Belytschko T. - Introduction (779635), страница 38
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parent domain to current configuration: x = x ξ e , t( )2. parent domain to initial configuration: X = X(ξ e )3. initial configuration to the current configuration, i.e. the motion x = x (X,t ) ≡ φ( X, t)( )()The map X = X ξ e corresponds to x = x ξ e , 0 . These maps are illustrated in Fig. 4.1 for atriangular element where a space-time plot of a two-dimensional triangular element is shown.The motion in each element is described by a composition of these maps(x = x (X,t ) = x ξe (X ),t)( )x ( X, t) = x ξe , t oξe (X ) in Ωe4-17(4.4.28)T.
Belytschko, Lagrangian Meshes, December 16, 1998( )( )where ξ e (X) = X−1 ξ e . For the motion to be well defined and smooth, the inverse map X−1 ξ e( )must exist and the function x = x ξ e , t must be sufficiently smooth and meet certain conditions ofregularity so that x( )−1(ξ ,t ) exists; these conditions are given in Section 4.4.8.eThe inverse mapx −1 ξe ,t is usually not constructed because in most cases it cannot be obtained explicitly, soinstead the derivatives with respect to the spatial coordinates are obtained in terms of the derivativeswith respect to the parent coordinates by implicit differentiation.The motion is approximated byxi (ξ, t) = xiI (t )N I (ξ)orx (ξ,t ) = x I( t) N I (ξ)(4.4.29)where we have dropped the supercript e on the element coordinates.
As can be seen in the above,the shape functions NI (ξ) are only functions of the parent element coordinates; the timedependence of the motion resides entirely in the nodal coordinates. The above represents a timedependent mapping between the parent domain and the current configuration of the element.Writing this map at time t = 0 we obtainXi (ξ) = xi ( ξ,0) = xiI (0) NI (ξ) = X iI NI (ξ)orX(ξ) = XI NI (ξ)(4.4.30)It can be seen from (4.4.30) that the map between the material coordinates and the elementcoordinates is time invariant in a Lagrangian element.
If this map is one-to-one and onto, then theelement coordinates can in fact be considered surrogate material coordinates in a Lagrangianmesh, since each material point in an element then has a unique element coordinate label. Toestablish a unique correspondence between element coordinates and the material coordinates in Ω 0 ,the element number must be part of the label. This does not apply to meshes which are notLagrangian, as will be seen in Chapter 7. The use of the initial coordinates X as materialcoordinates in fact originates mainly in analysis; in finite element methods, the use of elementcoordinates as material labels is more natural.As before, since the element coordinates are time invariant, we can express thedisplacements, velocities and accelerations in terms of the same shape functionsui (ξ ,t ) = uiI ( t)N I ( ξ)u(ξ, t) = uI (t ) NI (ξ)(4.4.31)u˙ i(ξ, t) = vi (ξ,t ) = viI ( t) NI (ξ)u(ξ, t) = v(ξ,t ) = v I (t ) N I (ξ )(4.4.32)v˙ (ξ, t ) = ˙viI ( t )N I ( ξ)˙v( ξ,t ) = ˙v I (t ) NI (ξ)(4.4.33)where we have obtained (4.4.32) by taking material time derivative of (4.4.31) and we haveobtained (4.4.33) by taking the material time derivative of (4.4.32).
The time dependence, asbefore, resides entirely in the nodal variables, since the element coordinates are independent oftime.4-18T. Belytschko, Lagrangian Meshes, December 16, 1998timetcurrentconfiguration, Ω231x = φ(X ,t)x ξi ,tξ2x2parentelement,ξ123y∆3initialconfiguration, Ω 101 X ξi Fig. 4.1. Initial and current configurations of an element and their relationships to the parent element.4.4.6. Derivatives of Functions. The spatial derivatives of the velocity field are obtained byimplicit differentiation because the function x( ξ,t ) is generally not explicitly invertible; i.e. it is notpossible to write closed-form expressions for ξ in terms of x .
By the chain rule,∂vi ∂vi ∂xk=∂ξ j ∂xk ∂ξ jorv,ξ = v, x x, ξ(4.4.34)The matrix ∂xk ∂ξ j is the Jacobian of the map between the current configuration of the elementand the parent element configuration. We will use two symbols for this matrix: x,ξ and Fξ , whereFijξ = ∂xi ∂ξ j . The second symbol is used to convey the notion that the Jacobian with respect tothe element coordinates can be viewed as a deformation gradient with respect to the parent elementconfiguration. In two dimensionsx,ξx ,ξ ( ξ,t ) ≡ Fξ ( ξ,t ) = 1y, ξ1x,ξ 2 y, ξ2 (4.4.35)As indicated in (4.4.35), the Jacobian of the map between the current and parent configurations is afunction of time.Inverting (4.4.34), we obtain4-19T.
Belytschko, Lagrangian Meshes, December 16, 1998∂vi −1 ∂vi ∂ξ k F = or∂ξk ξkj ∂ξk ∂x j Lij =L = v, x = v,ξ x,ξ−1 = v,ξ Fξ−1(4.4.36)Thus computation the derivatives with respect to ξ involves finding the inverse of the Jacobianbetween the current and parent element coordinates; the matrix to be inverted in the twodimensional case is given in (4.4.35).
Similarly for the shape functions NI , we haveNIT, x = N TI, ξ x−, ξ1 = NIT, ξ Fξ−1(4.4.37)where the transpose appears in the matrix expressions because we consider NI , x and NI , ξ to becolumn matrices and the matrix on the RHS of the above must be a row matrix. The determinantof the element Jacobian Fξ ,( )Jξ = det x, ξ(4.4.38)is called the element Jacobian determinant; we append the subscript to distinguish it from thedeterminant of the deformation gradient, J. Substituting (4.4.37) into (4.4.36) givesLij = viI∂NI −1F∂ξk ξkjorL = v I N TI, ξ x,−ξ 1(4.4.39)The rate-of-deformation is obtained from the velocity gradient by using (3.3.10).4.4.7. Integration and Nodal Forces. Integrals on the current configuration are related tointegrals over the reference domain and the parent domain by∫ g(x )dΩ = ∫ g (X )JdΩ0 = ∫ g(ξ) Jξ d∆ΩeΩe0∆and∫ g(X)dΩ 0 = ∫ g( ξ)Jξ d∆0Ω 0e(4.4.40)∆where J and Jξ are the determinants of the Jacobians between the current configurations and thereference and parent element configurations, respectively.
Part of equations is identical to(3.2.18). The other part is obtained in the same way by using the map between the current andparent configurations. The above is consitent with our conventionWhen the internal nodal forces are computed by integration over the parent domain,(4.4.11) is tranformed to the parent element domain by (4.4.40), givingfiIint =∂NI∫ ∂xΩejσ ji dΩ = ∫∆∂N Iσ J d∆∂x j ji ξ(4.4.41)The external nodal forces and the mass matrix can similarly be integrated over the parent domain.4.4.8. Conditions on Parent to Current Map. The finite element approximation to themotion x (ξ,t ) , which maps the parent domain of an element onto the current domain of the4-20T.
Belytschko, Lagrangian Meshes, December 16, 1998element, is subject to the same conditions as φ( X,t ) , as given in Section 3.3.6, except that nodiscontinuities are allowed. These conditions are:1. x (ξ,t ) must be one-to-one and onto;2. x (ξ,t ) must be at least C 0 in space;3. the element Jacobian determinant must be positive, i.e.( )Jξ ≡ det x, ξ > 0 .(4.4.43)These conditions insure that x (ξ,t ) is invertible.( )We now explain why the condition det x,ξ > 0 is necessary.
We first use the chain rule toexpress x,ξ in terms of F and X, ξ :∂xi∂x ∂Xk∂X= i= Fik k∂ξ j ∂Xk ∂ξ j∂ξ jorx,ξ = x,X X,ξ = FX,ξ(4.4.44a)We can also write the above asFξ = F ⋅Fξ0(4.4.44b)which highlights the fact that the deformation gradient with respect to the parent elementcoordinates is the product of the standard deformation gradient and the initial deformation gradientwith respect to the parent element coordinates.
The determinant of the product of two matrices isequal to the product of the determinants, so( )( )det x,ξ =det (F) det X,ξ ≡ JJξ0(4.4.45)We assume that the elements in the initial mesh are properly constructed so that Jξ0 = Jξ (0) > 0 forall elements; otherwise the initial mapping would not be one-to-one. If Jξ( t) ≤ 0 at any time thenby (4.4.45), J ≤ 0 . By the conservation of matter ρ = ρ 0 J so J ≤ 0 implies ρ ≤ 0 , which isphysically impossible.
Therefore it is necessary that Jξ( t) > 0 for all time. In some calculations,excessive mesh distortion can result in severely deformed meshes in which Jξ ≤ 0. This implies anegative density, so such calculations violate the physical principle that mass is always positive.4.4.9. Simplifications of Mass Matrix.When the same shape functions are used for allcomponents, it is convenient to take advantage of the form of the mass matrix (4.4.20) by writingit as˜MijIJ = δij MIJ(4.4.46)where4-21T.
Belytschko, Lagrangian Meshes, December 16, 1998˜ = ρN N dΩMIJ∫ I J˜ = ρNT NdΩM∫Ω(4.4.47)ΩThen the equations of motion (4.4.22) become˜ v˙ + f int = f extMIJ iJiIiI(4.4.48)This form is advantageous when the consistent mass matrix is used with explicit time integration,since the order of the matrix which needs to be inverted is reduced by a factor of nSD .We next show that the mass matrix for a Lagrangian mesh is constant in time. If the shapefunctions are expressed in terms of parent element coordinates, then( )MijIJ =δ ij ∫ ρN I N J det x, ξ d∆ =δ ij ∫ ρ NI NJ dΩ∆(4.4.49)Ω( )Since det x,ξ and the density are time dependent, this mass matrix appears to be time dependent.To show that the matrix is in fact time independent, we transform the above integral to theundeformed configuration by (3.2.18), givingMijIJ = δij ∫ ρ N I NJ JdΩ 0(4.4.50)Ω0From mass conservation, (B4.1.1) it follows that ρJ = ρ 0 .
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