An Efficient Implicit High-order Flux Reconstruction Method by Automatic Differentiation using Operator Overloading for Flow Simulation on Unstructured Hexahedral Grids

The least square-based finite difference-finite volume (LSFD-FV) method has been developed and successfully applied to solve various two-dimensional flow problems with a high-order of accuracy. In this paper, the extension of the LSFD-FV method to the three-dimensional (3D) case on unstructured grids is presented. Different from other existing high-order methods, the LSFD-FV method combines the good features of the least square-based finite difference (LSFD) scheme for derivative approximation and the finite volume (FV) discretization for conservation of physical laws. Within each control cell, a high-order polynomial resultant from a Taylor series expansion is used to approximate the solution in the FV discretization of governing equations, where the derivatives are approximated by the LSFD scheme. As a result, the mesh-free nature of the LSFD scheme endows the LSFD-FV method with the flexibility on unstructured grids. Additionally, the straightforward algorithm of the LSFD scheme and the direct utilization of the Taylor series polynomial make the LSFD-FV method user-friendly and easy to implement. Furthermore, the inviscid and viscous fluxes are simultaneously evaluated by lattice Boltzmann and gas kinetic flux solvers in this work, which avoids additional and complicated treatment for the viscous discretization. Accuracy studies on unstructured hexahedral and tetrahedral grids validate the third-order of accuracy of the 3D LSFD-FV method. Various 3D incompressible and compressible numerical experiments are also presented. The results show that the proposed method enjoys the higher accuracy than the conventional second-order method on the same mesh. When the same accuracy is achieved, the present high-order method has superior computational efficiency to the second-order counterpart.

The goal of this dissertation is two-fold. The first part concerns the development of a numerical method for solving Maxwell's equations on unstructured hexahedral grids that employs both high order spatial and high order temporal discretizations. The second part involves the use of this method as a computational tool to perform high fidelity simulations of various electromagnetic devices such as optical transmission lines and photonic crystal structures to yield a level of accuracy that has previously been computationally cost prohibitive. This work is based on the initial research of Daniel White who developed a provably stable, charge and energy conserving method for solving Maxwell's equations in the time domain that is second order accurate in both space and time. The research presented here has involved the generalization of this procedure to higher order methods. High order methods are capable of yielding far more accurate numerical results for certain problems when compared to corresponding h-refined first order methods , and often times at a significant reduction in total computational cost. The first half of this dissertation presents the method as well as the necessary mathematics required for its derivation. The second half addresses the implementation of the method in a parallel computational environment, its validation using benchmark problems, and finally its use in large scale numerical simulations of electromagnetic transmission devices.

An efficient implicit lower-upper symmetric Gauss-Seidel (LU-SGS) solution algorithm has been developed for a high order multi-domain spectral difference method on unstructured hexahedral grids. The LU-SGS solver is preconditioned by the block element matrix, and the system of equations is then solved with an exact LU decomposition approach. In addition, the effects of several parameters on the convergence rate in the implicit scheme have been investigated for both external and internal flows. The implicit scheme has shown a speed up factor of more than an order of magnitude relative to the multi-stage explicit Runge-Kutta scheme for several demonstration problems.

¶A code for the simulation of atmospheric flows in 3D is presented. The underlying mathematical model is fully compressible, it takes gravity into account but Coriolis forces, turbulence and viscosity are neglected. The general numerical code consists of a finite volume discretization on unstructured hexahedral grids in 3D. The code is presently being investigated on applications to the calculation of atmospheric gravity waves on a mesh which has a structured type and is locally refined near the orography. We develop two schemes, the main difference between them lies in the different discretizations for the mass fluxes. We show that both schemes resolve typical structures of gravity waves in potential flow, linear hydrostatic motion and nonlinear non-hydrostatic regime. We compare advantages and disadvantages of the developed schemes.

Automatic differentiation approach for solving one-dimensional flow and heat transfer problems

An implicit non-linear lower-upper symmetric Gauss-Seidel (LU-SGS) solution algorithm has been developed for a high-order spectral difference Navier-Stokes solver on unstructured hexahedral grids. The non-linear LU-SGS solver is preconditioned by the block element matrix, and the system of equations is then solved with a LU decomposition approach. The large sparse Jacobian matrix is computed numerically, resulting in extremely simple operations for arbitrarily complex residual operators. Several viscous test cases were performed to evaluate the performance. The implicit solver has shown speedup of 1 to 2 orders of magnitude over the multi-stage Runge-Kutta time integration scheme.

A collocated-grid spectral difference method for compressible flows

The canine nasal cavity contains a complex airway labyrinth, dedicated to respiratory air conditioning, filtering of inspired contaminants, and olfaction. The small and contorted anatomical structure of the nasal turbinates has, to date, precluded a proper study of nasal airflow in the dog. This study describes the development of a high-fidelity computational fluid dynamics (CFD) model of the canine nasal airway from a three-dimensional reconstruction of high-resolution magnetic resonance imaging scans of the canine anatomy. Unstructured hexahedral grids are generated, with large grid sizes ((10-100) x 10(6) computational cells) required to capture the details of the nasal airways. High-fidelity CFD solutions of the nasal airflow for steady inspiration and expiration are computed over a range of physiological airflow rates. A rigorous grid refinement study is performed, which also illustrates a methodology for verification of CFD calculations on complex unstructured grids in tortuous airways. In general, the qualitative characteristics of the computed solutions for the different grid resolutions are fairly well preserved. However, quantitative results such as the overall pressure drop and even the regional distribution of airflow in the nasal cavity are moderately grid dependent. These quantities tend to converge monotonically with grid refinement. Lastly, transient computations of canine sniffing were carried out as part of a time-step study, demonstrating that high temporal accuracy is achievable using small time steps consisting of 160 steps per sniff period. Here we demonstrate that acceptable numerical accuracy (between approximately 1% and 15%) is achievable with practical levels of grid resolution (approximately 100 x 10(6) computational cells). Given the popularity of CFD as a tool for studying flow in the upper airways of humans and animals, based on this work we recommend the necessity of a grid dependence study and quantification of numerical error when presenting CFD results in complicated airways.

Efficient nonlinear solvers for Laplace–Beltrami smoothing of three-dimensional unstructured grids

We present a clear and general method for constructing hierarchical vector bases of arbitrary polynomial degree for use in the finite element solution of Maxwell's equations. Our focus in this paper is on unstructured hexahedral grids with elements of higher order geometry (i.e. curved elements). Hierarchical bases enable p-refinement methods, where elements in a mesh can have different degrees of approximation, to be easily implemented. This can prove to be quite useful as sections of a computational domain can be selectively refined in order to achieve a greater error tolerance without the cost of refining the entire domain. We present a specific procedure for computing a hierarchical 1-form basis of arbitrary polynomial degree as well as the corresponding hierarchical degrees of freedom.

A high-order spectral difference (SD) method has been developed with graphics processing units (GPUs) using compute unified device architecture (CUDA). It solves the three-dimensional Navier-Stokes equations on unstructured hexahedral grids with RungeKutta time integration. The method is efficient since operations are completed in a onedimensional fashion and the equations are solved in differential form, removing explicit surface and volume integral calculations. Additionally, solution and flux reconstructions are completed locally per cell, increasing the parallelization of the implementation. Due to this efficiency, the application of GPU computing is appealing. This paper presents the SD method implementation with GPU CUDA computing and presents accuracy studies with isotropic vortex propagation and Couette flow, verifies the high-order accuracy of the solver with a numerical sensitive aero-acoustic problem, and compares the developed solver and a high-order finite difference solver with a case presented in the 1st International Workshop on High-Order CFD Methods. Finally, the GPU solver is compared to a similar central processing unit (CPU) solver, where speed-ups ranging from 20-40x faster are illustrated.

The Galerkin method with discontinuous basis functions is adapted for solving the Euler and Navier-Stokes equations on unstructured hexahedral grids. A hybrid multigrid algorithm involving the finite element and grid stages is used as an iterative solution method. Numerical results of calculating the sphere inviscid flow, viscous flow in a bent pipe, and turbulent flow past a wing are presented. The numerical results and the computational cost are compared with those obtained using the finite volume method.

The article presents the numerical investigation of non–Newtonian effects of steady blood flow in complete idealized 3–D bypass models, whose native artery is either coronary or femoral with average physiological parameters. Considering the blood to be a generalized Newtonian fluid, the shear–dependent viscosity is described by two well–known macroscopic non–Newtonian models (the Carreau–Yasuda model and the modified Cross model). The results were obtained by own developed computational software based on the pseudo–compressibility approach and on the cell–centred finite volume method defined on unstructured hexahedral grids. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

: We apply the Generalized Minimal Residual (GMRES) iterative equation solution technique to a set of full, unsymmetric matrix systems generated by a standard boundary element method. The test problems chosen produced well conditioned matrices. The GMRES technique, when used without preconditioning and with a sufficient number of trial vectors, solved the matrix system using as few as 23% of the operations required by a direct Gauss reduction. The class of partial LU decomposition preconditioners tested degraded the condition number of the matrices, and consequently did not reduce the GMRES solution time. In general, the GMRES technique does not appear to be of practical interest compared to the direct reduction unless other factors (availability of a good approximation to the final solution, etc.) intervene.

In this contribution, we use an adaptive unstructured grid algorithm for the solution of steady laminar compressible viscous flows over compression corners. The spatial discretisation is achieved by means of general assemblies of triangular or quadrilateral cells, while the temporal discretisation is accomplished in a fully implicit fashion. The unknowns are associated with the cell vertices [1,2] and a first order upwind algorithm results from the use of the flux difference splitting method of Roe [3]. A higher order extension is achieved by using the MUSCL concepts of van Leer [4]. This requires a monotonic linear reconstruction of the solution on a general unstructured grid and this is accomplished by the use of variational recovery [5] with the incorporation of the slope limiting [6]. The solution of the implicit equation system is obtained by a point implicit relaxation process.