List of Symbols In the following the meaning of the important symbols in the text with the corresponding physical units are listed. Some letters are multiply used (how- ever, only in different contexts), in order to keep as far as possible the standard notations as they commonly appear in the literature. Matrices, dyads, higher order tensors A general system matrix B procedure matrix for iterative methods C iteration matrix for iterative methods 2 C, Cij N/m material matrix 2 E, Eijkl N/m elasticity tensor G, Gij Green-Lagrange strain tensor L general lower triangular matrix I unit matrix J, Jij Jacobi matrix P preconditioning matrix 2 P, Pij N/m 2nd Piola-Kirchhoff stress tensor S, Sij strain rate tensor S, Sij stiffness matrix e e S , Sij unit element stiffness matrix k k S , Sij element stiffness matrix 2 T, Tij, T˜ij N/m Cauchy stress tensor U general upper triangular matrix δij Kronecker symbol , ij Green-Cauchy strain tensor ijk permutation symbol sgs 2 τij N/m subgrid-scale stress tensor test 2 τij N/m subtest-scale stress tensor 308 List of Symbols Vectors a, ai m material coordinates b, bi load vector ˜ b, bi N/kg volume forces per unit mass e e b , bi unit element load vector k k b , bi element load vector c m/s translating velocity vector d, di Nms moment of momentum vector ei, eij Cartesian unit basis vectors f, fi N/kg volume forces per mass unit h, hi N/ms heat flux vector 2 j, ji kg/m s mass flux vector n, ni unit normal vector p, pi Ns momentum vector t, ti unit tangent vector 2 t, ti N/m stress vector u, ui m displacement vector v, vi m/s velocity vector g rmg v , vi m/s grid velocity w,˜wi m/s relative velocity vector x, xi m spatial coordinates ϕ, ϕi test function vector ω, ωi 1/s angular velocity vector Scalars (latin upper case letters) A m2 cross sectional area n Bi Bernstein polynomial of degree n B Nm2 flexural stiffness C Courant number Cs Smagorinsky constant Cg dynamic Germano parameter D diffusion number D kg/ms diffusion coefficient 2 D0 m unit triangle 2 Di m general triangle 3 G, Gi 1/m , 1/m filter function E N/m2 elasticity modulus i i Egrid,Ejump error indicators EP efficiency for P processors par EP parallel efficiency for P processors num EP numerical efficiency for P processors last EP load balancing efficiency for P processors List of Symbols 309 Fc flux through face Sc C Fc convective flux through face Sc D Fc diffusive flux through face Sc G N/m2s production rate of turbulent kinetic energy H mheight I m4 axial angular impulse J Jacobi determinant K Nm plate stiffness L mlength M Nm bending moment NB s data transfer time e Nj shape function in unit element i Nj local shape function Nj global shape function Nit number of iterations P Nm potential energy Pa Nm/s power of external forces 2 Q0 m unit square 2 Qi m general quadrilateral Q Nm/s power of heat supply Q N transverse force R kg/m3s mass source RT 1/s data transfer rate R Nm/kgK specific gas constant S m2 bzw. m surface or boundary curve Sc control volume face SP speed-up for P processors T K temperature T˜ K reference temperature TL s latency time for data transfer TK s data transfer time Tv turbulence degree TH higher order terms 3 2 V , Vi m bzw. m (control) volume or (control) area 3 V0 m reference volume W Nm total energy of a body W N/m2 strain energy density function Scalars (latin lower case letters) a m/s speed of sound c species concentration cp Nm/kgK specific heat capacity at constant pressure cv Nm/kgK specific heat capacity at constant volume 310 List of Symbols d m plate thickness e Nm/kg specific internal energy n eP total numerical error at point P and time tn f general source term f N/m3 force density g scalar source term g m/s2 acceleration of gravity h m measure for grid spacing fl N/m longitudinal load fq N/m lateral load k Nm/kg turbulent kinetic energy kL N boundary force (bar) l m turbulent length scale m kg mass m˙ c kg/s mass flux through face Sc p N/m2 pressure q Nm/skg heat source p, p N/m2 pressure correction s Nm/kgK specific internal entropy t stime u m/s velocity component in x-direction uτ m/s wall shear stress velocity u+ normalized tangential velocity v m/s velocity component in y-direction vn m/s normal component of velocity vt m/s tangential component of velocity v¯ m/s characteristic velocity w m deflection wi weights for Gauß quadrature x m spatial coordinate y m spatial coordinate y+ normalized wall distance Scalars (greek letter) α general diffusion coefficient αφ under-relaxation factor for φ α Nm/kg thermal expansion coefficient αnum numerical (artificial) diffusion α˜ N/Kms heat transfer coefficient β flux-blending parameter βc artificial compressibility parameter Γ domain boundary γ interpolation factor List of Symbols 311 δ m wall distance Nm/s dissipation of turbulent kinetic energy εtol error tolerance η,˜η m spatial coordinate θ K temperature deviation θ control parameter for θ-method κ N/Ks heat conductivity κ condition number of a matrix κ K´arm´an constant λ N/m2 Lam´e constant λP aspect ratio of grid cell λmax spectral radius μ N/m2 Lam´e constant μt kg/ms turbulent viscosity μ kg/ms dynamic viscosity ν Poisson number ν m2/s kinematic viscosity Ω problem domain ξ, ξ˜ m spatial coordinate ξc grid expansion ratio Π Nm strain energy ρ kg/m3 density 3 ρ0 kg/m reference density τ N/m2 stiffness 2 τw N/m wall shear stress n τP truncation error at point P and time tn φ scalar transport quantity φ filtered or averaged quantity φ φ small scale portion or fluctuation of φ ϕ virtual displacement ψ N/m2s specific dissipation function ψ general conservation quantity ψ m2/s velocity potential ω relaxation parameter for SOR method Others Ma Mach number Re Reynolds number Nu Nußelt number Pe Peclet number Peh grid Peclet number 2 δSc morm length or area of control volume face Sc δV m3 or m2 volume or area of V 312 List of Symbols Δt s time step size Δx m spatial grid spacing Δy m spatial grid spacing F discretization rule H function space for test functions Ih 2h interpolation operator I2h h restriction operator L spatial discretization operator S iteration method References 1. O. Axelsson und V.A. Barker Finite Element Solution of Boundary Value Problems Academic Press, Orlando, 1984 (for Chap. 7) 2. K.-J. Bathe Finite-Element Procedures Prentice Hall, New Jersey, 1995 (for Chaps. 5 and 9) 3. D. Braess Finite Elements 2nd edition, University Press, Cambridge, 2001 (for Chaps. 5 and 9) 4. W. Briggs Multi-Grid Tutorial 2nd edition, SIAM, Philadelphia, 2000 (for Chap. 12) 5. T.J. Chung Computational Fluid Mechanics Cambridge University Press, 2002 (for Chaps. 3, 4, 6, 8, 10, 11, and 12) 6. H. Eschenauer, N. Olhoff, and W. Schnell Applied Structural Mechanics Springer, Berlin, 1997 (for Chaps. 2, 5, and 9) 7. G.E. Farin Curves and Surfaces for Computer Aided Geometric Design: A Practical Guide 5th edition, Academic Press, London, 2001 (for Chap. 3) 8. J. Ferziger und M. Peri´c Computational Methods for Fluid Dynamics 3rd edition, Springer, Berlin, 2001 (for Chaps. 4, 6, 8, 10, and 11) 9. C.A.J. Fletcher Computational Techniques for Fluid Dynamics (Vol. 1, 2) Springer, Berlin, 1988 (for Chaps. 4, 6, and 10) 10. W. Hackbusch Multi-Grid Methods and Applications Springer, Berlin, 1985 (for Chap. 12) 11. W. Hackbusch Iterative Solution of Large Sparse Systems of Equations Springer, Berlin, 1998 (for Chap. 7) 314 References 12. C. Hirsch Numerical Computation of Internal and External Flows (Vol. 1, 2) Wiley, Chichester, 1988 (for Chaps. 4, 6, 7, 8, and 10) 13. K.A. Hoffmann und S.T. Chang Computational Fluid Dynamics for Engineers I, II Engineering Education System, Wichita, 1993 (for Chaps. 3, 6, and 8) 14. G.A. Holzapfel Nonlinear Solid Mechanics Wiley, Chichester, 2000 (for Chap. 2) 15. P. Knupp und S. Steinberg Fundamentals of Grid Generation CRC Press, Boca Raton, 1994 (for Chap. 3) 16. R. Peyret (Editor) Handbook of Computational Fluid Mechanics Academic Press, London, 1996 (for Chaps. 10 and 11) 17. S.B. Pope Turbulent Flows University Press, Cambridge, 2000 (for Chap. 11) 18. P. Sagaut Large Eddy Simulation for Incompressible Flows 2nd edition, Springer, Berlin, 2003 (for Chap. 11) 19. J. Salen¸con Handbook of Continuum Mechanics Springer, Berlin, 2001 (for Chap. 2) 20. H.R. Schwarz Finite Element Methods Academic Press, London, 1988 (for Chaps. 5 and 9) 21. L.R. Scott, T. Clark, and B. Bagheri Scientific Parallel Computing Princeton University Press, 2005 (for Chap. 12) 22. R. Siegel and J.R. Howell Thermal Radiation Heat Transfer 4th edition, Taylor & Francis, New York, 2002 (for Chap. 2) 23. J.H. Spurk Fluid Mechanics Springer, Berlin, 1997 (for Chap. 2) 24. J. Stoer und R. Bulirsch Introduction to Numerical Analysis 3rd edition, Springer, Berlin, 2002 (for Chaps. 6 and 7) 25. S. Timoschenko und J.N. Goodier Theory of Elasticity McGraw Hill, New York, 1970 (for Chap.