Belytschko T. - Introduction (779635), страница 74
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A schematic diagramrepresenting these descriptions is shown in Fig. 7.1 and a summary of the kinematics for ageneral ALE formulation with Lagrangian and Eulerian formulations shown as specialcases is given in Table 7.1.Spatial Domain Ω•xˆf ( c , t )f( X , t )•χˆReference Domain Ω•y( X , t )XMaterial Domain Ω 0Fig 7.1 Mappings between Lagrangian, Eulerian, ALE descriptionsDescriptionMaterialMotionMeshDisplacementVelocityAccelerationGeneral ALEx = f( X,t)x = ˆf ( c ,t)MaterialMeshMaterialu = x− Xˆ = x− cuv = u,t[ X ]Lagrangianx = f( X,t)x = f( X,t)( c = X, fˆ = f )u = x− Xˆu = x − X = uv = u,t[ X ]Eulerianx = f( X,t)x = I(x)( c = x, ˆf = I )u = x− Xˆu = x − x = 0v = u,t[ X ]Meshvˆ = uˆ ,t[ χ]vˆ = uˆ ,t[ X] = vvˆ = uˆ ,t[ x] = 0Materiala = v,t[ X]a = v,t[ X]a = v,t[ X]W.K.Liu, Chapter 76aˆ = vˆ ,t[χ ]aˆ = vˆ ,t[ X] = aaˆ = vˆ ,t[x] = 0Table 7.1 Kinematics for a general ALE formulation with Lagrangian and Eulerianformulations shown as special cases.Mesh7.2.2 Time DerivativesIn the balance laws, the material time derivative of a function appears.
For a givenscalar function, f = f (x,t) = F(X,t), the material time derivative, and the spatialderivative or spatial gradient of f which appears in continuum conservation laws aredefined as:D∂F(X,t )f˙ =F(X,t) == f (x(X,t ),t),t [X ]Dt∂t [X ](7.2.14a)∂f≡ f ,i or grad x f ≡ grad f∂xi(7.2.14b)andrespectively.
The subscript x denotes partial differentiation with respect to x. These twoimportant shorthand notations will be used subsequently.7.2.3 Convective VelocityAlthough functions f(x, t) are usually given in terms of x and t, it is convenient inALE mechanics to express the function f in terms of c in the finite element formulationsince the initial input coordinates c are fixed in the finite element mesh.In general, by composition of mapping, f can be expressed as a function of X and t,denoted by F; a function of x and t, denoted by f; or a function of c and t, denoted by fˆ .That isf = F( X,t) = f( x, t) = ˆf ( c ,t)(7.2.15)These are different functions, which represent the same field.
The material time derivativecan be expressed for the different descriptions as follow:Df= F˙ = material time derivative= F,t[X ] (X,t)Dt= f ,t[ x] +∂f ∂xi= f ,t[ x] + f ,i vi∂xi ∂t [ X ](X , t)(7.2.16a)(x, t)(7.2.16b)W.K.Liu, Chapter 77∂fˆ ∂χ i∂ ˆf= fˆ,t[χ ] += ˆf ,t[ χ] +w∂χ i ∂t [ X]∂χ i i( c , t)(7.2.16c)where wi is the particle velocity in the referential coordinates and may be defined explicitelyaswi =∂χi∂t(7.2.16d)[ X]. In Eq. (7.2.16c), the variable c is not defined explicitly in terms of X and t through thecomponents of x, but is given in terms of the material motion f and also of the meshmotion ˆf .
That is:x j = φ j ( X, t) = φˆ j ( c ,t)(7.2.17)Differentiating with respect to time while holding X fixed gives:x j,t[X] = v j =∂ φˆ j∂t+[ χ]∂ φˆ j ∂χ i∂χ i ∂t= vˆ j +[X]∂x j ∂χ i∂χ i ∂t [ X](7.2.18)The second term on the right hand side can be rearranged to yield:∂x j ∂χ i∂x j=w = v j − vˆ j ≡ c j∂χ i ∂t [ X] ∂χ i i(7.2.19)where c j are the components of the convective velocity c. Applying the chain rule to Eq.(7.2.16c) and employing Eq. (7.2.19) yields:Df∂f ∂x j ∂χi= F˙ = fˆ,t[ χ] += fˆ,t[χ ] + f , j c jDt∂x j ∂χ i ∂t [ X](7.2.20a)or in vector notation:Df= ˆf ,t[ χ] + c ⋅ grad f = fˆ,t[χ ] + c ⋅ ∇ x fDt(7.2.20b)It can be shown that Eq.
(7.2.20a) reduces to Eqs. (7.2.16a) and (7.2.16b) whenc = X( c = 0) and c = x( c = v) , respectively. The former is known as the Lagrangiandescription, whereas the latter is the Eulerian description. Equation (7.2.20) is the materialtime derivative of f in a referential (i.e., ALE) description.W.K.Liu, Chapter 78ExampleThe comparison of the Lagrangian, Eulerian, and ALE descriptions is pictorially depictedin Fig. 7.2 by a 4-node one dimensional finite element mesh. The finite element nodes andthe material points are denoted by circles( ) and solid dots( ), respectively. Thenormalized coordinates are: X1 = 0, X2 = 1, X3 = 2 , and X 4 = 3; and normalied time isbetween 0 and 1.
In Chapter 3, the Lagrangian and Eulerian descriptions were described asshown in Figs. 7.2(a) and (c). To illustrate the ALE description, as shown in Fig. 7.2(b),the motion of the material points is described by:x = φ (X,t) = (1− X 2 )t + X t 2 + X(7.2.13a)In order to regulate the mesh motion, the four mesh nodes are spaced uniformly based onthe end points of the material motion, that is, φ (X1 ,t) and φ (X4 ,t). Therefore, the meshmotion can be described by a linear Lagrange polynomial:χ − χ1χ − χ4x = φˆ (χ,t) =φ (X1 ,t) +φ (X4 ,t)χ 4 − χ1χ1 − χ 4(7.2.13b)Combining Eqs.
(7.2.13a) and (b) yields :x=χ − χ1χ − χ4(1 − X12 ) t + X1( t 2 +1) ] +(1 − X42 ) t + X4 ( t 2 +1) ][[χ4 − χ 1χ1 − χ4Therefore, we have:material displacement:u = x − X = (1− X 2 )t + Xt 2 + X − X = (1 + X − X 2 ) tmaterial velocity:v=∂u∂t=( 1 + X − X2 )Xmaterial acceleration:a=∂v∂t=0Xmesh displacement:χ − χ1χ − χ4uˆ = x − χ =( 1− X12 )t + X1( t 2 +1) ] +(1 − X4 2 )t + X 4 ( t 2 + 1) ] − χ[[χ 4 − χ1χ1 − χ4mesh velocity:∂ uˆχ − χ1χ − χ4vˆ ==(1 − X12 ) + 2X1t ] +( 1− X 42 ) + 2X4 t ][[∂t χ χ 4 − χ1χ1 − χ 4mesh acceleration:W.K.Liu, Chapter 79∂vˆ2X (χ − χ1 ) 2X4 (χ − χ 4 )aˆ == 1+∂t χχ4 − χ1χ1 − χ 4The ALE mapping from the material domain to the reference domain is given by:χ = ψ( X, t) =[(1 − X 2 )t + X( t2 +1)]( χ4 − χ1 ) + χ1φ( X1, t) − χ4 φ( X4 ,t)φ( X1 ,t) − φ( X4 ,t)The particle velocity and acceleration in the referential coordinates may then be computedusing Eq.
(7.2.16d) and its time deravitive, respectively.Comparing the two motions above, even though both motions give the same range of x, thetwo mappings are quite different as shown in Eqs. (7.2.13a,b) and Fig.7.2b.t(a) Lagrangian Descriptiont(b) ALE Descriptiont1t1x, X, χx, X, χ00X1X2X3X1X4t (c) Eulerian DescriptionX3X4NodesMaterial Pointst1x, X, χNodal TrajectoryMaterial Point Trajectory0X1X2X2X3X4Fig 7.2 Comparison of Lagrangian, Eulerian, and ALE descriptions7.4 Updated ALE Balance Laws in Referential DescriptionTo derive the updated ALE balance laws analogous to those of the Lagrangiandescription, it is convenient to first use the Lagrangian equations given in Chapter 3 andthen apply Eq. (7.2.20) to the material time derivatives to obtain the ALE conservationlaws. Consequently, the only difference between the updated Lagrangian and updated ALEformulations is in the material time derivative terms.
For completeness, the total ALEformulations are given in Appendix 7.1.W.K.Liu, Chapter 7107.4.1 Conservation of Mass (Equation of Continuity) in ALEThe continuity equation is given by:ρ˙ + ρv j,j = 0(7.4.1)Applying the material time derivative operator Eqs. (7.2.20) to Eq. (7.4.1), the continuityequation becomes:ρ,t[χ ] + ρ, j c j + ρv j, j = 0(7.4.2a)or in vector form:ρ,t[χ ] + c ⋅ grad ρ + ρ∇⋅ v = 0(7.4.2b)where ∇⋅ v is the divergence of v in index free notation.An alternate way of deriving the continuity equation is to employ the Reynolds transporttheorem (given in Chapter 3) and using the divergence theorem to give: ∂ρ∫Ω ∂t+x∂(ρvi ) dΩ = 0∂xi (7.7.14b)Assuming there are no discontinuities in the linear momentum, an application of the chainrule yields∫Ω ∂ρ∂ρ∂vi +v+ρdΩ = 0i ∂t x∂xi∂xi (7.7.14c)Observing that the first two terms yield the material time derivative of ρ and hence usingEq.
(7.2.20), Eq. (7.7.14c) becomes:∫Ω ∂ρ∂ρ∂vi +c+ρdΩ = 0i ∂t∂xi∂xi χ(7.7.14d)and since Ω is arbitrarily chosen, it follows that:∂ρ∂ρ∂v+ ci+ρ i = 0in Ω(7.7.15)∂t χ∂xi∂x iwhich is identical to Eq.(7.4.2). It is noted that if there is a discontinuity, we cannot applythe chain rule to the linear momentum since there is a jump in ρvi hence we have to employthe conservative form, Eq.(7.7.14b) instead of the non-conservative form, Eq.(7.4.1)W.K.Liu, Chapter 7117.4.2 Conservation of Linear Momentum in ALEThe conservative form of the momentum equation is given as:D(ρ vi ) + (ρ vi )v j,j = σ ji, j + ρbiDt(7.4.6a)It was shown in Chapter 3 that if there are no discontinuities, then the non-conservativeform of the momentum equation can be obtained by applying the chain rule to ρvi . With thehelp of the continuity equation, Eq (7.4.1), we obtainρ v˙i = σ ji, j + ρbiSimilarly, after applying the material time derivative operator Eqs.(7.2.20) to Eq.(7.4.6a),the momentum equation becomes:{}ρ vi,t[ χ] + c j vi, j = σ ji , j + ρbi(7.4.8a)or, in index free notation:{}ρ v, t [ χ ] + c⋅ grad v = div( σ) + ρb(7.4.8b)It is a simple exercise that by applying the material time derivative operator to the energyequation derived in Chapter 3, and show that the non-conservative form of the energyequation is:ρ E˙ = (vi σ ij ),j + bi vi + (kij θ, j ),i + ρs7.5 Formal Statement of the Updated ALE Governing Equations in NonConservative Form (Strong Form) in Referential DescriptionIn the equations given below, (7.6.2), kij and vi are the components of the thermalconductivity matrix and convective heat transfer coefficients, respectively; θ 0 is theambient temperature; bi are the components of the body force; and s is the heat source.
Theobjective of the initial/boundary-value problem is to find the following functions:andu(X , t)material displacement(7.6.1a)σ (x, t)Cauchy stress tensor(7.6.1b)θ(x, t)thermodynamic temperature(7.6.1c)ˆ ( c , t)umesh displacement(7.6.1d)W.K.Liu, Chapter 712ρ(x, t)density(7.6.1e)such that they satisfy the following field and state equations shown in Box 7.1:Strong FormDescriptionofUpdatedALEGoverningEquationsinReferentialContinuity Equationρ˙ + ρv k,k = 0orρ,t[χ ] + ρ,i ci + ρv k,k = 0(7.6.2a)orρ (vi,t[χ ] + vi, j c j ) = σ ji , j + ρbi(7.6.2b)Momentum Equationsρ v˙i = σ ji ,j + ρbiEnergy Equationρ E˙ = (vi σ ij ),j + bi vi + (kij θ, j ),i + ρsorρ(E,t[χ ] + E,i ci ) = (vi σij ), j + bi vi + (kij θ, j ),i + ρs(7.6.2c)Equations of Statesupplemented by the constitutive equations given in Chapter 6.Natural Boundary Conditionsti (x,t) = n j (x,t)σ ji (x,t)on ∂Γ tx(7.6.2g)qi (x,t) = −kij (θ,x,t )θ , j (x,t) + vi (θ,t)(θ − θ 0 )on ∂Γ tθx(7.6.2h)on ∂Γ gx(7.6.2i)Essential Boundary Conditionsui (x,t) = ui (x,t)θ(x,t) = θ (x,t)on∂Γ gθx(7.6.2j)Initial Conditionsˆ0u( X,0) = u0 , ˆu( c , 0) = uv( X, 0) = v0 , ˆv( c , 0) = ˆv0(7.6.2k)(7.6.2l)Mesh Motionˆ ( c , t) = a given representation except, perhaps, on part of the boundary.uW.K.Liu, Chapter 713(7.6.2m)Box 7.1 Strong Form of Updated ALE Governing Equations in Referential DescriptionPrior to developing the weak form and Petrov-Galerkin finite element discretization of theALE continuity and momentum equations outlined above, it is most instructive to digressbriefly and formally acquaint the reader with the general Petrov-Galerkin method.
In doingso, it hoped that the necessity and power of such an approach will become apparent andthat an intuitive feeling for the physics involved will be brought to light. The followingsection is designed to fulfill this requirement after which our development of the ALEequations will resume.7.14 Introduction to the Petrov-Galerkin MethodIn this section, streamline upwinding by a Petrov Galerkin method(SUPG) isformulated. Prior to the development of this method, the need for an upwinding schemewas motivated by an examination of the classical advection-diffusion equation. Theadvection-diffusion equation is a useful model for studying the momentum since itcorresponds to a linearization of the transport equation.
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