Energy-Dependent, Self-Adaptive Mesh h(p)-Refinement of a Constraint-Based Continuous Bubnov-Galerkin Isogeometric Analysis Spatial Discretization of the Multi-Group Neutron Diffusion Equation with Dual-Weighted Residual Error Measures

References

• Ainsworth, M., and J. T. Oden. 1997. A posteriori error estimation in finite element analysis. Comput. Methods Appl. Mech. Eng. 142 (1–2):1–88. doi: 10.1016/S0045-7825(96)01107-3.
   Web of Science ®Google Scholar
• Ainsworth, M., and J. T. Oden. 2000. A posteriori error estimation in finite element analysis. Pure and applied mathematics. John Wiley & Sons, Inc. doi: 10.1002/9781118032824.
   Google Scholar
• Akhras, H. A., T. Elguedj, A. Gravouil, and M. Rochette. 2017. Towards an automatic isogeometric analysis suitable trivariate models generation - Application to geometric parametric analysis. Comput. Methods Appl. Mech. Eng. 316:623–45. doi: 10.1016/j.cma.2016.09.030.
   Web of Science ®Google Scholar
• Argonne National Laboratory. 1977. Argonne code center: Benchmark problem book. Report No. ANL-7416. Argonne National Laboratory.
   Google Scholar
• Babuška, I., and B. Q. Guo. 1992. The h, p, and h – p version of the finite element method; basis theory and applications. Adv. Eng. Softw. 15 (3–4):159–74. doi: 10.1016/0965-9978(92)90097-Y.
   Web of Science ®Google Scholar
• Balay, S., S. Abhyankar, M. Adams, J. Brown, P. Brune, K. Buschelman, L. Dalcin, et al. 2018. PETSc users manual: Revision 3.9. Report No. ANL-95/11 Rev. 3.9. Argonne National Laboratory.
   Google Scholar
• Bangerth, W., and R. Rannacher. 2003. Adaptive finite element methods for differential equations. Lectures in Mathematics ETH Zurich. Springer. doi: 10.1007/978-3-0348-7605-6.
   Google Scholar
• Bazilevs, Y., V. M. Calo, J. A. Cottrell, J. A. Evans, T. J. R. Hughes, S. Lipton, M. A. Scott, and T. W. Sederberg. 2010. Isogeometric analysis using T-splines. Comput. Methods Appl. Mech. Eng. 199 (5–8):229–63. doi: 10.1016/j.cma.2009.02.036.
   Web of Science ®Google Scholar
• Becker, E. B., G. F. Carey, and J. T. Oden. 1981. Finite elements: An introduction. Prentice-Hall Inc. ISBN 10: 0133170578/ISBN 13: 9780133170573.
   Google Scholar
• Boltzmann, L. E. 1872. Weitere Studien über das Wärmegleichgewicht unter Gasmolekülen. Sitzungsberichte Der Mathematisch-Naturwissenschaftlichen Classe Der Kaiserlichen Akademie Der Wissenschaften 66:275–370. doi: 10.1007/978-3-322-84986-1_3.
   Google Scholar
• Bonito, A., R. A. Devore, and R. H. Nochetto. 2013. Adaptive Finite Element Methods for Elliptic Problems with Discontinuous Coefficients. SIAM J. Numer. Anal. 51 (6):3106–34. doi: 10.1137/130905757.
   Web of Science ®Google Scholar
• Buffa, A., G. Sangalli, and R. Vázquez. 2014. Isogeometric methods for computational electromagnetics: B-spline and T-spline discretisations. Comput. Phys. 257:1291–320. doi: 10.1016/j.jcp.2013.08.015.
   Google Scholar
• Ceze, M., and K. J. Fidkowski. 2013. Anisotropic hp-adaptation framework for functional prediction. AIAA J. 51 (2):492–509. doi: 10.2514/1.J051845.
   Google Scholar
• Coox, L., F. Greco, O. Atak, D. Vandepitte, and W. Desmet. 2017. A robust patch coupling method for NURBS-based isogeometric analysis of non-conforming multipatch surfaces. Comput. Methods Appl. Mech. Eng. 316:235–60. doi: 10.1016/j.cma.2016.06.022.
   Web of Science ®Google Scholar
• Cottrell, J. A., T. J. R. Hughes, and Y. Bazilevs. 2009. Isogeometric analysis: Toward integration of CAD and FEA. John Wiley & Sons, Inc. ISBN: 978-0-470-74873-2.
   Google Scholar
• Cottrell, J., T. Hughes, and A. Reali. 2007. Studies of refinement and continuity in isogeometric structural analysis. Comput. Methods Appl. Mech. Eng. 196 (41–44):4160–83. doi: 10.1016/j.cma.2007.04.007.
   Web of Science ®Google Scholar
• Cox, M. G. 1972. The numerical evaluations of B-splines. IMA J. Appl. Math. 10 (2):134–49. doi: 10.1093/imamat/10.2.134.
   Google Scholar
• Cummins, W. E., and R. Matzie. 2018. Design evolution of PWRs: Shippingport to generation III+. Prog. Nucl. Energy 102:9–37. doi: 10.1016/j.pnucene.2017.08.008.
   Web of Science ®Google Scholar
• de Boor, C. 1972. On calculating with B-splines. J. Approximation Theory 6 (1):50–62. doi: 10.1016/0021-9045(72)90080-9.
   Google Scholar
• Demkowicz, L. 2004. Projection based interpolation. Report No. ICES 04-03. University of Texas at Austin, Austin.
   Google Scholar
• Demkowicz, L. 2007. Computing with hp-adaptive finite elements: One and two dimensional elliptic and maxwell problems. Vol. 1 of Chapman & Hall/CRC Applied Mathematics and Nonlinear Science Series. Chapman & Hall/CRC. doi: 10.1201/9781420011685.
   Google Scholar
• Demkowicz, L., J. Kurtz, D. Pardo, M. Paszynski, W. Rachowicz, and A. Zdunek. 2008. Computing with hp-adaptive finite elements: three dimensional elliptic and maxwell problems with applications. Vol. 2 of Chapman & Hall/CRC Applied mathematics and nonlinear science series. Chapman & Hall/CRC. doi: 10.1201/9781420011692.
   Google Scholar
• Demkowicz, L., P. Monk, L. Vardapetyan, and W. Rachowicz. 2000. De Rhamm diagram for hp finite element spaces. Math. Comput. Appl. 39 (7–8):29–38. doi: 10.1016/S0898-1221(00)00062-6.
   Google Scholar
• Demkowicz, L., W. Rachowicz, and P. Devloo. 2002. A fully automatic hp-adaptivity. J. Sci. Comput. 17 (1–4):117–42. doi: 10.1023/A:1015192312705.
   Web of Science ®Google Scholar
• Donea, J., and A. Huerta. 2003. Finite element methods for flow problems. John Wiley & Sons, Inc. doi: 10.1002/0470013826.
   Google Scholar
• Ern, A., and J.-L. Guermond. 2004. Theory and practice of finite elements. Vol. 159 of applied mathematical sciences. Springer. doi: 10.1007/978-1-4757-4355-5.
   Google Scholar
• Farrell, P. E., and J. R. Maddison. 2011. Conservative interpolation between volume meshes by local Galerkin projection. Comput. Methods Appl. Mech. Eng. 200 (1–4):89–100. doi: 10.1016/j.cma.2010.07.015.
   Web of Science ®Google Scholar
• Finlayson, B. A. 1972. The method of weighted residuals and variational principles with application in fluid mechanics, heat and mass transfer. Vol. 87 of mathematics in science and engineering. Academic Press. doi: 10.1137/1.9781611973242.
   Google Scholar
• Gahalaut, K. P. S., J. K. Kraus, and S. K. Tomar. 2013. Multigrid methods for isogeometric discretisation. Comput. Methods Appl. Mech. Eng. 253 (100):413–25. doi: 10.1016/j.cma.2012.08.015.
   PubMedGoogle Scholar
• Gaston, D., C. Newman, G. Hansen, and D. Lebrun-Grandié. 2009. MOOSE: A parallel computational framework for coupled systems of nonlinear equations. Nucl. Eng. Des. 239 (10):1768–78. doi: 10.1016/j.nucengdes.2009.05.021.
   Web of Science ®Google Scholar
• Goffin, M. A., C. M. J. Baker, A. G. Buchan, C. C. Pain, M. D. Eaton, and P. N. Smith. 2013. Minimising the error in eigenvalue calculations involving the Boltzmann transport equation using goal-based adaptivity on unstructured meshes. Comput. Phys. 242:726–52. doi: 10.1016/j.jcp.2012.12.035.
   Google Scholar
• Guo, B., and I. Babuška. 1986a. The h – p version of the finite element method, part 1: The basic approximation results. Comput. Mech. 1 (1):21–41. doi: 10.1007/BF00298636.
   Google Scholar
• Guo, B., and I. Babuška. 1986b. The h – p version of the finite element method, part 2: General results and applications. Comput. Mech. 1 (3):203–20. doi: 10.1007/BF00272624.
   Google Scholar
• Hall, S. K., M. D. Eaton, and M. M. R. Williams. 2012. The application of isogeometric analysis to the neutron diffusion equation for a pincell problem with an analytic benchmark. Ann. Nucl. Energy 49:160–9. doi: 10.1016/j.anucene.2012.05.030.
   Web of Science ®Google Scholar
• Hansen, K. F., and C. M. Kang. 1973. Finite element methods for reactor physics analysis. Adv. Nucl. Sci. Eng. 51 (4):456–95. doi: 10.13182/NSE73-A23278.
   Web of Science ®Google Scholar
• Hartmann, R. 2007. Adjoint consistency analysis of discontinuous galerkin discretizations. SIAM J. Numer. Anal. 45 (6):2671–96. doi: 10.1137/060665117.
   Web of Science ®Google Scholar
• Hébert, A. 1985. Application of the hermite method for finite element reactor calculations. Nucl. Sci. Eng. 91 (1):34–58. doi: 10.13182/NSE85-A17127.
   Web of Science ®Google Scholar
• Hébert, A. 2019. A user guide for trivac version-4. Report No. IGE-293. Ecole Polytechnique de Montréal - Institut de Génie Nucléaire.
   Google Scholar
• Henson, V. E., and U. M. Yang. 2002. BoomerAMG: A parallel algebraic multigrid solver and preconditioner. Appl. Numer. Math. 41 (1):155–77. doi: 10.1016/S0168-9274(01)00115-5.
   Web of Science ®Google Scholar
• Hughes, T. J. R., J. A. Cottrell, and Y. Bazilevs. 2005. Isogeometric analysis: CAD, finite elements, NURBS, exact geometry and mesh refinement. Comput. Methods Appl. Mech. Eng. 194 (39–41):4135–95. doi: 10.1016/j.cma.2004.10.008.
   Web of Science ®Google Scholar
• Jiao, X., and M. T. Heath. 2004. Common-refinement-based data transfer between non-matching meshes in multiphysics simulations. Numer. Methods Eng. 61 (14):2402–27. doi: 10.1002/nme.1147.
   Web of Science ®Google Scholar
• Kagan, P., A. Fischer, and P. Z. Bar-Yoseph. 2003. Mechanically based models: Adaptive refinement for B-spline finite element. Numer. Methods Eng. 57 (8):1145–75. doi: 10.1002/nme.717.
   Web of Science ®Google Scholar
• Lathouwers, D. 2011a. Goal-oriented spatial adaptivity for the SN equations on unstructured triangular meshes. Ann. Nucl. Energy 38 (6):1373–81. doi: 10.1016/j.anucene.2011.01.038.
   Web of Science ®Google Scholar
• Lathouwers, D. 2011b. Spatially adaptive eigenvalue estimation for the SN equations on unstructured triangular meshes. Ann. Nucl. Energy 38 (9):1867–76. doi: 10.1016/j.anucene.2011.05.013.
   Web of Science ®Google Scholar
• Latimer, C., J. Kópházi, M. D. Eaton, and R. G. McClarren. 2020. A geometry conforming isogeometric method for the self-adjoint angular flux (SAAF) form of the neutron transport equation with a discrete ordinate (S N) angular discretisation. Ann. Nucl. Energy 136:107049. doi: 10.1016/j.anucene.2019.107049.
   Web of Science ®Google Scholar
• Lewis, E. E., and W. F. Miller. Jr. 1984. Computational methods of neutron transport. John Wiley & Sons, Inc. ISBN-10‏: ‎0471092452; ISBN-13‏: ‎978-0471092452.
   Google Scholar
• Martin, W. R., and J. J. Duderstadt. 1977. Finite element solutions of the neutron transport equation with applications to strong heterogeneities. Nucl. Sci. Eng. 62 (3):371–90. doi: 10.13182/NSE77-A26979.
   Web of Science ®Google Scholar
• Martineau, R., D. Andrs, R. Carlsen, D. Gaston, J. Hansel, F. Kong, A. Lindsay, C. Permann, A. Slaughter, E. Merzari, et al. 2020. Multiphysics for nuclear energy applications using a cohesive computational framework. Nucl. Eng. Des. 367:110751. doi: 10.1016/j.nucengdes.2020.110751.
   Web of Science ®Google Scholar
• Melenk, J. M., A. Parsania, and S. Sauter. 2013. General DG-methods for highly indefinite helmholtz problems. J. Sci. Comput. 57 (3):536–81. doi: 10.1007/s10915-013-9726-8.
   Web of Science ®Google Scholar
• Metcalf, M., J. Reid, and M. Cohen. 2011. Modern fortran explained. Numerical Mathematics and Scientific Computation. Oxford University Press. doi: 10.1093/oso/9780198811893.001.0001.
   Google Scholar
• Mitchell, W. F., and M. A. McClain. 2014. A comparison of hp-adaptive strategies for elliptic partial differential equations. ACM Trans. Math. Softw. 41 (1):1–39. doi: 10.1145/2629459.
   Web of Science ®Google Scholar
• Mylonakis, A. G., M. Varvayanni, N. Catsaros, P. Savva, and D. G. E. Grigoriadis. 2014. Multi-physics and multi-scale methods used in nuclear reactor analysis. Ann. Nucl. Energy 72:104–19. doi: 10.1016/j.anucene.2014.05.002.
   Web of Science ®Google Scholar
• Nitsche, J. A. 1971. Über ein Variationsprinzip zur Lösung von Dirichlet-Problemen bei Verwendung von Teilrämen, die keinen Randbedingungen unterworfen sind. In Abhandlungen aus dem mathematischen Seminar der Universität Hamburg. Vol. 36, pp. 9–15. Springer. doi: 10.1007/BF02995904
   Google Scholar
• Oden, J. T., and J. N. Reddy. 2011. An introduction to the mathematical theory of finite elements. Dover Publications. doi: 10.1002/nme.1620110715.
   Google Scholar
• Owens, A. R. 2017. Discontinuous isogeometric analysis methods for the first-order form of the neutron transport equation with discrete oridnate angular discretisation. PhD thesis, Imperial College London.
   Google Scholar
• Owens, A. R., J. A. Welch, J. Kópházi, and M. D. Eaton. 2017. An adaptive, hanging-node, discontinuous isogeometric analysis method for the first-order form of the neutron transport equation with discrete ordinate (SN) angular discretisation. Comput. Methods Appl. Mech. Eng. 318:215–41. doi: 10.1016/j.cma.2017.01.036.
   Web of Science ®Google Scholar
• Owens, A. R., J. Kópházi, and M. D. Eaton. 2017. Energy dependent mesh adaptivity of discontinuous isogeometric discrete ordinate methods with dual weighted residual error estimators. Comput. Phys. 335:352–86. doi: 10.1016/j.jcp.2017.01.035.
   Google Scholar
• Park, H. 2006. Coupled space-angle adaptivity and goal-oriented error control for radiation transport calculations. PhD thesis, Georgia Institute of Technology. http://hdl.handle.net/1853/13944
   Google Scholar
• Piegl, L., and W. Tiller. 1997. The NURBS book: Monographs in visual communication. Springer. ISBN 10: 3540615458; ISBN 13: 9783540615453.
   Google Scholar
• Rachowicz, W., J. T. Oden, and L. Demkowicz. 1989. Toward a universal h-p adaptive finite element strategy, Part 3. Design of h-p meshes. Comput. Methods Appl. Mech. Eng. 77 (1–2):181–212. doi: 10.1016/0045-7825(89)90131-X.
   Web of Science ®Google Scholar
• Ragusa, J. 2008. A simple Hessian-based 3D mesh adaptation technique with applications to the multigroup diffusion equations. Ann. Nucl. Energy 35 (11):2006–18. doi: 10.1016/j.anucene.2008.06.008.
   Web of Science ®Google Scholar
• Reed, W. H. 1971. New difference schemes for the neutron transport equation. Nucl. Sci. Eng. 46 (2):309–14. doi: 10.13182/NSE46-309.
   Web of Science ®Google Scholar
• Rivière, B. 2008. Discontinuous Galerkin methods for solving elliptic and parabolic equations: Theory and Implementation. Frontiers in Applied Mathematics. SIAM. doi: 10.1137/1.9780898717440.
   Google Scholar
• Salari, K., and P. Knupp. 2000. Code verification by the method of manufactured solutions. Report No. SAND2000-1444. Sandia National Laboratories. doi: 10.2172/759450.
   Google Scholar
• Sangalli, G., and M. Tani. 2016. Isogeometric preconditioners based on fast solvers for the Sylvester equation. SIAM J. Sci. Comput. 38 (6):A3644–A3671. doi: 10.1137/16M1062788.
   Web of Science ®Google Scholar
• Schunert, S., Y. Wang, R. Martineau, and M. D. DeHart. 2015. A new mathematical adjoint for the modified SAAF-SN equations. Ann. Nucl. Energy 75:340–52. doi: 10.1016/j.anucene.2014.08.028.
   Web of Science ®Google Scholar
• Semenza, L. A., E. E. Lewis, and E. C. Rossow. 1972. The application of the finite element method to the multigroup neutron diffusion equation. Nucl. Sci. Eng. 47 (3):302–10. doi: 10.13182/NSE72-A22416.
   Web of Science ®Google Scholar
• Theler, G. 2013. Unstructured grids and the multigroup neutron diffusion equation. Sci. Technol. Nucl. Install. 2013:1–26. doi: 10.1155/2013/641863.
   Web of Science ®Google Scholar
• Turner, J. A., K. Clarno, M. Sieger, R. Bartlett, B. Collins, R. Pawlowski, R. Schmidt, and R. Summers. 2016. The virtual environment for reactor applications (VERA): Design and architecture. Comput. Phys. 326:544–68. doi: 10.1016/j.jcp.2016.09.003.
   Google Scholar
• Wang, Y. 2009. Adaptive mesh refinement solution techniques for the multigroup S N transport equation using a higher-order discontinuous finite element method. PhD thesis, Texas A&M University. https://hdl.handle.net/1969.1/ETD-TAMU-2009-05-641
   Google Scholar
• Wang, Y., and J. C. Ragusa. 2007. Goal-oriented hp-mesh adaptation for 1-D multigroup diffusion equations. In Proceedings of the Joint International Toplical Meeting on Mathematics & Computation and Supercomputing in Nuclear Applications. American Nuclear Society.
   Google Scholar
• Wang, Y., and J. Ragusa. 2005. Adaptive hp-mesh refinement applied to 1-D, one-group diffusion problems. Trans. Am. Nucl. Soc. 93:585.
   Google Scholar
• Wang, Y., and J. Ragusa. 2009a. Application of hp adaptivity to the multigroup diffusion equations. Nucl. Sci. Eng. 161 (1):22–48. doi: 10.13182/NSE161-22.
   Web of Science ®Google Scholar
• Wang, Y., and J. Ragusa. 2009b. On the convergence of DGFEM applied to the discrete ordinates transport equation for structured and unstructured triangular meshes. Nucl. Sci. Eng. 163 (1):56–72. doi: 10.13182/NSE08-72.
   Web of Science ®Google Scholar
• Wang, Y., and J. Ragusa. 2011. Standard and goal-oriented adaptive mesh refinement applied to radiation transport on 2D unstructured triangular meshes. Comput. Phys. 230 (3):763–88. doi: 10.1016/j.jcp.2010.10.018.
   Google Scholar
• Wang, Y., W. Bangerth, and J. Ragusa. 2009. Three-dimensional h-adaptivity for the multigroup neutron diffusion equations. Prog. Nucl. Energy 51 (3):543–55. doi: 10.1016/j.pnucene.2008.11.005.
   Web of Science ®Google Scholar
• Warsa, J. S. 2008. A continuous finite element-based, discontinuous finite element method for S N transport. Nucl. Sci. Eng. 160 (3):385–400. doi: 10.13182/NSE160-385TN.
   Web of Science ®Google Scholar
• Welch, J. A. 2018. Isogeometric analysis for second-order forms of the neutron transport equation with applications to reactor physics. PhD thesis. Imperial College London.
   Google Scholar
• Welch, J. A., J. Kópházi, A. R. Owens, and M. D. Eaton. 2017a. A geometry preserving, conservative, mesh-to-mesh isogeometric interpolation algorithm for spatial adaptivity of the multigroup, second-order even-parity form of the neutron transport equation. Comput. Phys. 347:129–46. doi: 10.1016/j.jcp.2017.06.015.
   Google Scholar
• Welch, J. A., J. Kópházi, A. R. Owens, and M. D. Eaton. 2017b. Isogeometric analysis for the multigroup neutron diffusion equation with applications in reactor physics. Ann. Nucl. Energy 101:465–80. doi: 10.1016/j.anucene.2016.11.015.
   Web of Science ®Google Scholar
• Wilson, S. G. 2021. Self-adaptive isogeometric spatial discretisations of the first and second-order forms of the neutron transport equation with dual-weighted residual error measures and diffusion acceleration. PhD thesis, Imperial College London.
   Google Scholar
• Wilson, S. G., J. Kópházi, A. R. Owens, and M. D. Eaton. 2018. Interior penalty schemes for discontinuous isogeometric methods with an application to nuclear reactor physics. In Proceedings of the International Conference of Nuclear Engineering (ICONE26), Vol. 3. Nuclear Fuel and Material, Reactor Physics and Transport Theory. ASME. doi: 10.1115/ICONE26-81322
   Google Scholar