Belytschko T. - Introduction (779635), страница 82

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Therefore,together with the constitutive equation, Eq. (7.12.5b), and Eq. (7.13.7a) , we can expressEq. (7.13.6) as:−∫Ω∂(δvi )µ  ∂ (δvi ) ∂(δv j )   ∂vi ∂v j PdΩ + ∫+  ∂x + ∂x  dΩΩ 2  ∂x∂x i∂xjii j(7.13.8)Now, substitute Eqs. (7.13.5) and (7.13.8) into (7.13.2), the weak form for themomentum equation and associated boundary condition is obtained:∫Ω δvi ρ∂vi∂vdΩ + ∫ δvi ρc j i dΩΩ∂t χ∂x jW.K.Liu, Chapter 7−∫Ω58∂(δvi )µ  ∂ (δvi ) ∂(δv j )   ∂vi ∂v j PdΩ + ∫++dΩΩ 2  ∂x∂x i∂x i   ∂x j ∂xi j− ∫ δviρgi dΩ − ∫ΩΓtδvi t j dΓ = 0(7.13.9)The weak forms for the continuity equation and the free surface update equation aresimply obtained by taking the inner product with δp and δxi , respectively.We may now state a suitable weak form for the momentum equation.W.K.Liu, Chapter 759Weak Form for Newtonian FluidGiven density, ρ , bulk modules, B, and Cauchy stress function, σ , defined in Table 7.2,respectively, find v ∈U v , p ∈U p and x ∈U x such that for every δv ∈U0v , δp ∈U 0p andδx ∈U0x :Continuity Equation [Eq.

(7.12.2a)]1 ∂p1∂p∂v∫Ω δp B ∂t χ dΩ+ ∫Ω δp B ci ∂x i dΩ + ∫Ω δp ∂xii dΩ = 0(7.13.10a)Momentum Equation [Eq. (7.12.2b)]∫Ω δvi ρ+∫Ω∂vi∂v∂(δvi )dΩ + ∫ δvi ρc j i dΩ − ∫pdΩΩΩ∂t χ∂x j∂xiµ  ∂(δvi ) ∂(δv j )   ∂vi ∂v j ++dΩ−2  ∂x j∂xi   ∂x j ∂x i ∫Ω δviρgidΩ − ∫Γtδvi t j dΓ = 0(7.13.10b)Free Surface Update EquationThe mesh rezoning equation [Eq. (7.12.2c)]:∂x∂x∫Ωˆ δx i ∂ti χ dΩˆ + ∫Ωˆ (δ jk − α jk ) v kδxi ∂χij dΩˆ − ∫Ωˆ δx ividΩˆ = 0(7.13.10c)or the mesh updating equation [Eq. (7.12.2d)]:∫ΩˆNSD v − vˆ∂x ijj ˆ ji ˆˆδx idΩ − ∫ ˆ δxi ∑JdΩ−δxi vi −∫ˆˆiiΩΩ∂t χJj =1Jˆ  ˆw dΩ = 0 (7.13.10d)Jˆii i j ≠1Constitutive Equationσ ij = − pδij + 2µDij(7.13.11e)Table 7.3 Weak Form of Slightly Compressible Viscous Flow with Moving BoundaryProblemW.K.Liu, Chapter 7607.13.1 Galerkin Approximation of Slightly Compressible Viscous Flow with MovingBoundariesTo obtain the semidiscrete equations by the Galerkin approximation, Eq.

(7.13.10a)through (7.13.10d) is replaced by the finite functions p h , v h and x h . That is:p → ph , x → x h and v → vh(7.13.11)In particular, we wish to separate these functions into two parts, the unknown parts wxh ,w hp and wvh and the prescribed boundary parts (essential boundary conditions) xh , p h andvh , so that:p h = w hp + p h(7.13.12a)v h = wvh + v h(7.13.12b)x h = wxh + xh(7.13.12c)Given ρ , B, σ as before, find vih = wvhi + vih , p h = w hp + p h and xih = w hxi + x ih ,hhhhhwhere wvh ∈U v0 , w hp ∈U 0p and w hx ∈U 0x , such that for every δvh ∈U0v , δph ∈U 0p andhδx h ∈U0x :Galerkin Form of Continuity Equation∫Ωδp hhh1 ∂w p1 ∂w pdΩ + ∫ δph cidΩ+ΩB ∂tB ∂x i∫Ω δphχ= − ∫ δp hΩ∂vidΩ∂x ihh1 ∂p1 ∂pdΩ − ∫ δp h cidΩΩB ∂tB ∂xi(7.13.13a)χGalerkin Form of Momentum Equation∫Ωδvih ρ∂whvi∂tχdΩ + ∫Ωδvih ρc j∂whvi∂x jdΩ − ∫Ω∂(δvih )PdΩ∂xiW.K.Liu, Chapter 7+∫Ωµ261 ∂(δvih ) ∂(δv hj )   ∂wvhi ∂wvhj  ∂x + ∂x   ∂x + ∂x  dΩ−ji ji = ∫ δvih h j dΓ − ∫Γ−∫Ωµ2Ωδvih ρ∂vihdΩ−∂t χ∫Ωδvhi ρc j∫Ω δvi ρgi dΩh∂vihdΩ∂x j ∂(δvih ) ∂(δv hj )   ∂vih ∂v hj  ∂x + ∂x   ∂x + ∂x  dΩji ji(7.13.13b)Galerkin Form of Free Surface Update EquationThe mesh rezoning equation:∫hh ∂wx iˆ δx iΩ∂t= ∫ ˆ δxihΩχˆ +dΩ∫ˆΩ()δ jk − α jk vk δx hi∂whxi∂χ jˆ − δx h v dΩdΩ∫ˆ i i ˆΩh∂xihh ∂x iˆ +ˆdΩδ−αvδxdΩjkjk k i∫ˆΩ∂t χ∂χ j()(7.13.13c)or the mesh updating equation:∫hh ∂wx iˆ δx iΩ∂t= − ∫ ˆ δx ihΩχNSD v − vˆjj ˆ ji ˆhhˆdΩ − ∫ ˆ δxi ∑JdΩ−δxi vi −∫ˆˆiiΩΩJj =1Jˆ  ˆw dΩJˆii i j ≠1∂xihˆdΩ∂t χ(7.13.13d)7.13.2 Element Matrices for Slightly Compressible Viscous Flow with Moving BoundariesThe discrete forms of the continuity, momentum and mesh updating equations arepresented next.

First, we define:NEQvhv =wvhh+v =∑A=1NUMNPvN Av (x)v A (t)+∑N vA (x)v Ah (t)A= NEQv+1(7.13.14a)W.K.Liu, Chapter 762NEQphp =w hph+p =∑NUMNPpNAp (x) p A (t)+A=1vˆ =whvˆh+ vˆ =∑NEQvˆ∑(7.13.14b)NUMNP vˆNAvˆ (x) vˆ A (t) +A=1δ vˆ h =NAp (x) p hA (t)A= NEQp+1NEQvˆh∑∑N Avˆ (x) vˆ Ah (t)(7.13.14c)A= NEQvˆ +1N vAˆ (X)c vAˆ (t)(7.13.14d)A=1hδp =NEQp∑N Ap (X)c Ap (t)(7.13.14e)NAv (X)c vA (t)(7.13.14f)A=1δvh =NEQv∑A=1where N Ap , N vA and N vAˆ are the continuous element shape function for pressure, velocityand mesh velocity, respectively.(ii) Mixed Formulation:Without any loss of generality, the free surface is assumed perpendicular to the χ 3direction.

The cofactors are∂x ∂x∂x ∂xJˆ13 = 2 3 − 2 3∂χ1 ∂χ 2 ∂χ 2 ∂χ1(7.13.20a)∂x ∂x∂x ∂xJˆ23 = 1 3 − 1 3∂χ 2 ∂χ1 ∂χ1 ∂χ 2(7.13.20b)∂x ∂x1 ∂x1 ∂x2Jˆ33 = 1−∂χ 2 ∂χ 2 ∂χ 2 ∂χ1(7.13.20c)It should be noted that x3 is the only unknown that defines the free surface which isassumed material (i.e. w3 = 0).(8d) Show that by substituting Eqs. (7.13.20) into Eq. (7.11.22b) yields:∂x31 ∂x∂x  ∂x+ ˆ 33  (v1 − vˆ1 ) 2 − (v2 − vˆ2 ) 1  3∂t χ J ∂χ 2∂χ 2  ∂χ 1W.K.Liu, Chapter 71 ∂x∂x  ∂x+ ˆ 33  −(v1 − vˆ1 ) 2 + (v2 − vˆ2 ) 1  3 = 0J ∂χ 1∂χ 1  ∂χ 263(7.13.21)(8e) Show that the convective term is:Lˆ A =∫Ω χe∂xiNˆ A cˆmdΩχ∂χ m(7.13.22a)by defining:1 ∂x∂x cˆ1 = ˆ 33  (v1 − vˆ1 ) 2 − (v2 − vˆ2 ) 1 J ∂χ 2∂χ 2 (7.13.22b)1 ∂x∂x cˆ2 = ˆ 33  −(v1 − vˆ1 ) 2 + (v2 − vˆ2 ) 1 J ∂χ1∂χ1 (7.13.22c)cˆ3 = 0(7.13.22d)7.15.

Numerical Example7.15.1 Elastic-plastic wave propagation problemAn elastic-plastic wave propagation problem is used to assess the ALE approach inconjunction with the regular fixed mesh method. The problem statement, given in Fig.1,represents a 1-D infinitely long, elastic-plastic hardening rod. Constant density andisothermal conditions are assumed to simplify the problem. Thus only the momentumequation and constitutive equation are considered for this problem. It should be noted thatthis elastic-plastic wave propagation problem does not require an ALE mesh and theproblem was selected because it provides a severe test of the stress update procedure andbecause of the availability of an analytic solution.

The problem is solved using 400elements which are uniformly spaced with a mesh size of 0.1. The mesh is arranged so thatno reflected wave will occur during the time interval under consideration. Materialproperties and computational parameters are also depicted in Fig.7.16.1. Four stages areinvolved in this problem:(1) t ∈[0, t1 ], the mesh is fixed, and a square wave is generated at the origin;(2) t ∈[t1 , t 2 ], the mesh is fixed and the wave travels along the bar;(3) t ∈[t2 , t3 ], two cases are studied:case A: the mesh is moved uniformly to the left-hand side with a constant speed − vˆ∗ ;case B: same as Case A except the mesh is moved to the right;W.K.Liu, Chapter 764(4) t = t 3 , the stress is reported as a function of spatial coordinates in Figs. 7.16.2 and7.16.3 for Case A and Case B, respectively.For both cases, the momentum and stress transport are taken into account by employing thefull upwind method for elastic and elastic-plastic materials.

The results are compared to :(1) Regular Galerkin method runs, in which all of the transport items are handled by theexact integration;(2) Fixed mesh runs, in which the finite element mesh is fixed in space and the results arepretty close to the analytic solutions.The results according to several time step size are reported in Table 7.16.1. The wavearrival time for both methods, with or without upwinding technique, agree well with thefixed mesh runs. However, the scheme without upwinding technique causes severeunrealistic spatial oscillations in Case A because of the significant transport effects.

Thenew method proposed here eliminates these oscillations completely. Base on these studies,it is found that the transport of stresses as well as yield stress ( and back stresses ifkinematic hardening) plays an important role in ALE computations for path-dependentmaterials, and the proposed update procedure is quite accurate and effective.W.K.Liu, Chapter 765ρ = 1 E = 104 E / ET = 3 σ y0 = 75 σ 0 = −100∆ x = ∆ χ = 0.1 vˆ ∗ = 0.25 E / ρ t1 = 45 t2 = 240 t 3 = 320(×10 −3 )1.t ∈[0,t1 ] mesh fixed, wave generated2.t ∈[t1 ,t 2 ] mesh fixed, wavetravellingχ =0x=0χ =0x=03 A. t ∈[t 2 ,t3 ] Case A: mesh moving with vˆ = − vˆ ∗χ =0x=03B. t ∈[t 2 ,t3 ] Case B: mesh moving with vˆ = + vˆ∗χ =0x=04. t = t 3report stress vs.

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