Belytschko T. - Introduction (779635), страница 4
Текст из файла (страница 4)
Equation (1.4.4) is called a Lagrangian (material) description for itexpresses the dependent variable in terms of the Lagrangian (material) coordinate.Equation (1.4.8) is called an Eulerian (spatial) description, for it expresses thedependent variable as a function of the Eulerian (spatial) coordinates. Henceforthin this book, we will not use different symbols for these different functions, butkeep in mind that if the same field variable is expressed in terms of differentindependent variables, then the functions must be different. In other words, asymbol for a dependent field variable is associated with the field, not the function.1-11T. Belytschko, Introduction, December 16, 1998tLagranian DescriptionB(t1)t1x, X0B(t = 0)tALE DescriptionB(t1)t1x, X0B(t = 0)tEulerian DescriptionB(t1)t1x, X0B(t = 0)NodalTrajectoryNodeMaterial PointTrajectoryMaterial PointFig.
1.1 Space time depiction of a one dimensional Lagrangian, Eulerian, and ALE (arbitraryLagrangian Eulerian) elements.1-12T. Belytschko, Introduction, December 16, 1998The differences between Lagrangian and Eulerian meshes are most clearlyseen in terms of the behavior of the nodes. If the mesh is Eulerian, the Euleriancoordinates of nodes are fixed, i.e. the nodes are coincident with spatial points. Ifthe mesh is Lagrangian, the Lagrangian (material) coordinates of nodes are timeinvariant, i.e. the nodes are coincident with material points. This is illustrated inFig.
1.1 for the mapping given by Eq. (1.4.3). In the Eulerian mesh, the nodaltrajectories are vertical lines and material points pass across element interfaces.In the Lagrangian mesh, nodal trajectories are coincident with material pointtrajectories, and no material passes between elements. Furthermore, elementquadrature points remain coincident with material points in Lagrangian meshes,whereas in Eulerian meshes the material point at a given quadrature point changeswith time. We will see later that this complicates the treatment of materials inwhich the stress is history-dependent.The comparative advantages of Eulerian and Lagrangian meshes can beseen even in this simple one-dimensional example.
Since the nodes are coincidentwith material points in the Lagrangian mesh, boundary nodes remain on theboundary throughout the evolution of the problem. This simplifies the impositionof boundary conditions in Lagrangian meshes. In Eulerian meshes, on the otherhand, boundary nodes do not remain coincident with the boundary. Therefore,boundary conditions must be imposed at points which are not nodes, and as weshall see later, this engenders significant complications in multi-dimensionalproblems. Similarly, if a node is placed on an interface between two materials, itremains on the interface in a Lagrangian mesh, but not in an Eulerian mesh.In Lagrangian meshes, since the material points remain coincident withmesh points, the elements deform with the material.
Therefore, elements in aLagrangian mesh can become severely distorted. This effect is apparent in a onedimensional problem only in the element lengths: in Eulerian meshes, the elementlength are constant in time, whereas in Lagrangian meshes, element lengthschange with time. In multi-dimensional problems, these effects are far moresevere, and elements can get very distorted. Since element accuracy degradeswith distortion, the magnitude of deformation that can be simulated with aLagrangian mesh is limited. Eulerian elements, on the other hand, are unchangedby the deformation of the material, so no degradation in accuracy occurs becauseof material deformation.To illustrate the differences between Eulerian and Lagrangian meshdescriptions in two dimensions, a two dimensional example will be considered.TIn two dimensions, the spatial coordinates are denoted by x = [ x , y] and thematerial coordinates by X = [X, Y]T .
The deformation mapping is given byx = φ(X, t )(1.4.9)where φ(X, t ) is a vector function, i.e. it gives a vector for every pair of theindependent variables. For every pair of material coordinates and time, thisfunction gives the pair of spatial coordinates corresponding to the current positionof the material particles. Writing out the above expression givesx = φ1( X , Y , t )(1.4.10)y = φ2 ( X , Y , t )1-13T.
Belytschko, Introduction, December 16, 1998As an example of a motion, consider pure shear in which the map is given byx = X + tYy=Y(1.4.11)Loriginal configurationdeformed configurationEFig. 1.2 Two dimensional shearing of a block showing Lagrangian (L) and Eulerian (E) elements.In a Lagrangian mesh, the nodes are coincident with material (Lagrangian)points, so the nodes remain coincident with material points, sofor Lagrangian nodes, X I =constant in timeFor an Eulerian mesh, the nodes are coincident with spatial (Eulerian) points, sowe can writefor Eulerian nodes, x I = constant in timePoints on the edges of elements behave similarly to the nodes: in Lagrangianmeshes, element edges remain coincident with material lines, whereas in Eulerianmeshes, the element edges remain fixed in space.To illustrate this statement we show Lagrangian and Eulerian meshes forthe shear deformation given by Eq.
(11) in Fig. 1.2. As can be seen from thefigure, a Lagrangian mesh is like an etching on the material: as the material isdeformed, the etching deforms with it. An Eulerian mesh is like an etching on asheet of glass held in front of the material: as the material deforms, the etching isunchanged and the material passes across it.The advantages and disadvantages of the two types of meshes are similarto those in one dimension.
In a Lagrangian mesh, element edges and nodes whichare initially on the boundary remain on the boundary, whereas in Eulerian meshesedges and nodes which are initially on the boundary do not remain on theboundary. Thus, in Lagrangian meshes, element edges (lines in two dimensions,surfaces in three dimensions) remain coincident with boundaries and materialinterfaces. In Eulerian meshes, element sides do not remain coincident withboundaries or material interfaces. Hence tracking methods or approximate1-14T. Belytschko, Introduction, December 16, 1998methods, such as volume of fluid approaches, have to be used for treating movingboundaries treated by Eulerian meshes; such as volume of fluid methodsdescribed in Section 5.?.
Furthermore, an Eulerian mesh must be large enough toenclose the material in its deformed state. On the other hand, since Lagrangianmeshes deform with the material, and they become distorted in the simulations ofsevere deformations. In Eulerian meshes, elements remain fixed in space, so theirshapes never change.A third type of mesh is an arbitrary Lagrangian Eulerian mesh, in whichthe nodes are programmed to move so that the advantages of both Lagrangian andEulerian meshes can be exploited. In this type of mesh, the nodes can beprogrammed to move arbitrarily, as shown in Fig.
1.1. Usually the nodes on theboundaries are moved to remain on the boundaries, while the interior nodes aremoved to minimize mesh distortion. This type of mesh is described and discussedfurther in Chapter 7.REFERENCEST. Belytschko (1976), Methods and Programs for Analysis of Fluid-StructureSystems," Nuclear Engineering and Design, 42 , 41-52.T. Belytschko and T.J.R. Hughes (1983), Computational Methods for TransientAnalysis, North-Holland, Amsterdam.K.-J. Bathe (1996), Finite Element Procedures, Prentice Hall, Englewood Cliffs,New Jersey.R.D.
Cook, D.S. Malkus, and M.E. Plesha (1989), Concepts and Applications ofFinite Element Analysis, 3rd ed., John Wiley.M.A. Crisfield (1991), Non-linear Finite Element Analysis of Solids andStructure, Vol. 1, Wiley, New York.T.J.R. Hughes (1987), The Finite Element Method, Linear Static and DynamicFinite Element Analysis, Prentice-Hall, New York.T.J.R.
Hughes (1996), personal communicationM. Kleiber (1989), Incremental Finite element Modeling in Non-linear SolidMechanics, Ellis Horwood Limited, John Wiley.J.T. Oden (1972), Finite elements of Nonlinear Continua, McGraw-Hill, NewYork.O.C. Zienkiewicz and R.L. Taylor (1991), The Finite Element Method, McGrawHill, New York.Z.-H. Zhong (1993), Finite Element Procedures for Contact-Impact Problems,Oxford University Press, New York.1-15T. Belytschko, Introduction, December 16, 1998GLOSSARY. NOTATIONVoigt Notation. In finite element implementations, Voigt notation is oftenuseful; in fact almost all linear finite element texts use Voigt notation. In Voigtnotation, second order tensors such as the stress, are written as column matrices,and fourth order tensors, such as the elastic coefficient matrix, are written assquare matrices. Voigt notation is quite awkward for the formulation of theequations of continuum mechanics.
![](https://s.studizba.com/z.php?f=/templates/si/i/noavatar.png&w=100&h=100&t=1"){kind=avatar}
