STABILIZED FINITE ELEMENT METHODS TO DEAL WITH INCOMPRESSIBILITY IN SOLID MECHANICS IN FINITE STRAINS

Transcription

1 European Congress on Computational Methods in Applied Sciences and Engineering ECCOMAS 2012) J. Eberhardsteiner et.al. eds.) Vienna, Austria, September 10-14, 2012 STABILIZED FINITE ELEMENT METHODS TO DEAL WITH INCOMPRESSIBILITY IN SOLID MECHANICS IN FINITE STRAINS D. Al Akhrass 1 2, S. Drapier 1, J. Bruchon 1, and S. Fayolle 2 1 Ecole des Mines de Saint-Etienne Centre SMS et LCG UMR CNRS , cours Fauriel Saint-Etienne - France {alakhrass,drapier,bruchon}@emse.fr 2 EDF R&D 1, avenue du général de Gaulle Clamart - France sebastien.fayolle@edf.fr Keywords: incompressibility, Orthogonal Sub Grid Scale method, mini-element, large strains Abstract. In this paper, methods to deal with incompressibility in solid mechanics are presented. A mixed formulation involving pressure and displacement fields is used at small strains and a continuous linear interpolation is considered for both fields, associated with a stabilization technique based on a sub-scale method namely the orthogonal sub-grid scale method. The performances of different elements such as P2/P1, P1+/P1, P1/P1 with OSGS are then compared. The previous developments have then been adapted to the finite strains framework with a model based on logarithmic strains, for which we propose a three-fields finite element formulation.

2 1 INTRODUCTION In the solid mechanics framework, simulations of forming process exhibit many difficulties such as nonlinearities due to large strains, contact, and nonlinear constitutive laws. To ensure accurate results, it is necessary to deal with the phenomena called incompressibility or quasiincompressibility. It is well-known that the standard displacement-based finite element method performs poorly in incompressible situations, producing too stiff solutions and oscillations of the stress field. Over the years, and particularly in the 90s, different strategies were proposed to reduce or avoid volumetric locking and pressure oscillations in finite element solutions. Different methods to deal with incompressibility have been developed such as under-integrated elements, Enhanced Assumed Strain methods EAS) [1], [2], B-bar and F-bar methods [3], or mixed formulations. Classically, a three-fields mixed finite element formulation can be derived from the mechanical potential in which the unknowns are the displacement, pressure and volumetric strain fields. In elastoplasticity, the volumetric strain can often be condensed and a two fields mixed finite element formulation is obtained. This is a popular and efficient way to deal with incompressibility. However, it is well known that the approximation orders must lead to finite element formulations fitting the Ladyjenskaia-Brezzi-Babuska LBB) stability condition [4]. Such suitable elements are for instance the very popular P2/P1 element with a quadratic approximation for displacement and a linear approximation for pressure. To decrease the computation cost, it is convenient to keep a linear approximation for both fields. However it is necessary to stabilize this element. To achieve the LBB condition, the so-called Mini-element [5] or P1+/P1 element enhances the displacement approximation with a bubble function. Unfortunately it works only for simplicial finite elements. In the last fifteen years, efforts have then been made to find other efficient stabilization methods and the variationnal multiscale method emerged [6]. This method considers that the continuous displacement field can be split in two components: one coarse corresponding to the finite element scale, and a finer one, corresponding to different scales of resolution. In this work, the OSGS method is considered [7], [8]. In this paper, we describe the OSGS method at small strains and present some results. We then present a three-fields finite element formulation at large strains, with a hyperelastic model based on logarithmic strains. 2 THE OSGS METHOD AT SMALL STRAINS We remind that the formulation of the mechanical problem to deal with incompressible or quasi-incompressible behavior can be written in a mixed format considering the hydrostatic pressure p as an independent unknown, additional to the displacement field u. Let us consider a bounded region of R 3, and its boundary. Therefore, if inertial effects are neglected, the strong form of the mixed problem to be solved in can be formulated as { divσ D ) + p f = 0 divu) = p κ 1) where κ is the bulk modulus and σ D the deviatoric part of the Cauchy tensor. System 1 is subjected to appropriate Diritchlet and Neumann boundary conditions in terms of prescribed displacements u = ū on u, and prescribed tractions, t = σ.n on t respectively. The weak form of the problem can be expressed as 2

3 Find u, p) V P such as σ D u) + pi) : δud t 0.δudΓ Γ t divu p κ )δpd = 0 δu, δp) V P f.δud = 0 2) where V is the displacement space, V is the space of the functions of V which are equal to zero on the domain s boundary Γ u where displacements are prescribed, and P the pressure space. One can also rewrite the formulation by introducing the bilinear form B and the linear form L Find u, p) V P such as Bu, p), δu, δp)) = Lδu, δp), δu, δp) V P 3) The use of low order elements is convenient to reduce the computations cost, which is why we consider linear elements for both fields. The problem associated with this approach is that the stability Ladyjenskaia-Brezzi-Babuska LBB) condition is not respected. Sub-grid scales methods represent a set of methods to circumvent the LBB condition in order to use linear elements. The orthogonal sub-grid scale method OSGS) is such an efficient alternative that allows to use continuous linear interpolations for both displacements and pressure fields. The main advantage of this method is that, contrarily to the mini-element which allows only the use of triangles and tetrahedral elements, quadrangles and hexahedral elements can also be used. We propose to describe its principle and to present some results at small strains, obtained with the industrial code Code Aster). 2.1 Description of the method The basic idea of the sub-grid scale approach is to approximate the effect of the component of the continuous solution which cannot be captured by the finite element solution and is the cause of the volumetric locking. For this, one considers that continuous displacement field can be approximated considering two components, a coarse one, and a finer one, corresponding to different scales or levels of resolution [6]. We consider that the displacement field can be split in the finite element displacement field u h and a finer component ũ which will be approximated so that we can write u, p) = u h, p h ) + ũ, 0) 4) with ũ defined in a space Ṽ to be specified. Only the displacements are stabilized, so that the discretized form of the problem writes Find u h, p h ) V h P h and ũ Ṽ such as { Buh, p h ), δu, δp h )) + Bũ, 0), δu, δp h )) = Lδu, δp h ) δu, δp h ) V P h Bu h, p h ), δũ, 0)) + Bũ, 0), δũ, 0)) = Lδũ, 0)) δũ, 0) Ṽ 0 5) After some integrations by parts, and assuming that ũ is equal to zero on the domain s boundary, the first equation of 5) may be expressed as 3

4 σ D u h ) + σ D ũ)) : ɛδu h )d + p h divδu h ) = divu h p κ )δpd + ũ δpd = 0 t 0.δu h dγ t + Γ t and the second equation of 5) may be expressed as f.δu h d 6) div σ D ũ)))δũd = f.δũd + divσ D u h )) + p h )δũd. 7) Let us note PṼ the projection on space Ṽ, so that 7) may be written as PṼ div σ D ũ))) = PṼ divσ D u h )) + p h f) 8) Since we don t want to solve the finer scale problem 8), ũ has to be approximated. For this, we search a scalar τ such as div σ D ũ) ) τ 1 ũ 9) To have an expression of ũ that can be used, one has to define the space Ṽ. The Orthgonal Sub- Grid Scale method consists in choosing it as the space orthogonal to those of classical finite element, so that we have Ṽ = V h. By noting Ph the orthogonal projection on the space V h, 8) and 9) give, on each element with τ τ e on each element) : 10) can be expressed as ũ e = τ e P h div σ D ũ)) + p h f)). 10) where Π h is the projection of p h on P h. Let us consider ũ e = τ e p h π h ) 11) τ e = c 2µ h 2 ) 1, 12) where c is a numerical constant, h the charateristic length of the element and µ the shear modulus. Thus, by substituting ũ by its expression 11) as function of pressure gradient, the complete displacement decomposition 4) can be fed into the global equilibrium 11). Then, by calculating Π h by using the relation between the gradient of pressure and its projection, we get the stabilized problem 4

5 Find u h, p h, π h ) V h P h Π h such as σ D u h ) : ɛδu h )d h + p h divδu h ) t 0.δu h f.δu h = 0 h h Γ th Γ th divu h ) p nelem h h κ )δp hd h τ e p h π h ) δp h d h = 0 e=1 nelem τ e p h π h ) δπ h d = 0 e=1 u h, δp h, δπ h ) Vh P h Π h 13) with Π h = V h To solve these equations, we use the mixed finite element method. We denote N u, N p the shape functions respectively for the displacement and pressure fields, and U, P and Θ nodal variables, so that we have u h = nbnodes k=1 ndim n=1 N u k,nu k,n 14) p h = nbnodes k=1 N k P k 15) π h = nbnodes k=1 ndim n=1 N u k,nπ k,n 16) Due to some potential nonlinear dependences of the stresses over the displacements, the solution of the system of equations 13 requires the use of an appropriate incremental/iterative procedure such as the Newton Raphson method. Within such a procedure, the system of linear equations to be solved for the i + 1)th correction) equilibrium iteration of the n + 1)th prediction) time step is K uu K up K uπ K K pp K pπ K πu K πp K ππ n+1,i) δu δp δπ n+1,i+1) R u = R p R π n+1,i) 17) where δu, δp and δπ are the iterative corrections to the nodal values for the displacements, pressure and pressure gradient, respectively and R u, R p and R π the residual vectors. On each element, the tangent terms are expressed as K uu = B T : D : Bd 18) K unp = divn un )N p d = K pu) T 19) 5

6 K uπ = 0 = K uπ) T 20) K pp = 1 N p ) N p d τ ) e N p T N p d 21) κ K pπn = τ e N p N un d = K πp) T 22) K πnπm = τ e N un ) T N un d 23) where D is the tangent material stiffness tensor and B the classical derivation tensor D ijpq = σd ij u ε pq u 24) 2.2 First numerical results B ijkn = 1 2 N n x j δ ik + N n x i δ jk ) 25) The only drawback of this method is that the stability parameter depends on an unknown constant, that has to be determined with numerical tests for each case. The value of this parameter can have an influence on the quality of the result. As a result, it is important to determine it correctly. To avoid this problem, we determined an expression for the stability parameter based on the results obtained following a parametric study of the influence of the mesh structure and of the material properties. We drew to the conclusion that the mesh structure and the Poisson s ratio didn t have much influence onto the results quality. But we found out a proportional relation between the Young s modulus and the value of the constant. Then, the constant c expression was chosen as τ = h ) The results presented here were obtained with this choice of stabilization parameter. We propose here to consider two tests with elastic perfectly plastic behaviors : a block locally compressed, and a thick cylinder under pressure. The first benchmark proposed here consists of a square specimen 10 10m 2 ) in plane strains conditions. The loading condition consists of a prescribed pressure on the top plate of 1000P a in the vertical direction. The bottom plate is fixed. Young s modulus is E = 20MP a and the Poissons ratio is ν = A perfectly plastic model is assumed considering a yield stress σ Y = 150MP a. The proposed formulation, linear element for both pressure and displacements fields associated with OGSS method for tetrahedral elements P1/P1 and hexahedral elements Q1-Q1, is compared with the solution of the mixed quadrilateral displacements/ linear pressure element, referred to as P2/P1 element for tetrahedrals, and Q2/Q1 for hexahedrals. The pressure distribution at the gauss points is represented for hexahedral and tetrahedral elements in Figure 1. 6

7 a) b) c) d) Figure 1: Pressure distribution at Gauss points : with tetrahedral a) P2/P1 elements b) OSGS method ; with hexahedral c) Q2/Q1 elements d) OSGS method Figure 1 shows that the pressure distribution is regular for all types of elements. We can also note that the results obtained with OSGS method, for both hexahedral or tetrahedral elements are similar to those obtained with P2/P1 elements, which may be considered as a reference because this kind of elements fulfills the LBB condition and a stabilization is not necessary. The second benchmark consists of a thick elasto-plastic cylinder with an external radius of 200mm and an inner radius of 100mm. The loading condition consists of a prescribed pressure inside the cylinder of 0, 18GP a. Young s modulus is E = 210GP a and the Poisson s ratio is ν = 0.3. A perfectly plastic model is assumed considering a yield stress σ Y = 0, 24GP a. We compared the convergence performances for linear elements P1), quadratic elements P2), mixed elements with quadratic displacements and linear pressure P2/P1) and mixed linear elements with OSGS and with mini-element P1+/P1). For this, we calculated the absolute error in norm L2 of the stress with respect to the analytical solution. Figure 2: Error on stress L2-norm) as a function of mesh size 7

8 Figure 2 shows that, as expected, the convergence with linear elements is not good. The convergence obtained with mixed linear elements is good with OSGS method, and slightly better with mini-element. 3 A THREE-FIELDS MIXED FINITE ELEMENT FORMULATION TO DEAL WITH INCOMPRESSIBILITY AT LARGE STRAINS In order to generalize the three fields mixed finite element formulation to large deformations, we chose to use hyperelastic-based models to deal with finite strains. These models don t exhibit objectivity problems and dissipation in the elastic domain contrarily to the hypoelastic-based ones. We choose a model based on the logarithmic strain tensor, developped by Miehe, Apel and Lambrecht [9]. This model has been implemented in Code Aster and gives good results. Our goal is to make it work in quasi-incompressible cases. For this, we adapt it first to a threefields finite element formulation, and then to a two-fields stabilized finite element formulation. In this section, we describe the model and present the three-fields formulation. 3.1 Description of the large strains model The implementation of this model can be done in three steps : the first one corresponds to a geometric pre-processing in order to define the logarithmic strains, the second one consists in considering the constitutive model and the third one is a geometric post-processing in order to obtain the classical tensors and tangent moduli for engineering use. In the geometric pre-processing step, we define the logarithmic strain tensor denoted E as E = 1 2 lnf T F ) = 1 lnc) 27) 2 with C the right Green deformation tensor and F the deformation gradient. In the elastoplasticity case, we can use the Lee hypothesis F = F e F p 28) where F e is the elastic part of the deformation gradient, and F p its plastic part. Then, we obtain an additive decomposition of the logarithmic strain tensor in an elastic part E e and a plastic part E p, similar to the small strains case with E = E e + E p 29) E e = 1 2 lnf e F et ), E p = 1 2 lnf p F pt ) 30) The power with respect to the unit volume of the reference configuration of the material is defined as Pt) = P t) : Ḟ t) 31) 8

9 in terms of the first Piola stress tensor We can also express it as Pt) = T t) : Ėt) 32) where T is the stress tensor work-conjugate to the rate Ė of the logarithmic strains, so that we have T = P : de df 33) The second step consists in using a constitutive model to get the strain tensor T and the tangent constitutive moduli. This model is considered as a constitutive box with the logarithmic strains E and internal variables consisting of the logarithmic plastic strain tensor E p and some additional hardening variables, as inputs. The output of the box is the stress T and the tangent moduli E ep {E, E p, α} Model {T, E ep } 34) The tangent moduli governs the stress rate with respect to the logarithmic strain rate Ṫ = E ep : Ė 35) The third and last step is the geometric post-processing which consists in recovering the standard stress tensors and their associated tangent moduli, by geometric transformation. In fact, for the second Piola-Kirchhoff stress tensor S and it associated elastic-plastic tangent moduli, we get the formulations S = T : 2 E C, and Cep = 2 E ) T : E ep : 2 E ) ) + T : 4 2 E C C C 2 36) The Lagrangian elastic-plastic tangent moduli C ep governs the sensitivity of the second Piola- Kirchhoff stress 3.2 A three-field finite element formulation Ṡ = C ep : 1 2Ċ 37) We propose to adapt the finite strains model presented above, to a three-field finite element formulation involving displacements u, pressure p and volume change θ. In this goal, we enhance the deformation gradient F with a scalar field permitting to mesure the volume and that is related weakly to the volume variation J. Several relations are possible, we choose lnj) = θ 38) 9

10 with J the gradient deformation s determinant. In that case, the enhanced deformation gradient F can be written ) 1/3 expθ) F = F 39) J We then define the enhanced right Green strain tensor C C = F F T 40) The structure equilibrium is obtained by the minimum of potential energy Eu, θ) Eu, θ) = Ψ C)d 0 W ext u) 0 41) with W ext the potential of external efforts and Ψ the stored energy function expressed in terms of the mixed right Green deformation tensor We introduce the Lagrange mutliplier p, which permits to impose the weak relation between J and θ. We then look for saddle points of the Lagrangian Π Using the relation Π = Eu, θ) + plnj) θ)d ) S = 2 Ψ C 43) the first Lagrangian variation can be expressed as δπ = 0 [ ) ] 1 2 S : δ C δj + p J δθ + δp lnj) θ) d 0 44) We introduce the displacement Eulerian gradient δl δl = x δu 45) Using the above notation 45), the Lagrangian variation may be expressed as δπ = 0 [ T F S F ) ] D + p 1 : δld 0 + δθ 0 [ 1 3 tr T F S F ) ] p d 0 + δp lnj) θ) d 0 46) 0 where F S F T ) D denotes the deviatoric part of F S F T, and 1 the second rank identity tensor 10

11 Thus, the residuals are δπ) u = 0 δπ) θ = δθ 0 [ T F S F ) ] D + p 1 : δld 0 47) [ 1 3 tr T F S F ) ] p d 0 48) δπ) p = δp lnj) θ) d ) To solve these equations, we use the mixed finite element method. We approximate the continuous spaces with discretized adapted spaces, in which there is the approximated solutions u h, p h, θ h ). Domain 0 is discretized by the mesh T h 0 ). We denote N u, N p and N θ the shapes functions respectively for the displacement, pressure and volume change fields, and U, P and Θ the nodal variables. Thus, for each element, the residuals are given by δπ) u = δπ) θ = N θ [ T F S F ) ] D + P 1 : Bd 0 50) [ 1 3 tr T F S F ) ] P N p d 0 51) δπ) p = N p lnj) ΘN θ) d 0 52) where B is the second rank tensor which is defined in each node as Resolving the equation δl ij = δl ij = nbnodes k=1 nbnodes k=1 δu ki N u k x j 53) δu ki B kj δπ = 0 54) requires the use of an appropriate incremental/iterative procedure such as the NewtonRaphson method. Within such a procedure, the system of linear equations to be solved for the i + 1)th correction) equilibrium iteration of the n + 1)th prediction) time step is 11

12 K uu K up K uπ K K pp K pπ K πu K πp K ππ n+1,i) δu δp δθ n+1,i+1) δπ) u = δπ) p δπ) θ n+1,i) where δu, δp and δθ are the iterative corrections to the nodal values for the displacements, pressure and volume variation, respectively and R u, R p and R π the residual vectors. The tangent terms are expressed as K uu = + B T B T + B T B ) : where c may be expressed by [ B T B T T ) : F S F ) ]) D + p 1 d + T F S F ) D 2 de 3 BT : K up = N p 1 : Bd = [ 2 K uθ = 3 N θ B T T : F S F ) D + c D : 1] = 2 K θθ = 9 55) ) ) 2B T D D : c : Bde 56) e T 1 B : F S F ) ) D d K pu ) T 57) K θu ) T 58) ) [ N θ T T tr F S F ) + 1 : c : 1] N θ d 59) ) T K θp = N θ N p d = K θp 60) K pp = 0 61) c = F F C ep F T F T 62) and we remind that C ep is the tangent material stiffness, calculated in 37). This formulation has been implemented in the industrial code Code Aster and the results will be presented soon. 4 CONCLUSIONS This paper presents the formulation of mixed linear/linear elements with OSGS method for incompressible elasticity and plasticity. Numerical examples show that results are free of volumetric locking and pressure oscillations, and qualitatively comparable to those obtained with mini-element or mixed quadratic/linear elements. A three-fields mixed element formulation at large strains, with an hyperelastic model based on logarithmic strains has been proposed. The results will be presented soon. The reduction of the formulation to a two-fields mixed element formulation is in progress. We also want to extend the stabilization methods to the large strains model presented. 12

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