Belytschko T. - Introduction (779635), страница 73
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The norm in the Hilbert space H 1 is defined by()a xH1()()1 1 2= ∫0 a 2 x + a,2x x dx Just as for vector norms, the major utility of these norms is in measuring errors infunctions. Thus if the finite element solution for the displacement in a onedimensional problem is denoted by uh( x ) and the exact solution is u( x ) , thenthe error in the displacement can be measured byerror= u h( x ) −u( x )L2The error in the strain, i.e. the first derivative of the displacement, can bemeasured by the H 1 norm. While this norm also includes the error in thefunction itself, the error in the derivative almost always dominates.
On the otherhand, you could measure the error in the strain by the L 2 norm of the first6-79T. Belytschko & B. Moran, Solution Methods, December 16, 1998derivative. This is not a valid norm in mathematics, because it can vanish for anonzero function (just take a constant), so it is called a seminorm.These norms can be generalized to arbitrary domains in multi-dimensional spaceand to vector and tensors by just changing the integrals and integrands.
Thus theL 2 norm of the displacement on a domain is given by()uxL212= ∫ ui x ui x dΩΩ()()The definition of the H 1 norm is somewhat more puzzling??? since as given inmathematical tests it is not a true scalar (it is not invariant with rotation):()uxH112= ∫ ui x ui x + ui, j x ui , j x dΩΩ()()() ()In general, the precise space to which a norm pertains is not given.
Usually only anumber, or even nothing is given by the norm sign. The norm must then beinferred from the context.In linear stress analysis, the energy norm is often used to measure error. It isgiven by12energy norm= ∫ εij x Cijklε kl x dΩ Ω()()Its behavior is similar to that of the H 1 norm.6-80W.K.Liu, Chapter 71CHAPTER 7Arbitrary Lagrangian Eulerian Formulationsby W.K.LiuNorthwestern University@ Copyright 19977.1 IntroductionIn Chapter 3, the classical Lagrangian and Eulerian approaches to the description of motionin continuum mechanics were presented. In the Lagrangian approach, the independentvariables are taken to be the initial position, X , of a material point and time, t.
Thus themotion is given byx = f( X,t)`(7.1.1)In this expression, the quantity x is the position occupied at time t by the material pointwhich occupied the position X at time t=0. The quantity f is a mapping which describesthe motion in terms of the independent variables X and t, and x is the value of themapping for the values X and t. Recall that, in the Lagrangian description, the distinctionbetween the value x and the mapping f is often ignored and we write x = x(X,t). Ascalar field F, for example, may be represented byF = F(X,t)(7.1.2)In the Eulerian description, the independent variables are spatial position x and time t.
Ascalar field in the Eulerian description may then be given byf = f (x,t)(7.1.4)The field can be represented in terms of either the Eulerian or Lagrangian coordinates asfollowsf ( x,t) = f ( f( X,t), t) = F( X, t)(7.1.5)but in fluid mechanics, the mapping f may not be known and this interpretation is notparticularly useful.In Chapter 4, Lagrangian finite elements were discussed.
In Lagrangian finite elementimplementations, the finite element mesh convects with the material. The advantages ofLagrangian finite elements include the ease of tracking material interfaces and boundaries aswell as the more straight-forward treatment of constitutive equations. Among thedisadvantages of a Lagrangian formultation include the severe distortions that the elementsmay undergo as they deform with the material resulting in a deterioration of performancedue to ill-conditioning.
Nevertheless, Lagrangian finite elements prove extremely useful inlarge deformation problems in solid mechanics and are most widely used in solidmechanics. Eulerian finite elements are most often used in fluid mechanics for the samereasons that Eulerian representations of the equations of continuum mechanics are used,W.K.Liu, Chapter 72i.e., there is often no well-defined reference configuration and the motion from a referenceconfiguration is often not known explicitly. In Eulerian finite elements, the elements arefixed in space and material convects through the elements. Eulerian finite elements thusundergo no distortion due to material motion; however the treatment of constitutiveequations and updates is complicated due to the convection of material through theelements.
Eulerian elements may also lack resolution in the most highly deforming regionsof the body.The aim of ALE finite element formulations is to capture the advantages of both Lagrangianand Eulerian finite elements while minimizing the disadvantages. As the name suggests,ALE formulations are based on a description of the equations of continuum mechanicswhich is an arbitrary combination of the Lagrangian and Eulerian descriptions. The wordarbitrary here means that the description (or specific combination of Lagrangian andEulerian character) may be specified freely by the user.
Of course, a judicious choice of theALE motion is required if severe mesh distortions are to be eliminated. Suitable choices ofthe ALE motion will be discussed. Before introducing the ALE finite element formulation,it is useful to first consider some preliminary topics in continuum mechanics which werenot covered in Chapter 3 and which provide the basis for the subsequent finite elementimplementation of the ALE methodology.7.2 ALE Continuum Mechanics7.2.1 Mesh Displacement, Mesh Velocity, and Mesh AccelerationIn figure (7.1), the motion x = f( X,t) is indicated as a mapping of the body from thereference configuration Ω 0 to the current or spatial configuration Ω .
To introduce the ALEˆ as shown. We note thatformulation, we now consider an alternative reference region Ωthis region need not be an actual configuration of the body. Our objective is to show howthe governing equations and kinematics for the body may be referred to this referenceconfiguration and then how to use this description to formulate the ALE finite elements.ˆ , are mapped to points x in the spatial region, Ω viaPoints c in the reference region, Ωthe mappingx = ˆf ( c ,t)(7.2.6)This mapping ˆf will ultimately play an important role in the ALE finite elementformulation. At this point, it is regarded as an arbitrary mapping (although it will beˆ to the region Ω .
The left hand side of (7.2.6)assumed to be invertible) of the region Ωgives the mapping ˆf as a function of c and t. By virtue of (7.2.6), and (7.1.1), we havex = ˆf ( c ,t) = f ( X,t)(7.2.7)which states that x in the Eulerian representation, c in the ALE representation, and X inthe Lagrangian representation are mapped into x(spatial coordinates) at time t. It is notedthat even though the ALE mapping ˆf is different from the material mapping f , the spatialcoordinates x are the same.W.K.Liu, Chapter 73In particular, if c is chosen to be the Lagrangian coordinate X, ˆf becomes the materialmapping f so that Eq.(7.2.7) becomes Eq.(7.1.1).
A natural question arises: what is theALE mapping ˆf if c is chosen to be the spatial coordinate x ? In this situation it is intuitiveto think that Eq.(7.2.6) becomes:x = ˆf ( x,t )(7.2.8)Therefore, ˆf is an identity mapping and it is not a function of time. As a result, we maydefine the material and mesh velocities in the spatial coordinate form:v( x,t) =∂f ( X, t)∂t(7.2.9a)Xandv( x,t ) =∂ˆf ( c ,t)∂t(7.2.9b)cIt is noted that the right hand sides of Eqs.(7.2.9) are simply the definitions of material andmesh velocities, whereas the complete knowledge of the functions of the material and meshvelocites are often the solutions to the ALE continuum conservation equations.
It is alsounderstood that the mesh velocity, v(x,t), is equal to zero for an Eulerian description. Wenow assume that the two velocity equations are given so that with the definitions of thematerial motion, Eq.(7.1.1), and the mesh motion, Eq.(7.2.6), a set of first orderboundary value equations are obtained:∂f( X, t)∂t= v( f( X,t), t)(7.2.10a)= v( fˆ ( c ,t), t)(7.2.10b)Xand∂ˆf ( c , t)∂tχThe objective of Eqs.(7.2.10) is: given the material velocity function v(x,t), and the meshvelocity function v(x,t), find the material mapping f( X, t) and the ALE mapping ˆf ( c , t)such that Eqs.(7.2.10) are satisfied with the following initial conditions :f( X, 0) = X 0(7.2.11a)ˆf ( c , 0) = X0(7.2.11b)andWith the stated initial boundary value problem, the above raised questions regarding theALE mapping ˆf when c is chosen to be x can be answered by choosing c = x ( anEulerian description, implying v( x,t ) = 0 ), so thatW.K.Liu, Chapter 74∂ˆf ( x, t )∂I( x)v( ˆf ( c ,t), t) ==∂t x∂t=0(7.2.12)xHence, Eq.(7.2.10b) becomes∂ˆf ( c , t)∂t=c∂fˆ ( x,t)∂t=0(7.2.13)xand therefore,ˆf ( χ,t) = constant,is determined from the initial conditions.
By choosing x = X0 , Eq.(7.2.8) becomes theidentity mapping so thatx = ˆf ( x,t ) = I( x)(7.2.14)Thus ˆf is indeed an identity mapping when c = x .In the finite element implementation of the ALE formulation, a mesh is defined with respectˆ . The motion ˆf ( c , t) is used to describe the motion of the meshto the configuration Ωand, as mentioned earlier, is chosen so as to reduce the effects of mesh distortion.
For thisreason, we also refer to ˆf ( c , t) as the mesh motion. In this sense, we introduce the meshˆ throughdisplacement, uˆ , for points in Ωˆ ( c , t)x = ˆf ( c ,t) = c + u(7.2.8)Consistent with this terminology, we also introduce the mesh velocity and accelerationˆ as followsfields for points in Ω∂ˆf ( c ,t)vˆ =∂t=[c ]ˆ ( c , t)∂u∂t(7.2.9)[c]and∂ˆv( c ,t )aˆ =∂t=[c]∂ 2ˆu( c ,t)∂t 2(7.2.10)[c ]This expression for velocity could be written as∂xvˆ =∂t [ c ](7.2.11)However, in the interest of clarity in the ALE formulation, we refrain from this notation asit eliminates the distinction between the mapping or motion (in this case ˆf ) and the valueW.K.Liu, Chapter 75of the mapping, x.
For purely Eulerian or Lagrangian descriptions, however, thisdistinction in notation can be dropped with little loss of clarity.Referring to (7.2.7), it can be seen that points in the reference configuration can beidentified asc = y( X,t) = ˆf −1 o f ( X, t)(7.2.12)although we will have little occasion to use this relation. Instead, we will make use of thepreviously defined mappings and the chain rule where necessary.
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