A Generalised RBF Finite Difference Approach to Solve Nonlinear Heat Conduction Problems on Unstructured Datasets

Radial Basis Functions have traditionally been used to provide a continuous interpolation of scattered data sets. However, this interpolation also allows for the reconstruction of partial derivatives throughout the solution field, which can then be used to drive the solution of a partial differential equation. Since the interpolation takes place on a scattered dataset with no local connectivity, the solution is essentially meshless. RBF-based methods have been successfully used to solve a wide variety of PDEs in this fashion. Such full-domain RBF methods are highly flexible and can exhibit spectral convergence rates Madych & Nelson (1990). However, in their traditional implementation the fully-populated matrix systems which are produced lead to computational complexities of at least order-N2 with datasets of size N. In addition, they suffer fromincreasingly poor numerical conditioning as the size of the dataset grows, and also with increasingly flat interpolating functions. This is a consequence of ill-conditioning in the determination of RBF weighting coefficients (as demonstrated in Driscoll & Fornberg (2002)), and is described by Robert Schaback Schaback (1995) as the uncertainty relation; better conditioning is associated with worse accuracy, and worse conditioning is associated with improved accuracy. Many techniques have been developed to reduce the effect of the uncertainty relation in the traditional RBF formulation, such as RBF-specific preconditioners Baxter (2002); Beatson et al. (1999); Brown (2005); Ling & Kansa (2005), or adaptive selection of data centres Ling et al. (2006); Ling & Schaback (2004). However, at present the only reliable methods of controlling numerical ill-conditioning and computational cost as problem size increases are domain decomposition Hernandez Rosales & Power (2007); Wong et al. (1999); Zhang (2007); Zhou et al. (2003), or the use of locally supported basis functions Fasshauer (1999); Schaback (1997); Wendland (1995); Wu (1995). In this work the domain decomposition principle is applied, forming a large number of heavily overlapping systems that cover the solution domain. A small RBF collocation system is formed around each global data centre, with each collocation system used to approximate the governing PDE at its centrepoint, in terms of the solution value at surrounding collocation points. This leads to a sparse global linear system which may be solved using a variety of standard solvers. In this way, the proposed formulation emulates a finite difference method, with the RBF collocation systems replacing the polynomial interpolation functions used in traditional finite difference methods. However, unlike such polynomial functions RBF collocation is well suited to scattered data, and the method may be applied to both structured and unstructured datasets without modification. The method is applied here to solve the nonlinear heat conduction equation. In order to reduce the nonlinearity in the governing equation the Kirchhoff integral transformation is applied, and the transformed equation is solved using a Picard iterative process. The application of the Kirchhoff transform necessitates that the thermal property functions be transformed to Kirchhoff space also. If the thermal properties are a known and integrable function of temperature then the transformation may be performed analytically. Otherwise, an integration-interpolation procedure can be performed using 1D radial basis functions, as described in Stevens & Power (2010). In recent years a number of local RBF collocation techniques have been proposed, and applied a wide variety of problems (for example; Divo & Kassab (2007); Lee et al. (2003); Sarler & Vertnik (2006); Wright & Fornberg (2006)). A more comprehensive review of such methods is given in Stevens et al. (2009). Unlike most local RBF collocation methods that are used in the literature, the technique described here utilises the Hermitian RBF collocation formulation (see section 2 for more details), and allows both the PDE-boundary and PDE-governing operators to be included within in the local collocation systems. This inclusion of the governing PDE within the basis functions is shown in Stevens et al. (2009) to significantly improve the accuracy and stability of solutions obtained for linear transport problems. Additionally, the incorporation of information about the convective velocity field into the basis functionswas shown to have a stabilising effect, similar to traditional upwinding methods but without the requirement to alter the stencil configuration based on the local convective field. The standard approach to the solution of linear and nonlinear heat conduction problems is the use of finite difference and finite volume methods with simple polynomial interpolants Bejan (1993); Holman (2002); Kreith & Bohn (2000). Due to the dominance of diffusion in most cases, central differencing techniques are commonly used to compute the heat fluxes. However, limiter methods (such as the unconditionally stable TVD schemes) may be used for nonlinear heat conduction problems where the effective convection term, which results from the non-zero variation of thermal conductivity with temperature, can be expected to approach the magnitude of the diffusive term (see, for example, Shen & Han (2002)). Full-domain RBF methods have also been examined for use with nonlinear heat conduction problems (see Chantasiriwan (2007)), however such methods are restricted to small dataset sizes, due to the computational cost and numerical conditioning experienced by full-domain RBF techniques on large datasets. The present work demonstrates how local RBF collocation may be used as an alternative to traditional finite difference and finite volume methods, for nonlinear heat conduction problems. The described method retains freedom from a volumetric mesh, while allowing solution over unstructured datasets. A central stencil configuration is used in each case, and the solution is stabilised via the inclusion of the governing and boundary PDEs within the local collocation systems (\u201cimplicit upwinding\u201d), rather than by adjusting the stencil configuration based on the local solution field (\u201ctraditional upwinding\u201d). The method is validated using a transient numerical example with a known analytical solution (see section 4), and the ability of the formulation to handle strongly nonlinear problems is demonstrated in the solution of a food freezing problem (see section 5)

PERCEPTIONS OF THE ROLE OF PRINCIPAL SUPERVISORS IN NASSAU COUNTY, NEW YORK

Research reveals that school leadership is the second most important in-school factor impacting student achievement, with greater impacts seen when principals’ leadership practice is focused on learning. Research also shows that leadership practices in the area of principal supervision and supporting principal leadership have a positive impact on student achievement. This qualitative phenomenological study utilized concepts from sociocultural learning theory to frame understandings of the lived experiences of principal supervisors in Nassau County, New York. The rationale for this study stems from a moral imperative to address gaps in learning and achievement in disadvantaged groups through effective educational leadership practice. The purposefully selected sample included ten superintendents who were identified from the 56 public school districts in Nassau County, New York. Interview transcripts from superintendents responsible for supervising elementary principals practicing in K-6 and K-5 schools were coded to identify significant statements, resulting in over 425 individual coding references. The analysis and interpretation of the findings were organized into four analytic categories found within the conceptual framework: (a) leadership support, (b) role of principal supervisor, (c) student achievement outcomes, and (d) principal leadership. Findings show that principal supervisors’ practice includes robust support for principals’ professional learning and an emphasis on fostering responsive relationships with principals. Superintendents often support principals by including other central office administrators in supervision. Participants understand the importance of principal leadership and its impact on student achievement outcomes, especially principals’ practice that emphasizes a focus on teaching and learning. The lens of sociocultural learning theory informs the extent to which these practices are performed. Along with these findings, this research identified that a specific role for supervising principals is not clearly defined and practitioners do not use a standard or framework that defines leadership practice focused on learning in their principal supervision work. Recommendations from this study have valuable implications for central office leaders who want to support principals’ leadership through research-based principal supervision practice that can ultimately impact student achievement outcomes and reduce persistent achievement gaps for students in Nassau County public schools

RANS closure approximation by artificial neural networks

Turbulence modelling remains a challenge for the simulation of turbomachinery flows. Reynolds Averaged Navier-Stokes (RANS) equations will still be used for high-Reynolds number flows for several years and so there is interest in improving their prediction capability. Machine learning techniques offer several strategies which could be exploited for this purpose. In this work, an approach to improve the Spalart-Allmaras model is investigated. In particular, the model is used to predict the flow around the T106c low pressure gas turbine cascade. As a first step, an Artificial Neural Network (ANN) is trained on the data generated by the original model. Then, an optimisation procedure is applied in order to find the weights of the network which minimise the error between the predicted results and the available experimental data. The new model is tested at different Reynolds numbers on the T106c cascade and on a wind turbine airfoil in post-stall conditions. Significant improvements are observed in the condition chosen for the optimisation. Future work will be devoted to the generalisation of the approach by including multiple working conditions optimisations and adding new physical variables as inputs of the ANN

Adaptive CFD schemes for aerospace propulsion

The flow fields which can be observed inside several components of aerospace propulsion systems are characterised by the presence of very localised phenomena (boundary layers, shock waves,...) which can deeply influence the performances of the system. In order to accurately evaluate these effects by means of Computational Fluid Dynamics (CFD) simulations, it is necessary to locally refine the computational mesh. In this way the degrees of freedom related to the discretisation are focused in the most interesting regions and the computational cost of the simulation remains acceptable. In the present work, a discontinuous Galerkin (DG) discretisation is used to numerically solve the equations which describe the flow field. The local nature of the DG reconstruction makes it possible to efficiently exploit several adaptive schemes in which the size of the elements (h-adaptivity) and the order of reconstruction (p-adaptivity) are locally changed. After a review of the main adaptation criteria, some examples related to compressible flows in turbomachinery are presented. An hybrid hp-adaptive algorithm is also proposed and compared with a standard h-adaptive scheme in terms of computational efficiency