# Carl Ollivier-Gooch

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## Graduate Student Supervision

##### Doctoral Student Supervision (Jan 2008 - May 2021)

Combustion technology has been applied in human society for millennia and, since the industrial revolution, has become an integral part of most energy supply chains. Simulation is an important tool in modern combustor design; this thesis aims to improve the quality of combustion simulation tools, and thereby facilitate the design of improved combustors. More specifically, it aims to examine how generalizing and/or relaxing the definition of conditional filtering – a common technique in turbulence-chemistry interaction modelling – can produce novel turbulence and combustion models. The work begins with an extension of the Conditional Source-term Estimation (CSE) model for turbulence-chemistry interaction modelling. A novel variation of the algorithm, termed CSE with Geometric Conditioning Variables (CSE-GCV) is proposed as a method of circumventing the theoretical and practical issues associated with traditional CSE ensemble division. In CSE-GCV, the concept of the conditional filter is generalized by introducing geometric (position-based) variables as conditioning variables. CSE-GCV is tested and found to be workable; a theoretical analysis demonstrates that CSE-GCV also generally has the advantage of reduced computational complexity compared to traditional CSE. In a separate study, the stabilization procedure employed in traditional dynamic sub-filter modelling for Large Eddy Simulation (LES) is re-interpreted as a form of conditional filtering based on position. This re-interpretation is used as the starting point for a "conditional dynamic" sub-filter model in which the stabilization procedure is based on filtering conditionally on scalar fields. Both the traditional and conditional models are applied to a turbulent flame; results suggest that the two models perform similarly, although performance of both is sub-optimal in the case considered. The final, two-part, study is based around the suggestion that, with sufficient conditioning, conditionally-filtered fields should be independent of position. It is found that assuming this uniformity produces a novel turbulence-chemistry interaction model, termed the Uniform Conditional State (UCS) model, in which the conditional scalar dissipation model is the key un-closed parameter. The UCS model is applied to a series of turbulent non-premixed flames, and is found to predict their properties to good accuracy, with details showing some sensitivity to the conditional scalar dissipation model.

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The purpose of this thesis is to develop a framework in which one can detect and automatically improve the numerical stability of cell-centered finite-volume calculations on unstructured meshes through optimization schemes that modify the mesh, the solution reconstruction, or the boundary conditions. In this process, eigenanalysis and the gradients of the eigenvalues with respect to different parameters are used to ensure energy stability of the system, consequently resulting in convergence. First, gradients of eigenvalues with respect to the local changes in the mesh are calculated to find directions and magnitudes of mesh movements that will make the Jacobian of a semi-discrete system of equations negative semi-definite. These mesh movement vectors are used to modify the mesh locally. The numerical results show that the proposed methods are able to locate the problematic parts of the mesh responsible for instabilities for several physical problems and to correct the instability. Secondly, I develop a mathematical method, introduced by Haider et al., to measure the stability impact of the reconstruction for high order and nonlinear problems, regardless of the solution. Second order and third order accurate advection and Burgers and Euler problems are used to present detailed practical results and discussion around the use of the local reconstruction map for stability analysis. This method shows that increasing the stencil size will lead to more stable problems. An empirical study is performed which sheds light on connections between the mesh properties and the stability of the reconstruction. I also propose a systematic approach to optimize both the shape and the size of the reconstruction stencil for better numerical stability through eigenvalue analysis. In this approach, one can directly optimize the solution reconstruction stencil for every control volume to obtain better numerical stability and convergence properties for steady state problems.The rightmost eigenpairs of the spatially discretized system of equations are used to obtain an optimized boundary condition which will ensure the energy stability of the system. Lastly, the sensitivity of the rightmost eigenvalues to the solution is measured to investigate the effect of using surrogate solutions for the purpose of linearizing the semi-discretized Jacobian.

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A numerical estimation of discretization error is performed for solutions to steady and unsteady models for compressible flow. An accurate and reliable estimate of discretization error is useful in obtaining a more accurate defect corrected solution, as well as a tight uncertainty bound as error bars. The error estimation procedure is performed by solving an auxiliary problem, known as the error transport equation (ETE), solved on the same mesh as the original model equations. Unlike unsteady adjoint methods for error in functionals, the ETE requires only one other set of equations to be solved, agnostic to the choice and number of output functionals, including common aerodynamic quantities such as lift or drag. Furthermore, co-advancing the ETE in time only requires the storage of local solutions in time and not the entire history, reducing memory requirements. This method of error estimation is performed in the context of higher order finite-volume methods on unstructured meshes. Approaches based on solving the ETE can be found in the literature for uniform or smooth meshes, but this has not been well studied for unstructured meshes. Such meshes necessarily have nonsmooth geometric features, which create many difficulties in accurate error estimation. These difficulties in accurate discretizations of the ETE are investigated, including the discretization of the ETE source term, which is critical to error estimate accuracy. It was found that the proposed schemes by the ETE approach can be more efficient and robust compared to solving the higher order problem. The choices of discretization schemes need to be made carefully, and these results demonstrate how it is possible, along with justification, to obtain asymptotically accurate, efficient, and robust error estimates that can be used with vast possibilities of model equations in practice.

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The design of aircraft depends increasingly on the use of Computational Fluid Dynamics (CFD) in which numerical methods are employed to obtain approximate solutions for fluid flows. One route to improve the numerical accuracy of CFD simulations is higher-order discretization methods. Moreover, finite volume discretizations are the method of choice in commercial CFD solvers and also in computational aerodynamics because of intrinsic conservative and shock-capturing properties. Considering that nearly all practical flows with aerodynamic applications are classified as turbulent, we develop a higher-order finite volume solver for the Reynolds Averaged Navier-Stokes (RANS) solution of turbulent compressible flows on unstructured meshes. Higher-order flow solvers must account for boundary curvature. Since turbulent flow simulations require anisotropic cells in shear layers, we use an elasticity analogy to project the boundary curvature into the interior faces and prevent faces from intersecting near curved boundaries. Furthermore, we improve the accuracy of solution reconstruction and output quantities on highly anisotropic cells with curvature using a local curvilinear coordinate system. A robust turbulence model for higher-order discretizations is fully coupled to the system of RANS equations and an efficient solution strategy is adopted for the convergence to the steady-state solution. We present our higher-order results for simple and complicated configurations in two dimensions. These results are verified by comparison against well-established numerical and experimental values in the literature. Our results show the advantages of higher-order methods in obtaining a more accurate solution with fewer degrees of freedom and also fast and efficient convergence to the steady-state solutions. Moreover, we propose an hp-adaptation algorithm for the unstructured finite volume solver based on residual-based and adjoint-based error indicators. In this approach, we enhance the local accuracy of the discretization via h-refinement or p-enrichment based on the smoothness of the solution. Mesh refinement is carried out by local cell division and introducing non-conforming interfaces in the mesh while order enrichment is obtained by local increase of the polynomial order in the reconstruction process. Our results show that this strategy leads to accuracy and efficiencyimprovements for several types of compressible flow problems.

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Error quantification for industrial CFD requires a new paradigm in which a robust flow solver with error quantification capabilities reliably produces solutions with known error bounds. Error quantification hinges on the ability to accurately estimate and efficiently exploit the local truncation error. The goal of this thesis is to develop a reliable truncation error estimator for finite-volume schemes and to use this truncation error estimate to improve flow solutions through defect correction, to correct the output functional, and to adapt the mesh. We use a higher-order flux integral based on lower order solution as an estimation of the truncation error which includes the leading term in the truncation error. Our results show that using this original truncation error estimate is dominated by rough modes and fails to provide the desired convergence for the applications of defect correction, output error estimation and mesh adaptation. So, we tried to obtain an estimate of the truncation error based on the continuous interpolated solution to improve their performance. Two different methods for interpolating were proposed: CGM's 3D surfaces and C¹ interpolation of the solution. We compared the effectiveness of these two interpolating schemes for defect correction and using C¹ interpolation of the solution for interpolating is more helpful compared to CGM, so we continued using C¹ interpolation for other purposes. For defect correction, although using the modified truncation error does not improve the order of accuracy, significant quantitative improvements are obtained. Output functional correction is based on the truncation error and the adjoint solution. Both discrete and continuous adjoint solutions can be used for functional correction. Our results for a variety of governing equations suggest that the interpolating scheme can improve the correction process significantly and improve accuracy asymptotically. Different adaptation indicators were considered for mesh adaptation and our results show that the estimate of the truncation error based on the interpolated solution is a more accurate indicator compared to the original truncation error. Adjoint-based mesh adaptation combined with modified truncation error provides even faster convergence of the output functional.

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The growth of computer power and storage capacity allowed engineers to tackle engineering design as an optimization problem. For transport aircraft, drag minimization is critical to increase range and reduce operating costs. Lift and geometric constraints are added to the optimization problem to meet payload and rigidity constraints of the aircraft. Higher order methods in CFD simulations have proved to be a valuable tool and are expected to replace current second order CFD methods in the near future; therefore, exploring the use of higher order CFD methods in aerodynamic optimization is of great research interest and is one goal of this thesis.Gradient-based optimization techniques are well known for fast convergence, but they are only local minimizers; therefore their results depend on the starting point in the design space. The gradient-independent optimization techniques can find the global minimum of an objective function but require vast computational effort; therefore, for global optimization with reasonable computational cost, a hybrid optimization strategy is needed.A new least-squares based geometry parametrization is used to describe airfoil shapes and a semi-torsional spring analogy mesh morphing tool updates the grid everywhere when the airfoil geometry changes during shape optimization.For the gradient based optimization scheme, both second and fourth order simulations have been used to compute the objective function; the adjoint approach, well known for its low computational cost, has been used for gradient computation and matches well with finite difference gradient. The gradient based optimizer have been tested for subsonic and transonic inverse design problems and for drag minimization without and with lift constraint to validate the developed optimizer. The optimization scheme used is Sequential Quadratic Programming (SQP) with the BFGS approximation of the Hessian matrix. A mesh refinement study is presented for an aerodynamically constrained drag minimization problem to show how second and fourth order optimal results behave with mesh refinement.A hybrid particle swarm / BFGS scheme has been developed for use as a global optimizer. It has been tested on a drag minimization problem with lift constraint; the hybrid scheme obtained a shock free profiles, while gradient-based optimization could not in general.

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Delaunay refinement is a mesh generation paradigm noted for offering theoretical guarantees regarding the quality of its output. As such, the meshes it produces are a good choice for numerical methods. This thesis studies the practical application of Delaunay refinement mesh generation to geometric domains whose boundaries are curved, in both two and three dimensions. It is subdivided into three manuscripts, each of them addressing a specific problem or a previous limitation of the method. The first manuscript is concerned with the problem in two dimensions. It proposes a technique to sample the boundary with the objective of improving its recoverability. The treatment of small input angles is also improved. The quality guarantees offered by previous algorithms are shown to apply in the presence of curves. The second manuscript presents an algorithm to construct constrained Delaunay tetrahedralizations of domains bounded by piecewise smooth surfaces. The boundary conforming meshes thus obtained are typically coarser than those output by other algorithms. These meshes are a perfect starting point for the experimental study presented in the final manuscript. Therein, Delaunay refinement is shown to eliminate slivers when run using non-standard quality measures, albeit without termination guarantees. The combined results of the last two manuscripts are a major stepping stone towards combining Delaunay refinement mesh generation with CAD modelers. Some algorithms presented in this thesis have already found application in high-order finite-volume methods. These algorithms have the potential to dramatically reduce the computational time needed for numerical simulations.

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High-order accurate methods have the potential to dramatically reduce the computational time needed for aerodynamics simulations. This thesis studies the discretization and efficient convergence to steady state of the high-order accurate finite-volume method applied to the simplified problem of inviscid and laminar viscous two-dimensional flow equations. Each of the three manuscript chapters addresses a specific problem or limitation previously experienced with these schemes. The first manuscript addresses the absence of a method to maintain monotonicity of the solution at discontinuities while maintaining high-order accuracy in smooth regions. To resolve this, a slope limiter is carefully developed which meets these requirements while also maintaining the good convergence properties and computational efficiency of the least-squares reconstruction scheme. The second manuscript addresses the relatively poor convergence properties of Newton-GMRES methods applied to high-order accurate schemes. The globalization of the Newton method is improved through the use of an adaptive local timestep and of a line search algorithm. The poor convergence of the linear solver is improved through the efficient assembly of the exact flux Jacobian for use in preconditioning and to eliminate the additional residual evaluations needed by a matrix-free method. The third manuscript extends the discretization method to the viscous fluxes and boundary conditions. The discretization is demonstrated to achieve the expected order of accuracy. The fourth-order scheme is also shown to be more computationally efficient than the second-order scheme at achieving grid-converged values of drag for two-dimensional laminar flow over an airfoil.

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##### Master's Student Supervision (2010 - 2020)

The use of unstructured meshes in the simulation of a computational field to solve a real world problem is ubiquitous. To resolve the flow near viscous boundaries of a geometry, we need to have good resolution in thin shear layers close to the surface. Having a structured mesh along the boundary is an option. However, we lose the ability to tackle arbitrary geometries that way. Hence, we develop techniques to get a nearly-structured mesh near the boundary where it's needed while retaining topological flexibility by being unstructured.A plethora of 3D boundary layer mesh generation techniques start off from a discretization of the surface. A majority of these techniques either use surface inflation or iterative point placement normal to the surface to generate the advancing layer 3D mesh. Generating good boundary layer meshes in 3D depends on the quality of the underlying surface discretization. We introduce a technique to generate advancing layer surface meshes, which will provide a better starting point for 3D anisotropic mesh generation. The technique takes an input triangulation of the surface, which is fairly easy to get, even for complex geometries. Surface segments are identified and these segments are meshed independently using an advancing-layer methodology. For each surface segment, a mesh is generated by advancing layers from the identified boundaries to the surface interior while deforming the existing triangulation. As the mesh generation technique introduced here is a closed advancing layer method, we have a valid surface mesh throughout the meshing process.The method introduced to generate advancing layer meshes produces semi-structured quad-dominant meshes with the ability to have local control over the aspect ratio of mesh elements at the boundary curves of the surface. Semi-structured 2D anisotropic meshes in the boundary layer regions have been shown to have superior fluid flow simulation results. Point placement in layers, local reconnection, front recovery, front collision handling and smoothing techniques used in the study help produce a valid surface mesh at each step of mesh generation. We demonstrate the ability of the meshing algorithm to tackle fairly complex geometries and coarse initial surface discretization.

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High-order accurate numerical discretization methods are attractive for their potential to significantly reduce the computational costs compared to the traditional second-order methods. Among the various unstructured higher-order discretization schemes, the k-exact reconstruction finite volume method is of interest for its straightforward mathematical formulation, and its compatibility with the current lower-order industrial solvers. However, current three-dimensional finite volume solvers are limited to the solution of inviscid and laminar viscous flow problems. Since three-dimensional turbulent flows appear in many industrial applications, the current thesis takes the first step towards the development of a three-dimensional higher-order finite volume solver for the solution of both inviscid and viscous turbulent steady-state flow problems. The k-exact finite volume formulation of the governing equations is rederived in a dimension-independent manner, where the negative Spalart-Allmaras turbulence model is employed. This one-equation model is reasonably accurate for many flow conditions, and its simplicity makes it a good starting point for the development of numerical algorithms. Then, the three-dimensional mesh preprocessing steps for a finite volume simulation are presented, including higher-order accurate numerical quadrature, and capturing the boundary curvature in highly anisotropic meshes. Also, the issues of k-exact reconstruction in handling highly anisotropic meshes are reviewed and addressed. Since three-dimensional problems can require much more memory than their two-dimensional counter-parts, solution methods that work in two dimensions might not be feasible in three dimensions anymore. As an attempt to overcome this issue, a practical and parallel scalable method for the solution of the discretized system of nonlinear equations is presented. Finally, the solution of four three-dimensional test problems are studied: Poisson’s equation in a cubic domain, inviscid flow over a sphere, turbulent flow over a flat plate, and turbulent flow over an extruded NACA 0012 airfoil. The solution is verified, and the resource consumption of the flow solver is measured. The results demonstrate the benefit and practicality of using higher-order methods for obtaining a certain level of accuracy.

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In this thesis, we present a new procedure for mesh adaptation for wakes. The approachstarts by tracking the wake centerline with an initial isotropic unstructured mesh. A vertex-centered finite volume method is used, and the velocity field is obtained from solution reconstruction. The velocity data is integrated numerically using an adaptive fourth-orderRunge-Kutta method. We insert the wake centerline into the existing unstructured meshas an internal boundary and use a metric-based anisotropic mesh adaptation to generateanisotropic cells in regions with large second derivatives of flow variables. In the secondstep, the problem is solved on adapted mesh and a new wake centerline is tracked. Wethen move the previous wake centerline (which is now a part of adapted mesh) to matchthe centerline obtained from the adapted mesh. To move the wake centerline, a solid mechanics analogy is used and the linear elasticity equation is solved on the adapted mesh.As a result, the displacement is propagated throughout the mesh and the already adaptedregions along the wake centerline are preserved. The process is then followed for subsequentcycles of anisotropic mesh adaptation to obtain a more accurate approximation of the wakecenterline. As an alternate strategy for obtaining an anisotropic mesh in the wake, we take the first geometry, together with the captured wake centerline from an unstructured triangular mesh,as an initial geometry to produce a quad dominant mesh, using an advancing layer method. The correctness of the streamline tracking algorithm is verified using an analytical velocity field. The mesh morphing approach is tested using the method of manufactured solutions,demonstrating that the linear finite element solution is second-order accurate. The resultsof laminar flow test cases for the attached and separated flow are presented and compared with some well-established numerical results in the literature. Our results show that the advancing layer mesh is more efficient in resolving the wake. In the end, one case for turbulentsubsonic flow is considered. For turbulent flow, a cell-centered finite volume method is usedand we only track the wake centerline at different angles of attack.

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High-order accurate numerical discretization methods are attractive for their potential to significantly reduce the computational costs compared to the traditional second-order methods. Among the various unstructured higher-order discretization schemes, the k-exact reconstruction finite volume method is of interest for its straightforward mathematical formulation, and its compatibility with the current lower-order industrial solvers. However, current three-dimensional finite volume solvers are limited to the solution of inviscid and laminar viscous flow problems. Since three-dimensional turbulent flows appear in many industrial applications, the current thesis takes the first step towards the development of a three-dimensional higher-order finite volume solver for the solution of both inviscid and viscous turbulent steady-state flow problems. The k-exact finite volume formulation of the governing equations is rederived in a dimension-independent manner, where the negative Spalart-Allmaras turbulence model is employed. This one-equation model is reasonably accurate for many flow conditions, and its simplicity makes it a good starting point for the development of numerical algorithms. Then, the three-dimensional mesh preprocessing steps for a finite volume simulation are presented, including higher-order accurate numerical quadrature, and capturing the boundary curvature in highly anisotropic meshes. Also, the issues of k-exact reconstruction in handling highly anisotropic meshes are reviewed and addressed. Since three-dimensional problems can require much more memory than their two-dimensional counter-parts, solution methods that work in two dimensions might not be feasible in three dimensions anymore. As an attempt to overcome this issue, a practical and parallel scalable method for the solution of the discretized system of nonlinear equations is presented. Finally, the solution of four three-dimensional test problems are studied: Poisson’s equation in a cubic domain, inviscid flow over a sphere, turbulent flow over a flat plate, and turbulent flow over an extruded NACA 0012 airfoil. The solution is verified, and the resource consumption of the flow solver is measured. The results demonstrate the benefit and practicality of using higher-order methods for obtaining a certain level of accuracy.

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Mesh quality plays an important role in improving the accuracy of the numerical simulation. There are different quality metrics for specific numerical cases. A regular mesh consisting of the equilateral triangles is one of them and is expected to improve the error performance. In this study, Engwirda’s frontal-Delaunay scheme and Marcum’s advancing front local reconnection scheme are described along with the conventional Delaunay triangulation. They are shown to improve the mesh regularity effectively. Even though several numerical test cases show that more regular meshes barely improve the error performance, the time cost in the solver of regular meshes is smaller than the Delaunay mesh. The time cost decrease in the solver pays off the additional cost in the mesh generation stage. For simple test cases, more regular meshes obtain lower errors than conventional Delaunay meshes with similar time costs. For more complicated cases, the improvement in errors is small but regular meshes can save time, especially for a high order solver. Generally speaking, a regular mesh does not improve the error performance as much as we expect, but it is worth generating.

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We present a new procedure to adaptively produce anisotropic metric-orthogonal meshes in this thesis. The approach is based on mesh optimization techniques: point insertion designed to improve mesh alignment as well as conforming to a metric, swapping in the metric space, point movement defined by target elements from a metric, and point deletion based on quality and metric length. These techniques are intended to produce quasi-structured meshes which have the advantages of flexibility for complex geometries of unstructured meshes and the directional accuracy of structured meshes. The result is reliable alignment for anisotropic meshes reducing our previous work's reliance on smoothing for good alignment. The methodology is implemented in 2D and extended to 3D. Examples of analytical metrics and error estimation metrics on a numerical simulation of a flow are shown.

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The overarching goal of CFD is to compute solutions with low numerical error. For finite-volume schemes, this error originates as error in the flux integral. For diffusion problems on unstructured meshes, the diffusive flux (computed from reconstructed gradients) is one order less accurate than the reconstructed solution. Worse, the gradient errors are not smooth, and so no error cancellation accompanies the flux integration, reducing the flux integral to zero order for second-order schemes. Our aim is to compute the gradient and flux more accurately at the cell boundaries and hence obtain a better flux integral for a slight increase in computational cost. We propose a novel reconstruction method and flux discretization to improve diffusive flux accuracy on cell-centred, isotropic unstructured meshes. Our approach uses a modified least-squares system to reconstruct the solution to second-order accuracy in the H₁ norm instead of the prevalent L₂ norm, thus ensuring second-order accurate gradients. Either circumcentres or containment centres are chosen as the control-volume reference points based on a criteria to facilitate calculation of second-order gradients at flux quadrature points using a linear interpolation scheme along with a high-accuracy jump term to enhance stability of the system. Numerical results show a significant improvement in the order of accuracy of the computed diffusive flux as well as the flux integral. When applied to a channel flow advection-diffusion problem, the scheme resulted in an increased order of accuracy for the flux integral along with gains in solution accuracy by a factor of two. The characteristics of the new scheme were studied through stability, truncation error and cost analysis. The increase in computational costs were modest and affordable. The behaviour of the scheme was also tested by implementing a variation of it within the ANSYS Fluent discretization framework.

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In this thesis, we develop a stability analysis model for the unstructured finite volume method. This model employs the matrix method to implement stability analysis. For the full discretization, where the temporal discretization employs backward Euler time-stepping, a linearization is used to construct the model. Both for the full discretization and semi-discretization, the stability condition is expressed in terms of eigenvalues. The validity of the stability analysis model is verified for linear cases and nonlinear cases. The analysis in this thesis also explains the phenomena that the defined energy can locally increase in the energy stability analysis method. This model can be applied to both linear problems and nonlinear problems; in this thesis, we focus on the 2D Euler equations. We also develop a stabilization methodology. This methodology is aimed at optimizing the eigenvalues of the Jacobian matrix by changing the coordinates of interior vertices of a mesh. Specifically, for an unstable spatial discretization, we can shift the unstable eigenvalues into stability region by changing the mesh. The stabilization methodology is verified by numerical cases as well. The success of this stabilization relies on a developed method to change the eigenvalues of a matrix in a quantitative and controllable way. This method is a general approach to optimize the eigenvalues of a matrix, which means it can be applied to other systems as well.

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The purpose of this thesis is three-fold. First, we devise a memory model for unstructured mesh data for efficient use of memory on parallel shared memory architectures with the purpose of lowering the synchronization overhead between threads and also excluding the probability of occurring race conditions. Second, we present a new thread-parallel edge and face swapping algorithm for two and three dimensional meshes using OpenMP for shared memory architectures. We show how removing the conflicts from the reconfiguration procedure by applying a vertex locking strategy can result in a near linear speed-up with parallel efficiency of close to one on two threads and 70% with sixteen threads on shared-memory processors. Finally, we derive a parallel mesh generation and refinement module for shared memory architectures based on pre-existing serial modules — GRUMMP — by implementing Chernikov and Chrisochoides’ parallel insertion algorithm along with the two above tools. Experiments show a worst case parallel efficiency of 60% for parallel refinement with 16 threads.

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Accuracy of numerical solution is of paramount importance for any CFD simulation. The error in satisfying the continuous partial differential equations by their discrete form results in truncation error and it has a direct influence on discretization errors. Discretization error, which is the difference in the numerical solution and the exact solution to a CFD problem, is generally the largest source of numerical errors. Understanding the relationship between the discretization and truncation error is crucial for reducing numerical errors. Studies have been carried out to understand the truncation-discretization error relationship in the interior regions of a computational domain but fewer for the boundary regions. The effect of different solution reconstruction methods, face gradient averaging schemes and boundary condition implementation methods on boundary truncation error in specific and overall truncation and solution error are the subject of research for this thesis. To achieve the goals laid out, the error has to be quantified first and then tests performed to compare the schemes. The Poisson's equation is chosen as the model diffusion equation. The truncation error coefficients, analogous to the analytical coefficients of the spatial derivatives in Taylor series expansion of truncation error, are quantified using error metrics. The solution error calculation is made possible by a careful selection of exact solutions and their appropriate source terms for Poisson's equation. Analytic tests are performed on a family of topologically regular meshes to test and verify the theoretical implementation of different schemes and to eliminate schemes performing poorly from consideration for numerical tests. The numerical tests are performed on unstructured triangular, mixed and pure quad meshes to extend the accuracy assessment for general meshes. The results obtained from both the tests are utilized to arrive at schemes where the overall truncation error and discretization error can be minimized simultaneously.

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An adaptive method for producing anisotropic quasi-structured meshes is presented in this thesis. Current anisotropic adaptation schemes produce meshes without any regular structure which can hurt accuracy and efficiency of the solution. By modifying the anisotropic adaptation schemes, producing aligned, quasi-structured meshes is possible which means that the accuracy and efficiency of the flow solution are improved. By using quasi-structured meshes, we can get the advantages of flexibility of unstructured meshes for complex geometries and accuracy of the high directional qualities of the structured meshes at the same time. The construction of the quasi-structured meshes from initial isotropic unstructured meshes is accomplished by assigning metrics to vertices based on the error estimation methods. The metrics are used to communicate the desired anisotropy to the meshing program. After assigning a metric to each vertex, the mesh is refined anisotropically using four mesh quality improvements operations to produce high quality anisotropic quasi-structured meshes: swapping to choose the diagonal of the quadrilateral formed by two neighboring triangles which results the maximum quality, inserting vertices for large triangles, vertex removal to eliminate small edges and vertex movements to optimize the location of the vertices so that quasi-structured meshes are created. The idea in the optimization process is to smooth a vertex location by seeking so that the final mesh contains target elements dictated by the metrics assigned in the three vertices of that triangle. The final, high quality mesh is produced by using these operations iteratively based on the metrics assigned to each vertex in an adaptive, solution-based process.

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Numerical experiments have proved that numerical errors are at least as large as other sources of error in numerical simulation of fluid flows. Approximating the continuous partial differential equations that govern the behavior of a fluid with discrete relations results in truncation error which is the initial source of numerical errors. Reducing numerical error requires the ability to quantify and reduce the truncation error. Although the truncation error can be easily found for structured mesh discretizations, there is no generic methodology for the truncation error analysis of unstructured finite volume discretizations. In this research, we present novel techniques for the analysis and quantification of the truncation error produced by finite volume discretization on unstructured meshes. These techniques are applied to compare the truncation error produced by different discretization schemes commonly used in cell-centered finite volume solvers. This comparison is carried out forfundamental scalar equations that model the fluid dynamic equations. These equations model both the diffusive and convective fluxes which appear in the finite volume formulation of the fluid flow equations. Two classes of tests are considered for accuracy assessment. Analytical tests on topologically perfect meshes are done to find the general form of the truncation error. Moreover, these tests allow us to eliminate from consideration thoseschemes that do not perform well even for slightly perturbed meshes. Given the results of the analytic tests, we define a truncation error metric based on the coefficients associated with the spatial derivatives in the series expansion of the truncation error. More complexnumerical tests are conducted on the remaining schemes to extend the accuracy assessment to general unstructured meshes consisting of both isotropic and anisotropic triangles. These results will guide us in the choice of appropriate discretization schemes for diffusive andconvective fluxes arising from discretization of real governing equations.

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The quenching of hydrocarbon flames on cold surfaces is considered to be a potential source of unburned hydrocarbon (UHC) emissions from internal combustion engines, but its contribution to emissions has been difficult to determine due to the strong coupling between physics, chemistry and flame/wall geometry. This is particularly problematic for high pressure direct injection (HPDI) engines where high pressures, inhomogeneous mixtures and complex piston geometry are present.In this work, a computational model is implemented to determine the distance at which hydrocarbon flames quench on cold walls during numerical simulation. This model accounts for variable pressure, temperature, gas mixture and the geometry conditions. The model presented in this work is an extension of the experimental work done by Boust et. al. with stoichiometric premixed flames at low pressures. The validation of this model for high pressure and diffusion flames is presented and shows that the correct trends in heat flux and order of magnitude of quench distance are observed. This model is further refined for engine simulation and enhanced by a two-zone mass diffusion model to account for post-quench oxidation of boundary fuel.A selection of engine cases are simulated for a variety of different conditions to determine the spatial distribution and temporal evolution of unburned fuel cold surfaces. It was found that wall quenching on the piston contributes up to 50% of the total UHC during the combustion cycle, the majority of which is oxidized during the expansion stroke; the final contribution is at most 10% but frequently near or less than 1%. As the injection pressure was increased, quenching on the piston surface became more extensive, through the quenching thickness itself decreased. UHC from wall quenching occurs more readily for higher load conditions due to the richer mixtures and incomplete mixing. Altered engine timing introduced coupled effects of changed flame/wall interaction and combustion characteristics. The data obtained from the model can be used to evaluate attempts to reduce UHC by changing combustion chamber geometry.

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## Publications

- A posteriori stability analysis and improvement for finite volume methods on unstructured meshes (2017)

AIAA SciTech Forum - 55th AIAA Aerospace Sciences Meeting, - An hp-adaptive unstructured finite volume solver for compressible aerodynamic flows (2017)

AIAA SciTech Forum - 55th AIAA Aerospace Sciences Meeting, - An hp-adaptive unstructured finite volume solver for compressible flows (2017)

International Journal for Numerical Methods in Fluids, 85 (10), 563-582 - Applications of the unsteady error transport equation on unstructured meshes (2017)

23rd AIAA Computational Fluid Dynamics Conference, 2017, - Higher-order unstructured finite volume RANS solution of turbulent compressible flows (2017)

Computers and Fluids, 143, 32-47 - Mesh adaptation using C1interpolation of the solution in an unstructured finite volume solver (2017)

International Journal for Numerical Methods in Fluids, - Mesh optimization to improve the stability of finite-volume methods on unstructured meshes (2017)

Computers and Fluids, 156, 590-601 - Methods for the evaluation of candidate meshes for the geometry and mesh generation workshop (2017)

23rd AIAA Computational Fluid Dynamics Conference, 2017, - Output error correction and mesh adaptation for unstructured mesh finite volume method (2017)

AIAA SciTech Forum - 55th AIAA Aerospace Sciences Meeting, - Reconstruction map stability analysis for cell centered finite volume methods on unstructured meshes (2017)

AIAA SciTech Forum - 55th AIAA Aerospace Sciences Meeting, - The impact of unstructured mesh generation approaches on errors (2017)

23rd AIAA Computational Fluid Dynamics Conference, 2017, - Towards higher order discretization error estimation by error transport using unstructured finite-volume methods for unsteady problems (2017)

Computers and Fluids, 154, 245-255 - A truncation error based mesh adaption metric for CFD (2016)

54th AIAA Aerospace Sciences Meeting, - An efficient implicit unstructured finite volume solver for generalised Newtonian fluids (2016)

International Journal of Computational Fluid Dynamics, 30 (3), 201-217 - Improving finite-volume diffusive fluxes through better reconstruction (2016)

Computers and Fluids, 139, 216-232 - Inserting a surface into an existing unstructured mesh (2016)

International Journal for Numerical Methods in Engineering, 106 (6), 484-500 - Smoothed truncation error in functional error estimation and correction using adjoint methods in an unstructured finite volume solver (2016)

Computers and Fluids, 140, 406-421 - The impact of unstructured mesh generation approach on truncation error (2016)

54th AIAA Aerospace Sciences Meeting, - Accuracy of discretization error estimation by the error transport equation on unstructured meshes (2015)

53rd AIAA Aerospace Sciences Meeting, - Accuracy of discretization error estimation by the error transport equation on unstructured meshes - Nonlinear systems of equations (2015)

22nd AIAA Computational Fluid Dynamics Conference, - Comparison of truncation error estimators for defect correction and output error estimation on unstructured mesh finite volume method (2015)

22nd AIAA Computational Fluid Dynamics Conference, - Higher-order unstructured finite volume methods for turbulent aerodynamic flows (2015)

22nd AIAA Computational Fluid Dynamics Conference, - Accuracy analysis of unstructured finite volume discretization schemes for diffusive fluxes (2014)

Computers and Fluids, 101, 220-232 - Higher-order least-squares reconstruction for turbulent aerodynamic flows (2014)

11th World Congress on Computational Mechanics, WCCM 2014, 5th European Conference on Computational Mechanics, ECCM 2014 and 6th European Conference on Computational Fluid Dynamics, ECFD 2014, , 7557-7568 - Truncation and discretization error for diffusion schemes on unstructured meshes (2014)

52nd AIAA Aerospace Sciences Meeting - AIAA Science and Technology Forum and Exposition, SciTech 2014, - Truncation and discretization error for difiusion schemes on unstructured meshes (2014)

52nd Aerospace Sciences Meeting, - Accuracy assessment of finite volume discretizations of convective fluxes on unstructured meshes (2013)

51st AIAA Aerospace Sciences Meeting including the New Horizons Forum and Aerospace Exposition 2013, - Anisotropic mesh adaptation: Recovering quasi-structured meshes (2013)

51st AIAA Aerospace Sciences Meeting including the New Horizons Forum and Aerospace Exposition 2013, - Eigenanalysis of truncation and discretization error on unstructured meshes (2013)

21st AIAA Computational Fluid Dynamics Conference, - Higher-order finite volume solution reconstruction on highly anisotropic meshes (2013)

21st AIAA Computational Fluid Dynamics Conference, - High order aerodynamic optimization using new hybrid sequential quadratic programing-particle swarm intelligence technique (2012)

50th AIAA Aerospace Sciences Meeting Including the New Horizons Forum and Aerospace Exposition, - Optimal mesh adaption for Burgers' equation (2012)

42nd AIAA Fluid Dynamics Conference and Exhibit 2012, - A comparison of Higher-Order Methods on a Set of Canonical Aerodynamics Applications (2011)

20th AIAA Computational Fluid Dynamics Conference 2011, - CONSTRUCTING CONSTRAINED DELAUNAY TETRAHEDRALIZATIONS OF VOLUMES BOUNDED BY PIECEWISE SMOOTH SURFACES (2011)
- Effect of pulse frequency and cylinder diameter on pressure screen rotor performance (2011)

Appita Annual Conference, , 89-96 - Influence of cylinder design and other factors on capacity and power consumption in a pressure screen (2011)

Appita Journal, 64 (1), 55-61 - Numerical calculation of quench distance for laminar premixed flames under engine relevant conditions (2011)

SAE Technical Papers, - Tetrahedral mesh generation using Delaunay refinement with non-standard quality measures (2011)

International Journal for Numerical Methods in Engineering, 87 (8), 795-820 - 3D phase-field simulations of interfacial dynamics in Newtonian and viscoelastic fluids (2010)

Journal of Computational Physics, 229 (2), 498-511 - An Interoperable, Data-Structure-Neutral Component for Mesh Query and Manipulation (2010)

ACM Transactions on Mathematical Software, 37 (3), 1-28 - EXPERIMENTAL MEASUREMENT OF PRESSURE PULSES FROM A PULP SCREEN ROTOR (2010)

Proceedings of the Asme Fluids Engineering Division Summer Conference - 2010 - Vol 1, Pts a-C, 1 (PARTS), 763-772 - Globalized matrix-explicit Newton-GMRES for the high-order accurate solution of the Euler equations (2010)
- High Performance Multi-Element Foil Pulp Screen Rotor - Pilot and Mill Trials (2010)

Journal of Pulp and Paper Science, 36 (3-4), 124-128 - Higher order two dimensional aerodynamic optimization using unstructured grids and adjoint sensitivity computations (2010)

48th AIAA Aerospace Sciences Meeting Including the New Horizons Forum and Aerospace Exposition, - Two-dimensional Delaunay-based anisotropic mesh adaptation (2010)

Engineering with Computers, 26 (4), 407-418 - A generalized framework for high order anisotropic mesh adaptation (2009)

Computers & Structures, 87 (11-12), 670-679 - A Numerical Investigation into the Effectiveness of Multi-Element Pressure Screen Rotor Foils (2009)
- Accuracy preserving limiter for the high-order accurate solution of the Euler equations (2009)

Journal of Computational Physics, 228 (23), 8693-8711 - AIAA 2009-1174 assessing validity of mesh refinement sequences, with application to DPW-III meshes (2009)

47th AIAA Aerospace Sciences Meeting including the New Horizons Forum and Aerospace Exposition, - High-order accurate reconstruction-based unstructured mesh finite-volume schemes: Status and challenges (2009)

47th AIAA Aerospace Sciences Meeting including the New Horizons Forum and Aerospace Exposition, - Interoperable mesh components for large-scale, distributed-memory simulations (2009)

180, - Obtaining and Verifying High-Order Unstructured Finite Volume Solutions to the Euler Equations (2009)

Aiaa Journal, 47 (9), 2105-2120 - Unstructured high-order accurate finite-volume solutions of the Navier-Stokes equations (2009)

47th AIAA Aerospace Sciences Meeting including the New Horizons Forum and Aerospace Exposition, - A computational fluid dynamic simulation of the pressure pulses produced by a solid core pulp screen rotor (2008)

Appita Journal, 61 (6), 479-484 - A high-order accurate unstructured finite volume Newton-Krylov algorithm for inviscid compressible flows (2008)

Journal of Computational Physics, 227 (4), 2582-2609 - Delaunay-Based Anisotropic Mesh Adaptation (2008)

Proceedings of the 17th International Meshing Roundtable, IMR 2008, , 141-157 - Effect of discretization order on preconditioning and convergence of a high-order unstructured Newton-GMRES solver for the Euler equations (2008)

Journal of Computational Physics, 227 (4), 2366-2386 - High performance foil rotor improves De-ink pulp screening (2008)

Progress in Paper Recycling, 17 (4), 5-10 - Limiters for unstructured higher-order accurate solutions of the euler equations (2008)

46th AIAA Aerospace Sciences Meeting and Exhibit, - Toward interoperable mesh, geometry and field components for PDE simulation development (2008)

Engineering with Computers, 24 (2), 165-182 - A language-independent api for unstructured mesh access and manipulation (2007)

Proceedings - International Symposium on High Performance Computing Systems and Applications, - Accuracy assessment methodology for a high-order unstructured finite volume solver (2007)

Collection of Technical Papers - 18th AIAA Computational Fluid Dynamics Conference, 2, 1071-1091 - Effect of discretization order on preconditioning and convergence of a higher-order unstructured Newton-Krylov solver for inviscid compressible flows (2007)

Collection of Technical Papers - 45th AIAA Aerospace Sciences Meeting, 13, 8884-8905 - Interoperable mesh and geometry tools for advanced petascale simulations (2007)

78, U147-U152 - Matrix-explicit GMRES for a higher-order accurate inviscid compressible flow solver (2007)

Collection of Technical Papers - 18th AIAA Computational Fluid Dynamics Conference, 1, 362-372 - Numerical simulations of pressure pulses produced by a solid core pressure screen rotor (2007)

Appita Annual Conference, 1, 265-271 - On obtaining high-order finite-volume solutions to the euler equations on unstructured meshes (2007)

Collection of Technical Papers - 18th AIAA Computational Fluid Dynamics Conference, 2, 1762-1781 - A high-order accurate unstructured GMRES solver for the compressible Euler equations (2006)

, 435-440 - A high-order accurate unstructured Newton-Krylov solver for inviscid compressible flows (2006)

Collection of Technical Papers - 36th AIAA Fluid Dynamics Conference, 3, 1878-1895 - A mesh-database-independent edge- And face-swapping tool (2006)

Collection of Technical Papers - 44th AIAA Aerospace Sciences Meeting, 9, 6445-6455 - Phase-field simulations of interfacial dynamics in viscoelastic fluids using finite elements with adaptive meshing (2006)

Journal of Computational Physics, 219 (1), 47-67 - The TSTT mesh interface (2006)

Collection of Technical Papers - 44th AIAA Aerospace Sciences Meeting, 9, 6372-6387 - A finite-element phase-field method for simulating interfacial dynamics in complex fluids (2005)

AIChE Annual Meeting, Conference Proceedings, - Computational Study of the Flow Around a Ducted Tip Hydrofoil (2005)

American Society of Mechanical Engineers, Fluids Engineering Division (Publication) FED, 257 (2 B), 899-908 - Editorial (2005)
- Numerical Simulation and Experimental Measurement of Pressure Pulses Produced by a Pulp Screen Foil Rotor (2005)
- A toolkit for numerical simulation of PDEs. II. Solving generic multiphysics problems (2004)

Computer Methods in Applied Mechanics and Engineering, 193 (36-38), 3891-3918 - A toolkit for numerical simulation of PDEs: I Fundamentals of generic finite-volume simulation (2003)

Computer Methods in Applied Mechanics and Engineering, 192 (9-10), 1147-1175 - Coarsening unstructured meshes by edge contraction (2003)

International Journal for Numerical Methods in Engineering, 57 (3), 391-414 - A High-Order-Accurate Unstructured Mesh Finite-Volume Scheme for the Advection–Diffusion Equation (2002)
- Guaranteed-quality triangular mesh generation for domains with curved boundaries (2002)

International Journal for Numerical Methods in Engineering, 55 (10), 1185-1213 - Numerical experiments for flow around a ducted tip hydrofoil (2002)

American Society of Mechanical Engineers, Pressure Vessels and Piping Division (Publication) PVP, 448 (1), 289-290 - Guaranteed-quality simplical mesh generation with cell size and grading control (2001)

Engineering with Computers, 17 (3), 269-286 - A cost/benefit analysis of simplicial mesh improvement techniques as measured by solution efficiency (2000)

International Journal of Computational Geometry & Applications, 10 (4), 361-382 - Quasi-ENO Schemes for Unstructured Meshes Based on Unlimited Data-Dependent Least-Squares Reconstruction (1997)
- 3.0.co;2-9" target="_blank">Tetrahedral mesh improvement using swapping and smoothing (1997)

International Journal for Numerical Methods in Engineering, 40 (21), 3979-4002 - MULTIGRID ACCELERATION OF AN UPWIND ENTER SOLVER ON UNSTRUCTURED MESHES (1995)

Aiaa Journal, 33 (10), 1822-1827 - Multigrid acceleration of an upwind Euler solver on unstructured meshes (1995)

AIAA journal, 33 (10), 1822-1827 - YIELD SURFACE AND STRAIN-HARDENING PROPERTIES OF CORDOVA CREAM LIMESTONE (1989)

International Journal of Rock Mechanics and Mining Sciences & Geomechanics Abstracts, 26 (5), 373-380

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