Belytschko T. - Introduction (779635), страница 84

Просмтор этого файла доступен только зарегистрированным пользователям. Но у нас супер быстрая регистрация: достаточно только электронной почты!

Текст из файла (страница 84)

(7.4.19a) reduces to:W.K.Liu, Chapter 7{70}ρ E,t[χ ] + E, j c j = (σ ij vi ),j + b j v j + (kij θ, j ),i + ρs(7.4.19b)or, in index free notation:{}ρ E,t[χ ] + c ⋅ grad E = div(v ⋅ σ) + v⋅ b + div(k ⋅ grad θ ) + ρs(7.4.19c)(3c) Show that the above equations can be specified in the Lagrangian description bychoosing: ∂x χ = X; φˆ = φ ; c = 0; J = det ∂X(7.4.20a)and they are given by:ρE,t[χ ] = (σ ij vi ),j + b j v j + (kij θ, j ),i + ρs(7.4.20b)or, in index free notation:ρE,t[χ ] = div(v ⋅ σ) + v ⋅ b + div(k ⋅grad θ) + ρs(7.4.20c)(3d) Similarly, show that the Eulerian energy equation is obtained by choosing: ∂x χ = x; φˆ = 1; c = v; vˆ = 0; J = det=1 ∂X (7.4.21a)and they are given by:{}ρ E,t[χ ] + E, j v j = (σ ij vi ),j + b j v j + (kij θ, j ),i + ρs(7.4.21b)or{}ρ E,t[χ ] + v ⋅ grad E = div(v ⋅σ ) + v ⋅ b + div(k ⋅ grad θ ) + ρs(7.4.21c)Exercise 7.4Show Eqs (7.13.10a), (7.13.10c), and (7.13.10d).Exercise 7.5 Galerkin ApproximationShow the following Galerkin approximation by substituting these approximationfunctions, Eqs (7.13.12), into Eqs.

(7.13.10).Exercise 7.6 The Continuity Equation(6a) Show that:˙ + Lp (P) + GT v = f extpMp P(7.13.15a)W.K.Liu, Chapter 771where M p is the generalized mass matrices for pressure; Lp is the generalized convectiveterms for pressure; G is the divergence operator matrix; f extp is the external load vector; Pand v are the vectors of unknown nodal values for pressure and velocity, respectively; andP˙ is the time derivative of the pressure.(6b) Show that:PMAB=∫ΩeLPA = ∫ΩePGAB=∫Ωe1 p pN N dΩB A B(7.13.15b)1 p ∂pN cdΩB A k ∂xk(7.13.15c)N Ap∂NBdΩ∂x m(7.13.15d)Example 7.2 1D Advection-Diffusion Equation2Pe φ,x − φ ,xx = 0Pe = 1.5τ = 0.438∆x = 1Exercise 7.7 The Momentum Equation(7a) Show that:Ma + L(v) + K µ v − GP = f extv(7.13.16a)where M is the generalized mass matrices for velocity; L is the generalized convectiveterms for velocity; G is the divergence operator matrix; f extv is the external load vectorapplied on the fluid; K µ is the fluid viscosity matrix; P and v are the vectors of unknownnodal values for pressure and velocity, respectively; and P˙ and a are the time derivative ofthe pressure, and the material velocity, holding the reference fixed.(7b) Show that:MAB = ∫ΩeLA = ∫ΩeρNA NB dΩρN A cm∂vidΩ∂xm(7.13.16b)(7.13.16c)W.K.Liu, Chapter 7Kµ = ∫Ωe72BT DB dΩ(7.13.16d)whereB = [B1 LBa LB NEN ] ∂Na ∂x1BTa = 0 02µ00D=0000µ0000(7.13.17a)∂Na∂x 2∂Na∂x100∂Na∂x2000002µ0000002µ00∂N a∂x30000µ00∂Na∂x3∂Na∂x 200000µ∂Na ∂x 3 0∂Na ∂x1 (7.13.17b)(7.13.17c)Exercise 7.8 The Mesh Updating Equation(8a) Show that:ˆ vˆ + Lˆ (x) − Mˆ v = f extxM(7.13.18a)ˆ is the generalized mass matrices for mesh velocity; Lˆ is the generalizedwhere Mextxconvective terms for mesh velocity; fis the external load vector; and vˆ is the vectors ofunknown nodal values for mesh velocity.(8b) Show that:ˆMˆ AB = ∫ ˆ e ρ Nˆ A Nˆ B dΩΩ(7.13.18b)The convective term is defined as follows:(i) Lagrangian-Eulerian Matrix Method:Define:cˆi = (δ ij − α ij )v j(8c) Show that the convective term is:(7.13.19a)W.K.Liu, Chapter 773∂xi ˆLˆ A = ∫ ˆ e Nˆ A cˆmdΩΩ∂χ m(7.13.19b)Exercise 6δvi, show that the streamlineδx jupwind/Petrov-Galerkin formulation for the momentum equation is:Replacing the test function δvi by δvi + τρc j0=∫Ω+∫Ω+δvi ρ∂vi∂vdΩ + ∫ δvi ρc j i dΩ−Ω∂t χ∂x j∫Ωµ  ∂(δvi ) ∂(δv j )   ∂vi ∂v j ++dΩ−2  ∂x j∂xi   ∂x j ∂x i NUMEL∑ ∫Ωe =1eτ ρc jδviδx j ∂viρ ∂tχ+ ρc j∫Γ∂(δvi )PdΩ−∂x i∫Ωδvi hj dΓ⇐ Galerkin∂vi ∂σ ij−− ρgi  dΩ∂x j ∂x jxδvi ρgi dΩ⇐ StreamlineUpwindReferences:Belytschko, T.

and Liu, W.K. (1985), "Computer Methods for Transient Fluid-StructureAnalysis of Nuclear Reactors," Nuclear Safety, Volume 26, pp. 14-31.Bird, R.B., Amstrong, R.C., and Hassager, 0. (1977), Dynamics of Polymeric Liquids,Volume 1: Fluid Mechanics, John Wiley and Sons, 458 pages.Huerta, A. (1987), Numerical Modeling of Slurry Mechanics Problems. Ph.D Dissertationof Northwestern University.Hughes, T.J.R., Liu, W.K., and Zimmerman, T.K. (1981), "LagrangianEulerian FiniteElement Formulation for Incompressible Viscous Flows”, Computer Methods in AppliedMechanics and Engineering, Volume 29, pp. 329-349.Hughes, T.J.R., and Mallet, M.

(1986), "A New Finite Element Formulation forComputational Fluid Dynamics: III.The Generalized Streamline Operator forMultidimensional AdvectiveDiffusive Systems," Computer Methods in AppliedMechanics and Engineering, Volume 58, pp. 305-328.Hughes, T.J.R., and Tezduyar, T.E. (1984), "Finite Element Methods for First-OrderHyperbolic Systems with Particular Emphasis on the Compressible Euler Equations”,Computer Methods in Applied Mechanics and Engineering, Volume 45, pp. 217-284.W.K.Liu, Chapter 774Hutter, K., and Vulliet, L. (1985), 'Gravity-Driven Slow Creeping Flow of aThermoviscous Body at Elevated Temperatures," Journal of Thermal Stresses, Volume 8,pp. 99-138.Liu, W.K., and Chang, H.

G. (1984), "Efficient Computational Procedures for LongTime Duration Fluid-Structure Interaction Problems," Journal of Pressure VesselTechnology, Volume 106, pp. 317-322.Liu, W.K., Lam, D., and Belytschko, T. (1984), "Finite Element Method forHydrodynamic Mass with Nonstationary Fluid," Computer Methods in AppliedMechanics and Engineering, Volume 44, pp. 177-211.Lohner, R., Morgan, K., and Zienkiewicz, O.C. (1984), "The Solution of NonlinearHyperbolic Equations Systems by the Finite Element Method," International Journal forNumerical Methods in Fluids, Volume 4, pp.

1043-1063.Belytschko, T. and Kennedy, J.M.(1978), ‘Computer models for subassemblysimulation’, Nucl. Engrg. Design, 49, 17-38.Liu, W.K. and Ma, D.C.(1982), ‘Computer implementation aspects for fluid-structureinteraction problems’, Comput. Methos. Appl. Mech. Engrg., 31, 129-148.Brooks, A.N. and Hughes, T.J.R.(1982), ‘Streamline upwind/Petrov-Galerkinformulations for convection dominated flows with particular emphasis on theincompressible Navier-Stokes equations’, Comput. Meths. Appl. Mech. Engrg., 32, 199259.Lohner, R., Morgan, K. and Zienkiewicz, O.C.(1984), ‘The solution of non-linearhyperbolic equations systems by the finite element method’, Int. J. Numer.

Meths. Fluids,4, 1043-1063.Liu, W.K.(1981) ‘Finite element procedures for fluid-structure interactions withapplication to liquid storage tanks’, Nucl. Engrg. Design, 65, 221-238.Liu, W.K. and Chang, H.(1985), ‘A method of computation for fluid structureinteractions’, Comput. & Structures, 20, 311-320.Hughes, T.J.R., and Liu, W.K.(1978), ‘Implicit-explicit finite elements in transientanalysis’, J.

Appl.Mech., 45, 371-378.Liu, W.K., Belytschko, T. And Chang, H.(1986), ‘An arbitrary Lagrangian-Eulerianfinite element method for path-dependent materials’, Comput. Meths. Appl. Mech. Engrg.,58, 227-246.Liu, W.K., Ong, J.S., and Uras, R.A.(1985), ‘Finite element stabilization matricesaunification approach’, Comput.

Meths. Appl. Mech. Engrg., 53, 13-46.Belytschko, T., Ong,S.-J, Liu, W.K., and Kennedy, J.M.(1984), ‘Hourglass control inlinear and nonlinear problems’, Comput. Meths. Appl. Mech. Engrg., 43, 251-276.W.K.Liu, Chapter 775Liu, W.K., Chang, H, Chen, J-S, and Belytschko, T.(1988), ‘Arbitrary LagrangianEulerian Petrov-Galerkin finite elements for nonlinear continua’, Comput. Meths. Appl.Mech.

Engrg., 68, 259-310.Benson, D.J.,(1989), ‘An efficient, accurate simple ALE method for nonlinear finiteelement programs’, Comput. Meths. Appl. Mech. Engrg, 72 205-350.Huerta, A. & Casadei, F.(1994), “New ALE applications in non-linear fast-transient soliddynamics”, Engineering Computations, 11, 317-345.Huerta, A. & Liu, W.K. (1988), “Viscous flow with large free surface motion”, ComputerMethods in Applied Mechanis and Engineering, 69, 277-324.W.K.Liu, Chapter 776---------- Backup of the previous version ------------In section 7.1, a brief introduction of the ALE is given. In section 7.2, the kinematics inALE formulation is described.

In section 7.3, the Lagrangian versus referential updates isgiven. In section 7.4, the updated ALE balance laws in referential description is described.In section 7.5, the strong form of updated ALE conservation laws in referenctialdescription is derived. In section 7.6, an example of dam-break is used to show theapplication of updated ALE. In section 7.7, the updated ALE is applied to the pathdependent materials extensively where the strong form, the weak form and the finiteelement decretization are derived. In this section, emphasize is focused on the stressupdate procedure.

Formulations for regular Galerkin method, Streamline-upwind/PetrovGalerkin(SUPG) method and operator splitting method are derived respectively. All thepath-dependent state variables are updated with a similar procedure. In addition, the stressupdate procedures in 1D case are specified with the elastic and elastic-plastic wavepropagation examples to demonstrate the effectiveness of the ALE method. In section 7.8,the total ALE method, the counterpart of updated ALE method, is studied.7.1 IntroductionThe theory of continuum mechanics (Malvern [1969], Oden [1972]) serves to establishan idealization and a mathematical formulation for the physical responses of a materialbody which is subjected to a variety of external conditions such as thermal and mechanicalloads.

Характеристики

Список файлов книги