D. Xiao

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.

Non-linear Petrov-Galerkin methods for reduced order hyperbolic equations and discontinuous finite element methods

A new Petrov-Galerkin approach for dealing with sharp or abrupt field changes in discontinuous Galerkin (DG) reduced order modelling (ROM) is outlined in this paper. This method presents a natural and easy way to introduce a diffusion term into ROM without tuning/optimising and provides appropriate modelling and stablisation for the numerical solution of high order nonlinear PDEs. The approach is based on the use of the cosine rule between the advection direction in Cartesian space-time and the direction of the gradient of the solution. The stabilization of the proper orthogonal decomposition (POD) model using the new Petrov-Galerkin approach is demonstrated in 1D and 2D advection and 1D shock wave cases. Error estimation is carried out for evaluating the accuracy of the Petrov-Galerkin POD model. Numerical results show the new nonlinear Petrov-Galerkin method is a promising approach for stablisation of reduced order modelling.

A non-intrusive reduced-order model for compressible fluid and fractured solid coupling and its application to blasting

This work presents the first application of a non-intrusive reduced order method to model solid interacting with compressible fluid flows to simulate crack initiation and propagation. In the high fidelity model, the coupling process is achieved by introducing a source term into the momentum equation, which represents the effects of forces of the solid on the fluid. A combined single and smeared crack model with the Mohr-Coulomb failure criterion is used to simulate crack initiation and propagation. The non-intrusive reduced order method is then applied to compressible fluid and fractured solid coupled modelling where the computational cost involved in the full high fidelity simulation is high. The non-intrusive reduced order model (NIROM) developed here is constructed through proper orthogonal decomposition (POD) and a radial basis function (RBF) multi-dimensional interpolation method.The performance of the NIROM for solid interacting with compressible fluid flows, in the presence of fracture models, is illustrated by two complex test cases: an immersed wall in a fluid and a blasting test case. The numerical simulation results show that the NIROM is capable of capturing the details of compressible fluids and fractured solids while the CPU time is reduced by several orders of magnitude. In addition, the issue of whether or not to subtract the mean from the snapshots before applying POD is discussed in this paper. It is shown that solutions of the NIROM, without mean subtracted before constructing the POD basis, captured more details than the NIROM with mean subtracted from snapshots.

Natural ventilation in cities: the implications of fluid mechanics

ABSTRACT Research under the Managing Air for Green Inner Cities (MAGIC) project uses measurements and modelling to investigate the connections between external and internal conditions: the impact of urban airflow on the natural ventilation of a building. The test site was chosen so that under different environmental conditions the levels of external pollutants entering the building, from either a polluted road or a relatively clean courtyard, would be significantly different. Measurements included temperature, relative humidity, local wind and solar radiation, together with levels of carbon monoxide (CO) and carbon dioxide (CO2) both inside and outside the building to assess the indoor–outdoor exchange flows. Building ventilation took place through windows on two sides, allowing for single-sided and crosswind-driven ventilation, and also stack-driven ventilation in low wind conditions. The external flow around the test site was modelled in an urban boundary layer in a wind tunnel. The wind tunnel results were incorporated in a large-eddy-simulation model, Fluidity, and the results compared with monitoring data taken both within the building and from the surrounding area. In particular, the effects of street layout and associated street canyons, of roof geometry and the wakes of nearby tall buildings were examined.

Non-intrusive model reduction for a 3D unstructured mesh control volume finite element reservoir model and its application to fluvial channels

A non-intrusive model reduction computational method using hypersurfaces representation has been developed for reservoir simulation and further applied to 3D fluvial channel problems in this work. This is achieved by a combination of a radial basis function (RBF) interpolation and proper orthogonal decomposition (POD) method. The advantage of the method is that it is generic and non-intrusive, that is, it does not require modifications to the original complex source code, for example, a 3D unstructured mesh control volume finite element (CVFEM) reservoir model used here. The capability of this non-intrusive reduced order model (NIROM) based on hypersurfaces representation has been numerically illustrated in a horizontally layered porous media case, and then further applied to a 3D complex fluvial channel case. By comparing the results of the NIROM against the solutions obtained from the high fidelity full model, it is shown that this NIROM results in a large reduction in the CPU computation cost while much of the details are captured.

Non-intrusive Reduced Order Modelling of Waterflooding in Geologically Heterogeneous Reservoirs

Production optimisation and history matching are two applications that require the engineer to run numerous flow simulations of flow in the subsurface. Each flow simulation can be very computationally intensive, especially if reservoir is geologically complex. In some cases it may not be feasible to perform the optimisation sufficiently quickly for it to be useful. This has driven the development of reduced order modelling (ROM) techniques. The problem with most ROMs is that they have to be hard-coded into the flow simulator and so cannot be used with the commercial simulators that are used by most oil companies. In addition most applications of ROMs have assumed that the geological description of the reservoir is known. This is generally not the case, indeed the aim of history matching is to adjust the geological model of the reservoir until the flow through the model replicates that which is observed. In this paper a non-intrusive reduced order model (NIROM) is presented that enables the engineer to vary the permeabilities within a heterogeneous reservoir for a fixed well pattern and then estimate the resulting waterflood performance. The NIROM uses a Smolyak sparse grid interpolation method, radial basis functions (RBF) and proper orthogonal decomposition (POD) is presented. ‘Non-intrusive’ means that the NIROM is implemented independently of the underlying flow model. Here we use it in conjunction with an unstructured mesh, control volume finite element (CVFEM), multiphase flow model. The NIROM is demonstrated using three reservoir models: one with four material layers, one with four baffles and one with eight baffles. The results compare well with those from a high fidelity full model and reduce the CPU time by a factor of a thousand.