Spectral/hp Element Methods for Computational Fluid Dynamics

by Karniadakis, George Em; Sherwin, Spencer J.

• ISBN13: 9780198528692
• ISBN10: 0198528698
• Edition: 2nd
• Format: Hardcover
• Copyright: 2005-08-11
• Publisher: Oxford University Press

Summary

Spectral methods have long been popular in direct and large eddy simulation of turbulent flows, but their use in areas with complex-geometry computational domains has historically been much more limited. More recently the need to find accurate solutions to the viscous flow equations around complex configurations has led to the development of high-order discretization procedures on unstructured meshes, which are also recognized as more efficient for solution of time-dependent oscillatory solutions over long time periods. Here Karniadakis and Sherwin present a much-updated and expanded version of their successful first edition covering the recent and significant progress in multi-domain spectral methods at both the fundamental and application level. Containing over 50% new material, including discontinuous Galerkin methods, non-tensorial nodal spectral element methods in simplex domains, and stabilization and filtering techniques, this text aims to introduce a wider audience to the use of spectral/hp element methods with particular emphasis on their application to unstructured meshes. It provides a detailed explanation of the key concepts underlying the methods along with practical examples of their derivation and application, and is aimed at students, academics and practitioners in computational fluid mechanics, applied and numerical mathematics, computational mechanics, aerospace and mechanical engineering and climate/ocean modelling.

Author Biography

George Em Karniadakis is a Professor at Brown University, Division of Applied Mathematics, 182 George Street, Providence, RI 02912, USA. Spencer Sherwin is Reader in computational Fluid Mechanics, Department of Aeronautics, Imperial College London, South Kensington Campus, London, SW7 2AZ, UK.

Table of Contents

1 Introduction	1	(15)
1.1 The basic equations of fluid dynamics	1	(6)
1.1.1 Incompressible flow	3	(3)
1.1.2 Reduced models	6	(1)
1.2 Numerical discretisations	7	(9)
1.2.1 The finite element method	7	(1)
1.2.2 Spectral discretisation	8	(1)
1.2.3 Why high-order accuracy in CFD?	9	(3)
1.2.4 Structured versus unstructured discretisation	12	(2)
1.2.5 What is hp convergence?	14	(2)
2 Fundamental concepts in one dimension	16	(65)
2.1 Method of weighted residuals	18	(3)
2.2 Galerkin formulation	21	(14)
2.2.1 Descriptive formulation	21	(4)
2.2.2 Two-domain linear finite element example	25	(4)
2.2.3 Mathematical formulation	29	(2)
2.2.4 Mathematical properties of the Galerkin approximation	31	(3)
2.2.5 Residual equation for the c° test and trial functions	34	(1)
2.3 One-dimensional expansion bases	35	(23)
2.3.1 Elemental decomposition: the h-type extension	36	(7)
2.3.2 Polynomial expansions: the p-type extension	43	(7)
2.3.3 Modal polynomial expansions	50	(4)
2.3.4 Nodal polynomial expansions	54	(4)
2.4 Elemental operations	58	(10)
2.4.1 Numerical integration	58	(6)
2.4.2 Differentiation	64	(4)
2.5 Error estimates	68	(5)
2.5.1 h-convergence of linear finite elements	68	(3)
2.5.2 L² error of the p-type interpolation in a single element	71	(1)
2.5.3 General error estimates for hp elements	72	(1)
2.6 Implementation of a one-dimensional spectral/hp element solver	73	(8)
2.6.1 Exercises	73	(5)
2.6.2 Convergence examples	78	(3)
3 Multi-dimensional expansion bases	81	(58)
3.1 Quadrilateral and hexahedral tensor product expansions	83	(8)
3.1.1 Standard tensor product extensions	84	(4)
3.1.2 Polynomial space of tensor product expansions	88	(3)
3.2 Generalised tensor product modal expansions	91	(30)
3.2.1 Coordinate systems	93	(7)
3.2.2 Orthogonal expansions	100	(8)
3.2.3 Modified C° expansions	108	(13)
3.3 Non-tensorial nodal expansions in a simplex	121	(10)
3.3.1 The Lagrange polynomial and the Lebesgue constant	123	(1)
3.3.2 Generalised Vandermonde matrix	124	(2)
3.3.3 Electrostatic points	126	(1)
3.3.4 Fekete points	127	(4)
3.4 Other useful tensor product extensions	131	(2)
3.4.1 Nodal elements in a prismatic region	131	(1)
3.4.2 Expansions in homogeneous domains	132	(1)
3.4.3 Cylindrical domains	133	(1)
3.5 Exercises: multi-dimensional elemental mass matrices	133	(6)
4 Multi-dimensional formulation	139	(112)
4.1 Local elemental operations	140	(45)
4.1.1 Integration within the standard region Ωst	141	(6)
4.1.2 Differentiation in the standard region Ωst	147	(6)
4.1.3 Operations within general-shaped elements	153	(7)
4.1.4 Discrete evaluation of the surface Jacobian	160	(4)
4.1.5 Elemental projections and transformations	164	(15)
4.1.6 Sum-factorisation/tensor product operations	179	(6)
4.2 Global operations	185	(30)
4.2.1 Global assembly and connectivity	186	(15)
4.2.2 Global matrix system	201	(2)
4.2.3 Static condensation/substructuring	203	(6)
4.2.4 Global boundary system numbering and ordering to enforce Dirichlet boundary conditions	209	(6)
4.3 Pre- and post-processing issues	215	(30)
4.3.1 Boundary condition discretisation	215	(1)
4.3.2 Elemental boundary transformation	216	(3)
4.3.3 Mesh generation for spectral/hp element discretisation	219	(2)
4.3.4 Global coarse meshing	221	(1)
4.3.5 High-order mesh generation	222	(12)
4.3.6 Particle tracking in spectral/lap element discretisations	234	(11)
4.4 Exercises: implementation of a two-dimensional spectral/hp element solver for a global projection problem using a C° Galerkin formulation	245	(6)
5 Diffusion equation	251	(47)
5.1 Galerkin discretisation of the Helmholtz equation	252	(6)
5.2 Numerical examples	258	(3)
5.3 Temporal discretisation	261	(4)
5.3.1 Forward multi-step schemes	262	(2)
5.3.2 Backward multi-step schemes	264	(1)
5.4 Eigenspectra and iterative solution of weak Laplacian	265	(11)
5.4.1 Time-step restriction and maximum eigenvalue growth	265	(1)
5.4.2 Iterative solution and preconditioners	266	(10)
5.5 Non-smooth domains	276	(17)
5.5.1 Laplace equation in two-dimensional domains	278	(1)
5.5.2 Laplace equation in three-dimensional domains	279	(4)
5.5.3 Poisson equation	283	(2)
5.5.4 Helmholtz equation	285	(1)
5.5.5 Singular basis	285	(2)
5.5.6 Eigenpair representation: Steklov formulation	287	(5)
5.5.7 Singularities and Stokes flow	292	(1)
5.6 Exercises: implementation of a two-dimensional spectral/hp element solver for a Helmholtz problem using a C° Galerkin formulation	293	(5)
6 Advection and advection-diffusion	298	(61)
6.1 Linear advection equation	299	(3)
6.1.1 Dispersion and diffusion errors	299	(3)
6.2 Galerkin and discontinuous Galerkin discretisations	302	(13)
6.2.1 Galerkin discretisation of the linear advection equation	303	(7)
6.2.2 Discontinuous Galerkin method	310	(5)
6.3 Eigenspectrum of weak advection operator	315	(13)
6.3.1 Numerical evaluation of the time-step restriction	316	(2)
6.3.2 Eigenspectrum of the weak advection operator in Ωst	318	(10)
6.4 Semi-Lagrangian formulation for advection-diffusion	328	(13)
6.4.1 Strong semi-Lagrangian method	330	(3)
6.4.2 Auxiliary semi-Lagrangian method	333	(3)
6.4.3 Convergence, efficiency, and stability of semi-Lagrangian schemes	336	(5)
6.5 Wiggles and high order: stabilisation techniques	341	(18)
6.5.1 Filters and relaxation	344	(4)
6.5.2 Spectral vanishing viscosity (SVV)	348	(4)
6.5.3 Over-integration of the viscous Burgers equation	352	(2)
6.5.4 Superconsistent collocation for advection-diffusion	354	(5)
7 Non-conforming elements	359	(41)
7.1 Interface conditions and implementation	361	(3)
7.2 Iterative patching	364	(7)
7.2.1 One-dimensional discretisation	364	(2)
7.2.2 Two-dimensional discretisation	366	(2)
7.2.3 Variational formulation	368	(1)
7.2.4 Interpretation of the relaxation procedure	369	(2)
7.3 Constrained approximation	371	(1)
7.4 Mortar patching	372	(9)
7.4.1 Projection and non-conforming spaces	373	(1)
7.4.2 The discrete second-order problem	374	(4)
7.4.3 Implementation	378	(2)
7.4.4 Condition number of the Laplacian	380	(1)
7.5 Discontinuous Galerkin method (DGM)	381	(19)
7.5.1 An inconsistent formulation	381	(2)
7.5.2 Local discontinuous Galerkin method (LDG)	383	(1)
7.5.3 The Baumann-Oden discontinuous Galerkin method	384	(1)
7.5.4 A unified formulation	384	(3)
7.5.5 Compactness of the stencil	387	(1)
7.5.6 Eigenspectrum	388	(2)
7.5.7 Convergence rate	390	(1)
7.5.8 Examples and comparisons	391	(4)
7.5.9 Stabilisation	395	(1)
7.5.10 Discontinuous Galerkin versus mixed formulation	396	(3)
7.5.11 Which DGM version to use?	399	(1)
8 Algorithms for incompressible flows	400	(55)
8.1 Variational formulation	400	(4)
8.2 Coupled methods for primitive variables	404	(10)
8.2.1 The Uzawa algorithm	404	(5)
8.2.2 Substructured Stokes system	409	(5)
8.3 Splitting methods for primitive variables	414	(21)
8.3.1 First-order schemes	414	(4)
8.3.2 High-order schemes	418	(15)
8.3.3 The inf-sup condition	433	(1)
8.3.4 Comparisons and recommendations	434	(1)
8.4 Velocity -vorticity formulation	435	(8)
8.4.1 Semi-discrete equations	438	(1)
8.4.2 Influence matrix implementation	439	(2)
8.4.3 Penalty method implementation	441	(1)
8.4.4 Spatial discretisation	441	(2)
8.5 Least-squares method	443	(4)
8.5.1 Formulation	443	(2)
8.5.2 Performance	445	(2)
8.6 The gauge method	447	(1)
8.7 Discretisation of nonlinear terms	448	(7)
8.7.1 Spatial discretisation	448	(3)
8.7.2 Temporal discretisation: semi-Lagrangian method	451	(4)
9 Incompressible flow simulations: verification and validation	455	(59)
9.1 Exact Navier-Stokes solutions	455	(8)
9.1.1 Moffatt eddies	455	(3)
9.1.2 Wannier flow	458	(1)
9.1.3 Kovasznay flow	458	(5)
9.1.4 Triangular duct flow	463	(1)
9.1.5 The Taylor vortex	463	(1)
9.2 BiGlobal stability analysis of complex flows	463	(8)
9.2.1 Formulation of the linearised eigenproblem	465	(2)
9.2.2 Iterative solution of the eigenproblem	467	(1)
9.2.3 Floquet analysis	468	(1)
9.2.4 Applications of BiGlobal stability	469	(2)
9.3 Direct numerical simulations DNS	471	(19)
9.3.1 Under-resolution and diagnostics	472	(11)
9.3.2 Stabilisation at high Reynolds number	483	(7)
9.4 Large-eddy simulations LES	490	(12)
9.4.1 Governing equations and filters	491	(4)
9.4.2 Subgrid models	495	(7)
9.5 Dynamic (dDNS) versus static DNS	502	(12)
9.5.1 p-refinement and p-threads	503	(4)
9.5.2 The three-step Texas algorithm	507	(3)
9.5.3 Non-conforming spectral element refinement	510	(4)
10 Hyperbolic conservation laws	514	(71)
10.1 Conservative formulation	515	(10)
10.1.1 Cell-averaging procedure	515	(4)
10.1.2 Reconstruction procedure	519	(1)
10.1.3 Interfacial constraint	520	(1)
10.1.4 Non-oscillatory approximation	521	(4)
10.2 Monotonicity	525	(6)
10.2.1 Flux-corrected transport (FCT)	525	(3)
10.2.2 Local projection limiting	528	(3)
10.3 Euler equations	531	(13)
10.3.1 One-dimensional equations	531	(6)
10.3.2 Two-dimensional equations	537	(3)
10.3.3 Discontinuous Galerkin method	540	(4)
10.4 Shallow-water equations	544	(9)
10.4.1 Governing equations	545	(1)
10.4.2 Discontinuous Galerkin formulation	545	(3)
10.4.3 Examples	548	(1)
10.4.4 Boussinesq equations	549	(4)
10.5 Navier-Stokes equations	553	(17)
10.5.1 Mixed and discontinuous Galerkin formulations	554	(5)
10.5.2 Convergence and simulations	559	(3)
10.5.3 A penalty formulation	562	(3)
10.5.4 Moving domains	565	(3)
10.5.5 Stability and over-integration	568	(2)
10.6 Shock-fitting techniques	570	(4)
10.7 Magneto-hydrodynamics (MHD)	574	(11)
10.7.1 Governing equations	575	(1)
10.7.2 B = 0 constraint	576	(1)
10.7.3 A discontinuous Galerkin MHD solver	577	(3)
10.7.4 Convergence and simulations	580	(5)
A Jacobi polynomials	585	(9)
A.1 Useful formulae for Jacobi polynomials	585	(3)
A.2 Askey hypergeometric orthogonal polynomials	588	(6)
A.2.1 Examples	591	(3)
B Gauss-type integration	594	(5)
B.1 Jacobi formulae	595	(2)
B.2 Evaluation of the zeros of Jacobi polynomials	597	(2)
C Collocation differentiation	599	(4)
C.1 Jacobi formulae	600	(3)
D C° continuous expansion bases	603	(22)
D.1 Modal basis	603	(8)
D.1.1 Two-dimensional expansions	603	(2)
D.1.2 Three-dimensional expansions	605	(6)
D.2 Nodal basis	611	(14)
D.2.1 Tensorial expansions	611	(1)
D.2.2 Non-tensorial expansions	612	(13)
E Characteristic flux decomposition	625	(4)
E.1 One dimension	625	(1)
E.2 Two dimensions	626	(1)
E.3 Three dimensions	627	(2)
References	629	(24)
Index	653