A.G. Buchan

Non-linear model reduction for the Navier-Stokes equations using residual DEIM method

This article presents a new reduced order model based upon proper orthogonal decomposition (POD) for solving the Navier-Stokes equations. The novelty of the method lies in its treatment of the equations non-linear operator, for which a new method is proposed that provides accurate simulations within an efficient framework. The method itself is a hybrid of two existing approaches, namely the quadratic expansion method and the Discrete Empirical Interpolation Method (DEIM), that have already been developed to treat non-linear operators within reduced order models. The method proposed applies the quadratic expansion to provide a first approximation of the non-linear operator, and DEIM is then used as a corrector to improve its representation. In addition to the treatment of the non-linear operator the POD model is stabilized using a Petrov-Galerkin method. This adds artificial dissipation to the solution of the reduced order model which is necessary to avoid spurious oscillations and unstable solutions. A demonstration of the capabilities of this new approach is provided by solving the incompressible Navier-Stokes equations for simulating a flow past a cylinder and gyre problems. Comparisons are made with other treatments of non-linear operators, and these show the new method to provide significant improvements in the solutions accuracy.

The Inner-Element Subgrid Scale Finite Element Method for the Boltzmann Transport Equation

Abstract This paper presents a new multiscale radiation transport method based on a Galerkin finite element spatial discretization of the Boltzmann transport equation. The approach incorporates a discontinuous subgrid scale (SGS) solution within the continuous finite element representation of the spatial variables. While the conventional discontinuous Galerkin (DG) method provides accurate and numerically stable solutions that suppress unphysical oscillations, the number of unknowns is relatively high. The key advantage of the proposed SGS approach is that the solutions are represented within the continuous finite element space, and therefore, the number of unknowns compared with DG is relatively low. The applications of this method are explored using linear finite elements, and some of the advantages of this new discretization over standard Petrov-Galerkin methods are demonstrated. The numerical examples are chosen to be demanding steady-state mono-energetic radiation transport problems that are likely to form unphysical oscillations within numerical scalar flux solutions. The numerical examples also provide evidence that the SGS method has a thick diffusion limit. This method is designed to work under arbitrary angular discretizations, so solutions using both spherical harmonics and discrete ordinates are presented.

A POD reduced order model for resolving angular direction in neutron/photon transport problems

This article presents the first Reduced Order Model (ROM) that efficiently resolves the angular dimension of the time independent, mono-energetic Boltzmann Transport Equation (BTE). It is based on Proper Orthogonal Decomposition (POD) and uses the method of snapshots to form optimal basis functions for resolving the direction of particle travel in neutron/photon transport problems. A unique element of this work is that the snapshots are formed from the vector of angular coefficients relating to a high resolution expansion of the BTEs angular dimension. In addition, the individual snapshots are not recorded through time, as in standard POD, but instead they are recorded through space. In essence this work swaps the roles of the dimensions space and time in standard POD methods, with angle and space respectively.It is shown here how the POD model can be formed from the POD basis functions in a highly efficient manner. The model is then applied to two radiation problems; one involving the transport of radiation through a shield and the other through an infinite array of pins. Both problems are selected for their complex angular flux solutions in order to provide an appropriate demonstration of the models capabilities. It is shown that the POD model can resolve these fluxes efficiently and accurately. In comparison to high resolution models this POD model can reduce the size of a problem by up to two orders of magnitude without compromising accuracy. Solving times are also reduced by similar factors.

Self-Adaptive Spherical Wavelets for Angular Discretizations of the Boltzmann Transport Equation

Abstract A new method for applying anisotropic resolution in the angular domain of the Boltzmann transport equation is presented. The method builds on our previous work in which two spherical wavelet bases were developed for representing the direction of neutral particle travel. The method proposed here enables these wavelet bases to vary their angular approximations so that fine resolution may be applied only to the areas of the unit sphere (representing the direction of particle travel) that are important. We develop an error measure that operates in conjunction with the wavelet bases to determine this importance. A procedure by which the angular resolution is gradually refined for steady-state problems is also given. The adaptive wavelets are applied to three test problems that demonstrate the ability of the wavelets to resolve complex fluxes with relatively few functions, and to achieve this a particular emphasis is placed on their ability to approximate particle streaming through ducts with voids. It is shown that the wavelets are capable of applying the appropriate resolution (as dictated by the error measure) to the directional component of the angular flux at all spatial positions. This method therefore offers a new and highly efficient adaptive angular approximation method.

Goal-based angular adaptivity applied to a wavelet-based discretisation of the neutral particle transport equation

A method for applying goal-based adaptive methods to the angular resolution of the neutral particle transport equation is presented. The methods are applied to an octahedral wavelet discretisation of the spherical angular domain which allows for anisotropic resolution. The angular resolution is adapted across both the spatial and energy dimensions. The spatial domain is discretised using an inner-element sub-grid scale finite element method. The goal-based adaptive methods optimise the angular discretisation to minimise the error in a specific functional of the solution. The goal-based error estimators require the solution of an adjoint system to determine the importance to the specified functional. The error estimators and the novel methods to calculate them are described. Several examples are presented to demonstrate the effectiveness of the methods. It is shown that the methods can significantly reduce the number of unknowns and computational time required to obtain a given error. The novelty of the work is the use of goal-based adaptive methods to obtain anisotropic resolution in the angular domain for solving the transport equation. Wavelet angular discretisation used to solve transport equation.Adaptive method developed for the wavelet discretisation.Anisotropic angular resolution demonstrated through the adaptive method.Adaptive method provides improvements in computational efficiency.

Adaptive Haar wavelets for the angular discretisation of spectral wave models

A new framework for applying anisotropic angular adaptivity in spectral wave modelling is presented. The angular dimension of the action balance equation is discretised with the use of Haar wavelets, hierarchical piecewise-constant basis functions with compact support, and an adaptive methodology for anisotropically adjusting the resolution of the angular mesh is proposed. This work allows a reduction of computational effort in spectral wave modelling, through a reduction in the degrees of freedom required for a given accuracy, with an automated procedure and minimal cost.

Reduced order borehole induction modelling

This article presents a new reduced order model based on proper orthogonal decomposition (POD) for solving the electromagnetic equation for borehole modelling applications. The method aims to accurately and efficiently predict the electromagnetic fields generated by an array induction tool – an instrument that transmits and receives electrical signals along different positions within a borehole. The motivation for this approach is in the generation of an efficient ‘forward model’ (which provides solutions to the electromagnetic equation) for the purpose of improving the efficiency of inversion calculations (which typically require a large number of forward solutions) that are used to determine surrounding material properties. This article develops a reduced order model for this purpose as it can be significantly more efficient to compute than standard models, for example, those based on finite elements. It is shown here how the POD basis functions are generated from the snapshot solutions of a high resolution model, and how the discretised equations can be generated efficiently. The novelty is that this is the first time such a POD model reduction approach has been developed for this application, it is also unique in its use of separate POD basis functions for the real and complex solution fields. A numerical example for predicting the electromagnetic field is used to demonstrate the accuracy of the POD method for use as a forward model. It is shown that the method retains accuracy whilst reducing the costs of the computation by several orders of magnitude in comparison to an established method.

Directions in Radiation Transport Modelling

Radiation transport modelling has come a long way in the last 50 years: 2D models have been replaced by 3D models; multi-group energy schemes have been replaced by continuous energy nuclear data representations in Monte Carlo models; accurate 3D geometrical representations are available, including import from CAD files. More exciting advances are on the horizon to increase the power of simulation tools. The advent of high performance computers is allowing bigger, higher fidelity models to be created, if the challenges of parallelization and memory management can be met. 3D whole core transport modelling is becoming possible. Uncertainty quantification is improving with large benefits to be gained from more accurate, less pessimistic estimates of uncertainty. Advanced graphical displays allow the user to assimilate and make sense of the vast amounts of data produced by modern modelling tools. Numerical solvers are being developed that use goal-based adaptivity to adjust the nodalisation of the system to provide the optimum scheme to achieve the user requested accuracy on the results, thus removing the need to perform costly convergence studies in space and angle etc. More use is being made of multi-physics methods in which radiation transport is coupled with other phenomena, such as thermal-hydraulics, structural response, fuel performance and/or chemistry in order to better understand their interplay in reactor cores.