### Browsing by Author "Wang, Lai"

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Item A comparative study of implicit Jacobian-free Rosenbrock-Wanner, ESDIRK and BDF methods for unsteady flow simulation with high-order flux reconstruction formulations(2019-04-09) Wang, Lai; Yu, MeilinWe conduct a comparative study of the Jacobian-free linearly implicit Rosenbrock-Wanner (ROW) methods, the explicit rst stage, singly diagonally implicit Runge-Kutta (ESDIRK) methods, and the second-order backward differentiation formula (BDF2) for unsteady flow simulation using spatially high-order flux reconstruction/correction procedure via reconstruction (FR/CPR) formulations. The pseudo-transient continuation is employed to solve the nonlinear systems resulting from the temporal discretizations with ESDIRK and BDF2. A Jacobian-free implementation of the restarted generalized minimal residual method (GMRES) solver is employed with a low storage element-Jacobi preconditioner to solve linear systems, including those in linearly implicit ROW methods and those from linearization of the nonlinear systems in ESDIRK and BDF2 methods. We observe that all ROW and ESDIRK schemes (from second order to fourth order) are more computationally efficient than BDF2, and ROW methods can potentially be more efficient than ESDIRK methods. However, the convergence tolerance of the GMRES solver for ROW methods needs to be sufficiently tight to preserve the nominal order of accuracy. In general, ESDIRK methods allow a larger physical time step size for unsteady flow simulation than ROW methods do.Item Comparison of ROW, ESDIRK, and BDF2 for Unsteady Flows with the High-Order Flux Reconstruction Formulation(Springer Nature, 2020-05-13) Wang, Lai; Yu, MeilinWe conduct a comparative study of the Jacobian-free linearly implicit Rosenbrock–Wanner (ROW) methods, the explicit first stage, singly diagonally implicit Runge–Kutta (ESDIRK) methods, and the second-order backward differentiation formula (BDF2) for the high-order flux reconstruction/correction procedure via reconstruction solution of the unsteady Navier–Stokes equations. Pseudo-transient continuation is employed to solve the nonlinear equation at each stage of ESDIRK (excluding the first stage) and each step of BDF2. A Jacobian-free implementation of the restarted generalized minimal residual method solver is employed with a low storage element-Jacobi preconditioner to solve the linear system at each stage of ROW and each pseudo time iteration of ESDIRK and BDF2. Several numerical experiments, including both laminar and turbulent flow simulations, are conducted to carry out the comparison. We observe that the multistage ROW2 and ESDIRK2 are more efficient than the multistep BDF2, and higher-order implicit time integrators are more efficient than lower-order ones. In general, the ESDIRK method allows a larger physical time step size for unsteady flow simulation than the ROW method when the element-Jacobi preconditioner is employed, especially for wall-bounded flows; and the ROW method can be more efficient than the ESDIRK method when the time step size is refined.Item Computational Study of Flapping Wing Response to Vertical Gusts at Low Reynolds Numbers(American Institute of Aeronautics and Astronautics, 2020-01-05) Poudel, Naresh; Wang, Lai; Yu, MeilinThis work presents a computational study of an oscillating NACA0012 airfoil’s response to vertical gusts at low Reynolds numbers. The gust is created by a cross-flow ducted floor jet and its interaction with a freestream flow causes the jet to bend downstream, thus creating a blockage effect and modifying the effective angle of attack (AoA) over an airfoil in the freestream flow. The interaction of the gust with the airfoil causes large unsteady forces, which exceed the peak static lift coefficient. As the gust becomes fully developed near the airfoil region, the airfoil exhibits a leading edge vortex formation and dynamic-stall-like phenomenon while remaining at a fixed zero degree AoA. The gust-wing interactions under dynamic pitching conditions are also studied by varying the reduced frequencies. The study shows that the effects of the gust can be mitigated by increasing the reducing frequency of the flapping wing. As a byproduct, larger lift and thrust will be produced.Item A dynamically load-balanced parallel p-adaptive implicit high-order flux reconstruction method for under-resolved turbulence simulation(2019-10-08) Wang, Lai; Gobbert, Matthias K.; Yu, MeilinWe present a dynamically load-balanced parallel p-adaptive implicit high-order flux reconstruction method for under-resolved turbulence simulation. The high-order explicit first stage, singly diagonal implicit Runge-Kutta (ESDIRK) method is employed to circumvent the restriction on the time step size. The pseudo transient continuation is coupled with the matrix-free restarted generalized minimal residual (GMRES) method to solve the nonlinear equations at each stage, except the first one, of ESDIRK. We use the spectral decay smoothness indicator as the refinement/coarsening indicator for p-adaptation. A dynamic load balancing technique is developed with the aid of the open-source library ParMETIS. The trivial cost, compared to implicit time stepping, of mesh repartitioning and data redistribution enables us to conduct p-adaptation and load balancing every time step. An isentropic vortex propagation case is employed to study the impact of element weights used in mesh repartitioning on parallel efficiency. We apply the p-adaptive solver for implicit large eddy simulation (ILES) of the transitional flows over a cylinder when Reynolds number (Re) is 3900 and the SD7003 wing when Re is 60000. Numerical experiments demonstrate that a significant reduction in the run time (up to 70%) and total number of solution points (up to 76%) can be achieved with p-adaptation.Item An efficient GPU-based h-adaptation framework via linear trees for the flux reconstruction method(2023-06-11) Wang, Lai; Witherden, Freddie; Jameson, AntonyIn this paper, we develop the first entirely graphic processing unit (GPU) based h-adaptive flux reconstruction (FR) method with linear trees. The adaptive solver fully operates on the GPU hardware, using a linear quadtree for two dimensional (2D) problems and a linear octree for three dimensional (3D) problems. We articulate how to efficiently perform tree construction, 2:1 balancing, connectivity query, and how to perform adaptation for the flux reconstruction method on the GPU hardware. As a proof of concept, we apply the adaptive flux reconstruction method to solve the inviscid isentropic vortex propagation problem on 2D and 3D meshes to demonstrate the efficiency of the developed adaptive FR method on a single GPU card. Depending on the computational domain size, acceleration of one or two orders of magnitude can be achieved compared to uniform meshing. The total computational cost of adaption, including tree manipulations, connectivity query and data transfer, compared to that of the numerical solver, is insignificant. It can be less than 2% of the total wall clock time for 3D problems even if we perform adaptation as frequent as every 10 time steps with an explicit 3-stage Runge--Kutta time integrator.Item A High-Order Dual-Time Stepping FR/CPR Method for Unsteady Incompressible Navier-Stokes Equations on Unstructured Moving Grids(AIAA SciTech Forum, 2016) Wang, Lai; Yu, MeilinA high-order accurate flux reconstruction/correction procedure via reconstruction (FR/CPR) method is developed to solve incompressible Navier-Stokes equations on unstructured moving grids. An artificial compressibility method is adopted to facilitate the common flux reconstruction on element interfaces. For unsteady flow simulations, a dual-time stepping method is used for temporal discretization. For simulations on deformable/moving grids, the Geometric Conservation Law (GCL) introduced in dynamic spatial coordinate transformation has been enforced. The newly developed method is verified with several steady and unsteady benchmark incompressible flow problems.Item An Implicit High-Order Preconditioned Flux Reconstruction Method for Low-Mach-Number Flow Simulation with Dynamic Meshes(2019-03-15) Wang, Lai; Yu, MeilinA fully implicit high-order preconditioned ux reconstruction/correction procedure via reconstruction (FR/CPR) method is developed to solve the compressible Navier{Stokes equations at low Mach numbers. A dual-time stepping approach with the second-order backward di erentiation formula (BDF2) is employed to ensure temporal accuracy for unsteady ow simulation. When dynamic meshes are used to handle moving/deforming domains, the geometric conservation law (GCL) is implicitly enforced to eliminate errors due to the resolution discrepancy between BDF2 and the spatial FR/CPR discretization. The large linear system resulting from the spatial and temporal discretizations is tackled with the restarted Generalized Minimal Residual (GMRES) solver in the PETSc (Portable, Extensible Toolkit for Scienti c Computation) library. Through several benchmark steady and unsteady numerical tests, the preconditioned FR/CPR methods have demonstrated good convergence and accuracy for simulating ows at low Mach numbers. The new ow solver is then used to study the e ects of Mach number on unsteady force generation over a plunging airfoil when operating in low-Mach-number ows. It is observed that weak compressibility has a signi cant impact on thrust generation but a negligible e ect on lift generation of an oscillating airfoil.Item An implicit P-multigrid flux reconstruction method for simulation of locally preconditioned unsteady Navier–Stokes equations at low Mach numbers(2019-08) Wang, Lai; Yu, MeilinWe develop a P-multigrid solver to simulate locally preconditioned unsteady compressible Navier–Stokes equations at low Mach numbers with implicit high-order methods. Specifically, the high-order flux reconstruction/correction procedure via reconstruction (FR/CPR) method is employed for spatial discretization and the high-order time integration is conducted by means of the explicit first stage, singly diagonally implicit Runge-Kutta (ESDIRK) method. Local preconditioning is used to alleviate the stiffness of the compressible Navier–Stokes equations at low Mach numbers and is only conducted in pseudo transient continuation to ensure the high-order accuracy of ESDIRK methods. We employ the element Jacobi smoother to update the solutions at different P-levels in the P-multigrid solver. Highorder spatiotemporal accuracy of the new solver for low-Mach-number flow simulation is verified with the isentropic vortex propagation when the Mach (Ma) number of the free stream is 0.005. The impact of the hierarchy of polynomial degrees on the convergence speed of the P-multigrid method is studied via several numerical experiments, including two dimensional (2D) inviscid and viscous flows over a NACA0012 airfoil at Ma = 0.001, and a three dimensional (3D) inviscid flow over a sphere at Ma = 0.001. The P-multigrid solver is then applied to coarse resolution simulation of the transitional flows over an SD7003 wing at 8◦ angle of attack when the Reynolds number is 60000 and the Mach number is 0.1 or 0.01.Item On the parallel implementation and performance study of high-order Rosenbrock-type implicit Runge-Kutta methods for the FR/CPR solutions of the Navier-Stokes equations(AIAA SciTech Forum, 2018) Wang, Lai; Yu, MeilinThe Rosenbrock-type implicit Runge-Kutta (ROIRK) methods only require one Jaco-bian matrix evaluation per time step rather than per stage as other types of implicit Runge-Kutta (IRK) methods need. This feature makes ROIRK attractive for numerical simulations using implicit methods. We present the parallel implementation of several matrix-based ROIRK methods with ﬂux reconstruction/correction procedure re-construction (FR/CPR) formulations for solving the 3D Navier-Stokes equations. In this study, METIS has been utilized to partition the mesh in the preprocessing. The complex-step derivative approximation is employed to evaluate the Jacobi matrix, ac-curate to machine zero. The GMRES solver in the PETSc library is used to iteratively solve the linear system. The ROIRK methods have demonstrated high order of ac-curacy in numerical simulations. The scalability study reveals that the matrix-based ROIRK methods have good parallel eﬃciency. With the block Jacobi preconditioner, it is observed that the linear systems resulting from ROIRK3-3 are stiﬀer than those from ROIRK2-2 and ROIRK4-6. This makes the scalability of ROIRK3-3 worse than ROIRK2-2 and ROIRK4-6 taking the number of stages into account.Item PARALLEL P-ADAPTIVE IMPLICIT HIGH-ORDER FLUX RECONSTRUCTION METHODS FOR UNDER-RESOLVED TURBULENT FLOW SIMULATION(2019-01-01) Wang, Lai; Yu, Meilin; Mechanical Engineering; Engineering, MechanicalA simplified flux reconstruction method, i.e., the compact direct flux reconstruction method, is developed using the compact finite difference approach within the standard element. It can be regarded as a differentiation form of the direct flux reconstruction method. Implicit high-order time integration methods are employed to achieve high-order spatiotemporal accuracy and circumvent the restriction on the time step size. To efficiently solve the nonlinear and linear systems resulting from the high-order discretization, we employ the Newton--Krylov solver as well as the p-multigrid solver. Specifically, the matrix-free implementation of the generalized minimal residual method is employed to significantly reduce memory consumption. The element-Jacobi preconditioner is used for the generalized minimal residual method and it also serves as a smoother for the p-multigrid solver. The impact of the polynomial hierarchy on the convergence speed of the V-cycle p-multigrid solver is discussed to reveal that the polynomial difference between two adjacent levels should be half of the polynomial degree on the finer level. The local preconditioning technique is employed to solve the Navier--Stokes equations at low Mach numbers. The local preconditioning can preserve the accuracy of high-order methods when the Mach number is small and accelerate the convergence of implicit methods. To further increase the efficiency of high-order methods when solving massive turbulent flows, a dynamic p-adaptation method is developed with a dynamic load balancing technique. When the p-adaptive implicit high-order method is applied to under-resolved turbulence simulation, it can significantly decrease the total number of solution points (up to 76%) as well as the run time (up to 70%). Overall, in this work, implicit high-order numerical methods with various acceleration techniques, namely, implicit time stepping, local preconditioning, p-multigrid solver, and p-adaptation, are developed and studied as our endeavor towards the efficient, accurate, and robust simulation of turbulent flows, including low-Mach-number problems.Item A Parallel Performance Study of the High-order Compact Direct Flux Reconstruction Method for Conservation Laws on Maya Cluster(2017) Wang, Lai; Yu, Meilin; Gobbert, Matthias K.The compact direct flux reconstruction method (CDFR) for conservation laws utilizes techniques from compact finite difference methods to directly approximate spatial derivatives of fluxes within standard elements. The CDFR scheme is a compact high-order method family which can be efficiently parallelized for high performance computing. In the present study, a parallel performance study of the 3rd-order CDFR scheme with a 3rd-order explicit Runge-Kutta scheme is conducted. The inviscid isentropic vortex propagation problem is adopted as a test case. The numerical performance studies have demonstrated that the CDFR method can efficiently solve conservation laws. The parallel performance study shows excellent observed speedup and efficiency. A comparison between different partition approaches of the mesh also demonstrates that optimized communication between processes can improve the parallel performance.Item A Preconditioned Flux Reconstruction/Correction Procedure via Reconstruction Formulation for Unsteady Low Mach Number Flows on Dynamic Unstructured Meshes(AIAA SciTech Forum, 2017) Wang, Lai; Yu, MeilinPreconditioning methods can signicantly decrease the condition number of the linear system resulted from the discretization of compressible Navier-Stokes equations at low Mach numbers by replacing the physical acoustic wave speeds with numerical ones. In the present study, the high-order accurate flux reconstruction/correction procedure via reconstruction (FR/CPR) method with low Mach number preconditioning is used to solve Navier-Stokes equations at low Mach numbers (Ma ~ O (10⁻³)). The dual time stepping method is used to handle un-steady flow simulations, wherein the second-order backward differentiation formula (BDF2) is adopted to discretize the temporal derivative with respect to the physical time. A simple modi cation of the preconditioning formulation is proposed to deal with dynamic meshes. Numerical results of several benchmark tests have demonstrated that the preconditioned FR/CPR method works well for low Mach number flows.Item Towards High-order Accurate Numerical Simulation of Unsteady Flow Physics over Domains with Large Deformation(AIAA, 2019) Yu, Meilin; Liu, Kan; Wang, LaiThis paper presents the development of a high-order flux reconstruction (FR) formulation for unsteady flow simulation with dynamic grid algorithms. Specifically, the high-order FR formulation for the Navier-Stokes equations in an arbitrary Lagrangian-Eulerian (ALE) format is developed for numerical simulation on moving domains. A hybrid moving grid algorithm consisting of algebraic grid smoothing and grid regeneration methods is developed to resolve domains with large deformation. The ‘dist-mesh’ technique is used for mesh regeneration, and local Lagrange interpolation within finite elements is used for flow field reconstruction. Several unsteady flow cases are studied to verify the effectiveness of the new method developed in this work.