A new implicit and compact optimization-based method is presented for high order derivative calculation for finite-volume numerical method on unstructured meshes. Highorder approaches to gradient calculation are often based on variants of the Least-Squares (L-S) method, an explicit method that requires a stencil large enough to accommodate the necessary variable information to calculate the derivatives. The new scheme proposed here is applicable for an arbitrary order of accuracy (demonstrated here up to 3rd order), and uses just the first level of face neighbors to compute all derivatives, thus reducing stencil size and avoiding stiffness in the calculation matrix. Preliminary results for a static variable field example and solution of a simple scalar transport (advection) equation show that the proposed method is able to deliver numerical accuracy equivalent to (or better than) the nominal order of accuracy for both 2nd and 3rd order schemes in the presence of a smoothly distributed variable field (i.e., in the absence of discontinuities). This new Optimization-based Gradient REconstruction (herein denoted OGRE) scheme produces, for the simple scalar transport test case, lower error and demands less computational time (for a given level of required precision) for a 3rd order scheme when compared to an equivalent L-S approach on a two-dimensional framework. For three-dimensional simulations, where the L-S scheme fails to obtain convergence without the help of limiters, the new scheme obtains stable convergence and also produces lower error solution when compared to a third order MUSCL scheme. Furthermore, spectral analysis of results from the advection equation shows that the new scheme is better able to accurately resolve high wave number modes, which demonstrates its potential to better solve problems presenting a wide spectrum of wavelengths, for example unsteady turbulent flow simulations.

Finite volume methods employing second-order gradient reconstruction schemes are often utilized to computationally solve the governing equations of fluid mechanics and transport. These schemes, while not as dissipative as first-order schemes, frequently produce oscillatory solutions in regions of discontinuities and/or unsatisfactory levels of dissipation in smooth regions of the variable field. Limiters are often employed to reduce the inherent variable over- and under-shoot; however, they can significantly increase the numerical dissipation of a solution, eroding a scheme’s performance in smooth regions. A novel gradient reconstruction scheme, which shows significant improvement over traditional second-order schemes, is presented in this work. Two implementations of this Optimization-based Gradient REconstruction (OGRE) scheme are examined: minimizing an objective function based on the mismatch between local reconstructions at midpoints or selected quadrature points between cell stencil neighbors. Regardless of the implementation employed, the resulting gradient calculation is a compact, implicit method that can be used with unstructured meshes by employing an arbitrary computational stencil. An adjustable weighting parameter is included in the objective function that allows the scheme to be tuned towards either greater accuracy or greater stability. To address over- and undershoot of the variable field near discontinuities, non-local, non-monotonic (NLNM) and local, non-monotonic (LNM) limiters have also been developed, which operate by enforcing cell minima and maxima on dependent variable values projected to cell faces. The former determines minimum and maximum values for a cell through recursive reference to the minimum and maximum values of its upwind neighbors. The latter determines these bounding values through examination of the extrema of values of the dependent variable projected from the face-neighbor cell into the original cell. Steady state test cases on structured and unstructured grids are presented, exhibiting the low-dissipative nature of the scheme. Results are primarily compared to those produced by existing limited and unlimited second-order upwind (SOU) and first-order upwind (FOU). Solution accuracy, convergence rate and computational costs are examined.

The construction of compact high-order Residual Distribution schemes for the discretizationof steady multidimensional advection-diffusion problems on unstructuredgrids is presented. Linear and non-linear scheme are considered. A piecewise continuouspolynomial approximation of the solution is adopted and a gradient reconstructionprocedure is used in order to have a continuous representation of both thenumerical solution and its gradient. It is shown that the gradient must be reconstructedwith the same accuracy of the solution, otherwise the formal accuracy ofthe numerical scheme is lost in applications in which diffusive effects prevail overthe advective ones, and when advection and diffusion are equally important. Thenthe method is extended to systems of equations, with particular emphasis on theNavier-Stokes and RANS equations. The accuracy, efficiency, and robustness of theimplicit RD solver is demonstrated using a variety of challenging aerodynamic testproblems.

The objective of this PhD research program is to develop an accurate and efficient heterogeneous three-dimensional computational model for simulating the drying of wood at temperatures below the boiling point of water. The complex macroscopic drying equations comprise a coupled and highly nonlinear system of physical laws for liquid and energy conservation. Due to the heterogeneous nature of wood, the physical model parameters strongly depend upon the local pore structure, wood density variation within growth rings and variations in primary and secondary system variables. In order to provide a realistic representation of this behaviour, a set of previously determined parameters derived using sophisticated image analysis methods and homogenisation techniques is embedded within the model. From the literature it is noted that current three-dimensional computational models for wood drying do not take into consideration the heterogeneities of the medium. A significant advance made by the research conducted in this thesis is the development of a three - dimensional computational model that takes into account the heterogeneous board material properties which vary within the transverse plane with respect to the pith position that defines the radial and tangential directions. The development of an accurate and efficient computational model requires the consideration of a number of significant numerical issues, including the virtual board description, an effective mesh design based on triangular prismatic elements, the control volume finite element discretisation process for the cou- pled conservation laws, the derivation of an accurate dux expression based on gradient approximations together with flux limiting, and finally the solution of a large, coupled, nonlinear system using an inexact Newton method with a suitably preconditioned iterative linear solver for computing the Newton correction. This thesis addresses all of these issues for the case of low temperature drying of softwood. Specific case studies are presented that highlight the efficiency of the proposed numerical techniques and illustrate the complex heat and mass transport processes that evolve throughout drying.

Cette thèse présente la construction de schémas distribuant le résidu (RD) d'ordre très élevés, pour la discrétisation d'équations d'advection-diffusion multidimensionnelles et stationnaires sur maillages non structurés. Des schémas linéaires ainsi que des schémas non linéaires sont considérés. Une approximation de la solution polynomiale par morceaux et continue sur chaque élément est adoptée, de plus une procédure de reconstruction du gradient que celle de la solution numérique est utilisée afin d'avoir une représentation continue de la solution numérique et de son gradient. Il est montré que le gradient doit être reconstruit avec la même précision de la solution, sans quoi la précision formel du schéma numérique est perdue dans les cas où les effets de diffusion prévalent sur les effets d'advection, et aussi quand l'advection et la diffusion sont également importants. Ensuite, la méthode est étendue à des systèmes d'équations, en particulier aux équations de Navier-Stokes et aux équations RANS. La précision, l'efficacité et la robustesse du solveur RD implicite sont démontrées sur plusieurs cas tests. / The construction of compact high-order Residual Distribution schemes for the discretizationof steady multidimensional advection-diffusion problems on unstructuredgrids is presented. Linear and non-linear scheme are considered. A piecewise continuouspolynomial approximation of the solution is adopted and a gradient reconstructionprocedure is used in order to have a continuous representation of both thenumerical solution and its gradient. It is shown that the gradient must be reconstructedwith the same accuracy of the solution, otherwise the formal accuracy ofthe numerical scheme is lost in applications in which diffusive effects prevail overthe advective ones, and when advection and diffusion are equally important. Thenthe method is extended to systems of equations, with particular emphasis on theNavier-Stokes and RANS equations. The accuracy, efficiency, and robustness of theimplicit RD solver is demonstrated using a variety of challenging aerodynamic testproblems.