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Article

Energy-Dependent, Self-Adaptive Mesh h(p)-Refinement of an Interior-Penalty Scheme for a Discontinuous Galerkin Isogeometric Analysis Spatial Discretization of the Multi-Group Neutron Diffusion Equation with Dual-Weighted Residual Error Measures

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References

  • Alvarez, G. B., A. F. D. Loula, E. G. D. do Carmo, and F. A. Rochinha. 2006. A discontinuous finite element formulation for Helmholtz equation. Comput. Methods Appl. Mech. Eng. 195 (33–36):4018–35. 10.1016/j.cma.2005.07.013.
  • Anderson, E., Z. Bai, C. Bischof, S. Blackford, J. Demmel, J. Dongarra, J. Du Croz, et al. 1999. LAPACK users’ guide. Technical Report, Society for Industrial and Applied Mathematics.
  • Argonne National Laboratory. 1977. Argonne Code Center: Benchmark problem book. Report No. ANL-7416, Argonne National Laboratory.
  • Arnold, D. N. 1982. An interior penalty finite element method with discontinuous elements. SIAM J. Numer. Anal. 19 (4):742–60. 10.1137/0719052.
  • Arnold, D. N., F. Brezzi, B. Cockburn, and L. D. Marini. 2002. Unified analysis of discontinuous Galerkin methods for elliptic problems. SIAM J. Numer. Anal. 39 (5):1749–79. 10.1137/S0036142901384162.
  • 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.
  • Bangerth, W., and R. Rannacher. 2003. Adaptive finite element methods for differential equations. In Lectures in mathematics ETH Zurich. Springer. doi: 10.1007/978-3-0348-7605-6.
  • Beer, G. 2015. Advanced numerical simulation methods: From CAD data directly to simulation results. Taylor & Francis Group, CRC Press.
  • Boltzmann, L. E. 1872. Weitere Studien über das Wärmegleichgewicht unter Gasmolekülen. Sitzungsberichte Der Mathematisch-Naturwissenschaftlichen Classe Der Kaiserlichen Akademie Der Wissenschaften 66:275–370. 10.1007/978-3-322-84986-1_3.
  • Brezzi, F., B. Cockburn, L. D. Marini, and E. Süli. 2006. Stabilization mechanisms in discontinuous Galerkin finite element methods. Computer Methods in Applied Mechanics and Engineering 195 (25–28):3293–310. 10.1016/j.cma.2005.06.015.
  • Brunero, F. 2012. Discontinuous Galerkin methods for isogeometric analysis. Master’s thesis, Università degli Studi di Milano. https://www.gs.jku.at/pubs/2012brunero-dipl.pdf.
  • Burman, E., A. Quarteroni, and B. Stamm. 2010. Interior penalty continuous and discontinuous finite element approximations of hyperbolic equations. J. Sci. Comput. 43 (3):293–312. 10.1007/s10915-008-9232-6.
  • Castillo, P. 2002. Performance of discontinuous Galerkin methods for elliptic PDEs. SIAM J. Sci. Comput. 24 (2):524–47. 10.1137/S106482750138833. 10.1137/S1064827501388339
  • Cottrell, J. A., T. J. R. Hughes, and Y. Bazilevs. 2009. Isogeometric analysis: Toward integration of CAD and FEA. John Wiley & Sons, Inc.
  • 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. 10.1016/j.cma.2007.04.007.
  • Cox, M. G. 1972. The numerical evaluations of B-splines. IMA J. Appl. Math. 10 (2):134–49. 10.1093/imamat/10.2.134.
  • de Boor, C. 1972. On calculating with B-splines. Journal of Approximation Theory 6 (1):50–62. 10.1016/0021-9045.(72)90080-9. 10.1016/0021-9045(72)90080-9
  • Demkowicz, L. 2004. Projection based interpolation. Report No. ICES 04-03, University of Texas at Austin.
  • 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.
  • Di Pietro, D. A. and A. Ern. 2012. Mathematical aspects of discontinuous Galerkin methods. Vol. 69 of Mathémathiques et applications. Springer. doi: 10.1007/978-3-642-22980-0.
  • Douglas, J., Jr, and T. Dupont. 1976. Interior penalty procedures for elliptic and parabolic Galerkin methods. Vol. 58 of Lecture notes in physics. Springer-Verlag. doi: 10.1007/BFb0120591.
  • Drosson, M., and K. Hillewaert. 2013. On the stability of the symmetric interior penalty method for the Spalart-Allmaras turbulence model. J. Comput. Appl. Math. 246:122–35. 10.1016/j.cam.2012.09.019.
  • Dryja, M. 2003. On discontinuous Galerkin methods for elliptic problems with discontinuous coefficients. Comput. Methods Appl. Math. 3 (1):76–85. 10.2478/cmam-2003-0007.
  • Epshteyn, Y., and B. Rivière. 2007. Estimation of penalty parameters for symmetric penalty Galerkin methods. J. Comput. Appl. Math. 206 (2):843–72. 10.1016/j.cam.2006.08.029.
  • 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.
  • 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. 10.1016/j.cma.2010.07.015.
  • 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. 10.1016/j.cma.2012.08.015.
  • Grote, M. J., A. Schneebeli, and D. Schötzau. 2007. Interior penalty discontinuous Galerkin method for Maxwell’s equations: Energy norm error estimates. J. Comput. Appl. Math. 204 (2):375–86. 10.1016/j.cam.2006.01.044
  • Hébert, A. 1985. Application of the Hermite method for finite element reactor calculations. Nucl. Sci. Eng. 91 (1):34–58. 10.13182/NSE85-A17127.
  • Hébert, A. 2019. A user guide for Trivac version-4 . Report No. IGE-293 , Ecole Polytechnique de Montréal - Institut de Génie Nucléaire .
  • Hébert, A. 2020. Applied reactor physics. 3rd ed. Presses Internationales Polytechnique.
  • Henson, V. E., and U. M. Yang. 2002. BoomerAMG: A Parallel algebraic multigrid solver and preconditioner. Appl. Numer. Math. 41 (1):155–77. 10.1016/S0168-9274.(01)00115-5. 10.1016/S0168-9274(01)00115-5
  • Hesthaven, J. S., and T. Warburton. 2010. Nodal discontinuous Galerkin methods: Algorithms. Analysis and applications. Vol. 54 of Texts in applied mathematics. Springer. doi: 10.1007/978-0-387-72067-8.
  • Hillewaert, K. 2013. Development of the discontinuous Galerkin method for high-resolution, large-scale CFD and acoustics in industrial geometries. PhD thesis, Université Catholique de Louvain. https://hdl.handle.net/2078.1/pul:29303100319670.
  • Kaplan, S. 1969. Variational methods in nuclear engineering. Adv. Nucl. Sci. Technol. 3 (135):185–221. 10.1016/B978-0-12-029305-6.50010-8.
  • Lewis, E. E., and W. F. Miller, Jr. 1984. Computational methods of neutron transport. John Wiley & Sons, Inc.
  • Li, B. Q. 2006. Discontinuous finite elements in fluid dynamics and heat transfer. Computational fluid and solid mechanics. Springer. doi: 10.1007/1-84628-205-5.
  • Melenk, J. M., A. Parsania, and S. Sauter. 2013. General DG-methods for highly indefinite Helmholtz problems. J. Sci. Comput. 57 (3):536–81. 10.1007/s10915-013-9726-8.
  • 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.
  • Nitsche, J. A. 1971. Über ein Variationsprinzip zur Lösung von Dirichlet-Problemen bei Verwendung von Teilrämen, die keinen Randbedingungen unterworfen sind. Vol. 36 of Abhandlungen aus dem mathematischen Seminar der Universität Hamburg. Springer. 10.1007/BF02995904
  • Oden, J. T., I. Babuŝka, and C. E. Baumann. 1998. A discontinuous hp finite element method for diffusion problems. Comput. Phys. 146 (2):491–519. 10.1006/jcph.1998.6032.
  • Oden, J. T., and J. N. Reddy. 2011. An introduction to the mathematical theory of finite elements. Dover Publications. doi: 10.1002/nme.1620110715.
  • Owens, A. R., J. Kópházi, and M. D. Eaton. 2017b. Optimal trace inequality constants for interior penalty discontinuous Galerkin discretisations of elliptic operators using arbitrary elements with non-constant Jacobians. Comput. Phys. 350:847–70. 10.1016/j.jcp.2017.09.020.
  • Owens, A. R., J. Kópházi, J. A. Welch, and M. D. Eaton. 2017a. Energy dependent mesh adaptivity of discontinuous isogeometric discrete ordinate methods with dual weighted residual error estimators. Comput. Phys. 335:352–86. 10.1016/j.jcp.2017.01.035.
  • 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.
  • Pazner, W., and T. Kolev. 2022. Uniform subspace correction preconditioners for discontinuous Galerkin methods with h(p)-refinement. Commun. Appl. Math. Comput. 4 (2):697–727. 10.1007/s42967-021-00136-3.
  • Percell, P., and M. F. Wheeler. 1978. A local residual finite element procedure for elliptic equations. SIAM J. Numer. Anal. 15 (4):705–14. 10.1137/0715047.
  • Piegl, L., and W. Tiller. 1997. The NURBS book: Monographs in visual communication. Springer.
  • 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.
  • Saad, Y. 2011. Numerical methods for large eigenvalue problems. 2nd ed., Classics in Applied Mathematics. SIAM. doi: 10.1137/1.9781611970739.
  • Salari, K., and P. Knupp. 2000. Code verification by the method of manufactured solutions . Report No. SAND2000-1444. Sandia National Laboratories.
  • Sangalli, G., and M. Tani. 2016. Isogeometric preconditioners based on fast solvers for the Sylvester equation. SIAM J. Sci. Comput. 38 (6):A3644–A3671. 10.1137/16M1062788.
  • 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. 10.1016/j.anucene.2014.08.028.
  • Shahbazi, K. 2005. An explicit expression for the penalty parameter of the interior penalty method. Comput. Phys. 205 (2):401–7. 10.1016/j.jcp.2004.11.017.
  • Theler, G. 2013. Unstructured grids and the multigroup neutron diffusion equation. Sci. Technol. Nucl. Install. 2013:1–26. 10.1155/2013/641863.
  • 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. 10.1016/j.pnucene.2008.11.005.
  • Wang, Y. 2009. Adaptive mesh refinement solution techniques for the multigroup SN 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.
  • Warburton, T., and J. S. Hesthaven. 2003. On the constants in hp-finite element trace inverse inequalities. Comput. Methods Appl. Mech. Eng. 192 (25):2765–73. 10.1016/S0045-7825.(03)00294-9. 10.1016/S0045-7825(03)00294-9
  • Warsa, J. S. 2008. A continuous finite element-based, discontinuous finite element method for S N transport. Nucl. Sci. Eng. 160 (3):385–400. 10.13182/NSE160-385TN.
  • Wheeler, M. F. 1978. An elliptic collocation-finite element method with interior penalties. SIAM J. Numer. Anal. 15 (1):152–61. 10.1137/0715010.
  • Wilson, S. G., M. D. Eaton, and J. Kópházi. 2024. 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 equations with dual-weighted residual error measures. Journal of Computational and Theoretical Transport 53:1–64. 10.1080/23324309.2024.2313460.
  • 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 Proc. of the International Conference of Nuclear Engineering (ICONE26), Vol. 3: Nuclear Fuel and Material, Reactor Physics and Transport Theory . ASME. 10.1115/ICONE26-81322
  • 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.