Mortar finite element methods allow for a flexible and efficient coupling of arbitrary .... that the tangential plane in each slave surface point x(1) locally defines an R2 ... denotes the Euclidean norm in R3, µ ⥠0 is the friction coefficient and ...... where the square matrix block Dâ associated with the slave corner nodes (i.e. the.
DUAL QUADRATIC MORTAR FINITE ELEMENT METHODS FOR 3D FINITE DEFORMATION CONTACT A. POPP1 , B.I. WOHLMUTH2 , M.W. GEE3 AND W.A. WALL1 Abstract. Mortar finite element methods allow for a flexible and efficient coupling of arbitrary non-conforming interface meshes and are meanwhile quite well-established in nonlinear contact analysis. In this paper, a mortar method for 3D finite deformation contact is presented. Our formulation is based on so-called dual Lagrange multipliers, which in contrast to the standard mortar approach generate coupling conditions that are much easier to realize, without impinging upon the optimality of the method. Special focus is set on second-order interpolation and on the construction of novel discrete dual Lagrange multiplier spaces for the resulting quadratic interface elements (8-node and 9-node quadrilaterals, 6-node triangles). Feasible dual shape functions are obtained by combining the classical biorthogonality condition with a simple basis transformation procedure. The finite element discretization is embedded into a primal-dual active set algorithm, which efficiently handles all types of nonlinearities in one single iteration scheme and can be interpreted as a semi-smooth Newton method. The validity of the proposed method and its efficiency for 3D contact analysis including Coulomb friction are demonstrated with several numerical examples. Key words. mortar finite element methods, dual Lagrange multipliers, contact, finite deformations, second-order interpolation
1. Introduction. Computational contact analysis in the regime of finite deformations has received much attention in recent years owing to its great relevance in many fields of engineering and the applied sciences. Among the most important challenges that have to be met with respect to finite element discretization is the question how to treat arbitrarily non-matching meshes at the interfaces between different bodies. Mortar methods, which were originally introduced as an abstract domain decomposition technique [2, 4], have proven to serve as a very convenient computational framework for contact analysis, especially when considering large deformations and sliding motions. The characteristic feature of mortar methods is the imposition of interface constraints in a weak sense instead of a strong, pointwise enforcement. In the context of contact analysis this allows for a variationally consistent treatment of non-penetration and frictional sliding constraints using Lagrange multipliers despite the inevitably non-matching interface meshes. One crucial ingredient of any mortar scheme is the definition of a suitable discrete Lagrange multiplier space. In the vast majority of publications, this space is simply based upon the trace space of the underlying finite element discretization, and this choice will be termed standard Lagrange multipliers in the following. Without claiming to be exhaustive, successful applications of such mortar methods using standard Lagrange multipliers for 2D and 3D contact analysis can be found in [3, 7, 8, 26, 27, 39]. An alternative approach, commonly known as dual Lagrange multipliers, was analyzed in [33] and heavily faciliates the treatment of mortar coupling conditions at the interface, while at the same time preserving the optimality of the method. The simplification is mainly achieved by chosing dual shape functions based on a biorthogonality condition, which avoids the necessity of solving a mass matrix 1 Institute for Computational Mechanics, Technische Universit¨ at M¨ unchen, Boltzmannstrasse 15, D–85748 Garching b. M¨ unchen, Germany. 2 Institute for Numerical Mathematics, Technische Universit¨ at M¨ unchen, Boltzmannstrasse 3, D–85748 Garching b. M¨ unchen, Germany. 3 Mechanics and High Performance Computing Group, Technische Universit¨ at M¨ unchen, Boltzmannstrasse 15, D–85748 Garching b. M¨ unchen, Germany.
1
2 system for the interface coupling problem. Extensions of the dual Lagrange multiplier approach to small deformation contact [16, 17], finite deformation mesh tying [25] and finite deformation contact [10, 23, 24] as well as efficient (multigrid) solution techniques [19, 36] have followed over the last years. However, mortar methods for 3D finite deformation contact have so far mainly been investigated in the context of first-order finite elements, although quadratic finite element interpolation offers several appealing features for contact analysis, such as better approximation of curved surfaces. Standard Lagrange multipliers have been successfully treated in [28]. On the other hand, the 3D second-order case turns out to be especially challenging for dual Lagrange multiplier interpolation. The only existing contributions are given in [20, 21], showing that unlike for standard Lagrange multipliers the extension of dual Lagrange multipliers to all types of quadratic interface meshes is a non-trivial task and requires special basis transformation procedures. However, investigations there are limited to the domain decomposition case and neither unilateral contact nor finite deformations are considered. In the present contribution, we therefore first propose two new procedures for defining suitable dual Lagrange multipliers for quadratic finite elements in 3D. Based upon ideas from [20, 21], we extend the focus to finite deformation contact analysis including Coulomb friction and also demonstrate the applicability of the newly developed dual shape functions in conjunction with our existing mortar framework [10, 23, 24]. As result, an algorithm is obtained that preserves all advantages of second-order mortar contact schemes using standard Lagrange multipliers [28], while at the same time adding the benefits of dual Lagrange multiplier interpolation. The remainder of this paper is organized as follows. A short overview of the finite deformation contact problem is given in Section 2, whereas mortar finite element discretization is described in Section 3. Then, Section 4 concentrates on discrete Lagrange multiplier spaces and introduces two new alternatives for second-order interpolation in 3D. The remaining parts of our contact algorithms, including the primal-dual active set strategy, are outlined in Section 5. The proposed methods are evaluated by several numerical examples in Section 6 and some conclusions are drawn in Section 7. (1)
(2)
2. Problem setting. Let the open sets Ω(1) , Ω(2) ⊂ R3 and Ωt , Ωt ⊂ R3 represent two bodies in the reference and current configuration, respectively. Retaining a common nomenclature in contact mechanics, the two bodies are referred to as slave (superscript (1)) and master (superscript (2)). The surfaces ∂Ω(i) , i = 1, 2 are (i) (i) (i) (i) divided into three disjoint boundary sets Γu , Γσ and Γc , where Γu is the Dirichlet (i) (i) boundary, Γσ is the Neumann boundary, and Γc represents the potential contact (i) (i) surface. Their counterparts in the current configuration are denoted as γu , γσ and (i) γc . A generalization of the problem statement to the case of multiple bodies or self contact is possible without conceptual differences. 2.1. Strong formulation. The boundary value problem of quasi-static finite deformation elasticity requires the displacement vectors u(i) = x(i) − X(i) , which describe the motion of the two deformable bodies from the reference configuration X(i) to the current configuration x(i) , to satisfy the following set of equations: (i)
Div (FS)
(i)
ˆ =0 +b 0 ˆ (i) u(i) = u (i)
(PN)
(i)
=ˆ t0
in Ω(i) × (0, T ) , on Γu(i) × (0, T ) , on Γσ(i) × (0, T ) , i = 1, 2 ,
(2.1)
3 where t ∈ [0, T ] plays the role of a pseudo-time. The material deformation gradient and the first and second Piola–Kirchhoff stress tensors are represented by F(i) , P(i) and S(i) , respectively. Prescribed displacements on the Dirichlet boundary are repre(i) ˆ (i) and prescribed tractions on the Neumann boundary by ˆ sented by u t0 . Moreover, (i) ˆ (i) denotes a body force per N(i) is the reference unit outward normal on Γσ , and b 0 (i) undeformed unit volume on Ω . Without loss of generality, we exemplarily consider an isotropic hyperelastic material behavior S=
∂Ψ ∂2Ψ , C= , ∂E ∂E2
(2.2)
represented by the strain energy function Ψ and the fourth order constitutive tensor C, which introduces a possibly nonlinear constitutive relationship between the second Piola–Kirchhoff stress tensor S and the Green–Lagrange strain tensor E = 1 T 2 F F − I . Specifically, the St.Venant–Kirchhoff model and a simple compressible Neo–Hookean model (see e.g. [14]) will be employed to characterize the material behavior in the examples in Section 6. In order to describe the non-penetration condition and the frictional sliding conditions of the two bodies, a scalar normal gap gn and a relative tangential velocity vector vτ are introduced as h i ˆ (2) (X(1) , t), t) , ˆ (2) (X gn = −n · x(1) (X(1) , t) − x (2.3) h i ˆ (2) (X(1) , t), t) , ˆ˙ (2) (X vτ = (I − n ⊗ n) · x˙ (1) (X(1) , t) − x (2.4) (1)
where n represents the current unit outward normal vector on the slave surface γc in (2) (1) x(1) , see Figure 2.1. Moreover, x ˆ(2) is the result of a smooth mapping P : γc → γc (2) (1) (2) ˙ ˆ of x onto the master surface γc along n. We point out that x is the velocity ˆ (2) , i.e. of the material contact point for X(1) at time t, and of the material point X ˆ (2) itself. therefore does not include a change of the contact point X ξ η Together with the two tangent vectors τ and τ , n forms an orthonormal basis (1) in x(1) . Note that the contact surface γc is a two-dimensional manifold, which means that the tangential plane in each slave surface point x(1) locally defines an R2 space embedded into the global R3 . Thus, the projection of the relative velocity into this manifold can be characterized via only two components vτξ and vτη as vτ = vτξ τ ξ + vτη τ η .
(2.5) (1)
Similar to the relative velocity, the contact traction tc on the slave surface can then be separated into normal and tangential parts, yielding ξ ξ η η t(1) c = pn n + t τ = pn n + tτ τ + tτ τ .
(2.6)
Altogether, we can summarize the classical Karush–Kuhn–Tucker (KKT) conditions of normal contact and the frictional conditions according to Coulomb’s law as gn ≥ 0 ,
pn ≤ 0 ,
pn gn = 0 ,
ψ := ktτ k − µ |pn | ≤ 0 ,
vτ + β tτ = 0 ,
(2.7) β ≥ 0,
ψβ = 0.
(2.8)
Here, k · k denotes the Euclidean norm in R3 , µ ≥ 0 is the friction coefficient and β ∈ R is a scalar parameter. Further details on contact kinematics and frictional sliding can be found in [18, 22, 38].
4
Fig. 2.1. Illustration of the notation for a two body 3D finite deformation contact problem.
2.2. Weak formulation. For the sake of simplicity, we restrict the derivation of a weak formulation to the frictionless contact problem here. The focus of this paper is on suitable discrete Lagrange multiplier spaces and the choice of these multipliers is completely independent of the precise tangential contact model. However, Coulomb friction is included in our actual implementation, and a suitable numerical example will be provided in Section 6. Moreover, it is pointed out that a more general problem description including the entire weak formulation of Coulomb friction can readily be found in the recent review article [35]. For frictionless sliding, the tangential part of the contact traction vanishes and thus the set of conditions (2.8) is simply replaced by tτ = 0 .
(2.9)
To start the derivation of a weak formulation of (2.1), the solution space U (i) and the weighting space V (i) are defined as n o (i) (i) (i) U (i) = uj ∈ H 1 Ω(i) , j = 1, 2, 3 | uj = u ˆj on Γu(i) , (2.10) n o (i) (i) V (i) = δuj ∈ H 1 Ω(i) , j = 1, 2, 3 | δuj = 0 on Γu(i) . (2.11) It is well-known that the KKT conditions (2.7) along with the frictionless sliding conditions (2.9) or any frictional contact model can be incorporated by defining a vector-valued Lagrange multiplier λ on the slave side of the contact interface. We point out that the choice which side carries the Lagrange mutlipliers is arbitrary at this point and only becomes important in the discrete setting. As for the definition of the contact traction in (2.6), we denote the normal and tangential parts of λ as λn n and λτ , respectively. This enables the definition of a convex cone of Lagrange multipliers [17] as n o (1) (1) M+ = λ ∈ M | λτ = 0 , hλn , δuj iγ (1) ≥ 0 , δuj ∈ W + , (2.12) c
5 (1)
where M is defined as dual space of the trace space W (1) of V (1) restricted to Γc . Moreover, h· , ·iγ (1) represents the duality pairing between M and W, which in turn c denote single scalar components of the corresponding o vector-valued spaces M and n (1) (1) + W. Finally, we have W = δuj ∈ W | δuj ≥ 0 . The weak solution of the contact problem can now be obtained from a mixed variational formulation, which is derived by applying the method of weighted residuals followed by integration by parts. Identifying the negative contact traction on the slave (1) side of the interface with the Lagrange multiplier vector λ = −tc and exploiting the balance of linear momentum across the contact interface yields the following saddlepoint like problem formulation: Find u(i) ∈ U (i) and λ ∈ M+ such that 2 X
(i)
(i)
∇δu , (FS)
Ω(i)
i=1
ˆ (i) − δu , b 0 (i)
+ δu(1) − (δu(2) ◦ P ), λ
(1)
Ω(i)
(i) − δu , ˆt0 (i)
(i)
(2.13)
Γσ
= 0 ∀ δu(i) ∈ V (i) ,
γc
supplemented by hδλn − λn , gn iγ (1) ≥ 0 c
∀ δλ ∈ M+ .
(2.14)
The constraint (2.14) will be treated by a primal-dual active set strategy (PDASS) or, to be more precise, by its reformulation as a semi-smooth Newton method as presented in great detail for both frictionless and frictional finite deformation contact in [10, 23, 24]. In this regard, it should be emphasized again that the normal gap gn is a nonlinear function of the displacements u(i) due to the fully nonlinear problem formulation. Spatial discretization of (2.13)–(2.14) by a mortar finite element method will be roughly outlined in Sections 3 and 4, whereas a detailed and comprehensive derivation is again to be found in [10, 23, 24]. For the sake of completeness, we would like to mention a notational detail: the last term in (2.13), which represents contact virtual work, is typically formulated in the current configuration for the considered finite deformation contact problems, while all other terms (representing internal and external virtual work) are formulated in the reference configuration. 3. Mortar finite element discretization. For the spatial discretization of the frictionless contact problem (2.13)–(2.14), standard isoparametric finite elements based on hexahedral, tetrahedral or mixed 3D meshes are employed. This defines (i) (i) the usual finite dimensional subspaces U h and V h being approximations of U (i) and V (i) , respectively. In this paper, we focus on second-order interpolation, which is characterized by 20-node or 27-node hexahedral elements as well as 10-node tetrahedral elements. Consequently, facets on the contact surfaces are typically 8-node or 9-node quadrilaterals and 6-node triangles. In a general form, displacement interpolation on the contact surfaces is given by (1)
(1) uh |Γ(1) c,h
=
n X
(2)
(1) Nk
k=1 (1)
(2)
(1) dk
,
(2) uh |Γ(2) c,h
=
n X
(2)
Nl
(2)
dl
(3.1)
l=1
with the shape functions Nk , Nl . The total numbers of slave nodes is n(1) and the total number of master nodes is n(2) . The discrete nodal displacements are denoted (1) (2) as dk and dl . In addition, an adequate discretization of the Lagrange multipliers λ
6 representing tractions on the contact interface is needed. This is based on a discrete + multiplier space M+ h being an approximation of M . Here, we distinguish between two possible families, namely standard and so-called dual Lagrange multipliers, where our focus will be on the latter. A general notation reads λh =
(1) m X
Φ j zj ,
(3.2)
j=1
with the shape functions Φj and the discrete nodal Lagrange multipliers zj . The total number of slave nodes carrying additional Lagrange multiplier degrees of freedom is m(1) . Typically, every slave node also serves as Lagrange multiplier node, i.e. m(1) = n(1) . However, in the context of quadratic finite elements it will be favorable to chose m(1) < n(1) in certain cases, see Sections 4.3 and 4.4.2. Standard Lagrange multipliers represent the classical approach for mortar methods [2, 4] and are usually taken from a (1) (1) finite dimensional subset W h ⊂ W (1) being the trace space of V h . Thus, standard mortar methods lead to identical shape functions for Lagrange multiplier and slave (1) displacement interpolation, i.e. Φj = Nj . It is important to point out that unlike for the general mortar setting, no modification of the Lagrange multiplier shape functions (1) at the boundaries of Γc is necessary for the contact problems considered here, as we (1) (1) assume that Γc ∩ Γu = ∅. In contrast, the dual approach is motivated by the observation that the Lagrange multipliers represent fluxes on the interface in the continuous setting. This duality argument is reflected by constructing dual Lagrange multiplier shape functions [33] (1) based on a biorthogonality condition with the displacements in W h , which assures that the Φj are again well-defined. Details on how to construct dual Lagrange multiplier shape functions in general, and for second-order finite element interpolation in 3D in particular, will be presented in Section 4. Moreover, we note that it has been shown in [30] in the context of a Laplace operator based model problem and standard Lagrange multipliers, that for conforming finite elements of polynomial degree p, piecewise polynomials of only degree p − 1 can be used instead of degree p in the defintion of the Lagrange multiplier space without losing the optimality of the discretization error. Piecewise polynomials of only degree p − 2, however, lead to a predicted deterioration of the discretization error by O(h1/2 ) measured in the energy norm. For quadratic finite elements, this means that only O(h3/2 ) instead of an optimal O(h2 ) convergence in the energy norm can be expected for such a choice of the discrete Lagrange multipliers. While the above observations apply to classical mortar domain decomposition, the contact case needs some further theoretical explanations. We stress that the solution of a contact problem is typically in H t (Ω(i) )3 with t < 52 and thus no better a priori estimates than O(h3/2 ) can be expected due to the regularity of the solution. Moreover, the inequality constraint of the contact formulation does not allow for higher-order a priori estimates, regardless of the selected discretization. These preliminary remarks motivate our approach to (1) use uniformly stable inf-sup pairings (W h , Mh ) having reproduction properties of order (2, 0) in Section 4.4. The idea of reducing the polynomial degree and thus the reproduction order for the definition of discrete Lagrange multipliers is one important ingredient for deriving feasible dual Lagrange multiplier shape functions. Substituting (3.1) and (3.2) into the weak form (2.13) now yields the discrete quasi-static force equilibrium fint (d) + B(d) z = fext ,
(3.3)
7 where all discrete nodal values of the displacements and the Lagrange multipliers are summarized into the global vectors d and z, respectively. Nonlinear internal forces fint (d) and external forces fext result from standard finite element discretization that is abundantly discussed in the literature and will not be repeated here. For defining the discrete mortar operator B(d) and thus the nonlinear vector of contact forces, all finite element nodes in the two domains Ω(1)h and Ω(2)h are split into three subsets: a subset S containing all n(1) potential slave side contact nodes, a subset M of all n(2) potential master side contact nodes and the set of all remaining nodes N . The global displacement vector can be sorted accordingly, T yielding d = (dS , dM , dN ) . Then B has the form B = [D
−M
T
0]
,
(3.4)
where the entries of the two mortar matrices D and M are given as Z (1) D[j, k] = Djk I3 = Φj Nk dγ I3 , j = 1, ..., m(1) , k = 1, ..., n(1) ,
(3.5)
(1)
γc,h
Z M[j, l] = Mjl I3 =
(2)
(1)
γc,h
Φj (Nl
◦ Ph ) dγ I3 ,
j = 1, ..., m(1) , l = 1, ..., n(2) . (3.6) (1)
(2)
Note that I3 ∈ R3×3 is the identity matrix and Ph : γc,h → γc,h defines a discrete approximation of the actual contact mapping P (see e.g. [7, 25]). In general, both D and M are of rectangular shape, however D becomes a square matrix for the common choice m(1) = n(1) . Similarly, discretized versions of the non-penetration conditions (2.14) and the frictionless sliding conditions incorporated in (2.12) can be derived. All details of this procedure can be found in our recent work [23, 24]. Here, we restrict the presentation to the final results, which comprise for every Lagrange multiplier slave node j = 1, ..., m(1) the discrete weak constraints g˜n,j ≥ 0
zn,j ≥ 0 ,
zn,j g˜n,j = 0 ,
zτ,j = 0.
(3.7)
It is easy to see that the constraints in (3.7) are simply discrete analogons of (2.7) and (2.9). Normal and tangential parts of the nodal Lagrange multipliers zj are defined just like the corresponding parts of the Lagrange multiplier vector λ in Section 2. Quantities with a tilde (such as g˜n,j ) indicate that they result from the mortar typical integration over the contact surface with a weighting introduced by the respective Lagrange multiplier shape function Φj . Equations (3.3) and (3.7) constitute the final discrete formulation of finite deformation contact with frictionless sliding in 3D. With some constraints still being stated as inequalities, the application of a suitable active set strategy beomces necessary, see Section 5 and [10, 17, 23, 24]. Let us also point out that a detailed description of mortar finite element discretization in the context of large sliding and Coulomb friction, including a frame-indifferent formulation of the discrete relative tangential velocity, can be found in [10, 39]. 4. Discrete Lagrange multiplier spaces. This section discusses different possibilities for the choice of the discrete Lagrange multiplier space M+ h and thus for the associated shape functions Φj in (3.2). In terms of accuracy, robustness and efficiency of the numerical method, this choice may be crucial. The three paragraphs 4.1–4.3 give a short overview of well-known discrete Lagrange multiplier spaces used for unilateral 3D contact problems in the literature and of their respective properties. Thereafter, two new alternatives of dual Lagrange multiplier spaces for second-order finite element interpolation in 3D will be proposed and analyzed in paragraph 4.4.
8 4.1. First-order interpolation: Standard Lagrange multipliers. Firstorder interpolation with standard Lagrange multipliers by far represents the most common discretization strategy in all mortar settings, ranging from classical domain decomposition for elasticity (e.g. [25]) to finite deformation contact (e.g. [26, 27]). (1) Multipliers are taken from a finite dimensional subset W h ⊂ W (1) of the trace space W (1) and thus every slave node also carries Lagrange multiplier degrees of freedom (m(1) = n(1) ). The shape functions Φj are identical to the displacement shape (1) functions Nj on the slave surface. For first-order finite elements in 3D, i.e. p = 1, this yields strictly positive linear (3-node triangles) or bilinear (4-node quadrilaterals) Lagrange multiplier shape functions of reproduction order pλ = 1. 4.2. First-order interpolation: Dual Lagrange multipliers. First-order interpolation based on dual Lagrange multipliers is also well-analyzed in the general mortar setting (e.g. [25, 33, 34]) and for unilateral contact (e.g. [11, 17, 24]). Dual shape functions Φj are readily constructed with the biorthogonality condition Z Z (1) (1) Φj Nk dγ = δjk Nk dγ (4.1) (1)
γc,h
(1)
γc,h
in the current configuration, where again m(1) = n(1) is chosen. Note that for distorted elements in general, and thus for finite deformation contact in particular, these shape functions depend on the deformation of the underlying finite element and thus can not be defined a priori, but result from simple element-local calculations (see e.g. [9, 11]). This yields non-positive, discontinuous Lagrange multiplier shape functions, and the reproduction order of the global Lagrange multiplier interpolation is reduced as compared with the standard case in Section 4.1. Yet, the element-local calculations assure that at least the constant polynomials are contained in the global Lagrange multiplier interpolation, i.e. pλ = 0. The interested reader is referred to [9] for the corresponding proof. However, it has been demonstrated in numerous investigations [25, 33] and also proven theoretically [30] that this choice has no negative influence on the optimality of the discretization error. Like for standard multipliers, one obtains O(h) convergence in the energy norm when dealing with Laplace operator based model problems, linear elasticity or frictionless contact problems. The main advantage that can be drawn from dual Lagrange multipliers is that the coupling conditions at the mortar interface are much easier to realize as compared with standard Lagrange multipliers. This is due to the fact that we obtain nodal displacement basis functions on the slave side of a mortar interface which have only local support [33]. Algebraically, this advantageous property of dual Lagrange multipliers can be observed by the mortar matrix D in (3.5) reducing to a diagonal matrix. 4.3. Second-order interpolation: Standard Lagrange multipliers. Consistent treatment of unilateral contact in the context of second-order finite elements and mortar methods is not straightforward in three dimensions. The reason lies in the use of weighted integrals for the variational formulation of the non-penetration condition in (2.14) combined with the inequality nature of this constraint. It is obvious, that the discrete non-penetration condition (3.7) should yield a positive weighted gap g˜n,j if the value of the unweighted physical gap function evaluated at slave node j is positive and vice versa. Otherwise, the numerical algorithm will generate non-physical gaps and penetrations, which either leads to unacceptable errors in the solution or to a non-converging active set strategy. Thus, we require the Lagrange multiplier shape
9 functions Φj to at least satisfy integral positivity, i.e. Z ! Φj dγ > 0.
(4.2)
(1)
γc,h
While the condition formulated above is readily satisfied by both standard and dual Lagrange multipliers for first-order interpolation, the second-order cases need further investigation. We have to distinguish between three different types of surface discretizations here: 9-node quadrilaterals (quad9 ), 8-node quadrilaterals (quad8 ) and 6-node triangles (tri6 ). Finite element shape functions for quad9 surfaces all satisfy condition (4.2) and can therefore be applied directly for standard Lagrange multiplier interpolation, which yields pλ = p = 2. However, certain finite element shape functions of quad8 and tri6 surfaces do not satisfy (4.2) and would thus not allow for an unmodified use in contact algorithms. As an example, the shape functions N1 associated with a corner node of a quad8 element and of a tri6 element both violate (4.2). Examining the standard reference quadrilateral (−1, 1)2 and the reference triangle with corners (0, 0), (1, 0) and (0, 1) in local coordinates ξ and η, we obtain Z Z 1 quad8 N1 (4.3) dγ = − (1 − ξ)(1 − η)(1 + ξ + η) dγ < 0 , 4 ele Zele Z N1tri6 dγ = (1 − ξ − η)(1 − 2ξ − 2η) dγ = 0 . (4.4) ele
ele
This argument motivates a simple but effective modification of standard Lagrange multiplier shape functions for mortar contact with second-order interpolation. Instead of choosing m(1) = n(1) and the shape functions Φj identical to the displacement shape functions Nj , a Lagrange multiplier interpolation based on strictly positive linear/bilinear polynomials (i.e. of reduced reproduction order pλ = 1) has been suggested and successfully applied to finite deformation contact analysis in [28]. Only the corner nodes carry discrete Lagrange multipliers in that case (m(1) < n(1) ). Moreover, this choice does not only satisfy (4.2), but pointwise non-negativity (Φj ≥ 0) which is even more preferable when dealing with the typical inequality constraints for contact. Note that in classical domain decomposition problems (e.g. linear elasticity with mortar mesh tying) only equality constraints have to be taken into account and thus the integral positivity condition (4.2) does not play a role as important as in the unilateral contact analysis. Numerical investigations in this context have demonstrated optimal spatial convergence O(h2 ) of the discretization error measured in the energy norm for both described choices, i.e. for quadratic/biquadratic (pλ = 2) and linear/bilinear (pλ = 1) standard Lagrange multiplier interpolation [28]. 4.4. Second-order interpolation: Dual Lagrange multipliers. In the following paragraph, we derive new dual Lagrange multiplier shape functions for 3D mortar contact analysis with second-order finite elements. Quite similarly to the standard multiplier case outlined above, we aim at deriving two suitable alternatives, i.e. quadratic/biquadratic dual Lagrange multipliers (m(1) = n(1) ) as well as a linear/bilinear version (m(1) < n(1) ). Having to combine both the biorthogonality condition (4.1) and the integral positivity requirement (4.2) makes this a non-trivial task, and to the best of our knowledge no comparable investigations exist in the literature. The basic idea is a generalization of the approach presented in [20] for plane interfaces and undistorted finite elements to a general finite deformation contact setting. It is easy to verify that a naive application of the biorthogonality condition (4.1)
10 to construct dual Lagrange multiplier shape functions would fail in the cases of quad8 and tri6 elements, because one would obtain Lagrange multiplier shape functions that do not satisfy (4.2). Moreover, for tri6 elements, the biorthogonality condition (4.1) does not even make sense when considering the result in (4.4). 4.4.1. Locally quadratic dual Lagrange multipliers. As a first step towards defining suitable quadratic dual shape functions, a simple basis transformation for the displacement shape functions Nj and nodal degrees of freedom on the slave surface is ˜j satisfy the integral introduced, which guarantees that the modified shape functions N positivity condition (4.2). As a consequence, dual Lagrange multiplier shape functions Φj constructed from these modified displacement shape functions and the standard biorthogonality condition (4.1) will also satisfy (4.2). This requirement still leaves a lot of freedom in arranging the actual basis transformation. Based on the usual node numbering in second-order finite elements, where first all corner nodes are listed and afterwards all edge nodes, one possible approach for tri6 elements is the following: T ˜ T N1 N1 1 ˜2 N2 0 N ˜ N 3 = N3 · 0 N4 α N ˜4 N5 0 N ˜5 ˜ N α 6 ele N6 ele |
0 1 0 α α 0
0 0 1 0 α α
0 0 0 0 0 0 1 − 2α 0 0 1 − 2α 0 0 {z :=Tele
0 0 0 , 0 0 1 − 2α }
(4.5)
with the transformation matrix Tele ∈ R6×6 and the index ele indicating a slave element. Computation of the inverse transformation matrix is straightforward, yielding 1 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0α . T−1 = (4.6) α 1 ele − 1−2α − 1−2α 0 0 0 1−2α α α 1 0 − 1−2α − 1−2α 0 0 1−2α α α 1 − 1−2α 0 − 1−2α 0 0 1−2α Investigating (4.5) and (4.6) in more detail, it becomes obvious that we have chosen the structure of the transformation matrix to be as simple as possible. As the shape functions associated with the edge nodes of tri6 R elements are strictly positive anyway, ˜j > 0 also for the corner nodes. To they can be beneficially used to guarantee ele N R ˜j > 0 for the edge nodes, we must guarantee that α < 1 . simultaneously preserve ele N 2 The proposed transformation is symmetric in the sense that edge nodes yield equal contributions to their two adjacent nodes, and the modified shape functions P6 corner ˜j = 1. This choice is by no means based on satisfy partition of unity, i.e. j=1 N restrictions, but it rather simplifies the biorthogonality construction later on. The quad8 case can be derived in analogy, with the numbers of edge and corner nodes simply changing from three to four. The effect of the proposed basis transformation for both tri6 and quad8 elements is illustrated in Figure 4.1. For the sake of completeness, we mention that no basis transformation is needed in the case of quad9 surfaces, because the corresponding shape functions Nj already satisfy (4.2), see [24]. Finally, the scalar α needs to be determined. Of course, α has to be chosen large enough to guarantee integral positivity of the modified displacement shape functions.
11
Fig. 4.1. Displacement shape functions for a quad8 element (left) and for a tri6 element (right). In each case, the unmodified shape functions N are illustrated in blue / dark grey and the modified ˜ in green / light grey for an exemplary corner node and edge node. shape functions N
Fig. 4.2. Displacement and Lagrange multiplier shape functions for a quad8 element (left) and ˜ are illustrated in green / for a tri6 element (right). In each case, the modified shape functions N light grey and the dual shape functions Φ in red / dark grey for both an exemplary corner node and an exemplary edge node.
It can easily be verified that in the case of undistorted finite elements, this requires α > 0 for tri6 elements and α > 18 for quad8 elements. A second criterion is elaborated 1 in [20], where the authors show that for undistorted finite elements with α = 12 1 (tri6 elements) and α = 5 (quad8 elements), respectively, it is even possible to have the piecewise linear polynomials included in the resulting dual Lagrange multiplier space M+ h . As mentioned in Section 4.2, the biorthogonality construction usually only guarantees that the constant polynomials are included in the dual Lagrange multiplier space (pλ = 0). However, with the focus of this contribution being on finite deformation contact, we are confronted with significant element distortions and not only with the special case of undistorted elements treated in [20]. This makes it impossible to chose α such that the piecewise linear polynomials are always included in the resulting dual Lagrange multiplier space. Moreover, mixed meshes with both tri6 and quad8 surface elements should also be considered and require a shared parameter α, anyway, in order to assure continuity of the global displacement interpolation. Based on these considerations and on our numerical experience, we suggest to choose the scalar α = 15 , which guarantees integral positivity of the modified displacement shape functions for all element distortions occurring in practice (and pλ = 1 for the special case of undistorted quad8 surface elements). Feasible dual Lagrange multiplier shape functions Φj are now readily obtained for all slave nodes (m(1) = n(1) ) from the common biorthogonality condition applied locally on each slave element to the basis transformed slave shape functions: Z Z (1) (1) ˜ ˜ (1) dγ , Φj Nk dγ = δjk N j, k = 1, ..., mele , (4.7) k ele
ele
(1)
where the number of Lagrange multiplier nodes is mele = 6 for tri6 elements and
12 (1)
mele = 8 for quad8 elements. On each distorted slave element a mass matrix system (1) (1) of size mele × mele has to be solved. The global basis functions are then obtained by a simple gluing process as is standard in finite element methods. Figure 4.2 illustrates the new dual shape functions for both tri6 and quad8 elements. In addition to the standard slave displacement interpolation in (3.1), we then obtain an alternative interpolation with basis-transformed quantities: (1)
(1) uh |Γ(1) c,h
=
n X
(1)
(1) Nk
k=1
(1) dk
=
n X
˜ (1) , ˜ (1) d N k k
(4.8)
k=1
˜ (1) are simply given based on nodal where the transformed nodal displacements d k connectivity information and the transformation matrix entries from (4.6) as ˜ (1) = 1 d(1) d k k ˜ (1) = d k
for corner nodes ,
adj X 1 α (1) dk − d(1) p 1 − 2α 1 − 2α p=1
for edge nodes ,
(4.9)
with adj representing the number of adjacent slave nodes to slave node k. Node-wise assembly of (4.9) finally yields ˜ S := T−1 dS , d
(4.10)
where dS is the sub-vector of all nodal displacements on the slave surface as introduced ˜ S is the corresponding vector with respect to the transformed basis. in Section 3, and d 1 1 1 1 Moreover, T ∈ R3n ×3n and T−1 ∈ R3n ×3n are the global transformation matrix and its inverse, respectively. In terms of numerical efficiency, it is important to point out that the transformation matrices are sparse matrices that can be defined once at t = 0 and that stay unaltered during the entire finite deformation process afterwards. Besides, the matrix entries of both T and T−1 can easily be computed from (4.9) and no actual matrix inversion needs to be carried out. If we apply the proposed basis transformation during discretization of the slave part of the contact virtual work in (2.13), we obtain (1) ˜T D ˜ T z = δdT T−T D ˜T z . (4.11) δuh , λh (1) = δ d S S γc,h
Thus, owing to the basis transformation and taking into account that m(1) = n(1) , the mortar matrix D is square and can be expressed as Z ˜ T−1 with D ˜ jk = ˜ (1) dγ , j, k = 1, ..., m(1) , D=D Φj N (4.12) k (1)
γc,h
˜ reduces to a diagonal matrix due to the biorthogonality relationship dewhere D fined in (4.7). While the actual mortar matrix D is no longer a diagonal matrix, it is important to stress that the proposed basis transformation nevertheless preserves all advantages of dual Lagrange multiplier interpolation. Due to the trivial inver˜ and T, the inverse of the mortar matrix D is readily obtained as sion of both D −1 −1 ˜ D = T D . Thus, the major benefit of dual Lagrange multipliers, namely the possibility to trivially condense all discrete Lagrange multiplier degrees of freedom
13 from the global system of equations, is retained also for second-order interpolation. Algebraically, this can be observed by D−1 still being a sparse matrix. It should also be emphasized that the proposed basis transformation effectively only promotes a beneficial multiplicative split of the (non-diagonal) mortar matrix ˜ and T. Therefore, the contact formulaD into two trivially invertible matrices D tion remains entirely non-invasive, i.e. independent of the particular finite element formulations in the computational domain. Examining the discontinuous dual shape functions Φj for second-order mortar finite element analysis (p = 2) defined in (4.7) in more detail, it can easily be verified that in general the resulting global Lagrange multiplier interpolation is of reproduction order pλ = 0, i.e. only constant polynomials are contained. 4.4.2. Locally linear dual Lagrange multipliers. In the last subsection, suitable dual Lagrange multipliers for second-order finite element interpolation have been derived based on a simple basis transformation procedure. Although the resulting dual shape functions are locally (bi-)quadratic, we have seen that in the general case of distorted elements it is only possible to assure the constant polynomials to be contained in the global Lagrange multiplier interpolation. i.e. pλ = 0. Thus, one may argue that reducing the total number of Lagrange multiplier degrees of freedom by using only locally linear dual shape functions would make mortar coupling more efficient for quadratic finite elements, while at the same time convergence of the discretization error should be unaffected. Moreover, for the standard Lagrange multiplier case, the corresponding procedure has already been outlined in Section 4.3, see also [28, 30]. This paragraph hence aims at deriving alternative dual Lagrange multipliers for second-order finite element interpolation, which are only based on locally linear shape functions. Only the corner nodes of quadratic finite elements, i.e. only 3 nodes for tri6 elements and 4 nodes for quad8 and quad9 elements, carry discrete Lagrange multiplier degrees of freedom in this case, and thus m(1) < n(1) . Using the same framework as in the last subsection, our element-wise basis transformation for tri6 elements is now defined as T ˜ T N1 N1 1 0 0 0 0 0 ˜ N2 0 1 0 0 0 0 N 2 ˜3 N3 N = · 0 0 1 0 0 0 , (4.13) N4 α α 0 1 0 0 N ˜4 N5 0 α α 0 1 0 N ˜5 ˜6 N6 ele α 0 α 0 0 1 N ele | {z } :=Tele
with the transformation matrix Tele ∈ R6×6 and the index ele indicating a slave element. Computation of the inverse is again straightforward, yielding 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 −1 . (4.14) Tele = −α −α 0 1 0 0 0 −α −α 0 1 0 −α 0 −α 0 0 1 Definition of the scalar α is easy this time. The simplest way of obtaining locally linear polynomials for the modified shape functions of the corner nodes is to set
14
Fig. 4.3. Displacement shape functions for a quad8 element (left) and for a tri6 element (right). In each case, the unmodified shape functions N are illustrated in blue / dark grey for both ˜ only differ from an exemplary corner node and an exemplary edge node. Modified shape functions N the unmodified ones for corner nodes and are illustrated in green / light grey.
Fig. 4.4. Displacement and Lagrange multiplier shape functions for a quad8 element (left) and ˜ are illustrated in green / for a tri6 element (right). In each case, the modified shape functions N light grey for both an exemplary corner node and an exemplary edge node. Dual shape functions Φ only exist for corner nodes and are illustrated in red / dark grey.
˜1 , N ˜2 and N ˜3 . α = 12 , as this results in the well-known linear hat functions for N Unlike before, only these corner nodes will be used in the biorthogonality construction later on and carry dual Lagrange multipliers. The only fundamental difference of the transformation matrix as compared with the one introduced in Section 4.4.1 fact P3 is the ˜j = 1 that partition of unity is now enforced for the corner nodes only, i.e. j=1 N P6 ˜ but j=1 N j > 1. Displacement shape functions of the edge nodes, on the other hand, remain unchanged in order to assure the invertibility of Tele . Finally, it can ˜j are quite simply the well-known be observed that the modified shape functions N hierarchical basis functions. Again, the case of quad8 elements can be derived in full analogy, with the numbers of edge and corner nodes simply changing from three to four. The effect of our second basis transformation for both tri6 and quad8 elements is illustrated in Figure 4.3. For the sake of completeness, we mention that the above approach is readily extendable to quad9 elements, too. While it is included in the actual implementation, we do not consider this case in detail here. As in Section 4.4.1, dual Lagrange multiplier shape functions Φj are now ob˜j via the well-known tained from the modified slave displacement shape functions N biorthogonality relationship (4.1). However, dual shape functions are only defined for the corner nodes, whereas edge nodes now do not carry Lagrange multipliers and in simple terms then play a similar role as master nodes. Local biorthogonality on each slave element then reads: Z Z (1) (1) ˜ ˜ (1) dγ , Φj Nk dγ = δjk N j, k = 1, ..., mele , (4.15) k ele
ele
(1)
where the number of Lagrange multiplier nodes is now only mele = 3 for tri6 elements
15 (1)
and mele = 4 for quad8 elements. Exemplary dual shape functions resulting from (4.15) are illustrated in Figure 4.4 for both tri6 and quad8 elements. It can be seen that the dual shape functions that we obtain are identical to the first-order case discussed in Section 4.2, which is not surprising in consideration of the fact that both cases are based on simple linear hat functions in the biorthogonality construction. ˜ (1) are given based on Similar as before, the transformed nodal displacements d k nodal connectivity information and the transformation matrix entries from (4.14) as ˜ (1) = 1 d(1) d k k ˜ (1) = 1 d(1) − d k k
for corner nodes , adj X
α d(1) p
for edge nodes ,
(4.16)
p=1
with adj representing the number of adjacent slave nodes to slave node k. At the global level, the basis transformation of all slave displacements (including both corner ˜ S := T−1 dS and retains all the nodes and edge nodes) can again be expressed as d properties described in Section 4.4.1. Splitting all slave side displacement degrees of ˜ ? ∈ R3m(1) and edge node displacements freedom into corner node displacements d S (1) (1) ˜ ◦ ∈ R3n −3m reveals the following substructure of the transformation matrix: d S ? ? ˜ Id 0 d d S ˜ dS = ˜ ◦ = (4.17) · S◦ = T−1 dS . A Id dS dS Therefore, discretization of the slave part of the contact virtual work in (2.13) yields ˜ ? )T (D ˜ ◦ )T (D ˜ ? )T z + (δ d ˜ ◦ )T z δu(1)h , λh (1) = (δ d S S γc,h h i ˜ ? )T + AT (D ˜ ◦ )T z + (δd◦ )T (D ˜ ◦ )T z , = (δd?S )T (D (4.18) S where the square matrix block D? associated with the slave corner nodes (i.e. the nodes actually carrying discrete Lagrange multipliers) can be expressed as Z ? ˜? +D ˜ ◦A ˜ (1) dγ , j, k = 1, ..., m(1) ˜ jk D? = D with D = Φj N k (1)
γc,h
◦ ˜ jk and D =
Z (1) γc,h
(1)
˜ dγ , Φj N k
j = 1, ..., m(1) , k = m(1) + 1, ..., n(1) .
(4.19)
˜ ? reduces to a diagonal matrix owing to the biorthogonality relationship Whereas D defined in (4.15), we observe that D? does not. Moreover, in contrast to Section 4.4.1 its inversion is also no longer trivial due to the coupling of corner nodes and edge node degrees of freedom represented by the matrix block A of T−1 . In order to nonetheless benefit from the major advantage of dual Lagrange multiplier interpolation, a different ˜ ? is diagonal, it becomes obvious that a strategy is needed here. Remembering that D trivial condensation of the discrete Lagrange multipliers is anyhow possible, namely when solving with respect to the basis transformed quantities. Thus, we simply follow the opposite approach as before and do not shift the contact terms to the unmodified basis dS but rather shift the internal/external force and stiffness terms to the modified ˜ S . Again, this step is performed at negligible computational cost, because the basis d ˜ S can be applied locally, i.e. at the element level before basis transformation dS := T d global assembly.
16 Finally, the resulting locally linear dual shape functions Φj , j = 1, ..., m(1) have the same global properties as the locally quadratic ones, i.e. they are discontinuous and in general only the constant polynomials are included in the global Lagrange multiplier interpolation (pλ = 0). However, whereas for the first alternative in 4.4.1 all slave nodes also carry discrete Lagrange multipliers (m(1) = n(1) ), the second alternative only allocates additional degrees of freedom to the slave corner nodes and thus works with a reduced number of constraints (m(1) < n(1) ). Assuming a regular interfacenot mesh, we readily obtain the estimates m(1) ≈ 14 n(1) for tri6 and quad9 elements and m(1) ≈ 13 n(1) for quad8 elements. 5. PDASS and solution algorithm. This section gives a short overview of the solution algorithm applied to the mortar finite element discretization of both frictionless and frictional contact, well-known as primal-dual active set stragey (PDASS). It has been demonstrated in [6, 13, 29] that the PDASS can equivalently be interpreted as a semi-smooth Newton method, which in the case of nonlinear problems suggests an integrated treatment of all nonlinearities (including the search for the active set and the stick / slip regions) within one single iteration loop. Successful applications to small deformation contact problems [1, 16, 17] and also to the finite deformation regime inlcuding Coulomb friction [10, 24] have confirmed the robustness and efficiency of such active set schemes. The essential idea of the semi-smooth Newton method in our context is a reformulation of the inequality constraints within so-called complementarity functions. These non-smooth equality conditions then allow for the inclusion of the active set decision (or the stick / slip set decision in the frictional case) into a standard Newton– Raphson procedure. As an example, the non-penetration constraints in (3.7) for every Lagrange multiplier node j = 1, ..., m(1) can equivalently be expressed as Cn,j (zj , d) = zn,j − max (0, zn,j − cn g˜n,j ) = 0 ,
cn > 0 ,
(5.1)
where the equivalence holds for arbitrary positive values of the complementarity parameter cn . Thus, cn represents a purely algorithmic parameter. While Cn,j is a continuous function, it is non-smooth and has no uniquely defined derivative at positions zn,j − cn g˜n,j = 0. Yet, it is well-known from literature on constrained optimization [13, 29] that the max-function is semi-smooth and therefore a Newton method with superlinear local convergence behavior can still be applied. A similar derivation for tangential contact conditions according to Coulomb’s law can be found in [10, 16]. As a result, the nonlinear contact problem defined by the discrete equilibrium of forces (3.3) and the discrete constraints (3.7) can be solved within one semi-smooth Newton scheme, where the active set A and the inactive set I = S \ A are updated after each iteration step according to the complementarity functions defined above. We emphasize that this solution algorithm requires a consistent linearization of all deformation-dependent quantities, such as e.g. the mortar matrices D and M. This linearization accounts for the better part of the implementational effort associated with a fully nonlinear mortar contact and has been described in detail in [10, 24]. Although this paper does not discuss the actual solution of the resulting linear systems of equations, we point out that dual Lagrange multipliers are very favorable in this context. The biorthogonality property reduces the mortar matrix inverse D−1 either to a diagonal matrix (see Section 4.2) or at least to a sparse matrix (see Section 4.4). Both cases have been shown to allow for a trivial condensation of the discrete Lagrange multiplier degrees of freedom from the final system. This in turn removes the typical saddle point type structure of the final linear systems and greatly
17 promotes the applicability of state-of-the-art iterative solvers and preconditioners, such as algebraic multigrid techniques [5]. 6. Numerical results. Three numerical examples are presented in detail to demonstrate the validity and efficiency of the proposed mortar contact formulation with dual Lagrange multipliers for second-order interpolation in 3D. The simulations are based on a parallel implementation of the contact algorithms described above in our in-house multiphysics research code BACI [32]. Before considering the actual contact case, spatial convergence will be analyzed in the context of mortar mesh tying for linear elasticity in Section 6.1. The classical Hertzian contact problem in Section 6.2 confirms the high quality of the results for the frictionless case, whereas the fully nonlinear case including Coulomb friction is investigated in Section 6.3. Typical contact patch tests with constant stress solutions were also conducted and are all passed to machine precision. This is not surprising since mortar discretizations of contact problems are well-known to satisfy patch tests on arbitrary non-matching interfaces by construction [22, 39], which constitutes one of their major advantages over traditional node-to-segment schemes. 6.1. Tied contact: Beam bending. As a first validation step, the proposed discrete dual Lagrange multipliers are analyzed for 3D linear elasticity. No unilateral contact is considered, yet, but classical domain decomposition with non-matching interface meshes (so-called tied contact). We simplify the fully nonlinear problem formulation given in Section 2 by introducing the small deformation assumption. Kine matics are then defined via the linearized strain tensor = 12 ∇ ⊗ u + (∇ ⊗ u)T . Material behavior is assumed to be linear-elastic based on Hooke’s law σ = C : , which relates the Cauchy stress tensor σ and the linearized strain tensor via a fourth-order constitutive tensor C. The components of C are given as Cijkl =
E Eν (δij δkl ) + (δik δjl + δil δjk ) , (1 + ν)(1 − 2ν) 2(1 + ν)
(6.1)
with Young’s modulus E and Poisson’s ratio ν. In the following, we consider the simple bending problem of a cuboid structure with dimensions lx × ly × lz (see left part of Figure 6.1), which is supported such that all rigid body modes are removed. The whole setup is symmetric with respect to the xy- and yz-planes and Dirichlet boundary conditions are given as ly ly lx , 0) = uz (0, , 0) = uz ( , 0, 0) ≡ 0 . 2 2 2 (6.2) Pure bending around the z-axis is then readily obtained by applying distributed loads fx = ±2p lyy to the two surfaces x = ± l2x . The analytical solution for this 3D bending problem of linear elasticity is well-known [31] to be ux (0, 0, 0) = uy (0, 0, 0) = uz (0, 0, 0) = ux (0,
ux =
2p xy , Ely
uy =
p (−x2 − νy 2 + νz 2 ) , Ely
uz = −
2pν yz . Ely
(6.3)
For our numerical simulation, geometry, material and loading parameters have been chosen as lx = 4, ly = 2, lz = 1, E = 1000, ν = 0.3 and p = 100. The deformed configuration and an exemplary finite element mesh are illustrated in the right part of Figure 6.1. A mortar interface, which has intentionally been given a curved shape to make the problem more general, cuts the structure into two non-matching parts.
18
Fig. 6.1. Tied contact beam bending example – problem setup and exemplary finite element mesh (left), deformed geometry and numerical solution for the displacement uy (right).
The convergence study carried out in the following analyzes the discretization error u − uh in the energy norm, which is defined as sZ ku − uh kenergy =
( − h ) : C : ( − h ) dΩ .
(6.4)
Ω
Both sub-domains are discretized with hexahedral and tetrahedral conforming finite elements of polynomial degree p, where either p = 1 (first-order elements) or p = 2 (second-order elements). Dual shape functions are used for the discrete Lagrange multipliers, and the focus is on our two different new sets of dual Lagrange multipliers for second-order interpolation defined in Section 4.4. Uniform mesh refinement is applied with the element size ratio of slave and master side being fixed at hhms = 23 . For comparison reasons, standard Lagrange multiplier interpolation according to Sections 4.1 and 4.3 has also been tested for all mesh types. The results for first-order finite elements in Figure 6.2 demonstrate that all cases considered converge asymptotically with the optimal order that can be expected in the given example, i.e. O(h). Analyzing the second-order cases (20-node or 27-node hexahedral elements and 10-node tetrahedral elements) in more detail, it can be seen that both linear and quadratic standard Lagrange multiplier interpolation according to Section 4.3 as well as locally quadratic dual Lagrange multiplier interpolation according to Section 4.4.1 (denoted by the ending ‘quad‘ in Figure 6.2) yield almost identical results. All of these second-order cases converge asymptotically with O(h2 ). Only the behavior for locally linear dual Lagrange multipliers as introduced in Section 4.4.2 (denoted by the ending ‘lin‘ in Figure 6.2) shows some marginal deterioration, especially in combination with the tet10 mesh. The reason for this slightly suboptimal behavior is not directly apparent and may be a topic of further investigation. All in all, a perceptible deterioration by O(h1/2 ), as predicted by the theory if the global Lagrange multiplier interpolation is only of reproduction order p − 2, can not be observed in the numerical results. This is in accordance with the results in [30], where different mortar methods have been analyzed for a Laplace operator problem. The interested reader is also refered to [28] for a very similar numerical example in the context of standard Lagrange multipliers only. Other error measures, e.g. the L2 norm of the discretization error, give equally conclusive results.
19
Fig. 6.2. Tied contact beam bending example – convergence of error in the energy norm with uniform mesh refinement for hexahedral meshes (left) and tetrahedral meshes (right) using standard Lagrange multipliers (top) and dual Lagrange multipliers (bottom).
Of course, the results obtained in this example are not directly transferable to unilateral contact problems and friction, because the regularity requirements for establishing a priori error estimates similar to [30] do not necessarily hold for these cases. Nevertheless, with the above validation of optimal convergence of the proposed dual Lagrange multipliers in the tied contact setting, we have at least gained a strong indicator for their applicability to contact analysis. 6.2. Frictionless contact with small deformations: Hertzian. In this paragraph, we analyze a 3D Hertzian contact problem consisting of an elastic hemisphere (R = 8, St.Venant–Kirchhoff material model with E = 200, ν = 0.3) and a rigid planar surface. A constant pressure p = 0.2 is applied to the top surface of the hemisphere and contact interaction is assumed to be frictionless. Analytical solutions for the contact traction distribution are well-known for Hertzian contact problems and can usually be characterized via the maximum normal contact traction λmax and n the radius ρ of the circular contact zone (see e.g. [31]). For the given set of parameters, we obtain λmax = 18.042 and ρ = 1.031 as analytical solution. The mortar n finite element discretization is based on 20-node hexahedral elements and our new locally quadratic dual Lagrange multipliers as proposed in Section 4.4.1. The setup, an exemplary mesh and an exemplary numerical solution for the contact tractions are illustrated in Figure 6.3. Uniform mesh refinement is applied to analyze spatial convergence of the numerical solution for the maximum contact traction λmax , see Figure 6.4, and of the n
20
Fig. 6.3. Hertzian contact with small deformations – problem setup and exemplary finite element mesh (left), one quarter of deformed geometry and normal contact traction solution λn (right).
Fig. 6.4. Hertzian contact with small deformations – convergence of the maximum normal contact traction λmax with mesh refinement for an external load p = 0.2 and for a reduced external n load p = 0.05. The small deviation from the analytical solutions λmax = 18.042 for p = 0.2 and n λmax = 11.366 for p = 0.05 is due to our fully nonlinear implementation. n
discretization error u − uh measured in the energy norm, see Figure 6.5. The finest mesh analyzed consists of approximately 3 million degrees of freedom and the relative error of λmax with respect to the analytical solution is 2.8%. This deviation is solely n due to the fact that while the analytical solution is based on the small deformation assumption, our implementation is fully nonlinear and thus the mortar matrices D and M as well as the weighted gaps g˜n,j are deformation-dependent. However, our implementation is still consistent, which becomes clear when the external load is exemplarily reduced to p = 0.05. Then, the difference between small deformation and finite deformation formulation and thus also the relative error of λmax further diminn ishes. Precisely, the analytical solution is given as λmax = 11.366 and ρ = 0.650 in n that case and the relative error for the finest mesh reduces to 1.7%, which can also be seen in Figure 6.4. The results of the convergence study of the discretization error are summarized in Figure 6.5, where only the case p = 0.2 is considered. As there indeed exists an analytical solution for the maximum contact traction λmax but not for the displacen ment field u, a numerical reference solution is computed based on a sufficiently fine
21
Fig. 6.5. Hertzian contact with small deformations – convergence of error in the energy norm with uniform mesh refinement for hexahedral meshes using dual Lagrange multipliers.
Fig. 6.6. Hertzian contact with small deformations – vertical closeup view of the active contact zone and visualization of the normal contact traction solution for different mesh sizes h when applying an external load p = 0.2 (top) or a reduced external load p = 0.05 (bottom). The respective analytical solution for the contact zone radius ρ is indicated with a light circle.
hex20 mesh. Asymptotically, we observe O(h3/2 ) convergence for second-order interpolation, which is in accordance with theoretical results for unilateral contact [37]. As discussed in Section 3, no better a priori estimates than O(h3/2 ) can be expected due to the reduced regularity of contact solutions. First-order interpolation based on hex8 elements is investigated for comparison purposes and yields optimal O(h) results. Numerical results for the active contact zone and for the normal contact traction distribution are illustrated in Figure 6.6 for different mesh sizes h. It can be very clearly seen how the circular shape of the contact zone and the parabolic traction profile are resolved more and more with mesh refinement. Furthermore, an excellent agreement of the numerical solution with the analytical reference value for the contact zone radius ρ is visually confirmed. All in all, the results demonstrate the unrestricted applicability of the proposed second-order dual Lagrange multipliers to unilateral contact analysis.
22
Fig. 6.7. Sliding example with finite deformations – problem setup and finite element mesh.
Is is worth noting that we cannot start the semi-smooth Newton scheme outlined in Section 5 with an empty active set A = ∅ in this example, since then the hemisphere would have no constraints in y-direction. There exist different possibilities to initialize the active set in such a case (e.g. only the lowest node of the hemisphere), but the convergence behavior of the active set strategy remains almost unaffected by this choice [17]. In all examples considered here, the semi-smooth Newton scheme needs less than 10 steps to locate the contact zone comprising up to 2165 active nodes. 6.3. Frictional contact with finite deformations: Sliding. The final example demonstrates the performance of the proposed algorithms in the most general contact case, including finite sliding and Coulomb friction as well as significant active set changes. Mortar finite element discretization of Coulomb friction case and a corresponding extension of the semi-smooth Newton approach presented in Section 5 is described in great detail in [10, 16]. We consider a test setup suggested in [15], which consists of two half-cylindrical structures oriented in y-direction as depicted in Figure 6.7. The inner and outer radii of the upper body (slave) and the lower body (master) are ri1 = 0.09, ro1 = 0.13, ri2 = 0.13, ro2 = 0.17 and the lengths of the two bodies are defined as l1 = 0.11 and l2 = 0.17. Initially, the two bodies are separated by a gap g = 0.01. The setup is made fully unsymmetric by not placing the upper body centrically on the lower body, but instead moving it a distance d = 0.02 in negative y-direction. Dirichlet conditions are applied as follows: The lower right surface (A) is completely fixed, whereas the upper right surface (B) is only fixed in the xy-plane and given a prescribed displacement uz = −0.1 in z-direction. The lower left surface (C) is fixed in the yz-plane, while the upper left surface (D) is fixed in y-direction and given a prescribed displacement uz = −0.12 in z-direction. Both surfaces (C) and (D) can move freely in x-direction. The prescribed displacements are applied in 80 load steps and a compressible Neo-Hookean material model is used for both bodies, with parameters E 1 = 120, ν 1 = 0.3, E 2 = 60 and ν 2 = 0.25. Figure 6.7 also shows the employed finite element mesh consisting of 20-node hexahedral elements. Moreover, locally quadratic dual Lagrange multipliers according to Section 4.4.1 are used. We investigate and compare the Coulomb friction case (with a coefficient of fric-
23
Fig. 6.8. Sliding example with finite deformations – deformation for the frictionless case (top) and for the frictional case with µ = 0.3 (bottom) with numerical solution for the displacement uz .
tion of µ = 0.3) and the frictionless case. Numerical results and some characteristic stages of deformation for both scenarios are shown in Figure 6.8. Due to tangential forces in the contact zone, there is considerably less relative sliding in the contact zone for the frictional case. Unfortunately, apart from slight differences in the final deformed configurations, this effect can hardly be seen in Figure 6.8. Thus, we provide an additional visualization of the normal and tangential contact tractions in Figure 6.9, which illustrates the effect of friction on the contact interface more clearly. The numerical efficiency of our semi-smooth Newton type PDASS is evaluated by monitoring the relative L2 norm of the total residual krkk2 /krk02 in iteration step k of a representative load step in Figure 6.10. Newton convergence results demonstrate that the PDASS allows for a very efficient solution of the considered fully nonlinear contact problem. It also becomes clear from Figure 6.10 that contact with friction is apparently harder to solve than the frictionless case and thus requires some more active set iterations as well as semi-smooth Newton steps. However, in all cases the correct active set and the correct stick and slip regions are found within only a few iteration steps. With the sets being fixed, the nonlinear iteration scheme then reduces to a standard (smooth) Newton method, and thus we obtain quadratic convergence in the limit owing to the underlying consistent linearization [10, 24]. Most importantly, the possibility to solve all sources of nonlinearities (including finite deformations, material and frictional contact itself) within one single nonlinear iteration loop constitutes a major advantage of the semi-smooth Newton method over a nested active set approach (see e.g. [11, 12]) or a nested augmented Lagrangian / Uzawa algorithm version of the penalty approach (see e.g. [26, 27, 28]). 7. Conclusion. In this paper a 3D mortar method for finite deformation contact including frictional sliding has been presented. Special focus has been set on quadratic finite element interpolation, i.e. possibly mixed meshes composed of hex27, hex20 and tet10 elements. The resulting contact algorithm for the first time successfully combines dual Lagrange multiplier ideas with second-order mortar finite elements in the fully nonlinear realm. To achieve this we have proposed two new approaches
24
Fig. 6.9. Sliding example with finite deformations – normal and tangential contact tractions for the frictionless case (top) and for the frictional case with µ = 0.3 (bottom).
Fig. 6.10. Sliding example with finite deformations – convergence behavior of the semi-smooth Newton method in terms of the relative L2 norm of the total residual for a representative load step.
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