - name: Thyagarajulu Gollapalli + orcid: 0000-0001-9394-4104 + affiliation: Monash University +publication_date: null +version: 1.0.0 +# The deposit writes archive_doi and repository_record_id when the note is +# published; leave them out until then. `doi` and `doi_registrant` were here +# once and are not fields the schema knows -- every note made from this +# template failed `pixi run validate` on all three of them. +license: CC-BY-4.0 +canonical_path: /annulus-and-spherical-stokes-benchmarks/ +legacy_paths: [] +# Facets, from vocabulary.yml. Both keys must be present even when empty: a +# note with no subject is normal -- many are purely about method. +subjects: + - mantle-convection + - planetary +methods: + - benchmarks-validation + - solvers + - finite-elements +ghost_tags: + - Underworld Code +figures: 8 +source: native diff --git a/articles/annulus-and-spherical-stokes-benchmarks/references.bib b/articles/annulus-and-spherical-stokes-benchmarks/references.bib new file mode 100644 index 0000000..54979b8 --- /dev/null +++ b/articles/annulus-and-spherical-stokes-benchmarks/references.bib @@ -0,0 +1,70 @@ +% Pinned rather than resolved from doi.org at build time. Letting MyST fetch +% citations over the network makes the build depend on doi.org answering, which +% failed on CI for UWTN 2026-011 and would have published a broken citation. + +@article{Moresi_2025, + title = {Underworld3: Mathematically Self-Describing Modelling in Python for Desktop, HPC and Cloud}, + volume = {10}, + ISSN = {2475-9066}, + url = {http://dx.doi.org/10.21105/joss.07831}, + DOI = {10.21105/joss.07831}, + number = {112}, + journal = {Journal of Open Source Software}, + publisher = {The Open Journal}, + author = {Moresi, Louis and Mansour, John and Giordani, Julian and Knepley, Matt and Knight, Ben and Graciosa, Juan Carlos and Gollapalli, Thyagarajulu and Lu, Neng and Beucher, Romain}, + year = {2025}, + month = aug, + pages = {7831} +} + +% No DOI: a preprint submitted to Computers & Geosciences, cited in the original +% post by its author-hosted PDF. The URL is the only persistent handle there is. +@unpublished{ThieulotPuckett_2018, + title = {Incompressible Stokes flow in an annulus: an analytical solution and numerical benchmark}, + author = {Thieulot, Cedric and Puckett, Elbridge Gerry}, + year = {2018}, + note = {Preprint submitted to Computers \& Geosciences}, + url = {https://www.math.ucdavis.edu/~egp/PUBLICATIONS/JOURNAL_ARTICLES/SUBMITTED/CAPT-EGP-2018.pdf} +} + +@article{Kramer_2021, + title = {Analytical solutions for mantle flow in cylindrical and spherical shells}, + volume = {14}, + ISSN = {1991-9603}, + url = {http://dx.doi.org/10.5194/gmd-14-1899-2021}, + DOI = {10.5194/gmd-14-1899-2021}, + number = {4}, + journal = {Geoscientific Model Development}, + publisher = {Copernicus GmbH}, + author = {Kramer, Stephan C. and Davies, D. Rhodri and Wilson, Cian R.}, + year = {2021}, + month = apr, + pages = {1899--1919} +} + +@article{Thieulot_2017, + title = {Analytical solution for viscous incompressible Stokes flow in a spherical shell}, + volume = {8}, + ISSN = {1869-9529}, + url = {http://dx.doi.org/10.5194/se-8-1181-2017}, + DOI = {10.5194/se-8-1181-2017}, + number = {6}, + journal = {Solid Earth}, + publisher = {Copernicus GmbH}, + author = {Thieulot, Cedric}, + year = {2017}, + month = nov, + pages = {1181--1191} +} + +@book{Boffi_2013, + title = {Mixed Finite Element Methods and Applications}, + ISBN = {9783642365195}, + ISSN = {0179-3632}, + url = {http://dx.doi.org/10.1007/978-3-642-36519-5}, + DOI = {10.1007/978-3-642-36519-5}, + series = {Springer Series in Computational Mathematics}, + publisher = {Springer Berlin Heidelberg}, + author = {Boffi, Daniele and Brezzi, Franco and Fortin, Michel}, + year = {2013} +} diff --git a/classification.yml b/classification.yml index 3ad124a..a658a5f 100644 --- a/classification.yml +++ b/classification.yml @@ -312,3 +312,8 @@ moving-the-mesh-without-remaking-it: article_type: technical-note subjects: [mantle-convection, earthquakes-faults] methods: [meshing, solvers, parallel-hpc] + +annulus-and-spherical-stokes-benchmarks: + article_type: technical-note + subjects: [mantle-convection, planetary] + methods: [benchmarks-validation, solvers, finite-elements] From 0328d445d6c77cba718038ba666380a2c126ddef Mon Sep 17 00:00:00 2001 From: lmoresi Date: Thu, 13 Aug 2026 21:29:09 +1000 Subject: [PATCH 2/2] Give the note a photographic banner and put the science figure back in the text Louis's call, and the right way round. The combined analytical-fields image was serving as the banner, which meant it could not also be a captioned figure where its author put it -- it would have rendered twice. A photograph in the header frees the figure to go back to the Benchmark Suite section with the caption gthyagi wrote for it. Nine figures again. The banner is credited through banner_credit in metadata.yml, not in the markdown. Writing the credit into the body does not survive: build-pdf strips the banner block and build-html regenerates it, so a hand-written credit is destroyed and replaced by a plain one. banner_credit is the field the regeneration reads, which is how Unsplash's attribution requirement survives a rebuild. Underworld development team with AI support from Claude Code --- ...annulus-and-spherical-stokes-benchmarks.md | 12 ++++++++---- .../figures/banner.jpg | Bin 423995 -> 291218 bytes .../figures/benchmark-fields.jpg | Bin 0 -> 423995 bytes .../metadata.yml | 3 ++- 4 files changed, 10 insertions(+), 5 deletions(-) create mode 100644 articles/annulus-and-spherical-stokes-benchmarks/figures/benchmark-fields.jpg diff --git a/articles/annulus-and-spherical-stokes-benchmarks/annulus-and-spherical-stokes-benchmarks.md b/articles/annulus-and-spherical-stokes-benchmarks/annulus-and-spherical-stokes-benchmarks.md index db98069..e01c700 100644 --- a/articles/annulus-and-spherical-stokes-benchmarks/annulus-and-spherical-stokes-benchmarks.md +++ b/articles/annulus-and-spherical-stokes-benchmarks/annulus-and-spherical-stokes-benchmarks.md @@ -30,7 +30,7 @@ exports: article_version: 1.0.0 software_version: underworld3 0.0.0 --- -
+
This post introduces a suite of annulus and spherical-shell Stokes benchmarks reproduced using Underworld3 [@Moresi_2025]. These benchmark problems have @@ -123,9 +123,13 @@ suites test curved geometries, pressure treatment, mesh approximation, boundary-condition implementation, smooth and singular forcing, and convergence behaviour in both two- and three-dimensional Stokes flow. -The figure at the head of this post collects the analytical benchmark fields -for all four cases: the Thieulot–Puckett annulus, the Kramer annulus, the -Thieulot spherical shell, and the Kramer spherical shell. +```{figure} figures/benchmark-fields.jpg +:alt: Four panels of analytical benchmark fields, showing density distributions for the Thieulot-Puckett annulus, the Kramer annulus, the Thieulot spherical shell and the Kramer spherical shell. + +Analytical benchmark fields used in the annulus and spherical-shell Stokes +benchmark suite. The panels collect the Thieulot–Puckett annulus, Kramer +annulus, Thieulot spherical-shell, and Kramer spherical-shell cases. +``` ## What Do We Measure? 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