Belytschko T. - Introduction (779635), страница 17

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Governing Equations for Eulerian Formulationcontinuity equation (mass conservation):∂ρ ∂ (ρv )+=0∂t∂xmomentum equation ∂v∂v  ∂ ( Aσ )ρA + v  =+ ρAb ∂t∂x ∂x ∂v∂v ρ  + v  = ∂σ + ρb ∂t∂x  ∂xstrain measure (rate-of-deformation):Dx = v, xconstitutive equation in rate form:Dσ=σ ,t ( x,t ) +σ , x ( x, t)v( x,t ) = StσD ( Dx ( X, t ),σ ( X, t ),etc., t ≤ t )Dt(B2.4.1)(B2.4.2)(B2.4.3)(B2.4.4)energy conservation equationsame as beforeThe governing equations are summarized in Box 2.4.

In comparison with the updatedLagrangian formulation we have just discussed, four points are noteworthy:1. The mass conservation equation is now written as a partial differential equation;the form used with Lagrangian meshes is not applicable because it applies onlyto material points.2. The material time derivative for the velocity in the momentum equation has beenwritten out in terms of the spatial time derivative and transport term.3. The constitutive equation is expressed in rate form; the total form cannot be usedsince the stress and rate of deformation are functions of material coordinates in ahistory-dependent material.4.

The boundary conditions are now imposed on spatial points which do not movewith time.The continuity equation has been written as a partial differential equation because it isnot possible to obtain an integral form such as Eq. (2.2.4) when the density is a function ofspatial coordinates. Therefore, the continuity equation must be treated as a separate partialdifferential equation, although there are approximations which enable the continuityequation to be omitted when the density changes little, as for a liquid or solid; these arediscussed in Chapter 7.The constitutive equation needs to be expressed in terms of material coordinates forhistory-dependent materials, so it is treated in rate form in this formulation. It is thus aseparate partial differential equation.In the general case, boundary conditions are required for the density, velocity andstress.

As will be seen in Chapter 7, the boundary conditions for the density and stress inan Eulerian mesh depend on whether the material is flowing in or out at the boundary. Inthis introductory exposition, we consider only boundaries where there is no flow. Theboundary points are then Lagrangian, and the density and stress can be determined at thesepoints by the Lagrangian mass conservation equation, Eq. (2.2.10) and the constitutive2-54T. Belytschko, Chapter 2, December 16, 1998equation, respectively. Therefore, there is no need for boundary conditions for thesevariables.2 .

1 0 WEAK FORMS FOR EULERIAN MESH EQUATIONSIn the Eulerian formulation, we have 3 unknowns or dependent variables: the densityρ(x, t), the velocity v(x, t) and the stress σ(x, t). The rate-of-deformation can easily beeliminated from the momentum equations by substituting (B2.4.3) into the constitutiveequation (B2.4.4). Therefore, we will need three sets of discrete equations. A weak formsof the momentum equation, the mass conservation equation and the constitutive equationwill be developed.

We will construct continuous solutions to the governing equations.The equations given in Box 2.4 can in fact have discontinuous solutions, withdiscontinuities in the density, stress and velocity, as when a shock occurs in the flow.However, we will take the approach of smearing any discontinuities over several elementswith a continuous function; this approach is called shock fitting or shock smearing. Thetrial and test functions will therefore be continuous functions of space.We consider first the weak form of the continuity equation.

The trial functions for thedensity are denoted by ρ( x,t ) , the test functions by δρ( x ) The test functions and the trialfunctions for the continuity equation must be piecewise continuously differentiable, so{ρ( x,t ) ∈R, R = ρ( x,t ) ρ( x,t ) ∈C 0 ( x), ρ( x, t) = ρ on Γρ{}(2.10.1)}δρ( x ) ∈R 0 , R = δρ ( x )δρ( x ) ∈C0 ( x ), δρ ( xa ) = 0, δρ( x b ) = 0(2.10.2)In this Section, we do not consider problems with prescribed densities on the boundaries.The weak form of the continuity equation is obtained by multiplying it by the testfunction δρ( x ) and integrating over the domain. This gives∫xbxaδρ( ρ ,t + ( ρv) , x ) dx = 0∀δρ ∈R 0(2.10.3)Only first derivatives with respect to the spatial variable of the density and velocity appearin the weak form, so there is no need for integration by parts. The consequence ofintegrating by parts are interesting and is examined in the Exercises.The weak form of the constitutive equation is obtained the same way.

We express thematerial derivative in terms of a spatial derivative and a transport term, givingσ ,t + σ , x v− S( v, x ,etc) = 0(2.10.4)The test and trial functions, δσ( x) and σ( x, t) , respectively, are subject to the samecontinuity and end conditions as for the density in the continuity equation, i.e., we letσ ∈ℜ , δσ ∈ℜ 0 . The weak form of the constitutive equation is then obtained bymultiplying it by the test function and integrating over the domain:xb∫xaδσ( σ, t +σ , x v −S( v, x ,etc)) dx = 02-55∀δσ ∈R 0(2.10.5)T. Belytschko, Chapter 2, December 16, 1998As in the continuity equation, there is no benefit in integrating by parts. Neither this weakform nor the weak continuity equation have a clear physical meaning.

They will be referredto as the weak continuity and constitutive equations.The weak form of the momentum equation is obtained by integrating the test functionδv( x) over the spatial domain. The procedure is identical to that in the updated Lagrangianformulation in Section 2.7. The test and trial functions are defined by Eqs (2.7.1) and(2.7.2). The resulting weak form is∫xbxaDv  δv,x Aσ −δv ρAb − ρA Dt   dx − (δvAt x ) Γ t = 0(2.10.6)or using (??)∫xbxa ∂vδv,x Aσ +δvρA ∂t +v , xv − b dx − (δvAt x ) Γ t = 0(2.10.7)Note that the limits of the integration are fixed in space.The weak form is identical to the principle of virtual power for the updated Lagrangianformulation except that the domain is fixed in space and the material time derivative isexpressed in its Eulerian form.

Thus the weak form of the momentum equation can bewrittenδ P =δ P int − δ P ext +δ P inert = 0 ∀δv ∈U 0(2.10.8)wherexbδPintxb= ∫ δv, xσAdx = ∫ δD xσAdx = ∫ δDxσΩxa(2.10.9)ΩxaxbδPext= ∫ δvρbAdx + (δvAt x ) Γxa(2.10.10)txbδPinert ∂v ∂v= ∫ δvρ  + v, x v Adx = ∫ δvρ  +v, xv  dΩ ∂t ∂txa(2.10.11)ΩAll of the terms are identical to the corresponding terms in the principle of virtual power forthe updated Lagrangian formulation, except that the limits of integration are fixed in spaceand the material time derivative in the inertial virtual power has been expressed in terms ofthe spatial time derivative and the transport term.

Similar expressions for the virtualpowers also hold on the element level.2.11. FINITE ELEMENT EQUATIONS2-56T. Belytschko, Chapter 2, December 16, 1998In a general Eulerian finite element formulation, approximations are needed for thepressure, stress and velocity. For each independent variable, test and trial functions areneeded. We will develop the equations for the entire mesh.

For simplicity, we considerthe case where the segment is 0 ≤ x ≤ L. As mentioned before, we consider the case wherethe end points are fixed in space and the velocities on these points vanish. There are thenno boundary conditions on the density or stress and the boundary conditions on thevelocity arev (0,t ) = 0,v (L, t ) = 0(2.11.1)The mapping between spatial and element parent coordinates is given byx = N I (ξ) x I(2.11.2)In contrast to the Lagrangian formulations, this mapping is constant in time since the nodalcoordinates x I are not functions of time. The trial and test functions are given bynNρ( x,t ) = ∑ NIρ ( x )ρ I( t)I =1nN∑σ( x, t) =I =1v( x, t) =NσI ( x )σ I (t )n N −1∑I= 2nNδρ (x ) = ∑ NIρ ( x )δρ I(2.11.3)I=1nNδσ ( x ) = ∑ NσI ( x)δσ I(2.11.4)I =1NI ( x)vI (t )δv( x) =n N −1∑I =2NI ( x )δvI(2.11.5)The velocity trial functions have been constructed so the velocity boundary condition isautomatically satisfied.Substituting the test and trial functions for the density into the weak continuityequation givesnN n N∑ ∑δρJ ∫0 ( N Jρ NIρ ρI , t + N Jρ (ρv), x )dx = 0L(2.11.6)I =1J = 1Since this holds for arbitrary δρJ at interior nodes, we obtainLρρL∫ 0 N I N J dxρ J ,t + ∫0NIρ ( ρv) , x dx = 0I =1 to nN(2.11.7)We define the following matricesL∫ ( N ) N dxMIJρ = ∫0 NIρ N Jρdx,Meρ =g Iρ = ∫0 NIρ ( ρv ), xdx,geρ = ∫ 0 N ρLρ TρΩeL( ) (ρv)T,x2-57dx(2.11.8)(2.11.9)T.

Belytschko, Chapter 2, December 16, 1998The discrete continuity equation can be then be written as∑ MIJρ ρ˙ J + gρI = 0 for I = 1 to n N ,or Mρ ˙ρ +gρ = 0(2.11.10)JThe matrices M ρ can be assembled from element matrices just like the mass matrix in themomentum equation. The column matrix g ρ is obtained by a scatter, or vector assembly.The matrix M ρ is time invariant and closely resembles the mass matrix. However, thecolumn matrix g ρ varies with time and must be computed in every time step. In mostcases, the element matrices are integrated in the parent coordinate system.The discrete form of the constitutive equation is obtained similarly. The result is∑ MIJσ ρ˙ J + gσI˙ + gσ = h= hσI for I =1 to nN , or Mσ σ(2.11.11)JwhereLMσIJ = ∫ N σI NσJ dx0LgσI = ∫ NσI vσ , x dx0andMσe = ∫andgσe = ∫ΩeΩe(Nσ )(Nσ )TTNσ dΩ(2.11.12)vσ , x dΩ(2.11.13)where the matrix relations on the right have been extracted from the indicial forms andimmediately specialized to elements by the procedure in Section 2.8.Currenttx(ξ)x(ξ)Original1Parent2xFig.

2.8 Eulerian element in current and original configurations, which are the same, and the mapping tothe parent element.Momentum Equation. The weak form of the momentum equation is identical to theweak form for the updated Lagrangian formulation except for the inertial term. Thereforethe expressions for the internal and external nodal forces are identical. The inertial nodal2-58ξT.

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