Belytschko T. - Introduction (779635), страница 39

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Hence (4.4.50) becomesMijIJ = δij ∫ ρ 0 NI N J dΩ 0 or MijIJ = δ ij ∫ ρ0N I NJ Jξ0 d∆Ω0(4.4.51)∆The compact form of the mass matrix, (4.4.47) can similarly be written as˜ =MIJ∫ ρ0 NI NJ dΩ0Ω0and˜ = I ρ N N dΩM IJ = IMIJ∫ 0 I J 0(4.4.52)Ω0In the above integrals, the integrand is independent of time, so the mass matrix is constant in time.It needs to be evaluated only at the beginning of a computation. The same result could be obtainedby computing the mass matrix by (4.4.49) at the initial time, i.e. in the intial configuration.

Themass matrix in (4.4.52) can be called total Lagrangian since it is evaluated in the reference(undeformed) configuration. We take the view here and subsequently that the discrete equationsshould be evaluated in whatever configuration is most convenient.4.5. IMPLEMENTATIONIn the implementation of the finite element equations developed in the previous Section,two approaches are popular:1.

the indicial expressions are directly treated as matrix equations;2. Voigt notation is used, as in linear finite element methods, so the square stress andstrain matrices are converted to column matrices.4-22T. Belytschko, Lagrangian Meshes, December 16, 1998Box 4.3Discrete Equations and Internal Nodal Force Algorithmfor the Updated Lagrangian FormulationEquations of Motion (discrete momentum equation)MijIJv˙ jJ + f iIint = fiIext for ( I,i ) ∉ΓviInternal Nodal ForcesfiIint = ∫ B Ij σ ji dΩ=Ω∂N∫ ∂x jI σ ji dΩ orΩ(B4.8.1)(f ) = ∫ Bin t TITI σdΩ(B4.8.2)ΩTf intI = ∫ BI {σ}dΩ in Voigt notationΩExternal Nodal ForcesfiIext = ∫ N Iρbi dΩ+Ω∫ N t dΓf Iext = ∫ NI ρbdΩ+ ∫ N I e i ⋅ tdΓorI iΓtiΩMass Matrix (total Lagrangian)MijIJ =δ ij ∫ ρ0 NI NJ dΩ0 = δij ∫ ρ0 N I N J Jξ0d∆Ω0M IJ(B4.8.3)Γti(B4.8.4)∆˜ = I ρ N N dΩ= IMIJ∫ 0 I J 0(B4.8.5)Ω0Internal nodal force computation for element1.

f int = 02. for all quadrature points ξQ[ ] [ ( ) ]L = [L ] = [v B ] = v B ; Li. compute B Ij = ∂N I ξ Q ∂x j for all Iii.iii. D =ij12iIIjTI Iij=(LT + L)∂NIv∂x j iIiv. if needed compute F and E by procedures in Box 4.7v. compute the Cauchy stress σ or the PK2 stress S or by constitutiveequationvi. if S computed, compute σ by σ = J −1FSFTintTvii. f intI ← fI +B I σJξ wQ for all Iend loopwQ are quadrature weights4-23T. Belytschko, Lagrangian Meshes, December 16, 1998Each of these methods has advantages, so both methods will be described. In Box 4.8 the discreteeqautions are summarized in both forms.

The internal force computations is then given for thematrix-indicial form.4.5.1.Indicial to Matrix Expressions. The conversion of indicial expressions to matrixfrorm is somewhat arbitrary and depends on individual preferences. In this book, we have tried tointerpret single-index variables as column matrices in most cases; the details are somewhat differentwhen there is a preference for row matrices.

To illustrate this procedure, consider the expressionfor the velocity gradient, Eq. (3.3.7) and (4.4.5):Lij =∂vi∂N= viI I∂x j∂x j(4.5.1)The above expression can be put into the form of a matrix product if we associate the index I witha column number in v and a row number in ∂N I ∂x j .

To simplify the writing of a matrixexpression, we define a matrix B byB jI =∂NI∂x j[ ] [B = B jI = ∂NI ∂x jor](4.5.2)where j is the row number in the matrix. The velocity gradient can then be expressed in terms ofthe nodal displacements by (4.5.1) and (4.5.2) by[ Lij ] = [viI ][B Ij ] = [ viI ][B jI ]TL = vB Tor(4.5.3)so, because of the implicit sum on I, the indicial expression corresponds to a matrix product.We can also often write the expression (4.5.1) without expressing the sum on I in matrixform. The B matrix is then subdivided into B I matrices, each associated with node I:B = [B1 ,B2 , B3 , ...,Bm ]{ } I = NI , xwhere BIT = B j(4.5.3b)For each node I, the B I matrix is a column matrix.

Then the expression for the velocity gradientcan be written as a sum of tensor products, a product of a column matrix with a row matrix, asshown belowv xI L = v I BIT =   NI , xv yI [vxI N I, xNI,y =  vyI N I, x]vxI NI , y vyI NI , y (4.5.4)To put the internal force expression (4.4.11) in matrix form, we first rearrange the terms sothat adjacent terms correspond to matrix products. This entails interchanging the row and columnnumber on the internal forces a shown below(f )int TiI∂NIσ ji dΩ= ∫ B IjT σ jidΩ∂xjΩΩ= fIiint = ∫The above can be put in the following matrix form4-24(4.5.5)T. Belytschko, Lagrangian Meshes, December 16, 1998[ fiIint ] = [ f Iiint ] = ∫ [∂N IT][ ]∂x j σ ji dΩ=Ω∫ [B jI ] [σ ji ]dΩTΩ(f ) = ∫ Bin t TITI σdΩ(4.5.6)ΩFor example, in two dimensions this gives[ fxIfyI]int=∫ [ NI , xΩσ xx σ xy NI , y dΩσ xy σ yy ](4.5.7)There are many other ways of converting indicial expressions to matrix form but the above isconvenient because it adheres to the convention of treating single index matrices as columnmatrices, which is customary in the finite element literature.

The expression for all nodal forcescan be obtained by using the B matrix as defined in (4.5.3b), which gives(f int )T= ∫ B T σdΩΩ4.5.2.Voigt notation. An alternate implementation which is widely used in linear finiteelement analysis is based on Voigt notation, see Appendix B. Voigt notation is useful forcomputing tangent stiffness matrices in Newton methods, See Chapter 6. In Voigt notation thestresses and rate-of-deformation are expressed in column vectors, so in two dimensions{D}T = [ Dx Dy 2D xy ]{σ}T = [σ x σ y σ xy ](4.5.11)We define the B I matrix so it relates the rate-of-deformation to the nodal velocities by{δD} = BI δv I{D} = B I vI(4.5.12)where the summation convention as usual applies to repeated indices.

The elements of the B Imatrix are obtained so as to meet the definition (4.5.12); this is illustrated in the followingexamples. Note that a matrix is enclosed in brackets only when this is needed to distinguish amatrix from its usual form as a square matrix; matrices and tensors of third order or higher whichbecome square matrices are written simply as boldface.The expression for the internal force vector can be derived in this notation by using thedefinition of the virtual internal power in terms of the nodal forces and nodal velocities and in termsof the stresses and rate of deformation, Eq. (4.3.19).

Since {D}T {σ} gives the internal power perunit volume (the column matrices were designed with this in mind), it follows thatTδ P int = δv TI f inI t = ∫ {δD} {σ}dΩ(4.5.13)ΩSubstituting (4.5.12) into the above and invoking the arbitrariness of {δv} givesf inI t = ∫ BTI {σ} dΩ(4.5.14)Ω4-25T. Belytschko, Lagrangian Meshes, December 16, 1998As will be shown in the examples, Eq. (4.5.14) gives the same expression for the internal nodalforces as Eq.

(4.5.6): Eq. (4.5.14) uses the symmetric part of the velocity gradient, whereas thecomplete velocity gradient has been used in Eq. (4.5.6). However, since the Cauchy stress issymmetric, the two expression are equivalent; this is verified in the following examples.It is sometimes convenient to place the displacement, velocities and nodal forces for anelement or a complete mesh in a single column matrix. We will then use the symbol d for thecolumn matrix of all nodal displacements, ˙d for the column matrix of nodal velocities and {f } forthe column matrix of nodal forces, i.e. u1  u2 d=  .

um  v1  ˙d =  v2 . vm  f1  f 2f = . fm (4.5.15)where m is the number of nodes. The correspondence between the two matrices is given byda = uiI where a = ( I −1)* nSD + i(4.5.16)Note that we use a different symbol for the column matrix of all nodal displacements and nodalvelocities because the symbols u and v refer to the displacement and velocity vector fields in thecontinuum mechanics description.In this notation, we can write the counterpart of Eq. (4.5.12) as{D} = Bd˙where B = [B1 , B2 ,..., Bm ](4.5.18)where the brackets around D indicate that the tensor is stored in colum matrix form; we do not putbrackets around B because this is always a rectangular matrix. The nodal forces are given by thecounterpart of Eq.

(4.5.14):{f }int = ∫ BT {σ}dΩ(4.5.19)ΩOften we omit the brackets on the nodal force, since the presence of a single term in Voigt notationalways indicates that the entire equation is in Voigt notation. The Voigt form can also be obtainedby rewriting (4.5.5) as∂N Iδ riσ jidΩ∂xjΩf Irint = ∫(4.5.20)Then defining the B matrix byBijIr =∂N Iδ∂x j ri(4.5.21)4-26T.

Belytschko, Lagrangian Meshes, December 16, 1998and converting the indices (i,j) by the kinematic Voigt rule to a and the indices (I.r) by the matrixcolumn vector rule givesTf aint = ∫ Bbaσ bdΩ or f int = ∫ BT{ σ}σdΩΩΩ(4.5.22)More detail and techniques for translating indicial notation to Voigt notation can be found inAppendix B.4.5.4.Numerical Quadrature. The integrals for the nodal forces, mass matrix and otherelement matrices can generally not be evaluated in closed form, and are instead integratednumerically (often called numerical quadrature).

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