Print Email Facebook Twitter Goal-Oriented Angular Adaptive Algorithm using Sensitivity Analysis for the Transport Equation and Boltzmann-Fokker-Planck Equation Title Goal-Oriented Angular Adaptive Algorithm using Sensitivity Analysis for the Transport Equation and Boltzmann-Fokker-Planck Equation Author Koeze, D.J. Contributor Vuik, C. (mentor) Faculty Electrical Engineering, Mathematics and Computer Science Department Numerical Analysis Programme Applied Mathematics Date 2012-09-30 Abstract In this work we examined the discretised form of Boltzmann-like transport, i.e. the neutron trans- port equation and the Boltzmann-Fokker-Plank (BFP) equation with the discontinuous Galerkin method and polynomial basis functions. In particular we examined an adaptive algorithm, which bases its decision of where to refine on the adjoint problem, as in sensitivity analysis. In this way an accurate detector response should be obtained in an efficient manner. The goal-orietend criterion uses the adjoint solution as a measure for importance to the detector response. This refinement technique is compared to traditional methods, which base refinement on the change in the solution of a local test refinement, and to the discrete ordinates method. Problems with one spatial and one angular dimension are used to test the adaptive algorithm. In previous work the same problems were solved with first order polynomials in the spatial di- rection and zeroth order polynomials in the angular direction. We saw then that constant patches, zeroth order polynomials in the angular part of the domain, could not represent the angular flux in diffusive materials accurately. We furthermore saw that the quality of the error estimate with the global adjoint approximation was reasonable, while with the local adjoint approximation it was poor. In this work we employed first order polynomials in both the spatial and angular domains. Linear patches, the first order polynomials in the angular domain, provide a better approximation of the angular flux using less unknowns. This can be seen when comparing both the constant and linear patches to the discrete ordinates method. Also the diffusive materials are now much easier to approximate, which has as effect that traditional refinement gives smaller errors than with constant patches. The quality of the error estimate with linear patches, however, is poor for almost every test case. A cause can be found in the approximation of the exact adjoint solution that is used in the estimate. This approximation is done by refining all the current mesh uniformly one level. It appears the space of the deeper level is only slightly larger than the coarser level on which the forward problem is solved, in the sense that the adjoint solution on that deeper level does not yield information not contained in the coarser level. The solution we propose is to solve the adjoint problem with higher order basis functions than those of the forward problem. Finally we present some results on the Boltzmann-Fokker-Planck equation discretised with discontinuous Galerkin. The discrete operator one obtains is very similar to the discrete transport operator, with extra bands added within a spatial element. To solve this system of equations one could use a Gauss-Seidel iteration, which means one sometimes uses outdated angular flux values of neighbours, or Kylov subspace methods, as the matrix operator is not computed explicitly. Subject Neutron Transport Equationsensitivity analysisadaptive algorithmDiscontinuous Galerkin method To reference this document use: http://resolver.tudelft.nl/uuid:3e96836d-39f6-46d2-a2a0-f3962cc9d803 Part of collection Student theses Document type master thesis Rights (c) 2012 Koeze, D.J. Files PDF thesis.pdf 791.02 KB Close viewer /islandora/object/uuid:3e96836d-39f6-46d2-a2a0-f3962cc9d803/datastream/OBJ/view