Symmetric Gauss-Seidel multigrid solution of the Euler equations on structured and unstructured grids

The efficient symmetric Gauss-Seidel (SGS) algorithm for solving the Euler equations of inviscid, compressible flow on structured grids, developed in collaboration with Jameson of Stanford University, is extended to unstructured grids. The algorithm uses a nonlinear formulation of an SGS solver, implemented within the framework of multigrid. The earlier form of the algorithm used the natural (lexicographic) ordering of the mesh cells available on structured grids for the SGS sweeps, but a number of features of the method that are believed to contribute to its success can also be implemented for computations on unstructured grids. The present paper reviews, the features of the SGS multigrid solver for structured gr0ids, including its nonlinear implementation, its use of “absolute” Jacobian matrix preconditioning, and its incorporation of multigrid, and then describes the incorporation of these features into an algorithm suitable for computations on unstructured grids. The implementation on unstructured grids is based on the agglomerated multigrid method developed by Sørensen, which uses an explicit Runge-Kutta smoothing algorithm. Results of computations for steady, transonic flows past two-dimensional airfoils are presented, and the efficiency of the method is evaluated for computations on both structured and unstructured meshes.     

An aspect ratio agglomeration multigrid for unstructured grids

A robust aspect ratio‐based agglomeration algorithm to generate high quality of coarse grids for unstructured and hybrid grids is proposed in this paper. The algorithm focuses on multigrid techniques for the numerical solution of Euler and Navier–Stokes equations, which conform to cell‐centered finite volume special discretization scheme, combines vertex‐based isotropic agglomeration and cell‐based directional agglomeration to yield large increases in convergence rates. Aspect ratio is used as fusing weight to capture the degree of cell convexity and give an indication of cell stretching. Agglomeration front queue is established to propagate inward from the boundaries, which stores isotropic vertex and also high‐stretched cell marked with different flag according to aspect ratio. We conduct the present method to solve Euler and Navier–Stokes equations on unstructured and hybrid grids and compare the results with single grid as well as MGridGen, which shows that the present method is efficient in reducing computational time for large‐scale system equations. Copyright © 2013 John Wiley & Sons, Ltd.     

Implicit time‒accurate solutions on unstructured dynamic grids

In this paper an unstructured multigrid algorithm is used as an iterative solution procedure for the discrete equations arising from an implicit time discretization of the unsteady Euler equations on tetrahedral grids. To calculate unsteady flows due to oscillating boundaries, a novel grid movement algorithm is introduced in which an elliptic equation with a non‒linear diffusion coefficient is used to define the displacement of interior grid nodes. This allows large grid displacements to be calculated in a single step. The multigrid technique uses an edge‒collapsing algorithm to generate a sequence of grids, and a pseudo‒time‒stepping smoother. On the coarser grids, no grid motion is used. Instead, surface normals are rotated consistently and transfer/interpolation weights are based on the time‒averaged grid co‒ordinates. A 2D NACA0012 test case is used to validate the programme. 3D results are presented for the M6 wing and a full aircraft configuration. © 1997 John Wiley & Sons, Ltd.     

基于聚合多重网格方法的跨音速颤振的数值模拟

采用流固耦合方法对跨音速颤振进行了数值模拟。流体方面在非结构网格上用有限体积方法求解了Euler方程;结构方面则求解了后掠机翼典型剖面的结构模态方程。时间推进采用双时间步长：对每一真实时间步,都通过基于聚合多重网格方法的伪时间步推进,对流体和结构方程交替迭代．得到一个稳态的流固耦合的解。文章最后给出了NACA64A010翼型剖面的跨音速颤振边界．与相关文献的计算结果符合良好。     

A point implicit unstructured grid solver for the euler and Navier–Stokes equations

An upwind finite element technique that uses cell-centred quantities and implicit and/or explicit time marching has been developed for computing hypersonic laminar viscous flows using adaptive triangular grids. The approach is an extension to unstructured grids of the LAURA algorithm due to Gnoffo. A structured grid of quadrilaterals is laid out near a solid surface. For inviscid flows the method is stable at Courant numbers of over 100000. A first-order basic scheme and a higher-order flux-corrected transport (FCT) scheme have been implemented. This technique has been applied to the problem of predicting type III and IV shock wave interactions on a cylinder, with a view to simulating the pressure and heating rate augmentation caused by an impinging shock on the leading edge of a cowl lip of an engine inlet. The predictions of wall pressure and heating rates compare very well with experimental data. The flow features are distinctly captured with a sequence of adaptively generated grids.     

用混合网格及各向异性多重网格法求解三维可压紊流流动

采用混合网格求解紊流Navier Stokes方程。在物面附近采用柱状网格 ,其他区域则采用完全非结构网格。方程的求解采用Jamson的有限体积法 ,紊流模型采用两层Baldwin Lomax代数紊流模型。用各向异性多重网格法来加速解的收敛。数值算例表明 ,用混合网格及各向异性多重网格求解紊流流动是非常有效的     

A finite volume unstructured multigrid method for efficient computation of unsteady incompressible viscous flows

An unstructured non‐nested multigrid method is presented for efficient simulation of unsteady incompressible Navier–Stokes flows. The Navier–Stokes solver is based on the artificial compressibility approach and a higher‐order characteristics‐based finite‐volume scheme on unstructured grids. Unsteady flow is calculated with an implicit dual time stepping scheme. For efficient computation of unsteady viscous flows over complex geometries, an unstructured multigrid method is developed to speed up the convergence rate of the dual time stepping calculation. The multigrid method is used to simulate the steady and unsteady incompressible viscous flows over a circular cylinder for validation and performance evaluation purposes. It is found that the multigrid method with three levels of grids results in a 75% reduction in CPU time for the steady flow calculation and 55% reduction for the unsteady flow calculation, compared with its single grid counterparts. The results obtained are compared with numerical solutions obtained by other researchers as well as experimental measurements wherever available and good agreements are obtained. Copyright © 2004 John Wiley & Sons, Ltd.     

Multigrid third‐order least‐squares solution of Cauchy–Riemann equations on unstructured triangular grids

In this paper, a multigrid algorithm is developed for the third‐order accurate solution of Cauchy–Riemann equations discretized in the cell‐vertex finite‐volume fashion: the solution values stored at vertices and the residuals defined on triangular elements. On triangular grids, this results in a highly overdetermined problem, and therefore we consider its solution that minimizes the residuals in the least‐squares norm. The standard second‐order least‐squares scheme is extended to third‐order by adding a high‐order correction term in the residual. The resulting high‐order method is shown to give sufficiently accurate solutions on relatively coarse grids. Combined with a multigrid technique, the method then becomes a highly accurate and efficient solver. We present some results to demonstrate its accuracy and efficiency, including both structured and unstructured triangular grids. Copyright © 2006 John Wiley & Sons, Ltd.     

Spectral p‐multigrid discontinuous Galerkin solution of the Navier–Stokes equations

Discontinuous Galerkin (DG) methods are very well suited for the construction of very high‐order approximations of the Euler and Navier–Stokes equations on unstructured and possibly nonconforming grids, but are rather demanding in terms of computational resources. In order to improve the computational efficiency of this class of methods, a high‐order spectral element DG approximation of the Navier–Stokes equations coupled with a p‐multigrid solution strategy based on a semi‐implicit Runge–Kutta smoother is considered here. The effectiveness of the proposed approach in the solution of compressible shockless flow problems is demonstrated on 2D inviscid and viscous test cases by comparison with both a p‐multigrid scheme with non‐spectral elements and a spectral element DG approach with an implicit time integration scheme. Copyright © 2010 John Wiley & Sons, Ltd.     

High‐order accurate p‐multigrid discontinuous Galerkin solution of the Euler equations

Discontinuous Galerkin (DG) methods have proven to be perfectly suited for the construction of very high‐order accurate numerical schemes on arbitrary unstructured and possibly nonconforming grids for a wide variety of applications, but are rather demanding in terms of computational resources. In order to improve the computational efficiency of this class of methods a p‐multigrid solution strategy has been developed, which is based on a semi‐implicit Runge–Kutta smoother for high‐order polynomial approximations and the implicit Backward Euler smoother for piecewise constant approximations. The effectiveness of the proposed approach is demonstrated by comparison with p‐multigrid schemes employing purely explicit smoothing operators for several 2D inviscid test cases. Copyright © 2008 John Wiley & Sons, Ltd.     

A linear solver based on algebraic multigrid and defect correction for the solution of adjoint RANS equations

A new solution approach for discrete adjoint Reynolds‐averaged Navier–Stokes equations is suggested. The approach is based on a combination of algebraic multigrid and defect correction. The efficient interplay of these two methods results in a fast and robust solution technique. Algebraic multigrid deals very well with unstructured grids, and defect correction completes it in such a way that a required second‐order accuracy of the solution is achieved. The performance of the suggested solution approach is demonstrated on a number of representative benchmarks.Copyright © 2014 John Wiley & Sons, Ltd.     

The simulation of inviscid,compressible flows using an upwind kinetic method on unstructured grids

A kinetic flux-vector-splitting method has been used to solve the Euler equations for inviscid, compressible flow on unstructured grids. This method is derived from the Boltzmann equation and is an upwind, cell-centered, finite volume scheme with an explicit time-stepping procedure. The Delaunay triangulation has been used to generate the grids. The approach is demonstrated for three flow field simulations, namely the subsonic flow over a two-component high-lift aerofoil, the transonic flow over an aerofoil and the supersonic flow in a channel.     

A Mach‐uniform unstructured staggered grid method

A novel Mach‐uniform method to compute flows using unstructured staggered grids is discussed. The Mach‐uniform method is a generalization of the pressure‐correction approach for incompressible flows, and is valid for Mach numbers ranging from 0 (incompressible) to > 1 (supersonic). The primary variables (ρ u ,p and ρ) are updated sequentially. The grid consists of triangles. A staggered positioning of the variables is employed: the scalar variables are located at the centroids of the triangles, whereas the normal momentum components are positioned at the midpoints of the faces of the triangles. Discretization of the two‐dimensional flow equations on unstructured staggered grids is discussed. For the cell face fluxes there is a choice between first‐order upwind and central approximation. Flows around the NACA 0012 airfoil with freestream Mach numbers ranging from 0 to 1.2 are computed to demonstrate the Mach‐uniform accuracy and efficiency of the proposed method. Copyright © 2002 John Wiley & Sons, Ltd.     

Implicit meanflow–multigrid algorithms for Reynolds stress model computation of 3‐D anisotropy‐driven and compressible flows

The present paper investigates the multigrid (MG) acceleration of compressible Reynolds‐averaged Navier–Stokes computations using Reynolds‐stress model 7‐equation turbulence closures, as well as lower‐level 2‐equation models. The basic single‐grid SG algorithm combines upwind‐biased discretization with a subiterative local‐dual‐time‐stepping time‐integration procedure. MG acceleration, using characteristic MG restriction and prolongation operators, is applied on meanflow variables only (MF–MG), turbulence variables being simply injected onto coarser grids. A previously developed non‐time‐consistent (for steady flows) full‐approximation‐multigrid (s–MG) is assessed for 3‐D anisotropy‐driven and/or separated flows, which are dominated by the convergence of turbulence variables. Even for these difficult test cases CPU‐speed‐ups r_CPUSUP∈[3, 5] are obtained. Alternative, potentially time‐consistent approaches (unsteady u–MG), where MG acceleration is applied at each subiteration, are also examined, using different subiterative strategies, MG cycles, and turbulence models. For 2‐D shock wave/turbulent boundary layer interaction, the fastest s–MG approach, with a V(2, 0) sawtooth cycle, systematically yields CPU‐speed‐ups of 5±½, quasi‐independent of the particular turbulence closure used. Copyright © 2008 John Wiley & Sons, Ltd.     

Solution of the 2‐D shallow water equations with source terms in surface elevation splitting form

A vertex‐centred finite‐volume/finite‐element method (FV/FEM) is developed for solving 2‐D shallow water equations (SWEs) with source terms written in a surface elevation splitting form, which balances the flux gradients and source terms. The method is implemented on unstructured grids and the numerical scheme is based on a second‐order MUSCL‐like upwind Godunov FV discretization for inviscid fluxes and a classical Galerkin FE discretization for the viscous gradients and source terms. The main advantages are: (1) the discretization of SWE written in surface elevation splitting form satisfies the exact conservation property (??‐Property) naturally; (2) the simple centred‐type discretization can be used for the source terms; (3) the method is suitable for both steady and unsteady shallow water problems; and (4) complex topography can be handled based on unstructured grids. The accuracy of the method was verified for both steady and unsteady problems, including discontinuous cases. The results indicate that the new method is accurate, simple, and robust. Copyright © 2007 John Wiley & Sons, Ltd.     

An efficient edge‐based level set finite element method for free surface flow problems

We present an efficient technique for the solution of free surface flow problems using level set and a parallel edge‐based finite element method. An unstructured semi‐explicit solution scheme is proposed. A custom data structure, obtained by blending node‐based and edge‐based approaches is presented so to allow a good parallel performance. In addition to standard velocity extrapolation (for the convection of the level set function), an explicit extrapolation of the pressure field is performed in order to impose both the pressure boundary condition and the volume conservation. The latter is also improved with a modification of the divergence free constrain. The method is shown to allow an efficient solution of both simple benchmark cases and complex industrial examples. Copyright © 2012 John Wiley & Sons, Ltd.     

THE PARALLEL BLOCK ADAPTIVE MULTIGRID METHOD FOR THE IMPLICIT SOLUTION OF THE EULER EQUATIONS

A method capable of solving very fast and robust complex non-linear systems of equations is presented. The block adaptive multigrid (BAM) method combines mesh adaptive techniques with multigrid and domain decomposition methods. The overall method is based on the FAS multigrid, but instead of using global grids, locally enriched subgrids are also employed in regions where excessive solution errors are encountered. The final mesh is a composite grid with uniform rectangular subgrids of various mesh densities. The regions where finer grid resolution is necessary are detected using an estimation of the solution error by comparing solutions between grid levels. Furthermore, an alternative domain decomposition strategy has been developed to take advantage of parallel computing machines. The proposed method has been applied to an implicit upwind Euler code (EuFlex) for the solution of complex transonic flows around aerofoils. The efficiency and robustness of the BAM method are demonstrated for two popular inviscid test cases. Up to 19-fold acceleration with respect to the single-grid solution has been achieved, but a further twofold speed-up is possible on four-processor parallel computers.     

Improved multislope MUSCL reconstruction on unstructured grids for shallow water equations

In shallow water flow and transport modeling, the monotonic upstream‐centered scheme for conservation laws (MUSCL) is widely used to extend the original Godunov scheme to second‐order accuracy. The most important step in MUSCL‐type schemes is MUSCL reconstruction, which calculate‐extrapolates the values of independent variables from the cell center to the edge. The monotonicity of the scheme is preserved with the help of slope limiters that prevent the occurrence of new extrema during reconstruction. On structured grids, the calculation of the slope is straightforward and usually based on a 2‐point stencil that uses the cell centers of the neighbor cell and the so‐called far‐neighbor cell of the edge under consideration. On unstructured grids, the correct choice for the upwind slope becomes nontrivial. In this work, 2 novel total variation diminishing schemes are developed based on different techniques for calculating the upwind slope and the downwind slope. An additional treatment that stabilizes the scheme is discussed. The proposed techniques are compared to 2 existing MUSCL reconstruction techniques, and a detailed discussion of the results is given. It is shown that the proposed MUSCL reconstruction schemes obtain more accurate results with less numerical diffusion and higher efficiency.     

The implementation of an aeronautical CFD flow code onto distributed memory parallel systems

The parallelization of an industrially important in‐house computational fluid dynamics (CFD) code for calculating the airflow over complex aircraft configurations using the Euler or Navier–Stokes equations is presented. The code discussed is the flow solver module of the SAUNA CFD suite. This suite uses a novel grid system that may include block‐structured hexahedral or pyramidal grids, unstructured tetrahedral grids or a hybrid combination of both. To assist in the rapid convergence to a solution, a number of convergence acceleration techniques are employed including implicit residual smoothing and a multigrid full approximation storage scheme (FAS). Key features of the parallelization approach are the use of domain decomposition and encapsulated message passing to enable the execution in parallel using a single programme multiple data (SPMD) paradigm. In the case where a hybrid grid is used, a unified grid partitioning scheme is employed to define the decomposition of the mesh. The parallel code has been tested using both structured and hybrid grids on a number of different distributed memory parallel systems and is now routinely used to perform industrial scale aeronautical simulations. Copyright © 2000 John Wiley & Sons, Ltd.     

Calculation of three‐dimensional turbulent flow with a finite volume multigrid method

A numerical method for the efficient calculation of three‐dimensional incompressible turbulent flow in curvilinear co‐ordinates is presented. The mathematical model consists of the Reynolds averaged Navier–Stokes equations and the k–ε turbulence model. The numerical method is based on the SIMPLE pressure‐correction algorithm with finite volume discretization in curvilinear co‐ordinates. To accelerate the convergence of the solution method a full approximation scheme‐full multigrid (FAS‐FMG) method is utilized. The solution of the k–ε transport equations is embedded in the multigrid iteration. The improved convergence characteristic of the multigrid method is demonstrated by means of several calculations of three‐dimensional flow cases. Copyright © 1999 John Wiley & Sons, Ltd.