INCOMPATIBLE BUBBLES: A GENERALIZED VARIATIONAL FRAMEWORK INCLUDING INCOMPATIBLE MODES AND ENHANCED ASSUMED STRAINS

Transcription

1 European Congress on Computational Methods in Applied Sciences and Engineering ECCOMAS 2004 P. Neittaanmäki, T. Rossi, S. Korotov, E. Oñate, J. Périaux, and D. Knörzer (eds.) Jyväskylä, July 2004 INCOMPATIBLE BUBBLES: A GENERALIZED VARIATIONAL FRAMEWORK INCLUDING INCOMPATIBLE MODES AND ENHANCED ASSUMED STRAINS Ignacio Romero and Manfred Bischoff Departamento de Medios Continuos y Teoría de Estructuras E.T.S.I. Caminos, Universidad Politécnica de Madrid Profesor Aranguren s/n, Madrid, Spain iromero@mecanica.upm.es, web page: ignacio Lehrstuhl für Statik Technische Universität München D München, Germany bischoff@bv.tum.de, web page: Key words: Finite element technology, locking, enhanced strain, incompatible modes. Abstract. A finite element approach for the solution of two-dimensional linear elasticity problems, called the Incompatible Bubble method, is presented. It is based on an extension of the classical method of Incompatible Modes to a class of finite elements equivalent to the Enhanced Assumed Strain method. In the latter methods an ad-hoc modification of the Jacobian, namely its evaluation at the element center in the computation of element matrices associated with incompatible displacements or enhanced strains, respectively, ensures consistency, i. e. satisfaction of the patch test. The method of incompatible bubbles avoids this variational crime by ensuring consistency by a special geometry-dependent design of the incompatible displacement functions (the incompatible bubbles ). Numerical experiments confirm locking-free behavior of the resulting elements and their close relationship to existing concepts. 1

2 1 INTRODUCTION The four-node two-dimensional solid element by Wilson [9] (Q6) has been one of the first plane strain/stress finite elements avoiding shear locking and volumetric locking. It is based on the method of Incompatible Modes, employing additional higher order displacement functions (quadratic functions in the case of a bilinear element). The corresponding parameters are assumed to be incompatible across element edges thus facilitating their elimination on the element level. The global number of degrees of freedom and related numerical expense are thus identical to standard Galerkin elements. The problem of inconsistency (Q6 does not satisfy the patch test) has been resolved later by Taylor et al. [8], introducing an ad-hoc modification of the isoparametric map associated with the incompatible displacements. The idea is to evaluate the Jacobian at the element center in the computation of quantities associated with incompatible displacements. This applies to both the isoparametric map of incompatible displacements and the definition of the infinitesimal area element in the corresponding integrals. This modification ensures satisfaction of the patch test, hence consistency of the method. The resulting element is usually referred to as Qm6 in the literature. Much later, Simo et al. [6] presented an extension of this concept to what is known today as the Enhanced (Assumed) Strain method (EAS). Instead of using incompatible displacements, here additional strain parameters are introduced on the basis of a modified version of the three-field Hu-Washizu functional. The EAS method is more general than the method of Incompatible Modes, comprising a number of different element formulations, one of which, Q1E4, is equivalent to the classical Qm6. The present work is motivated mainly by the wish to shed some light on the mathematical interrelation of these meanwhile well-established concepts of element technology in structural mechanics to stabilization methods which have emerged mainly in the context of fluid dynamics and the variational multiscale approach. Graphically, the incompatible modes (or enhanced strains, respectively) can be interpreted as representing the small scales. More rigorously, it is known since [2] that there exist similarities between stabilized methods and the residual free bubble approach. Apparently, residual free bubbles and incompatible modes have in common that additional higher order functions on the element level are introduced, altering element behavior in a (hopefully) advantageous way. The method of Incompatible Bubbles, introduced in this paper, may be a first step towards more insight into the aforementioned interrelations. The main feature of the method is that an ad-hoc modification of the Jacobian is obsolete, still retaining consistency of the resulting finite elements. This is accomplished by a sophisticated choice of the incompatible bubble functions. As an intermediate result, we introduce an Extended Incompatible Modes method and prove its equivalence to the Enhanced Assumed Strain concept. To the authors best knowledge this has not been described earlier. 2

3 2 PROBLEM STATEMENT In this section we summarize the equations of the boundary value problem of linearized elasticity in two dimensions restricted to elastic, isotropic, homogeneous materials. The numerical methods employed throughout the article are applicable for more general materials as well as three dimensional domains. However, for the sake of conciseness, we will restrict our analysis to the class of problems described. Let be a polygonal open set in R 2 with points denoted by x. Its boundary Γ has outward normal vector n and can be partitioned into two disjoint sets Γ d and Γ n, so that Γ = Γ n Γ d and Γ n Γ d =. A displacement function u : R 2 is the solution to the linearized elasticity problem if it verifies the following boundary value problem with the definitions div σ + f = 0 in u = 0 on Γ d (1) σ n = t on Γ n ε = S [u] := 1 2 [ u + ( u)t ], σ = C : ε. (2) The second order tensor ε is the infinitesimal strain; σ, the Cauchy stress tensor; C the tensor of material elasticities; f are the external body forces, and t the tractions applied on Γ n. The notation u indicates the gradient of the vector u. 3 SUMMARY OF SOME EXISTING FINITE ELEMENT FORMULATIONS FOR THE ELASTICITY PROBLEM In this section we summarize three well established finite element formulations for the solution of the boundary value problem (1): the standard Galerkin formulation, the Incompatible Modes (IM) method, and the Enhanced Assumed Strain (EAS) method. For all the numerical methods employed in this article a finite element mesh needs to be defined. This mesh consists of a partition of the domain into n el quadrilateral elements, each of them occupying a subdomain, e = 1, 2,..., n el and connected at n node nodes. For each element, the isoparametric map χ e is an invertible mapping from the biunit square, denoted, to the element domain and it is defined by χ e (ξ, η) = 4 N i (ξ, η)x i e, (ξ, η) = [ 1, 1] [ 1, 1], (3) i=1 where the functions N i are the standard Lagrange shape functions in the biunit square, and x i e are the nodal coordinates of the ith node of element e (see, e.g., [4, chapter 3] for an elaboration of the interpolation process and the isoparametric concept). For each point x, there corresponds a unique point in the biunit square of coordinates (ξ, η) 3

4 and we write x = χ e (ξ, η). We denote F e the jacobian of the map χ e and j e = det(f e ), its determinant. Later, we will need to refer to the jacobian and its determinant evaluated at the center of the element and we will denote them as F e,o and j e,o, respectively. 3.1 STANDARD DISPLACEMENT FORMULATION We briefly describe the displacement formulation for problem (1). This is the standard and most basic finite element approximation for the problem of interest (see, i.e.[4]) Variational equations and finite element formulation The standard Galerkin formulation is based on the following variational principle, known as the principle of virtual work. Let V be the functional space: V = { v : R 2, v [H 1 ()] 2, v = 0 on Γ d }. (4) Then, the weak solution of problem (1) is the function u V such that σ : δε d = f δu d + t δu dγ, (5) Γ n for all δu V and δε = S [δu]. To construct the Galerkin approximation of the previous variational problem we define the approximation space { } n V h = v h : node R 2, v h (x) = n a (x)v a, v h = 0 on Γ d, (6) where the n a are the shape functions defined in terms of the coordinates x, that is n a e = n i e = N i χ 1 e, a being the global node with local number i in element e. The Galerkin approximation of the principle of virtual work is the function u h V h such that σ h δε h d = f δu h d + t δu h dγ, (7) Γ n for all δu h V h, with σ h = C : ε h, ε h = S [u h ] and δε h = S [δu h ]. The strain tensor can also be calculated with the chain rule as ε h = [ ] ξ (u h χ e ) F 1 S e, (8) where ξ is the gradient operator with respect to the coordinates (ξ, η). a=1 4

5 3.1.2 Matrix expression Let U e denote the vector of nodal displacements of the nodes belonging to element e and, with an abuse of notation, refer with the same symbols ε h to the matrix (Voigt) expression of the strain tensor. Then we have the following matrix expressions for the displacement and strain fields inside an element: u h (x) = N e (x)u e, ε h (x) = B e (x)u e. (9) e e The matrix N e contains the shape functions and the matrix B e is the discrete strain operator. We will use the symbol C to denote the matrix expression of the elasticity tensor C. Using the previous matrix identities, the contribution of element e to the global stiffness matrix of the problem and the external force vector can be derived from the principle of virtual work (7) by a standard argument, giving K STD e = B T e CB e d, F STD e = N T e f d + N T t e dγ. Γ n (10) 3.2 THE INCOMPATIBLE MODES FORMULATION In [9], a modification on the standard quadrilateral element was proposed with the goal of improving its performance in what it was referred to as bending dominated situations. The modification consisted in the addition of several non-conforming shape functions at the element level which resembled the bending modes of a quadrilateral. The corresponding extra degrees of freedom could be statically condensed to yield an element with the same number of degrees of freedom as the standard displacement element, but improved accuracy. The element, with the correction described in [8], performs very well in those bending dominated situations and, moreover, does not lock in the incompressible limit. See also [7] for a mathematical analysis of this formulation. For each element e, the bending modes are displacement functions m 1 e, m 2 e, m 3 e, m 4 e vanishing outside. Denoting ũ h the total displacement due to the incompatible modes, we can write ũ h 4 (x) = m i e e(x)ũ i e, (11) i=1 where ũ i e is the degree of freedom associated with the i-th incompatible mode of the element. In [9], the components of the bending modes in a Cartesian coordinate system are given explicitly as functions m i = m i e χ e on the biunit square. [ ] [ ] [ ] [ ] ξ m 1 = η, m 2 = 0 ξ 2, m 3 = 2 1, m 4 0 = 1 0 η 2. (12) 1 5

6 The infinitesimal strain field corresponding to the incompatible modes is calculated in each element with equation (8) giving ε h e = S [ũ h ] = [ ξ (ũ h χ e ) F 1 e ] S = 4 a=1 [ ξ m a F 1 e ] S ũ a e. (13) The previous expression can be written more compactly in matrix form as ε h (x) = B e (x)ũe, (14) e where Ũe is the column vector containing are the degrees of freedom of element e associated to the bending modes. The lack of continuity of the global interpolation functions across element boundaries indicates that the corresponding formulation will be non-conforming. To accommodate to this incompatibility, a different variational principle than (5) must be defined. For this, consider the functional spaces: { } M h = ũ h : R 2, ũ h 4 (x) = m i e e(x)ũ i e,, (15) i=1 W h = { u h : R 2, u h (x) = û h (x) + ũ h (x), with û h V h and ũ h M h}, and the following variational principle: Find u h W h such that n el e=1 σ h : δε h d = f δû h d + t δû h dγ, Γ n (16) for all δu h = δû h + δũ h W h, with σ h = C : S [û h + ũ h ] and δε h = S [δû h + δũ h ]. Remarks 1 1. The space W h of trial functions includes incompatible displacements and thus the integral of the internal virtual work can not be computed as defined in (5) and it must be replaced with the element-wise sum defined in (16). This is a well known example of a variational crime ([7]). 2. Note that, in contrast with the variational principle of the standard displacement formulation, the terms involving the virtual work of the external forces and tractions do not contain the full virtual displacement δu h but only its compatible part. By choosing in (16) a virtual displacement belonging to V h, a equation is obtained which is almost identical to the standard Galerkin variational equation (7), σ h δˆε h d = f δû h d + t δû h dγ, (17) Γ n 6

7 where and δˆε h = S [δû h ]. The only difference between this last equation and the standard Galerkin variational equation is the definition of the stress tensor σ h. Next, choosing in (16) a virtual displacement in M h with support only in element e, we obtain σ h δ ε d = 0, (18) where σ h is as before and δ ε = S [δũ h ] Matrix equations Equations (17) and (18) define the method of incompatible modes as described originally in [9]. In matrix form they read n el A e=1 [ ] B T e C(B e U e + B e Ũ e ) d N T e f d N T t e dγ = 0, Γ n BT e C(B e U e + B e Ũ e ) d = 0, where we have used the standard notation for the assembly operator. The second of these equations can be solved for the degrees of freedom Ũe by static condensation and introduced in the first one. Defining the auxiliary matrices L e = BT e CB e d, De = BT e C B e d, (20) the equivalent element stiffness matrix of the method and the external forces vector finally become: K IM e = B T e CB e d L T 1 e D e e, F IM e = N T e f d + N T t e dγ. Γ n (21) Remarks 2 1. The constant part of the incompatible shape functions m i, i = 1,..., 4 plays no role in the equations of the method because only their derivatives appear in the element stiffness matrix (through the B matrix). 2. The nonzero entries of the strain matrix B e consist of derivatives of the incompatible displacements with respect to the Cartesian coordinates (x 1, x 2 ). However, as currently defined, the resulting finite element formulation is not consistent for arbitrarily distorted meshes. To make the formulation consistent, Taylor [8] proposed to redefine the strains of the incompatible displacement as ε h ε := j e,o j e 7 (19) [ ξ (ũ h χ e )F 1 e,o] S, (22)

8 instead of (13). This ad hoc definition of the derivative, and therefore of the matrix B e, can find no justification within the variational framework furnished by the principle of virtual work, however it is sufficient to ensure the consistency of the method even with distorted meshes. This modification amounts to replacing the discrete strain matrix B e with a modified matrix B e such that ε (x) = B e(x)ũe. (23) A consequence of this remark is that the final form of the IM method corresponds to the matrix equations (19) where the strain matrix B e is replaced by B e. 3.3 THE ENHANCED STRAIN FORMULATION The class of Assumed Enhanced Strain methods proposed by Simo and Rifai [6] provided a variational framework in which to construct a general class of finite element formulations for elasticity. As a particular instance, it recovered the method of Incompatible Modes as described in the previous section, justifying the variational crimes previously identified. Moreover, it allowed the formulation of new finite elements with even better properties than the Incompatible Modes element. We refer to the cited reference for details on the formulation of the method, which we now summarize. To describe the Enhanced Strain method we consider the functional space (4) for the displacements together with two new spaces for the enhanced strains and stresses, respectively: E h = { ε h, } ε h : ε h d <, T h = { σ h, } σ h : σ h d <. (24) It is important to note that neither the stresses in T h nor the enhanced strains in E h need to be continuous fields. In particular, they can be defined element by element. The weak form of the Enhanced Strain method is obtained as a stationary point of a Hu-Washizu three-field functional and reads: Find the displacement u h V h, the enhanced strain ε h E h and the stresses σ h T h such that s (δu h ) : C : ( S u h + ε h ) d = f δu h d + t δu h dγ, Γ n δ ε h : C : ( S u h + ε h ) d = 0, (25) ε h : δσ h d = 0, for all δu h V h, δ ε h E h, δσ h T h. Conditions for stability and convergence of this method were identified in [6] and amount to certain conditions on the test spaces: Let S V h denote the space of strains of displacements in V h. Then s V h E h = {0}. 8

9 The space T h contains, at least, the functions which are constant in each. Stresses in T h are energy orthogonal to enhanced strains in E h, i.e. σ h : ε h d = 0 for all σ h T h, ε h E h. Under the conditions stated, the third equation in (25) is trivially verified. In order to express the remaining two equations in matrix form let us define the enhanced strain in an element e to be of the form: ε h (x) = G e (x)α e, (26) e where the precise definition of the matrix G e of enhanced strains is left open for the moment. With these definition, the matrix form of equations (25) becomes n el A e=1 [ ] B T e C(B e U e + G e α e ) d N T e f d N T t e dγ = 0, Γ n G T e C(B e U e + G e α e ) d = 0, As in the Incompatible Modes method, the second of these equations can be solved, in each element, for the enhanced strain parameters α e. Defining the matrices: L e = G T e CB e d, D e = G T e CG e d, (28) the equivalent element stiffness matrix of the Enhanced Strain method and the corresponding external forces vector are: K EAS e = B T e CB e d L T e D 1 e L e, F EAS e = N T e f d + N T t e dγ. Γ n (29) The equations of the Enhanced Assumed Strain method (29) are identical to those of the Incompatible Modes method (21) up to the definition of the matrices L e, D e in the first method, and L e, D e, in the second. In fact, the definitions of these matrices are also identical (compare (20) with (28)), and the only difference between them is that in the EAS method the matrix G e is used to define them whereas the matrix B e is employed in the IM formulation. As long as the three conditions imposed by stability and consistency of the method are not violated, any matrix basis for the space of enhanced strains can be chosen to define an EAS method. In particular, Simo and Rifai[6] proposed that any enhanced strain be of the form ( ε h χ e )(ξ, η) = j e,o j e (ξ, η) F T e,o E(ξ, η)f 1 e,o, (30) 9 (27)

10 for some strain field E defined on the biunit square. In matrix form, the previous equation reads (G e χ e )(ξ, η) = j e,o j e (ξ, η) Po e E(ξ, η) (31) where P o e is the matrix that transforms strains in the biunit square to strains in the physical domain, evaluated at the center of the biunit square. The matrix E(ξ, η) is a matrix of enhanced strain modes defined also on the biunit square. The design of EAS elements amounts then to the choice of the columns in E. In particular, the choice ξ E(ξ, η) = E 4 (ξ, η) := 0 η 0 0, (32) 0 0 ξ η gives a method that can be shown to be identical to the Incompatible Modes. However, the EAS method is more general for it includes elements that can not be recovered by adding more compatible or incompatible modes to the standard displacement element. The reason for this is that the infinitesimal strain operator mapping smooth displacements to symmetric second order tensors is not onto. For example, the following Enhanced Strain matrices were proposed in [6] and [1], respectively, and can not be recovered as strain matrices for displacement fields: ξ ξη ξ ξη 0 0 E 5 (ξ, η) := 0 η 0 0 ξη, E 7 (ξ, η) := 0 η ξη 0. (33) 0 0 ξ η ξ 2 η ξ η 0 0 ξη 4 AN EXTENDED INCOMPATIBLE MODES FORMULATION In this section we present a generalization of the Incompatible Modes formulation which we will refer to as Extended Incompatible Modes (XIM). In some cases, the proposed formulation recovers existing EAS elements, as the ones described in the previous section. There are two fundamental results in this section: a theorem that identifies the condition for equivalence of the two formulations and an explicit identification of nonconforming modes (or bubbles) that can verify the equivalence condition. The variational framework of the Extended Incompatible Modes method is the same as for the original Incompatible Modes formulation: a modified principle of virtual work that can accommodate the discontinuity of the interpolation functions across element edges and that ignores the non-conforming part of the external virtual work. See equation (16). The new ingredient of the method is the choice of an extended family of incompatible deformation modes such that the resulting method behaves essentially equally to any EAS element. As in the case of the IM formulation, the starting point of the proposed method is an additive decomposition of the displacement field and its variations into a compatible and an incompatible part. The displacement field thus obtained is of the form: u h (x) = û h (x) }{{} + ũ h (x) }{{} compatible incompatible 10. (34)

11 The compatible part belongs to the space V h defined in (6) and the incompatible part is defined to be, inside each element, n bub (ũ h χ e )(ξ, η) = F T e,o b a (ξ, η)ũ a e, (35) where F e,o is the Jacobian of the isoparametric map evaluated at the center of the element, b a, a = 1,..., n bub are incompatible displacement functions vanishing outside and ũ a e are the degrees of freedom associated with each of these modes. Precise definitions of these modes will be given in the examples of this section. As in the case of the IM formulation, the XIM method is not consistent for distorted meshes if the field of infinitesimal strains ε h associated with the incompatible modes is evaluated according to equation (8). The lack of consistency of the newly proposed formulation is corrected using the same ad hoc modification as in the IM formulation, as described in remark 2. This modification consists in replacing the standard definition of the strain field associated with the incompatible modes by expression (22). For the incompatible displacement (35) the modified strain field results in [ 4 ] ε h e ε := j e,o F T e,o S ξ m a ũ a e F 1 e,o. (36) j e By comparing the previous definition of the strain with the strain of the EAS method (equation (30)) we conclude that the matrix form of the strain field (36) can be written as ( ε h χ e )(ξ, η) = ( B e χ e )(ξ, η)ũe = j e,o j e (ξ, η) Po e H(ξ, η) Ũe, (37) where H is the matrix of strains in the biunit square defined by a=1 a=1 H(ξ, η)ũe = 4 S ξ m a (ξ, η)ũ a e (38) a= Matrix equations The methods of IM and XIM are based on the same variational principle and thus the matrix equations of the two methods are identical, up to the definitions of the discrete strain matrices of the non-conforming displacements. The equations of XIM can be summarized as: K XIM e = B T e CB e d L T e e L e = B,T e CB e d, De = 1 D L e e, F XIM e = B,T e C B e d. 11 N T e f d + N T t e dγ, Γ n (39)

12 4.1 Equivalence of XIM and EAS formulations The non-conforming formulation proposed in this section can be shown to be related to the EAS method as described in the following theorem: Theorem 1 Consider an EAS element with a matrix of m enhanced modes E(ξ, η) and an XIM element with m incompatible bubbles and strain matrix H(ξ, η) defined in (38). 1. If for each element e there exists a constant invertible matrix Q e such that E(ξ, η) = H(ξ, η)q e, (40) for all (ξ, η) in the biunit square, then the two formulations are identical. 2. Assume now that the integrals needed to evaluate the external force vector and the stiffness matrix of the formulation are evaluated using numerical quadrature over n int integration points (ξ 1, η 1 ), (ξ 2, η 2 ),..., (ξ nint, η nint ) in the biunit square. If for each element there exists a constant invertible matrix Q e such that the relation (40) holds for every integration point (ξ g, η g ), g = 1, 2,..., n int, then the two formulations give identical results when the integrals of the discrete equations of the method are computed using numerical quadrature. Proof: To prove equivalence of the two formulations we show that their force vectors and stiffness matrices are identical. By comparing the expressions of the force vector and stiffness matrix of the EAS and XIM formulations (equations (29) and (39), respectively) we observe that the only difference between the two methods is that the stiffness matrix of the first method uses the matrix G e (x) where the stiffness matrix of the second method employs the matrix B e(x). From the definition of these matrices in equations (31) and (37) it follows that the condition (40) of the theorem implies that G e (x) = B e(x)q e. From the definitions of the matrices L e, D e, L e and D e in equations (20) and (28) it follows that: L e = G T e CB e d = Q T e BT e CB e d = Q T L e e, e (41) D e = G T e CG e d = Q T e BT e C B e d Q e = Q T D e e Q e. Using these relations and the definition of the stiffness matrix of the Enhanced Strain method we can show that the stiffness matrices of this method coincides with the stiffness matrix of the Extended Incompatible Modes method: K EAS e = K STD e = K STD e = K STD e = K XIM e, L T e D 1 e L e L T e Q e Q 1 e L T 1 e D L e e 12 D 1 e Q T e Q T e L e (42)

13 which concludes the proof of the first part of the theorem. The second statement is proven identically replacing the integrals in the definition of the matrices and vectors by numerical quadrature. The previous theorem shows that an EAS method with matrix G e can be identically recovered from an XIM formulation if one can find compatible or incompatible bubbles (modes) whose strain matrix B e has the same range as G e. This result does not tell that any EAS method can be recovered as an XIM method, but it turns out that actually all of the EAS formulations used in practice have a corresponding XIM counterpart, at least when numerical quadrature is employed. Since the original IM element is known to be equivalent to the EAS method with modes (32) it is only left to show that the 5 and 7-modes EAS elements can be obtained from incompatible bubbles. This is shown in the next two examples Example 1 Consider the 5 parameter EAS formulation with E 5 matrix given by equation (33) 1 and consider the XIM element with 5 incompatible displacement bubbles b i, i = 1,..., 5. The components of the first four bubbles are given in equation (12), and those of the fifth mode are [ ] L b 5 2 (ξ)l 1 (η) (ξ, η) = L 1 (ξ)l 2. (43) (η) The notation L i (ξ) is used to denote the ith Legendre polynomial in the variable ξ (and likewise for η) which are L 1 (ξ) = ξ, L 2 (ξ) = 1 2 (3ξ2 1). (44) A direct calculation shows that the corresponding strain matrix H for these incompatible modes is: 2ξ ξη H = H 5 := 0 0 2η 0 3ηξ. (45) 3 0 2ξ 0 2η 2 (ξ2 η 2 ) It does not exist a matrix Q e such that E 5 (ξ, η) = H 5 (ξ, η)q e, for all (ξ, η) as required in the first part of the theorem to prove that the two methods described are equivalent. However, if numerical quadrature is employed to integrate the element matrices, and more specifically Gaussian quadrature with 2 2 integration points, such a matrix Q e exists. To see this, observe that the (3, 5) component of matrix H 5 is just L 2 (ξ) L 2 (η). Likewise, the (3, 5) component of matrix E 5 is 2 3 (L2 (ξ) L 2 (η)). However, Legendre polynomials of order two have the property that they are identically zero at Gauss quadrature points of order two. We conclude that both terms in the numerically integrated matrices H 5 and 13

14 E 5 are zero. At any quadrature point (ξ g, η g ) of the Gaussian 2 2 rule we have ξ g ξ g η g 2ξ g ξ g η g E 5 (ξ g, η g ) = 0 η g 0 0 ξ g η g, H 5 (ξ g, η g ) = 0 0 2η g 0 3η g ξ g. 0 0 ξ g η g 0 0 2ξ g 0 2η g 0 (46) For each of the quadrature points these two matrices verify E 5 (ξ g, η g ) = H 5 2 (ξ g, η g )Q e, with Q e = (47) Thus, by the previous theorem, the two methods will give identical results when numerical quadrature is employed. Example 2 Consider the EAS element with 7 enhanced strain modes and E 7 matrix given by equation (33) 2. Consider next the XIM element with 7 incompatible modes b i, i = 1,..., 7, where the first four are again the incompatible modes (12). The last three modes are defined as [ ] [ ] ] L 2 (ξ)l 1 (η) b 5 (ξ, η) = 0, b 6 (ξ, η) = 0 L 1 (ξ)l 2 (η), b 7 (ξ, η) = [ L 1 (ξ)l 2 (η) L 2 (ξ)l 1 (η) The strain matrix H 7 that corresponds to the incompatible modes defined is: 1 2ξ ξη 0 H(ξ, η) = H 7 2 (3η2 1) (ξ, η) := η 0 0 3ηξ 2 (3ξ2 1). (49) 1 0 2ξ 0 2η 2 (3ξ2 1 1) 2 (3η2 1) 6ξη As in the previous example, there is no matrix Q e that relates matrices E 7 (ξ, η) and H 7 (ξ, η) as required to prove equivalence. Nevertheless, as before, if Gaussian quadrature is employed with two integration points per direction, the matrix H 7 simplifies at any of these points (ξ g, η g ) to be: 2ξ g ξ g η g 0 0 H 7 (ξ g, η g ) := 0 0 2η g 0 0 3η g ξ g 0. (50) 0 2ξ g 0 2η g 0 0 6ξ g η g At these quadrature points the matrices E 7 (ξ g, η g ) and H 7 (ξ g, η g ) are related by a constant invertible matrix Q e, as can be easily verified. This shows that the EAS formulation with seven enhanced strain modes and the XIM method with seven modes are identical if numerical quadrature is employed to evaluate the integrals involved in the formulation. 14 (48)

15 5 A VARIATIONALLY CONSISTENT INCOMPATIBLE BUBBLES FOR- MULATION The variational crimes of the method of Incompatible Modes or its generalization XIM presented in section 4 can be justified when they are interpreted as particular cases of the Enhanced Strain formulation. However, it would be desirable to have methods of comparable accuracy that can be completely explained in the context of the principle of virtual work. In this section we propose a solution for this question, identifying the source of the crimes, discussing the remedy proposed by [8] and suggesting a new alternative, termed method of Incompatible Bubbles (IB). A general framework in which we could try to understand the method of Incompatible Bubbles is the subgrid scale formulations provided, for example, by the Variational Multiscale Method ([5]) and the related Residual Free Bubbles (e.g. [3]). In all these formulations the solution to the problem of interest is split into a coarse scale part (which we have denoted û h ) and a fine scale component (ũ h in the previous sections). Accepting this ansatz in the context of the principle of virtual work and an identical split of the admissible variations one is led to the equations: σ h δˆε h d = f δû h d + t δû h dγ, Γ n (51) σ h δ ε h d = f δũ h d + t δũ h dγ, Γ n for all coarse scale variations δû h and subgrid scale variations δũ h. In the IM and IB methods the fine scale part of the solution and its variations are chosen a priori to be a linear combination of bubbles, i.e. functions with support smaller or equal to an element. This choice makes the second equation in (51) decouple into n el independent ones of the form σ h δ ε h d = f δũ h d + t δũ h dγ. Γ n (52) In the method of Residual Free Bubbles, the subgrid scale functions are chosen to vanish at element boundaries. However, in the current analysis we propose to enlarge this set and accept bubbles with nonzero value at element edges. The resulting method is incompatible and certain modifications must be made to the equations in order to make the method convergent. The first modification imposed by the incompatibility of the test functions is that the first integral over the domain in equation (51) 1 must be replaced by a sum over all the element domains as in (16), to avoid singularities on the edges. The resulting method might not be consistent. An effective way to test consistency of the formulation is the patch test originally proposed by Irons and discussed, for example, in [7]. As argued in 15

16 [8], the patch test imposes the following restriction on incompatible formulations such as the one considered: for every incompatible displacement ũ h, ε h d = 0, (53) with ε h being the strains associated with ũ h, obtained either with the standard formula (2) or by a modified expression. One of such modifications that enforces this condition by construction is (22), proposed in [8]. We now explore a different approach which consists in choosing, for each element, incompatible bubbles in such a way that the patch test is automatically satisfied. Of course, the precise form of these bubbles can not be known a priori and will depend on the element shape, as it will be shown next. For this, consider a known basis b 1, b 2,..., b m for the bubbles in an element. These could be, for example, the vectors with Legendre polynomials components, as discussed in section 4. The idea consists in constructing, for each element, m modified bubbles given by: b i (x) = b i (x) x 1 vol( ) where vol( ) = b i,x 1 d x 2 vol( ) b i,x 2 d i = 1,..., m (54) d. By construction, each of these bubbles satisfies b i d = 0, (55) and thus, for every incompatible displacement of the form ũ h (x) = n bub a=1 ba (x)ũ a, defined over, the patch test condition (53) reads: n bub ε h d = S b a d ũ a = 0. (56) a=1 The procedure outlined above gives the guidelines to formulate elements with include incompatible displacement fields which, moreover, are consistent for arbitrarily distorted meshes and do not employ ad hoc definitions for the discrete strain operator. The resulting formulations can incorporate four, five or seven incompatible bubbles per quadrilateral, as defined in section 4. 6 NUMERICAL EXPERIMENTS We want to mention at the beginning of this section that all previously mentioned equivalences and interrelations between the various finite element formulations investigated in this paper have been verified numerically. This comprises standard tests like the patch test and comparison of element stiffness matrices for arbitrary geometries. As there is no use in merely listing identical numbers we do without a further documentation of the corresponding experiments. 16

17 6.1 Eigenvalue Analyses The original motivation to develop alternative formulations to standard Galerkin finite elements is to avoid locking. For two-dimensional solid elements volumetric locking is probably the most important phenomenon. Apart from that, shear locking and trapezoidal locking may occur which are not further discussed herein. They may become prominent when curved, thin-walled structures are analyzed using solid elements. Volumetric locking emanates from an over-constrained stiffness matrix and is especially pronounced for nearly incompressible materials along with plane strain conditions. A general tendency of an element to exhibit volumetric locking can be detected by an eigenvalue analysis of the element stiffness matrix. For a four-node bilinear element we expect three zero eigenvalues, representing the rigid body modes and five non-zero eigenvalues, corresponding to in-plane deformation modes. In a locking-free element only one of these eigenvalues tends to infinity in the incompressible limit ν 0.5. If there is more than one infinite eigenvalue the element tends to lock. We investigate the case of a non-rectangular element with nodal coordinates x 1 = 0.0, x 2 = 1.0, x 3 = 1.5, x 4 = 0.2, y 1 = 0.0, y 2 = 0.2, y 3 = 2.3, y 4 = 0.8. (57) As a first reference, the results of an eigenvalue analysis of the stiffness matrix obtained by the standard displacement element Q1 for material parameters E = 70.0, ν = 0.0 are reported. The vector of eigenvalues is λ Q1 = [ ]. As Poisson s ratio is zero these values correspond to an element behavior which is free from volumetric locking. The fact that the zero eigenvalues are not exactly zero and one is even negative can be traced back to round-off errors. Changing Poisson s ratio to ν = changes the numerical results to λ Q1 = [ ]. Obviously, the first three values increase dramatically. One of the corresponding eigenvectors (deformation modes) represents the volumetric mode. The other two are in-plane bending modes and the large values indicate volumetric locking. For the Extended Incompatible Mode elements we obtain λ XIM4 = [ ], λ XIM5 = [ ], λ XIM7 = [ ], pointing towards locking-free behavior in either case as two of the large eigenvalues, namely the ones corresponding the in-plane bending modes, are small now. The results 17

18 obtained with Q1E4, Q1E5 and Q1E7 are similar, apart from different round-off errors. The results obtained with Incompatible Bubble elements are λ IB = [ ], confirming their locking-free behavior. 6.2 Driven cavity problem The idea for this numerical experiment is borrowed from fluid mechanics and represents a tough test towards the tendency of finite elements to exhibit volumetric locking. A rectangular plate under plane strain conditions is subject to clamped boundary conditions at three edges, the displacements at the remaining edge are constrained in normal direction and a constantly distributed load acts in tangential direction. The system is investigated using a regular and a distorted mesh. Meshes and problem data are given in Figure 1. Figure 1: Driven cavity problem, meshes, loads and boundary conditions. The results for the horizontal displacement u at the center of the upper edge are documented in Figure 2. The most obvious observation is that the standard Galerkin formulation shows a strong dependence of the results on the bulk modulus κ = E. 3(1 2ν) Displacements tend to zero as the bulk modulus approaches infinity in the incompressible limit ν 0. The Enhanced Strain and Incompatible Bubble elements are locking-free, i. e. 18

19 Figure 2: Driven cavity problem, results. the aforementioned dependence on ν is avoided. Moreover, the results of Q1E4, Q1E7 and Q1IB are identical, as predicted by the theoretical considerations in the previous sections. For the case of a distorted mesh the results of the locking-free elements are no longer exactly coincident. However, the numerical results are so close that the corresponding curves are practically indistinguishable. Moreover it can be observed from comparison to the results obtained with the undeformed mesh, represented by a dashed line, that mesh distortion does not have a strong impact on the results in this case. 7 SUMMARY A new finite element formulation, termed Incompatible Bubbles method has been presented, derived on the basis of a generalization of the classical method of Incompatible Modes. The crucial difference to both the method of Incompatible Modes and the Enhanced Strain method is that an ad hoc modification of the element Jacobian is avoided while retaining consistency and satisfaction of the patch test. Numerical performance of these elements turns out to be practically identical to that of Enhanced Strain elements. Exact equivalence is attained for special element geometries. Though being of mainly theoretical merit this result may represent a first step towards more insight into the nature of Enhanced Strain elements and their interrelation to Residual Free Bubble methods, stabilized methods and the variational multiscale approach. 19

20 REFERENCES [1] U. Andelfinger and E. Ramm. EAS-elements for two-dimensional, three-dimensional, plate and shell structures and their equivalence to HR-elements. Int. Journal of Numerical Methods in Engineering, 36: , [2] F. Brezzi, L. P. Franca, T. J. R. Hughes, and A. Russo. b = g. Computer Methods in Applied Mechanics and Engineering, 145: , [3] F. Brezzi, D. Marini, and A. Russo. Applications of the pseudo residual free bubbles to the stabilization of convection-diffusion problems. Computer Methods in Applied Mechanics and Engineering, 166:51 63, [4] T. J. R. Hughes. The finite element method. Prentice-Hall Inc., Englewood Cliffs, NJ, [5] T. J. R. Hughes, G. R. Feijoo, L. Mazzei, and J.-B. Quincy. The variational multiscale method a paradigm for computational mechanics. Computer Methods in Applied Mechanics and Engineering, 166:3 24, [6] J. C. Simo and M. S. Rifai. A class of mixed assumed strain methods and the method of incompatible modes. Int. Journal of Numerical Methods in Engineering, 29(8): , [7] G. Strang and G. J. Fix. An analysis of the finite element method. Prentice Hall, Englewood Cliffs, N.J., [8] R. L. Taylor, P. J. Beresford, and E. L. Wilson. A nonconforming element for stress analysis. Int. Journal of Numerical Methods in Engineering, 10(6): , [9] E. L. Wilson, R. L. Taylor, W. P. Doherty, and J. Ghaboussi. Incompatible displacement models. In A. R. Robinson S. J. Fenves, N. Perrone and W. C. Schnobrich, editors, Numerical and Computer Models in Structural Mechanics, pages 43 57, New York, Academic Press. 20