Belytschko T. - Introduction (779635), страница 85
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Since a material body B defined as a continuum is a collection of material particlesp, the purpose of continuum mechanics is to provide governing equations which describethe deformations and motions of a continuum in space and time under thermal andmechanical disturbances.The mathematical model is achieved by labelling the points in the material body B bythe real number planes Ω , where Ω is the region (or domain) of the Euclidean space.Henceforth, the material body B is replaced by an idealized mathematical body, namely, theregion Ω . Instead of being interested in the atomistic view of the particles p, thedescription of the behavior of the body B will only pertain to the regions of Euclideanspace .Equations describing the behavior of a continuum can generally be divided into fourmajor categories: (1) kinematic, (2) kinetic (balance laws), (3) thermodynamic, and (4)constitutive.
Detailed treatments of these subjects can be found in many standard texts.The two classical descriptions of motion, are the Lagrangian and Eulerian descriptions.Neither is adequate for many engineering problems involving finite deformation especiallywhen using finite element methods. Typical examples of these are fluid-structure-solidinteraction problems, free-surface flow and moving boundary problems, metal formingprocesses and penetration mechanics, among others.Therefore, one of the important ingredients in the development of finite elementmethods for nonlinear mechanics involves the choice of a suitable kinematic description foreach particular problem.
In solid mechanics, the Lagrangian description is employedextensively for finite deformation and finite rotation analyses. In this description, thecalculations follow the motion of the material and the finite element mesh coincides with theW.K.Liu, Chapter 777same set of material points throughout the computation. Consequently, there is no materialmotion relative to the convected mesh. This method has its popularity because(1) the governing equations are simple due to the absence of convective effects, and(2) the material properties, boundary conditions, stress and strain states can be accuratelydefined since the material points coincide with finite element mesh and quadrature pointsthroughout the deformation. However, when large distortions occur, there aredisadvantages such as:(1) the meshes become entangled and the resulting shapes may yield negative volumes,and(2) the time step size is progressively reduced for explicit time-stepping calculations.On the other hand, the Eulerian description is preferred when it is convenient to modela fixed region in space for situations which may involve large flows, large distortions, andmixing of materials.
However, convective effects arise because of the relative motionbetween the flow of material and the fixed mesh, and these introduce numerical difficulties.Furthermore, the material interfaces and boundaries may move through the mesh whichrequires special attention.In this chapter, a general theory of the Arbitrary Lagrangian-Eulerian (ALE)description is derived. The theory can be used to develop an Eulerian description also. Thedefinitions of convective velocity and referential or mesh time derivatives are given.
Thebalance laws, such as conservation of mass, balances of linear and angular momentum andconservation of energy are derived within the mixed Lagrangian-Eulerian concept. Thedegenerations of the mixed description to the two classical descriptions, Lagrangian andEulerian, are emphasized. The formal statement of the initial/boundary-value problem forthe ALE description is also discussed.7.2 Kinematics in ALE formulation7.2.1 Mesh Displacement, Mesh Velocity and Mesh AccelerationIn order to complete the referential description, it is necessary to define the referentialmotion; this motion is called the mesh motion in the finite element formulation.The motion of the body B, which occupies a reference region Ωχ , is given byx = φˆ (χ,t) = χ + uˆ (χ,t) = φ(X,t)(7.2.7)This ALE referential(mesh) region Ω χ is specified throughout and its motion is defined bythe mapping function φˆ such that the motion of χ ∈Ω at time t is denoted by χ ∈Ω andχχuˆ (χ, t) is the mesh displacement in the finite element formulation.
It is noted that eventhought in general the mesh function φˆ is different from the material function φ , the twomotions are the same as given in Eq.(7.2.7). The corresponding velocity (mesh velocity)and acceleration (mesh acceleration) are defined as :vˆ =∂x= x ,t[ χ] = uˆ ,t[ χ]∂t [ χ]mesh velocity(7.2.8a)W.K.Liu, Chapter 778and∂ vˆaˆ == vˆ ,t[χ ]∂t [ χ]mesh acceleration(7.2.8b)The motion φˆ is arbitrary and the usefulness of the referential descripation will depend onhow this motion is chosen.Depending on the choice of χ , we can obtain the Lagranginan description by setting χ = Xand φˆ = φ , the Eulerian description by setting χ = x , and the ALE description by settingφˆ ≠ φ .
The general referential description is referred to as Arbitrary Lagrangian-Eulerian(ALE) in the finite element formulation. In this description, the function φˆ must bespecified such that the mapping between x and χ is one to one. With this assumption andby the composition of the mapping (denoted by a circle), a third mapping is defined suchthatχ = ψ(X,t) = φˆ −1 oφ (X,t)(7.2.10)Similarly, for this motion displacement, velocity and acceleration variables can bedefined.However, this is not necessary. These displacement, velocity and accelerationvariables can instead be defined with the aid of the chain rule and the appropriate mappings.The schematic set up of these descriptions in one-dimension is shown in Fig. 1, and asummary of the three descriptions is given in Table 7.1.Fig.7.2 is shown to compare the three descriptions further, where the 1D motion of thematerial is specified as:x = (1− X 2 )t + X t 2 + XFig 7.2 Comparsion of Lagranian, Eulerian, ALE descriptionReferences:W.K.Liu, Chapter 779Fried, I., and Johnson, A.
R., [1988]. "A Note on Elastic Energy Density Function forLargely Deformed Compressible Rubber Solids," Computer Methods in AppliedMechanics and Engineering, 69, pp. 53-64.Hughes, T. J. R., [1987]. The Finite Element Method, Linear Static and Dynamic FiniteElement Analysis, Prentice-Hall.Malvern, L. E., [1969]. Introduction to the Mechanics of a Continuous Medium, PrenticeHall.Noble, B., [1969]. Applied Linear Algebra, Prentice-Hall.Oden, J. T., [1972].
Finite Elements of Nonlinear Continua, McGraw Hill.REFERENCESBelytschko, T., and Kennedy, J.M. (1978), "Computer Models for SubassemblySimulation," Nuclear Engineering Design, Volume 49, pp. 17-38.Belytschko, T., Kennedy, J.M., and Schoeberie, D.F. (1980), 'QuasiEulerian FiniteElement Formulation for Fluid Structure Interaction, 11 Journal of Pressure VesselTechnology, American Society of Mechanical Engineers, Volume 102, pp. 62-69.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.Brugnot, G., and Pochet, R. (1981), "Numerical Simulation Study of Avalanches,"Journal of Glaciology, Volume 27, Number 95, pp. 7788.Brooks, A.N., and Hughes, T.J.R. (1982), "Streamline Upwind/PetrovGalerkinFormulations for Convection Dominated Flows with Particular Emphasis on theIncompressible Navier-Stokes Equations, " Computer Methods in Applied Mechanics andEngineering-, Volume 32, pp. 199-259.Carey, G.F., and Oden, J.T.
(1986), Finite Elements: Fluid Mechanics, Volume VI of theTexas Finite Element Series, Prentice Hall, 323 pages.Chen, C., and Armbruster, J.T. (1980), "Dam-Break Wave Model: Formulation andVerification," Journal of the Hydraulics Division,American Society of Civil Engineers, Volume 106, Number HY5, pp.747-767.Donea, J. (1983), "Arbitrary Lagrangian-Eulerian Finite Element Methods," ComputationalMethods for Transient Analysis, Edited by T.
Belytschko and T.J.R. Hughes, ElvesierScience Publishers, pp. 473-516.Donea, J. (1984), "A Taylor-Galerkin Method for Convective Transport Problems,"International Journal for Numerical Methods in Engineering, Volume 20, pp. 101-119.W.K.Liu, Chapter 780Donea, J., Fasoli-Stella, P., and Giuliani, S. (1977), "Lagrangian and Eulerian FiniteElement Techniques for Transient Fluid Structure Interaction Problems," Transactions ofthe 4th International Conference on Structural Mechanics in Reactor Technology, PaperBl/2.Dressler, R.F. (1952), "Hydraulic Resistance Effect Upon the Dam-Break Functions,'Journal of Research of the National Bureau of Standards, Volume 49, Number 3, pp.
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