Spatial adaptivity of the SAAF and Weighted Least Squares (WLS) forms of the neutron transport equation using constraint based, locally refined, isogeometric analysis (IGA) with dual weighted residual (DWR) error measures

This paper describes a methodology that enables NURBS (Non-Uniform Rational B-spline) based Isogeometric Analysis (IGA) to be locally refined. The methodology is applied to continuous Bubnov-Galerkin IGA spatial discretisations of second-order forms of the neutron transport equation. In particular this paper focuses on the self-adjoint angular flux (SAAF) and weighted least squares (WLS) equations. Local refinement is achieved by constraining degrees of freedom on interfaces between NURBS patches that have different levels of spatial refinement. In order to effectively utilise constraint based local refinement, adaptive mesh refinement (AMR) algorithms driven by a heuristic error measure or forward error indicator (FEI) and a dual weighted residual (DWR) or goal-based error measure (WEI) are derived. These utilise projection operators between different NURBS meshes to reduce the amount of computational effort required to calculate the error indicators. In order to apply the WEI to the SAAF and WLS second-order forms of the neutron transport equation the adjoint of these equations are required. The physical adjoint formulations are derived and the process of selecting source terms for the adjoint neutron transport equation in order to calculate the error in a given quantity of interest (QoI) is discussed. Several numerical verification benchmark test cases are utilised to investigate how the constraint based local refinement affects the numerical accuracy and the rate of convergence of the NURBS based IGA spatial discretisation. The nuclear reactor physics verification benchmark test cases show that both AMR algorithms are superior to uniform refinement with respect to accuracy per degree of freedom. Furthermore, it is demonstrated that for global QoI the FEI driven AMR and WEI driven AMR produce similar results. However, if local QoI are desired then WEI driven AMR algorithm is more computationally efficient and accurate per degree of freedom

Generalized averaged Gaussian quadrature and applications

A simple numerical method for constructing the optimal generalized averaged Gaussian quadrature formulas will be presented. These formulas exist in many cases in which real positive GaussKronrod formulas do not exist, and can be used as an adequate alternative in order to estimate the error of a Gaussian rule. We also investigate the conditions under which the optimal averaged Gaussian quadrature formulas and their truncated variants are internal

MS FT-2-2 7 Orthogonal polynomials and quadrature: Theory, computation, and applications

Quadrature rules find many applications in science and engineering. Their analysis is a classical area of applied mathematics and continues to attract considerable attention. This seminar brings together speakers with expertise in a large variety of quadrature rules. It is the aim of the seminar to provide an overview of recent developments in the analysis of quadrature rules. The computation of error estimates and novel applications also are described

Manager’s and citizen’s perspective of positive and negative risks for small probabilities

So far „risk‟ has been mostly defined as the expected value of a loss, mathematically PL, being P the probability of an adverse event and L the loss incurred as a consequence of the event. The so called risk matrix is based on this definition. Also for favorable events one usually refers to the expected gain PG, being G the gain incurred as a consequence of the positive event. These “measures” are generally violated in practice. The case of insurances (on the side of losses, negative risk) and the case of lotteries (on the side of gains, positive risk) are the most obvious. In these cases a single person is available to pay a higher price than that stated by the mathematical expected value, according to (more or less theoretically justified) measures. The higher the risk, the higher the unfair accepted price. The definition of risk as expected value is justified in a long term “manager‟s” perspective, in which it is conceivable to distribute the effects of an adverse event on a large number of subjects or a large number of recurrences. In other words, this definition is mostly justified on frequentist terms. Moreover, according to this definition, in two extreme situations (high-probability/low-consequence and low-probability/high-consequence), the estimated risk is low. This logic is against the principles of sustainability and continuous improvement, which should impose instead both a continuous search for lower probabilities of adverse events (higher and higher reliability) and a continuous search for lower impact of adverse events (in accordance with the fail-safe principle). In this work a different definition of risk is proposed, which stems from the idea of safeguard: (1Risk)=(1P)(1L). According to this definition, the risk levels can be considered low only when both the probability of the adverse event and the loss are small. Such perspective, in which the calculation of safeguard is privileged to the calculation of risk, would possibly avoid exposing the Society to catastrophic consequences, sometimes due to wrong or oversimplified use of probabilistic models. Therefore, it can be seen as the citizen‟s perspective to the definition of risk