References¶
Every method that dualmesh implements is cited here to the work that introduced it, not to a textbook that repeats it and not to another code that uses it. Where the implementation follows a design decision taken by another project — the object model of MOOSE, the way OVITO organises its manual — the project is named as the influence and the underlying method is cited separately to its own source.
Each entry below was checked against the publisher’s record: a DOI landing page or the publisher-deposited metadata held by Crossref, the project’s own citation page for the software entries, or a library catalogue record for the books. Where two sources disagreed, the disagreement is noted in the entry. The checks were made in September 2026.
The method¶
J. N. Reddy, Computational Methods in Engineering: Finite Difference, Finite Volume, Finite Element, and Dual Mesh Control Domain Methods, 1st edition, CRC Press, Boca Raton, FL, 2024, 594 pp., ISBN 978-1-032-46637-8. DOI: 10.1201/9781003382812. The dual mesh control domain method is Chapter 5; the finite volume methods of Theory manual are Chapter 3. Every verification case in Verification that is marked “book” comes from this text.
J. N. Reddy, “A dual mesh finite domain method for the numerical solution of differential equations”, International Journal for Computational Methods in Engineering Science and Mechanics, 20(3):212–228, 2019. DOI: 10.1080/15502287.2019.1610987. The paper that introduces the method.
J. N. Reddy, N. Kim and M. Martinez, “A dual mesh control domain method for the solution of nonlinear Poisson’s equation and the Navier–Stokes equations for incompressible fluids”, Physics of Fluids, 32(9):093608, 2020. DOI: 10.1063/5.0026274. The first nonlinear application, and the first paper to use the name control domain rather than finite domain.
J. N. Reddy and P. Nampally, “A dual mesh finite domain method for the analysis of functionally graded beams”, Composite Structures, 251:112648, 2020. DOI: 10.1016/j.compstruct.2020.112648.
J. N. Reddy, P. Nampally and A. R. Srinivasa, “Nonlinear analysis of functionally graded beams using the dual mesh finite domain method and the finite element method”, International Journal of Non-Linear Mechanics, 127:103575, 2020. DOI: 10.1016/j.ijnonlinmec.2020.103575. The source of the von Kármán beam formulation used by the beams of the solid mechanics module.
P. Nampally and J. N. Reddy, “Bending analysis of functionally graded axisymmetric circular plates using the dual mesh finite domain method”, Latin American Journal of Solids and Structures, 17(7):e302, 2020. DOI: 10.1590/1679-78256218.
J. N. Reddy and M. Martinez, “A dual mesh finite domain method for steady-state convection–diffusion problems”, Computers & Fluids, 214:104760, 2021. DOI: 10.1016/j.compfluid.2020.104760.
J. N. Reddy, P. Nampally and N. Phan, “Dual mesh control domain analysis of functionally graded circular plates accounting for moderate rotations”, Composite Structures, 257:113153, 2021. DOI: 10.1016/j.compstruct.2020.113153. The source of the von Kármán circular plate formulation.
P. Nampally, E. Ruocco and J. N. Reddy, “Bending analysis of functionally graded rectangular plates using the dual mesh control domain method”, International Journal for Computational Methods in Engineering Science and Mechanics, 22(5):425–437, 2021. DOI: 10.1080/15502287.2021.1890279.
Z. Jiao, T. Heblekar, G. Wang, R. Xu, W. Chen and J. N. Reddy, “Analysis of plane elasticity problems using the dual mesh control domain method”, Computer Methods in Applied Mechanics and Engineering, 416:116342, 2023. DOI: 10.1016/j.cma.2023.116342.
T. Y. Heblekar, J. N. Reddy and A. R. Srinivasa, “Analysis of nonlinear problems using the Dual Mesh Control Domain Method with arbitrary meshes”, Computer Methods in Applied Mechanics and Engineering, 427:117044, 2024. DOI: 10.1016/j.cma.2024.117044. The paper that takes the method to unstructured meshes and to Newton linearisation, which is what dualmesh implements.
Z. Jiao, T. Heblekar, G. Wang, R. Xu and J. N. Reddy, “Static, free vibration, and buckling analysis of functionally graded plates using the dual mesh control domain method”, Computers & Structures, 305:107575, 2024. DOI: 10.1016/j.compstruc.2024.107575.
P. Areias, A. R. Srinivasa, F. Moleiro and J. N. Reddy, “Finite strain analysis with the dual mesh control domain method”, International Journal for Numerical Methods in Engineering, 126(1):e7654, 2025. DOI: 10.1002/nme.7654.
T. Y. Heblekar and J. N. Reddy, “An improved dual mesh control domain formulation for the unsteady flow of viscous incompressible fluids”, Physics of Fluids, 37(3):033119, 2025. DOI: 10.1063/5.0259696. The transient formulation for the Navier–Stokes equations.
T. Y. Heblekar, J. N. Reddy and A. R. Srinivasa, “p-Refinement in the dual-mesh control domain method”, Computing in Science & Engineering, 28(1):64–73, 2026. DOI: 10.1109/MCSE.2025.3641847.
T. Y. Heblekar, J. N. Reddy and A. R. Srinivasa, “A spectral dual mesh control domain framework for nonlinear fluid flow and coupled multi-field problems”, International Journal of Non-Linear Mechanics, 186:105350, 2026. DOI: 10.1016/j.ijnonlinmec.2026.105350.
T. Heblekar, J. N. Reddy and A. Srinivasa, “Dual-mesh control-domain analysis of nonlinear problems in mechanics”, Theoretical and Applied Mechanics (Belgrade), 2026, online first. DOI: 10.2298/TAM260601007H. Volume, issue and pages were not assigned at the time of checking.
Finite volume methods¶
S. V. Patankar, Numerical Heat Transfer and Fluid Flow, Hemisphere Publishing, Washington, DC, 1980. The cell-centred finite volume method in the form Chapter 3 of [Reddy2024] presents it.
H. Jasak, Error Analysis and Estimation for the Finite Volume Method with Applications to Fluid Flows, PhD thesis, Imperial College London, 1996. The non-orthogonal correction and the deferred correction treatment of it, in the form used by the two finite volume discretisations of dualmesh.
I. Demirdžić and S. Muzaferija, “Numerical method for coupled fluid flow, heat transfer and stress analysis using unstructured moving meshes with cells of arbitrary topology”, Computer Methods in Applied Mechanics and Engineering, 125(1–4):235–255, 1995. DOI: 10.1016/0045-7825(95)00800-G.
T. J. Barth and D. C. Jespersen, “The design and application of upwind schemes on unstructured meshes”, AIAA Paper 89-0366, 27th Aerospace Sciences Meeting, Reno, NV, 9–12 January 1989. DOI: 10.2514/6.1989-366. The least-squares gradient reconstruction used by the cell-centred method.
T. J. Barth, “Recent developments in high order K-exact reconstruction on unstructured meshes”, AIAA Paper 93-0668, 31st Aerospace Sciences Meeting and Exhibit, Reno, NV, 11–14 January 1993. DOI: 10.2514/6.1993-668.
Finite elements¶
B. M. Irons, “Engineering applications of numerical integration in stiffness methods”, AIAA Journal, 4(11):2035–2037, 1966. DOI: 10.2514/3.3836. The isoparametric map, which dualmesh uses for the geometry of every element and of every control domain patch.
G. Bedrosian, “Shape functions and integration formulas for three-dimensional finite element analysis”, International Journal for Numerical Methods in Engineering, 35(1):95–108, 1992. DOI: 10.1002/nme.1620350106. The rational basis of the pyramid element.
M. G. Duffy, “Quadrature over a pyramid or cube of integrands with a singularity at a vertex”, SIAM Journal on Numerical Analysis, 19(6):1260–1262, 1982. DOI: 10.1137/0719090. The collapsed-coordinate map by which a triangle, a tetrahedron, a prism or a pyramid is integrated as a square or a cube with some corners merged.
J. Douglas, Jr. and T. Dupont, “Galerkin approximations for the two point boundary problem using continuous, piecewise polynomial spaces”, Numerische Mathematik, 22(2):99–109, 1974. DOI: 10.1007/BF01436724. The nodal superconvergence of the Galerkin method: the finite element solution of a two-point boundary value problem is more accurate at the nodes than anywhere else, which is why quadratic elements give fourth-order nodal values. (The publisher’s deposited metadata lists only the first author; both are confirmed by the zbMATH and EUDML records of the same article.)
J.-P. Aubin, “Behavior of the error of the approximate solutions of boundary value problems for linear elliptic operators by Galerkin’s and finite difference methods”, Annali della Scuola Normale Superiore di Pisa, Classe di Scienze, Serie 3, 21(4):599–637, 1967. With [Nitsche1968], the duality argument that gives the Galerkin method one more order of accuracy in \(L^2\) than in the energy norm.
J. Nitsche, “Ein Kriterium für die Quasi-Optimalität des Ritzschen Verfahrens”, Numerische Mathematik, 11(4):346–348, 1968. DOI: 10.1007/BF02166687.
P. J. Roache, “Code verification by the method of manufactured solutions”, Journal of Fluids Engineering, 124(1):4–10, 2002. DOI: 10.1115/1.1436090. The method of manufactured solutions as a procedure for verifying the order of accuracy of a code.
K. Salari and P. Knupp, Code Verification by the Method of Manufactured Solutions, Sandia National Laboratories report SAND2000-1444, 2000. DOI: 10.2172/759450.
J. Barlow, “Optimal stress locations in finite element models”, International Journal for Numerical Methods in Engineering, 10(2):243–251, 1976. DOI: 10.1002/nme.1620100202. The Gauss points of an element are where the derivative of a finite element solution is most accurate. This is the result that explains why the dual mesh control domain method gains no order from quadratic elements: its control domain interfaces are at the midpoints between nodes, not at the Gauss points.
O. C. Zienkiewicz, R. L. Taylor and J. M. Too, “Reduced integration technique in general analysis of plates and shells”, International Journal for Numerical Methods in Engineering, 3(2):275–290, 1971. DOI: 10.1002/nme.1620030211.
T. J. R. Hughes, M. Cohen and M. Haroun, “Reduced
and selective integration techniques in the finite element analysis of
plates”, Nuclear Engineering and Design, 46(1):203–222, 1978.
DOI: 10.1016/0029-5493(78)90184-X. The source of the
reduced_integration parameter.
D. S. Malkus and T. J. R. Hughes, “Mixed finite element methods — reduced and selective integration techniques: a unification of concepts”, Computer Methods in Applied Mechanics and Engineering, 15(1):63–81, 1978. DOI: 10.1016/0045-7825(78)90005-1. The equivalence between selective reduced integration and a mixed formulation, which is why reduced integration of the penalty and shear terms is legitimate rather than a trick.
G. de Vahl Davis, “Natural convection of air in a square
cavity: a bench mark numerical solution”, International Journal for
Numerical Methods in Fluids, 3(3):249–264, 1983.
DOI: 10.1002/fld.1650030305. The benchmark of the coupled flow and heat
transfer test, examples/natural_convection.py.
T. J. R. Hughes, W. K. Liu and A. Brooks, “Finite element analysis of incompressible viscous flows by the penalty function formulation”, Journal of Computational Physics, 30(1):1–60, 1979. DOI: 10.1016/0021-9991(79)90086-X. The penalty formulation used by the fluids module.
Solvers¶
R. E. Wengert, “A simple automatic derivative evaluation program”, Communications of the ACM, 7(8):463–464, 1964. DOI: 10.1145/355586.364791. Forward-mode automatic differentiation, which is how dualmesh forms exact Jacobians.
A. Griewank and A. Walther, Evaluating Derivatives: Principles and Techniques of Algorithmic Differentiation, 2nd edition, SIAM, Philadelphia, PA, 2008, ISBN 978-0-89871-659-7. DOI: 10.1137/1.9780898717761.
M. R. Hestenes and E. Stiefel, “Methods of conjugate gradients for solving linear systems”, Journal of Research of the National Bureau of Standards, 49(6):409–436, 1952 (Research Paper 2379). Several secondary sources give the last page as 435; the scanned article ends on page 436.
H. A. van der Vorst, “Bi-CGSTAB: a fast and smoothly converging variant of Bi-CG for the solution of nonsymmetric linear systems”, SIAM Journal on Scientific and Statistical Computing, 13(2):631–644, 1992. DOI: 10.1137/0913035.
Y. Saad, “ILUT: a dual threshold incomplete LU factorization”,
Numerical Linear Algebra with Applications, 1(4):387–402, 1994.
DOI: 10.1002/nla.1680010405. The threshold preconditioner offered as
preconditioner = "ilut".
Y. Saad, Iterative Methods for Sparse Linear Systems, 2nd edition, SIAM, Philadelphia, 2003. ISBN 978-0-89871-534-7. DOI: 10.1137/1.9780898718003. Section 10.3.2, “Zero fill-in ILU (ILU(0))”, is the default preconditioner of the serial Krylov solvers and the default subdomain solver of the Schwarz preconditioners.
A. George, “Nested dissection of a regular finite element
mesh”, SIAM Journal on Numerical Analysis, 10(2):345–363, 1973.
DOI: 10.1137/0710032. With [LiptonRoseTarjan1979], the source of the
fill and operation counts of sparse direct factorisation in two and three
dimensions that decide when linear_solver = "automatic" factorises.
R. J. Lipton, D. J. Rose and R. E. Tarjan, “Generalized nested dissection”, SIAM Journal on Numerical Analysis, 16(2):346–358, 1979. DOI: 10.1137/0716027.
R. A. Nicolaides, “Deflation of conjugate gradients with applications to boundary value problems”, SIAM Journal on Numerical Analysis, 24(2):355–365, 1987. DOI: 10.1137/0724027. The coarse space of one constant per subdomain used by the two-level Schwarz preconditioner.
J. M. Tang, R. Nabben, C. Vuik and Y. A. Erlangga, “Comparison of two-level preconditioners derived from deflation, domain decomposition and multigrid methods”, Journal of Scientific Computing, 39(3):340–370, 2009. DOI: 10.1007/s10915-009-9272-6. Their operator A-DEF1, \(M^{-1}P + Q\), is how the two levels of the Schwarz preconditioner are combined.
J. W. Demmel, S. C. Eisenstat, J. R. Gilbert, X. S. Li and
J. W. H. Liu, “A supernodal approach to sparse partial pivoting”, SIAM
Journal on Matrix Analysis and Applications, 20(3):720–755, 1999.
DOI: 10.1137/S0895479895291765. The algorithm behind the direct solver:
Eigen’s own documentation states that SparseLU uses the main techniques
of the sequential SuperLU package, which is this paper.
X.-C. Cai and M. Sarkis, “A restricted additive Schwarz preconditioner for general sparse linear systems”, SIAM Journal on Scientific Computing, 21(2):792–797, 1999. DOI: 10.1137/S106482759732678X. The subdomain preconditioner of the distributed solver.
M. Dryja and O. B. Widlund, “Domain decomposition algorithms with small overlap”, SIAM Journal on Scientific Computing, 15(3):604–620, 1994. DOI: 10.1137/0915040. The two-level additive Schwarz method with a coarse space. The idea is older — it appears in a 1987 Courant Institute technical report by the same authors — but that report has no DOI and no primary record that could be checked, so the journal paper is cited instead.
A. Toselli and O. Widlund, Domain Decomposition Methods — Algorithms and Theory, Springer Series in Computational Mathematics, volume 34, Springer, Berlin, 2005, ISBN 978-3-540-20696-5. DOI: 10.1007/b137868. The analysis that explains why a coarse level is needed for the iteration count to stay bounded as ranks are added.
G. Karypis and V. Kumar, “A fast and high quality multilevel scheme for partitioning irregular graphs”, SIAM Journal on Scientific Computing, 20(1):359–392, 1998. DOI: 10.1137/S1064827595287997. METIS, used to partition a mesh when it is available at build time.
M. J. Berger and S. H. Bokhari, “A partitioning strategy for nonuniform problems on multiprocessors”, IEEE Transactions on Computers, C-36(5):570–580, 1987. DOI: 10.1109/TC.1987.1676942. Recursive coordinate bisection, the built-in partitioner used when METIS is absent.
Time integration and adaptivity¶
J. Crank and P. Nicolson, “A practical method for numerical evaluation of solutions of partial differential equations of the heat-conduction type”, Proceedings of the Cambridge Philosophical Society, 43(1):50–67, 1947. DOI: 10.1017/S0305004100023197. The journal has since been renamed Mathematical Proceedings of the Cambridge Philosophical Society, under which name the publisher’s record lists it.
L. F. Richardson, “The approximate arithmetical solution by finite differences of physical problems involving differential equations, with an application to the stresses in a masonry dam”, Philosophical Transactions of the Royal Society A, 210:307–357, 1911. DOI: 10.1098/rsta.1911.0009.
L. F. Richardson and J. A. Gaunt, “The deferred approach to the limit”, Philosophical Transactions of the Royal Society A, 226:299–361, 1927. DOI: 10.1098/rsta.1927.0008. The extrapolation that the error-controlled time stepper uses to estimate the error of a step by comparing one step with two half steps. The phrase “deferred approach to the limit” belongs to this 1927 paper with Gaunt, not to the 1911 paper, which is often miscited for it.
E. Hairer, S. P. Nørsett and G. Wanner, Solving Ordinary Differential Equations I: Nonstiff Problems, 2nd revised edition, Springer Series in Computational Mathematics, volume 8, Springer, Berlin, 1993, ISBN 978-3-540-56670-0. The step-size controller of the error-controlled stepper follows the standard form given here.
O. C. Zienkiewicz and J. Z. Zhu, “A simple error estimator and adaptive procedure for practical engineering analysis”, International Journal for Numerical Methods in Engineering, 24(2):337–357, 1987. DOI: 10.1002/nme.1620240206. The gradient recovery error indicator. (The word “engineering” is misspelt in the publisher’s deposited title; it is given correctly here.)
O. C. Zienkiewicz and J. Z. Zhu, “The superconvergent patch recovery and a posteriori error estimates. Part 1: The recovery technique”, International Journal for Numerical Methods in Engineering, 33(7):1331–1364, 1992. DOI: 10.1002/nme.1620330702.
O. C. Zienkiewicz and J. Z. Zhu, “The superconvergent patch recovery and a posteriori error estimates. Part 2: Error estimates and adaptivity”, International Journal for Numerical Methods in Engineering, 33(7):1365–1382, 1992. DOI: 10.1002/nme.1620330703. dualmesh implements the simpler averaging recovery of [ZienkiewiczZhu1987], not the patch recovery of these two papers; they are cited because they are the reference a reader looking for a better recovery should go to.
W. Dörfler, “A convergent adaptive algorithm for Poisson’s
equation”, SIAM Journal on Numerical Analysis, 33(3):1106–1124, 1996.
DOI: 10.1137/0733054. Bulk marking, implemented as
mark_by_error_fraction().
M.-C. Rivara, “Algorithms for refining triangular grids
suitable for adaptive and multigrid techniques”, International Journal for
Numerical Methods in Engineering, 20(4):745–756, 1984.
DOI: 10.1002/nme.1620200412. Longest-edge bisection, the conforming
refinement used by refine_marked().
M.-C. Rivara, “Local modification of meshes for adaptive and/or multigrid finite-element methods”, Journal of Computational and Applied Mathematics, 36(1):79–89, 1991. DOI: 10.1016/0377-0427(91)90227-B. The propagation-to-conformity form of the algorithm, which is the form implemented.
Physics and verification data¶
J. N. Reddy, Introduction to the Finite Element Method, 4th edition, McGraw-Hill Education, New York, NY, 2019, ISBN 978-1-259-86190-1. The third edition and earlier are titled An Introduction to the Finite Element Method; the fourth dropped the article.
J. N. Reddy, Theories and Analyses of Beams and Axisymmetric Circular Plates, CRC Press, Boca Raton, FL, 2022, ISBN 978-1-032-14739-0. DOI: 10.1201/9781003240846. The beam and circular plate theories used by the solid mechanics module.
J. N. Reddy, Theory and Analysis of Elastic Plates and Shells, 2nd edition, CRC Press, Boca Raton, FL, 2007, ISBN 978-0-8493-8415-8.
J. N. Reddy and D. K. Gartling, The Finite Element Method in Heat Transfer and Fluid Dynamics, 3rd edition, CRC Press, Boca Raton, FL, 2010, ISBN 978-1-4200-8598-3.
J. N. Reddy, “Analysis of functionally graded plates”, International Journal for Numerical Methods in Engineering, 47(1–3):663–684, 2000. DOI: 10.1002/(SICI)1097-0207(20000110/30)47:1/3<663::AID-NME787>3.0.CO;2-8. The power-law through-thickness variation used by the functionally graded material module.
U. Ghia, K. N. Ghia and C. T. Shin, “High-Re solutions for incompressible flow using the Navier-Stokes equations and a multigrid method”, Journal of Computational Physics, 48(3):387–411, 1982. DOI: 10.1016/0021-9991(82)90058-4. The lid-driven cavity data used in Verification.
Software that dualmesh builds on, or learns from¶
G. Guennebaud, B. Jacob and others, Eigen, 2010,
https://libeigen.gitlab.io. The linear algebra library. The project has
moved from eigen.tuxfamily.org, which now redirects; the citation above
is the one the project’s own BibTeX page currently asks for.
W. Jakob, J. Rhinelander and D. Moldovan, pybind11 — Seamless operability between C++11 and Python, 2017, https://github.com/pybind/pybind11. The citation the project asks for in its documentation FAQ.
N. Schlömer, meshio: Tools for mesh files, Zenodo.
DOI: 10.5281/zenodo.1173115. Mesh file reading and writing. The DOI is the
concept DOI naming all versions, which is what the project’s CITATION.cff
specifies.
L. Harbour, G. Giudicelli, A. D. Lindsay, P. German, J. Hansel, C. Icenhour, M. Li, J. M. Miller, R. H. Stogner, P. Behne, D. Yankura, Z. M. Prince, C. DeChant, D. Schwen, B. W. Spencer, M. Tano, N. Choi, Y. Wang, M. Nezdyur, Y. Miao, T. Hu, S. Kumar, C. Matthews, B. Langley, N. Nobre, A. Blair, C. MacMackin, H. Bergallo Rocha, E. Palmer, J. Carter, J. Meier, A. E. Slaughter, D. Andrš, R. W. Carlsen, F. Kong, D. R. Gaston and C. J. Permann, “4.0 MOOSE: enabling massively parallel multiphysics simulation”, SoftwareX, 31:102264, 2025. DOI: 10.1016/j.softx.2025.102264. The current reference for MOOSE, and the one the project’s citation page asks for. dualmesh follows MOOSE’s object model – named, registered kernels, boundary conditions and materials with validated, self-documenting parameters, assembled into one monolithic, fully coupled system with an automatically differentiated Jacobian. No MOOSE source code is used; the influence is on the design, and every algorithm that MOOSE also implements is cited above to its own source.
C. J. Permann, D. R. Gaston, D. Andrš, R. W. Carlsen, F. Kong, A. D. Lindsay, J. M. Miller, J. W. Peterson, A. E. Slaughter, R. H. Stogner and R. C. Martineau, “MOOSE: enabling massively parallel multiphysics simulation”, SoftwareX, 11:100430, 2020. DOI: 10.1016/j.softx.2020.100430. The previous framework paper, which [MOOSE2025] supersedes.
D. Gaston, C. Newman, G. Hansen and D. Lebrun-Grandié, “MOOSE: a parallel computational framework for coupled systems of nonlinear equations”, Nuclear Engineering and Design, 239(10):1768–1778, 2009. DOI: 10.1016/j.nucengdes.2009.05.021.
B. S. Kirk, J. W. Peterson, R. H. Stogner and G. F. Carey, “libMesh: a C++ library for parallel adaptive mesh refinement/coarsening simulations”, Engineering with Computers, 22(3–4):237–254, 2006. DOI: 10.1007/s00366-006-0049-3. The finite element library underneath MOOSE. dualmesh does not use it; the reference is given because a reader comparing the two frameworks will want it.
H. G. Weller, G. Tabor, H. Jasak and C. Fureby, “A tensorial approach to computational continuum mechanics using object-oriented techniques”, Computers in Physics, 12(6):620–631, 1998. DOI: 10.1063/1.168744. The code used for the independent cross-checks in Cross-verification against OpenFOAM.
A. Stukowski, “Visualization and analysis of atomistic simulation data with OVITO — the Open Visualization Tool”, Modelling and Simulation in Materials Science and Engineering, 18(1):015012, 2010. DOI: 10.1088/0965-0393/18/1/015012. Named as an influence on the organisation of this manual, in which every keyword has a page of its own that says what the keyword means, what it defaults to, and what happens when it is changed.
C. Geuzaine and J.-F. Remacle, “Gmsh: a three-dimensional finite element mesh generator with built-in pre- and post-processing facilities”, International Journal for Numerical Methods in Engineering, 79(11):1309–1331, 2009. DOI: 10.1002/nme.2579. One of the mesh generators whose output dualmesh reads, through meshio.