Carl Ollivier-Gooch

Convergence acceleration is a crucial tool for enhancing the efficiency and accuracy of numerical algorithms. Although many numerical flow solvers yield accurate solutions, they may take a long time to converge to the desired tolerance. To address this issue, this thesis proposes a convergence acceleration scheme that employs dynamic mode decomposition (DMD) to generate an over-relaxation update for flow solvers. The DMD analysis identifies the dominant solution modes causing the slow convergence of the flow solver and relaxes them towards their steady-state by extrapolating these modes to infinity and constructing a linear combination of extrapolated sums.The thesis also proposes an automation framework that leverages a machine learning pipeline to automate the application of this DMD acceleration scheme. The pipeline evaluates the DMD-generated over-relaxation updates and assesses whether they will result in convergence acceleration or not, thus ensuring a more robust convergence acceleration framework.We demonstrate that our technique accelerates the convergence of finite-volume computations and that it can achieve instantaneous convergence for linear problems and two to five orders of magnitude reduction in residual for laminar and turbulent problems. Also we show that the automation framework can successfully automate this process and eliminate the need for any input or control of the algorithm. We also demonstrate that the automation framework can apply this technique without any user intervention.The proposed convergence acceleration framework is entirely data-driven and can be integrated into a flow solver as a module which only needs a few solution updates as its input. This scheme is particularly useful for researchers who are satisfied with their flow solver's performance but seek to accelerate it without altering their code implementation by adopting more complex time-advance schemes and time stepping methods.

View record

Accuracy of flow simulations is a major concern in Computational Fluid Dynamics (CFD) applications. A possible outcome of inaccuracy in CFD results is missing a major feature in the flow field. Many methods have been proposed to reduce numerical errors and increase overall accuracy, but these are not always efficient or even feasible. In this study, a purely data-driven approach is proposed to assess flow simulations both in a qualitative and a quantitative manner. In this regard, Principal Component Analysis (PCA) has been performed on compressible flow simulations around an airfoil to map the high-dimensional CFD data to a lower-dimensional PCA subspace. A machine learning classifier based on the extracted principal components has been developed to detect the simulations that miss the separation bubble behind the airfoil. The evaluative measures indicate that the model is able to detect most of the simulations where the separation region is poorly resolved. Besides the classifier, a machine learning regressor has been trained on the same PCA subspace to predict the error in the output drag coefficient. The predictions reveal that the regression model estimates accurate errors with a tight uncertainty bound. Further, more efficient models built on top of fewer PCA modes have been implemented that show similar performance. In addition, the developed models were used to inspect simulations solved on a different mesh configuration from the one the models were trained on. This generalization framework gives rise to some challenges that are thoroughly discussed. Overall, the results demonstrate that machine learning models based on the principal components of the data set are promising tools to detect possible missing flow features and predict numerical errors in CFD.

View record

The use of computational fluid dynamics (CFD) is now a ubiquitous part of of the engineering design process, especially in aerodynamic applications. The CFD process requires a high-quality volume mesh on which to perform the simulation. However, mesh generation is typically a choke point for the process, and is often the most time-consuming step. This emphasizes the need for robust, automatic mesh generation tools that can reliably produce high-quality meshes. Advancing layer (AL) meshing is one method of mesh generation in which mesh elements are extruded off of the surface mesh one layer at a time. AL meshes are preferred for their unmatched cell alignment, orthogonality, and fraction of prismatic elements. However, two limitations of a standard AL scheme are the handling of complex corners and highly anisotropic surface meshes. This thesis, first, presents a solution to the problem of complex corners, where extruding a single vertex is inadequate. The solution presented is a robust method that essentially modifies the surface mesh by extruding in multiple directions at the corner, then creating adjacent cells, and filling holes. From here it is possible to extrude the mesh as usual. Second, an extension of the AL method is presented which enables extrusion from highly anisotropic surface meshes. Anisotropy is characterized by the use of highly stretched cells in specific areas where the flow physics produce large gradients in one direction but not in one or both of the others, a wing for example. Anisotropic cells allow for the desired mesh resolution but at a fraction of the cell count. The method presented extrudes layers while maintaining the anisotropic pseudo-structure and aspect ratio of the surface mesh and allows for a smooth transition back to regular isotropic extrusion. It is accomplished by exploiting the anisotropic pseudo-structure to decimate entire lines of vertices at once and adjusting the smoothing algorithm to maintain distance ratios between neighbouring vertices. This method demonstrates AL's ability to produce high-quality anisotropic meshes, opening a wide array of meshing possibilities.

View record

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.

View record

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.

View record

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.

View record

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.

View record

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.

View record

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.

View record

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.

View record

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.

View record

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.

View record

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.

View record

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.

View record

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.

View record

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.

View record