Carl Ollivier-Gooch


Research Interests

Algorithm Development for Computational Fluid Dynamics
Error and Stability Analysis for Unstructured Mesh Methods
Unstructured Mesh Generation
Applied Aerodynamics

Relevant Degree Programs


Research Methodology

Development of software for numerical simulation of PDE's
Unstructured mesh generation


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Graduate Student Supervision

Doctoral Student Supervision (Jan 2008 - Mar 2019)
Numerical estimation of discretization error on unstructured meshes (2018)

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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An adaptive higher-order unstructured finite volume solver for turbulent compressible flows (2017)

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 estimation and mesh adaptation paradigm for unstructured mesh finite volume methods (2017)

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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Aerodynamic optimization using high-order finite-volume CFD simulations (2011)

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 mesh generation of curve-bounded domains (2009)

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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Efficient high-order accurate unstructured finite-volume algorithms for viscous and inviscid compressible flows (2009)

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 (2010-2017)
A higher-order unstructured finite volume solver for three-dimensional compressible flows (2017)

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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The impact of mesh regularity on errors (2017)

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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Adaptive creation of orthogonal anisotropic triangular meshes using target matrices (2016)

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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Improving finite-volume diffusive fluxes through better reconstruction (2016)

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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Stability analysis and stabilization of unstructured finite volume method (2016)

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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Thread-parallel mesh generation and improvement using face-edge swapping and vertex insertion (2015)

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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Truncation error analysis for diffusion schemes in boundary regions of unstructured meshes (2015)

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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Anisotropic mesh adaptation : recovering quasi-structured meshes (2013)

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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Truncation error analysis of unstructured finite volume discretization schemes (2012)

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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Effects of cold wall quenching on unburned hydrocarbon emissions from a natural gas HPDI engine (2011)

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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Recent Tri-Agency Grants

The following is a selection of grants for which the faculty member was principal investigator or co-investigator. Currently, the list only covers Canadian Tri-Agency grants from years 2013/14-2016/17 and excludes grants from any other agencies.

  • Improvements to unstructured mesh finite volume methods for CFD: Novel adaptive techniques and improved robustness - Natural Sciences and Engineering Research Council of Canada (NSERC) - Collaborative Research and Development Grants - Project (2016/2017)
  • Numerical simulation of aircraft aerodynamics with error quantification - Natural Sciences and Engineering Research Council of Canada (NSERC) - Discovery Grants Program - Individual (2015/2016)
  • Error control and stability improvements for unstructured mesh finite volume methods in computational fluid dynamics - Natural Sciences and Engineering Research Council of Canada (NSERC) - Collaborative Research and Development Grants - Project (2013/2014)
  • High-fidelity aerodynamic simulation and shape optimization of aircraft - Natural Sciences and Engineering Research Council of Canada (NSERC) - Discovery Grants Program - Individual (2013/2014)



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