High-order simulation of the Caradonna and Tung rotor in hover

Abstract: In this paper, we validate a previously proposed highorder method for simulating unsteady flows for a helicopter rotor in hover. To demonstrate the performance and efficiency of this strategy, three cases have been studied. The first case involves a threedimensional circular cylinder at the onset of the shear-layer transition regime with a Reynolds number of Re = 1000 and Mach number of M = 0.2, while the second case examines the turbulent flow over a 3D SD 7003 airfoil undergoing heaving and pitching motions with a Reynolds number of Re = 10000 and Mach number of M = 0.1. These cases aim to illustrate the accuracy and efficiency of the routine when applied to transitional and turbulent flows. Finally, a hovering model of the Caradonna and Tung helicopter rotor with a tip Mach number of Mt = 0.526, Re = 2.358 × 106, an angular velocity of ? = 29.9237 radian per second, and blade pitching angle of ? = 8? is studied. This strategy is validated and compared against numerical and experimental reference data in terms of accuracy and computational cost, considering functional targets such as lift, drag, and thrust coefficients of the simulations. Results demonstrate that the algorithm can track regions of interest, such as boundary layers and wake regions, and yields a considerable speed-up when applied to parallel simulations. Qualitative and quantitative results showed equivalent levels of accuracy with significant speed-up when applied to parallel simulations. Hence, the proposed algorithm is an effective and accurate approach for simulating unsteady transitional and turbulent flows.Résumé de la communication présentée lors du congrès international tenu conjointement par Canadian Society for Mechanical Engineering (CSME) et Computational Fluid Dynamics Society of Canada (CFD Canada), à l’Université de Sherbrooke (Québec), du 28 au 31 mai 2023

GPU-accelerated discontinuous Galerkin methods on hybrid meshes: applications in seismic imaging

Seismic imaging is a geophysical technique assisting in the understanding of subsurface structure on a regional and global scale. With the development of computer technology, computationally intensive seismic algorithms have begun to gain attention in both academia and industry. These algorithms typically produce high-quality subsurface images or models, but require intensive computations for solving wave equations. Achieving high-fidelity wave simulations is challenging: first, numerical wave solutions may suffer from dispersion and dissipation errors in long-distance propagations; second, the efficiency of wave simulators is crucial for many seismic applications. High-order methods have advantages of decreasing numerical errors efficiently and hence are ideal for wave modelings in seismic problems. Various high order wave solvers have been studied for seismic imaging. One of the most popular solvers is the finite difference time domain (FDTD) methods. The strengths of finite difference methods are the computational efficiency and ease of implementation, but the drawback of FDTD is the lack of geometric flexibility. It has been shown that standard finite difference methods suffer from first order numerical errors at sharp media interfaces. In contrast to finite difference methods, discontinuous Galerkin (DG) methods, a class of high-order numerical methods built on unstructured meshes, enjoy geometric flexibility and smaller interface errors. Additionally, DG methods are highly parallelizable and have explicit semi-discrete form, which makes DG suitable for large-scale wave simulations. In this dissertation, the discontinuous Galerkin methods on hybrid meshes are developed and applied to two seismic algorithms---reverse time migration (RTM) and full waveform inversion (FWI). This thesis describes in depth the steps taken to develop a forward DG solver for the framework that efficiently exploits the element specific structure of hexahedral, tetrahedral, prismatic and pyramidal elements. In particular, we describe how to exploit the tensor-product property of hexahedral elements, and propose the use of hex-dominant meshes to speed up the computation. The computational efficiency is further realized through a combination of graphics processing unit (GPU) acceleration and multi-rate time stepping. As DG methods are highly parallelizable, we build the DG solver on multiple GPUs with element-specific kernels. Implementation details of memory loading, workload assignment and latency hiding are discussed in the thesis. In addition, we employ a multi-rate time stepping scheme which allows different elements to take different time steps. This thesis applies DG schemes to RTM and FWI to highlight the strengths of the DG methods. For DG-RTM, we adopt the boundary value saving strategy to avoid data movement on GPUs and utilize the memory load in the temporal updating procedure to produce images of higher qualities without a significant extra cost. For DG-FWI, a derivation of the DG-specific adjoint-state method is presented for the fully discretized DG system. Finally, sharp media interfaces are inverted by specifying perturbations of element faces, edges and vertices

Schnelle Löser für Partielle Differentialgleichungen

This workshop was well attended by 52 participants with broad geographic representation from 11 countries and 3 continents. It was a nice blend of researchers with various backgrounds

Towards industrial large eddy simulation using the FR/CPR method

NASA’s 2030 CFD Vision calls for the development of accurate and efficient scale-resolving simulations for turbulent flow, such as large eddy simulation (LES) and direct numerical simulation (DNS). This is primarily because the Reynolds-averaged Navier-Stokes (RANS) approach has failed to predict vortex-dominated flow involving large flow separations, e.g., flow through a jet engine or over aircraft near the edge of the flight envelope, i.e., during take-off and landing at high angles of attack. Although the DNS approach resolves all turbulence scales, it is too expensive in the foreseeable future for real world flow problems because of the disparate length and time scales in the flow. LES resolves the energetic large scales while modeling the smaller scales, so it provides a good compromise between accuracy and cost. As a result, LES is widely considered to be the method of choice for next generation CFD design tool. The major obstacle for LES is its considerable computational cost since unsteady 3D simulations need to be performed to obtain the mean flow quantities such as the drag and lift coefficients. In order to resolve the dominant scales in a turbulent flow, numerical methods used for LES should have low dissipation and dispersion errors. This means standard second order finite-volume methods are usually not accurate or efficient enough for LES applications. High-order methods (order of accuracy 2) have demonstrated their potential for LES and DNS in the past decade because of their low embedded numerical dissipation and dispersion errors. In the present study, we develop and demonstrate a recently developed high-order method, called flux reconstruction (FR) or correction procedure via reconstruction (CPR), for industrial LES. A major advantage of the FR/CPR method is its capability to handle unstructured mixed meshes, and its compactness and scalability, which is particularly desired on modern super-computers. We therefore address the following major pacing items in industrial LES in the present study: High-order methods Geometric flexibility Efficient time integration Efficient implementation on modern super computers Demonstration for real world application

Spectral/hp element methods: recent developments, applications, and perspectives

The spectral/hp element method combines the geometric flexibility of the classical h-type finite element technique with the desirable numerical properties of spectral methods, employing high-degree piecewise polynomial basis functions on coarse finite element-type meshes. The spatial approximation is based upon orthogonal polynomials, such as Legendre or Chebychev polynomials, modified to accommodate C0-continuous expansions. Computationally and theoretically, by increasing the polynomial order p, high-precision solutions and fast convergence can be obtained and, in particular, under certain regularity assumptions an exponential reduction in approximation error between numerical and exact solutions can be achieved. This method has now been applied in many simulation studies of both fundamental and practical engineering flows. This paper briefly describes the formulation of the spectral/hp element method and provides an overview of its application to computational fluid dynamics. In particular, it focuses on the use the spectral/hp element method in transitional flows and ocean engineering. Finally, some of the major challenges to be overcome in order to use the spectral/hp element method in more complex science and engineering applications are discussed

Toward Accurate, Efficient, and Robust Hybridized Discontinuous Galerkin Methods

Computational science, including computational fluid dynamics (CFD), has become an indispensible tool for scientific discovery and engineering design, yet a key remaining challenge is to simultaneously ensure accuracy, efficiency, and robustness of the calculations. This research focuses on advancing a class of high-order finite element methods and develops a set of algorithms to increase the accuracy, efficiency, and robustness of calculations involving convection and diffusion, with application to the inviscid Euler and viscous Navier-Stokes equations. In particular, it addresses high-order discontinuous Galerkin (DG) methods, especially hybridized (HDG) methods, and develops adjoint-based methods for simultaneous mesh and order adaptation to reduce the error in a scalar functional of the approximate solution to the discretized equations. Contributions are made in key aspects of these methods applied to general systems of equations, addressing the scalability and memory requirements, accuracy of HDG methods, and efficiency and robustness with new adaptation methods. First, this work generalizes existing HDG methods to systems of equations, and in so doing creates a new primal formulation by applying DG stabilization methods as the viscous stabilization for HDG. The primal formulation is shown to be even more computationally efficient than the existing methods. Second, by instead keeping existing viscous stabilization methods and developing a new convection stabilization, this work shows that additional accuracy can be obtained, even in the case of purely convective systems. Both HDG methods are compared to DG in the same computational framework and are shown to be more efficient. Finally, the set of adaptation frameworks is developed for combined mesh and order refinement suitable for both DG and HDG discretizations. The first of these frameworks uses hanging-node-based mesh adaptation and develops a novel local approach for evaluating the refinement options. The second framework intended for simplex meshes extends the mesh optimization via error sampling and synthesis (MOESS) method to incorporate order adaptation. Collectively, the results from this research address a number of key issues that currently are at the forefront of high-order CFD methods, and particularly to output-based hp-adaptation for DG and HDG methods.PHDAerospace EngineeringUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttps://deepblue.lib.umich.edu/bitstream/2027.42/137150/1/jdahm_1.pd