Belytschko T. - Introduction (779635), страница 36
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It is also called the fundamental theorem of variational calculus;sometimes we call it the function scalar product theorem since it is the counterpart of the scalarproduct theorem given in Chapter 2. We follow Hughes [1987, p.80] in proving (4.3.13). As afirst step we show that αi ( X) = 0 in Ω . For this purpose, we assume thatδvi (X ) =αi (X) f ( X)where(4.3.14)1. f (X) > 0 on Ω but f (X) = 0 on Γ int and f (X) = 0 on Γti2. f (X) is C −1Substituting the above expression for δvi into (4.3.13) gives∫ αi (X)α i (X )f (X)dΩ = 0(4.3.15)ΩThe integrals over the boundary and interior surfaces of discontinuity vanish because the arbitraryfunction f (X) has been chosen to vanish on these surfaces. Since f (X) > 0 , and the functionsf (X) and αi ( X) are sufficiently smooth, Equation (4.3.15) implies αi ( X) = 0 in Ω for i =1 tonSDTo show that the γ i ( X) = 0 , letδvi (X ) =γ i (X) f ( X)where(4.3.16)1.
f (x) > 0 on Γint ; f (x) = 0 on Γti ;2. f (x) is C −1Substituting (4.3.16) into (4.3.13) gives∫γi( x)γ i (x) f (x )dΓ = 0(4.3.17)Γintwhich implies γ i ( x) = 0 on Γint (since f (x) > 0 ).The final step in the proof, showing that βi (x ) = 0 is accomplished by using a functionf (x) > 0 on Γti . The steps are exactly as before. Thus each of the αi (x ), β i (x ), and γ i ( x) mustvanish on the relevant domain or surface. Thus Eq. (4.3.12) implies the strong form: themomentum equation, the traction boundary conditions, and the interior continuity conditions, Eqs.(4.3.3).Let us now recapitulate what has been accomplished so far in this Section.
We firstdeveloped a weak form, called the principle of virtual power, from the strong form. The strongform consists of the momentum equation, the traction boundary conditions and jump conditions.4-9T. Belytschko, Lagrangian Meshes, December 16, 1998The weak form was obtained by multiplying the momentum equation by a test function andintegrating over the current configuration.
A key step in obtaining the weak form is theelimination of the derivatives of the stresses, Eq. (4.3.5-6). This step is crucial since as a result,the stresses can be C-1 functions. As a consequence, if the constitutive equation is smooth, thevelocities need only be C0 .Equation (4.3.4) could also be used as the weak form. However, since the derivatives ofthe stresses would appear in this alternate weak form, the displacements and velocities would haveto be C1 functions (see Chapter 2); C1 functions are difficult to construct in more than onedimension.
Furthermore, the trial functions would then have to be constructed so as to satisfy thetraction boundary conditions, which would be very difficult. The removal of the derivative of thestresses through integration by parts also leads to certain symmetries in the linearized equations, aswill be seen in Chapter 6.
Thus the integration by parts is a key step in the development of theweak form.Next we started with the weak form and showed that it implies the strong form. This,combined with the development of the weak form from the strong form, shows that the weak andstrong forms are equivalent.
Therefore, if the space of test functions is infinite dimensional, asolution to the weak form is a solution of the strong form. However, the test functions used incomputational procedures must be finite dimensional. Therefore, satisfying the weak form in acomputation only leads to an approximate solution of the strong form.
In linear finite elementanalysis, it has been shown that the solution of the weak form is the best solution in the sense thatit minimizes the error in energy, Strang and Fix (1973). In nonlinear problems, such optimalityresults are not available in general.4.3.3. Physical Names of Virtual Power Terms. We will next ascribe a physical nameto each of the terms in the virtual power equation. This will be useful in systematizing thedevelopment of finite element equations.
The nodal forces in the finite element discretization willbe identified according to the same physical names.To identify the first integrand in (4.3.9), note that it can be written as∂ (δvi )σ ji = δLij σ ji = δDij + δWij σ ji = δDijσ ji =δD : σ∂x j()(4.3.18)Here we have used the decomposition of the velocity gradient into its symmetric and skewsymmetric parts and that δWijσ ij = 0 since δWij is skew symmetric while σ ij is symmetric.Comparison with (B4.1.4) then indicates that we can interpret δDijσ ij as the rate of virtual internalwork, or the virtual internal power, per unit volume.
Observe that w˙ int in (B4.1.4) is power per˙ int = D: σ is the power per unit volume. The total virtual internal power δ P int isunit mass, so ρwdefined by the integral of δDijσ ij over the domain, i.e.δ P int = ∫ δDijσ ijdΩ=Ω∂ (δvi )σ dΩ≡ ∫ δLijσ ij dΩ = ∫ δD :σdΩ∂x j ijΩΩΩ∫(4.3.19)where the third and fourth terms have been added to remind us that they are equivalent to thesecond term because of the symmetry of the Cauchy stress tensor.The second and third terms in (4.3.9) are the virtual external power:4-10T. Belytschko, Lagrangian Meshes, December 16, 1998δPnSDextnSD= ∫ δviρbi dΩ+ ∑ ∫ δv jt j dΓ= ∫ δv ⋅ρbdΩ + ∑ ∫ δvj e j ⋅ tdΓΩj=1 ΓtjΩ(4.3.20)j=1 ΓtjThis name is selected because the virtual external power arises from the external body forcesb(x, t) and prescribed tractions t ( x,t ) .The last term in (4.3.9) is the virtual inertial power∫δ P inert = δvi ρv˙ idΩ(4.3.21)Ωwhich is the power corresponding to the inertial force.
The inertial force can be considered a bodyforce in the d’Alembert sense.Inserting Eqs. (4.3.19-4.3.21) into (4.3.9), we can write the principle of virtual power asδ P =δ P int − δ P ext + δ P inert = 0 ∀δvi ∈U 0(4.3.22)which is the weak form for the momentum equation. The physical meanings help in rememberingthe weak form and in the derivation of the finite element equations. The weak form is summarizedin Box 4.2.BOX 4.2Weak Form in Updated Lagrangian Formulation:Principle of Virtual PowerIfσ ij is a smooth function of the displacements and velocities and vi ∈U , then ifδ P int − δ P ext + δ P inert = 0∀δvi ∈U 0(B4.2.1)then∂σ ji∂x j+ ρbi = ρ˙vi in Ω(B4.2.2)n j σ ji = ti on Γti(B4.2.3)nj σ ji = 0 on Γ int(B4.2.4)whereδ P int = ∫ δD:σdΩ = ∫ δDij σij dΩ =ΩΩ∂ (δvi )σ ij dΩ∂xjΩ∫4-11(B4.2.5)T.
Belytschko, Lagrangian Meshes, December 16, 1998δ P ext = ∫ δv⋅ρbdΩ+ΩnSD∑ ∫(j =1 Γt)δv ⋅e j t ⋅e j dΓ = ∫ δvi ρbi dΩ+ΩjnSD∑ ∫ δv j t jdΓj=1 Γtjδ P inert = ∫ δv ⋅ρv˙ dΩ= ∫ δvi ρv˙ idΩΩ(B4.2.6)(B4.2.7)Ω4.4 UPDATED LAGRANGIAN FINITE ELEMENT DISCRETIZATION4 .
4 . 1 Finite Element Approximation. In this section, the finite element equations for theupdated Lagrangian formulation are developed by means of the principle of virtual power. For thispurpose the current domain Ω is subdivided into elements Ω e so that the union of the elementscomprises the total domain, Ω = ∪Ω e . The nodal coordinates in the current configuration areedenoted by xiI , I = 1 to nN . Lower case subscripts are used for components, upper case subscriptsfor nodal values. In two dimensions, xiI = [ xI , yI ], in three dimensions xiI = [ xI , yI , z I ] .
Thenodal coordinates in the undeformed configuration are XiI .In the finite element method, the motion x ( X, t) is approximated byxi (X,t ) = N I ( X) xiI (t )orx ( X, t) = NI (X)x I (t )(4.4.1)where NI (X) are the interpolation (shape) functions and x I is the position vector of node I.Summation over repeated indices is implied; in the case of lower case indices, the sum is over thenumber of space dimensions, while for upper case indices the sum is over the number of nodes.The nodes in the sum depends on the entity considered: when the total domain is considered, thesum is over all nodes in the domain, whereas when an element is considered, the sum is over thenodes of the element.Writing (4.4.1) at a node with initial position X J we havex ( XJ , t) = x I ( t)N I (X J ) = x I (t )δ IJ = x J (t )(4.4.3)where we have used the interpolation property of the shape functions in the third term.
Interpretingthis equation, we see that node J always corresponds to the same material point X J : in aLagrangian mesh, nodes remain coincident with material points.We define the nodal displacements by using Eq. (3.2.7) at the nodesuiI (t ) = xiI (t ) − XiIoru I ( t) = x I (t ) −X I(4.4.4a)u (X,t ) = uI (t )N I ( X)(4.4.4b)The displacement field isui ( X, t ) = xi (X,t ) − Xi = uiI (t ) NI (X)4-12orT. Belytschko, Lagrangian Meshes, December 16, 1998which follows from (4.4.1), (4.4.2) and (4.4.3).The velocities are obtained by taking the material time derivative of the displacements,givingvi (X,t ) =∂ui ( X,t ) ˙= uiI ( t)N I (X) = viI (t ) NI (X)∂torv( X, t) = ˙u I (t ) NI (X)(4.4.5)where we have written out the derivative of the displacement on the left hand side to stress that thevelocity is a material time derivative of the displacement, i.e., the partial derivative with respect totime with the material coordinate fixed.
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