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A Hybrid High-Order discretization combined with Nitsche’s method for contact and Tresca friction in small
strain elasticity
Franz Chouly, Alexandre Ern, Nicolas Pignet
To cite this version:
Franz Chouly, Alexandre Ern, Nicolas Pignet. A Hybrid High-Order discretization combined with Nitsche’s method for contact and Tresca friction in small strain elasticity. SIAM Journal on Scientific Computing, Society for Industrial and Applied Mathematics, 2020, 42, pp.2300-2324.
�10.1137/19M1286499�. �hal-02283418v2�
A Hybrid High-Order discretization combined with Nitsche’s method for contact and Tresca friction in small strain elasticity
Franz Chouly
∗,2, Alexandre Ern
3, and Nicolas Pignet
1,31
EDF R&D / ERMES, Palaiseau, France and IMSIA - UMR EDF/CNRS/CEA/ENSTA 9219, Palaiseau, France
2
Universit´ e Bourgogne Franche-Comt´ e, Institut de Math´ ematiques de Bourgogne, 21078 Dijon, France
3
Universit´ e Paris-Est, CERMICS (ENPC), 77455 Marne-la-Vall´ ee Cedex 2, France and INRIA Paris, 75589 Paris, France
February 26, 2020
Abstract
We devise and analyze a Hybrid high-order (HHO) method to discretize unilateral and bilateral con- tact problems with Tresca friction in small strain elasticity. The nonlinear frictional contact conditions are enforced weakly by means of a consistent Nitsche’s technique with symmetric, incomplete, and skew- symmetric variants. The present HHO-Nitsche method supports polyhedral meshes and delivers optimal energy-error estimates for smooth solutions under some minimal thresholds on the penalty parameters for all the symmetry variants. An explicit tracking of the dependency of the penalty parameters on the material coefficients is carried out to identify the robustness of the method in the incompressible limit, showing the more advantageous properties of the skew-symmetric variant. 2D and 3D numerical results including comparisons to benchmarks from the literature and to solutions obtained with an industrial software, as well as a prototype for an industrial application, illustrate the theoretical results and reveal that in practice the method behaves in a robust manner for all the symmetry variants in Nitsche’s formulation.
Keywords. General meshes; Arbitrary order; Hybrid discretization; Nitsche’s method; Unilateral contact;
Tresca friction; Elasticity; Locking-free methods.
Mathematics Subject Classification. 65N12, 65N30, 74M15.
1 Introduction
Hybrid high-order (HHO) methods have been introduced for linear elasticity in [22] and for linear diffusion problems in [23]. HHO methods are formulated in terms of face unknowns that are polynomials of arbitrary order k ≥ 0 on each mesh face and in terms of cell unknowns which are polynomials of order l ∈ { k, k ± 1 } , with l ≥ 0, in each mesh cell. The devising of HHO methods hinges on two operators, both defined locally in each mesh cell: a reconstruction operator and a stabilization operator. The cell unknowns can be eliminated locally by static condensation leading to a global transmission problem posed solely in terms of the face unknowns. HHO methods offer various assets: they support polyhedral meshes, lead to local conservation principles, and optimal convergence rates. HHO methods have been bridged in [18] to Hybridizable Dis- continuous Galerkin methods [19] and to nonconforming Virtual Element Methods [5]. HHO methods have been extended to many other PDEs. Examples in computational mechanics include nonlinear elasticity [7], hyperelasticity with finite deformations [2], and elastoplasticity with small [4] and finite [3] deformations.
∗franz.chouly@u-bourgogne.fr
The goal of the present work is to devise, analyze and evaluate numerically a HHO method to approximate contact problems with Tresca friction in small strain elasticity. Either unilateral or bilateral contact can be considered. The main ingredient is to employ a Nitsche-type formulation to enforce the nonlinear frictional contact conditions. The present HHO-Nitsche method can be deployed on polyhedral meshes. As is classical with Nitsche’s technique, we can consider symmetric, incomplete and skew-symmetric variants.
Our main results, Theorem 4.6 and Corollary 4.8, provide for all symmetry variants quasi-optimal energy error estimates with convergence rates of order O (h
r) for solutions with regularity H
1+r, r ∈ (
12, k +1], where h is the mesh size and k ≥ 1 is the order of the polynomials for the cell and the face unknowns, except for the face unknowns located on the frictional contact boundary where polynomials of order (k + 1) are employed.
Note that the optimal order of convergence is O (h
k+1) obtained with r = k+1. These results are established under minimal thresholds for the penalty parameters weakly enforcing the contact and friction conditions, and do not require any assumption on the (a priori unknown) friction/contact set. Particular attention in the analysis is paid to the dependency of these parameters on the Lam´e parameters, showing that the skew- symmetric variant enjoys more favorable properties regarding robustness in the incompressible limit, at least from a theoretical viewpoint. Our 2D and 3D numerical tests include comparisons with benchmarks from the literature and with solutions obtained with the industrial software code aster [26]. We also consider a prototype of an industrial application featuring a notched plug in a rigid pipe. Our numerical tests indicate a more favorable dependency of the penalty parameters on the material parameters since robustness in the quasi-incompressible regime is observed in all considered situations.
Let us put our work in perspective with the literature. For most discretizations, Tresca friction creates additional difficulties in order to establish optimal convergence in comparison to the frictionless case (see, e.g., [34, 29, 25] and the references therein for frictionless contact). As a consequence convergence results addressing Tresca friction are quite rare. The rate O (h
r) for the energy error with a regularity H
1+r(Ω), r ∈ (0, 1], has been obtained in the 2D case for a mixed low-order finite element method (FEM) under some technical assumptions on the contact/friction set [34, Theorem 4.9] (this is the first optimal bound to the best of our knowledge). In the 3D case the rate O(h
min(12,r)) has also been reached without additional assumption [34, Theorem 4.10]. For the penalty method, the rate of O (h
12+r2+r2) with a regularity H
32+r(Ω), r ∈ (0,
12), and the quasi-optimal rate of O (h | log h |
12) with a regularity H
2(Ω) were established in [14]
without additional assumptions on the contact/friction set. This result has been improved recently in [24]
and optimal rates have been recovered if the penalty parameter is large enough. An important step forward for the discretization of contact problems was accomplished in [13] by combining Nitsche’s method with FEM. The FEM-Nitsche method differs from standard penalty techniques which are generally not consistent.
Moreover no additional unknown (Lagrange multiplier) is needed and therefore no discrete inf-sup condition must be fulfilled contrary to mixed methods. For contact problems with Tresca friction discretized with FEM-Nitsche, optimal energy-error convergence of order O(h
r) has been proved in [11] with the regularity H
1+r(Ω), r ∈ (
12, k], where k ≥ 1 is the polynomial degree of the Lagrange finite elements. To this purpose there is no need of any additional assumption on the contact/friction set. Note that the technical difficulties associated with the treatment of contact and friction condition when Nitsche’s technique is not employed, are not limited to FEM, but appear as well for other discretizations such as discontinuous Galerkin (DG) [32] or Virtual finite elements (VEM) [33].
The devising and analysis of HHO-Nitsche methods was started in [10] for the scalar Signorini problem.
Therein a face version and a cell version were analyzed, depending on the choice of the discrete unknown
used to formulate the penalty terms. The cell version used cell unknowns of order (k + 1) (these unknowns
can be eliminated by static condensation) and a modified reconstruction operator inspired from the unfitted
HHO method from [9], leading to energy error estimates of order O (h
r) with H
1+r-regularity, r ∈ (
12, k +
1]. Unfortunately, the modification of the reconstruction operator is not convenient in the context of
elasticity as it hampers a key commuting property with the divergence operator which is crucial in the
incompressible limit. This difficulty is circumvented in the present work by using face polynomials of
order (k + 1) on the faces located on the contact/friction boundary. The numerical analysis also involves
two novelties. Firstly, the error analysis, which adapts ideas from [15, 16] for FEM-Nitsche to HHO-
Nitsche, is more involved than [10] since it covers all the symmetry variants and since it hinges on a sharper
bound on the consistency error allowing for a sharper threshold on the penalty parameters, especially
in the case of the skew-symmetric variant. Secondly, for the first time concerning FEM-Nitsche as well,
we track explicitly the dependency of the penalty parameters on the Lam´e parameters for the various
symmetry variants. Furthermore the present study is completed with 2D and 3D numerical tests including
Figure 1: Geometry of the contact problem.
a prototype for an industrial application. Finally, let us mention that polyhedral discretizations for contact and friction problems have received some attention recently, as motivated by some numerical evidence illustrating their flexibility and accuracy. These discretizations use for instance VEM [35, 33], the weak Galerkin (WG) method [28] or the hybridizable discontinous Galerkin (HDG) methods [36], combined with different techniques to handle contact and friction (such as a direct approximation of the variational equality, node-to-node contact, penalty, Lagrange multipliers). The present work constitutes, to our knowledge, the first polyhedral discretization method for frictional contact problems using Nitsche’s technique.
This paper is organized as follows. The model problem is described in Section 2. The HHO-Nitsche method is introduced in Section 3, and the stability and error analysis is contained in Section 4. Numerical results are discussed in Section 5.
2 Model problem
Let Ω be a polygon/polyhedron in R
d, d ∈ { 2, 3 } , representing the reference configuration of an elastic body.
The boundary ∂Ω is partitioned into three nonoverlapping parts (see Fig 1): the Dirichlet boundary Γ
D, the Neumann boundary Γ
N, and the contact/friction boundary Γ
C. We assume meas(Γ
D) > 0 to prevent rigid body motions and meas(Γ
C) > 0 to ensure that contact is present. The small strain assumption is made, as well as plane strain if d = 2. The linearized strain tensor associated with a displacement field v : Ω → R
dis ε(v) :=
12(∇v + ∇v
T) ∈ R
dsym×d. Assuming isotropic behavior, the Cauchy stress tensor resulting from the strain tensor ε(v) is denoted by σ(v) and is given by
σ(v) = 2µε(v) + λ trace(ε(v))I
d∈ R
dsym×d, (1) where µ and λ are the Lam´e coefficients of the material satisfying µ > 0 and 3λ + 2µ > 0, and I
dis the identity tensor of order d. In what follows, we set κ := max(1,
2µλ). Let n be the unit outward normal vector to Ω. On the boundary we consider the following decompositions into normal and tangential components:
v = v
nn + v
tand σ
n(v) := σ(v) · n = σ
n(v)n + σ
t(v) where v
n:= v · n and σ
n(v) := σ
n(v) · n (so that v
t· n = 0 and σ
t(v) · n = 0).
The body is subjected to volume forces f ∈ L
2(Ω; R
d) in Ω and to surface loads g
N∈ L
2(Γ
N; R
d) on Γ
N, and it is clamped on Γ
D(for simplicity). The model problem consists in finding the displacement field u : Ω → R
dsuch that
∇· σ(u) + f = 0 in Ω,
u = 0 on Γ
D,
σ
n(u) = g
Non Γ
N, (3) and (4) hold true on Γ
C,
(2)
where the unilateral contact conditions on Γ
Care as follows:
(i) u
n≤ 0, (ii) σ
n(u) ≤ 0, (iii) σ
n(u) u
n= 0, (3)
whereas the Tresca friction conditions on Γ
Cread (see, e.g., [30, Equ. (10.8)]):
(iv) | σ
t(u) | ≤ s if u
t= 0, (v) σ
t(u) = − s u
t| u
t| if | u
t| > 0, (4) where s ≥ 0 is a given threshold and |·| stands for the Euclidean norm in R
d(or the absolute value depending on the context). More generally s can be a nonnegative-valued function on Γ
C.
Remark 2.1 (Variants). The case of frictionless contact is recovered by setting s := 0 in (4). The case of bilateral contact with Tresca friction can be considered by keeping (4) whereas (3) is substituted by the following equation:
u
n= 0 on Γ
C. (5)
In the case of unilateral contact with Tresca friction, nonzero tangential stress ( | σ
t(u) | > 0) can occur in regions with no-adhesion (u
n< 0), which is not expected physically (for Coulomb friction, σ
t(u) = 0 when u
n< 0). The setting of bilateral contact prevents such situations (since u
n= 0, there are no regions with no-adhesion). Therefore, the setting of bilateral contact with Tresca friction can be useful for the numerical simulation of specific situations that essentially correspond to persistent contact between elastic bodies with a high intensity of contact pressures (see, e.g. , [30, Section 10.3] or [31]).
We introduce the Hilbert space V
Dand the convex cone K such that V
D:=
v ∈ H
1(Ω; R
d) | v = 0 on Γ
D, K := { v ∈ V
D| v
n≤ 0 on Γ
C} ,
i.e., the Dirichlet condition on Γ
Dis explicitly enforced in the space V
Dand the non-interpenetration condition on Γ
Cis explicitly enforced in the cone K. We define the following bilinear form and the following linear and nonlinear forms:
a(v, w) := (σ(v), ε(w))
Ω= 2µ(ε(v), ε(w))
Ω+ λ( ∇· v, ∇· w)
Ω, (6)
`(w) := (f , w)
Ω+ (g
N, w)
ΓN, j(w) := (s, | w
t| )
ΓC, (7) for any v and w in V
D, where the notation ( · , · )
ωstands for the L
2-inner product on the set ω. The weak formulation of (2) as a variational inequality of the second kind is
Find u ∈ K such that
a(u, w − u) + j(w) − j(u) ≥ `(w − u), ∀ w ∈ K. (8) This problem admits a unique solution according, e.g., to [30, Theorem 10.2].
An important observation is that it is possible to reformulate the contact and friction conditions (3)-(4) as nonlinear equations. For any real number x ∈ R , let [x] := min(x, 0) denote its projection onto the closed convex subset R
−:= ( −∞ , 0]. Moreover, let [ · ]
αdenote the orthogonal projection onto B(0, α) ⊂ R
d, where B(0, α) is the closed ball centered at the origin 0 and of radius α > 0, i.e., for all x ∈ R
d, we have [x]
α:= x if | x | ≤ α and [x]
α:= α
|xx|if | x | > α. The following result has been pointed out in [20] (see also [11] for formal proofs).
Proposition 2.2 (Reformulation as nonlinear conditions) . Let γ
nand γ
tbe positive functions on Γ
C. The conditions (3)-(4) enforcing unilateral contact with Tresca friction can be reformulated as follows:
σ
n(u) = [τ
n(u)] , τ
n(u) := σ
n(u) − γ
nu
n, (9) σ
t(u) = [τ
t(u)]
s, τ
t(u) := σ
t(u) − γ
tu
t. (10)
3 HHO-Nitsche method
In this section we devise and analyze the HHO-Nitsche method to approximate the frictional contact problem
(8).
3.1 Meshes and discrete unknowns
Let ( T
h)
h>0be a mesh sequence, where for all h > 0, the mesh T
his composed of nonempty disjoint cells such that Ω = S
T∈Th
T. The mesh cells are conventionally open subsets in R
d(not necessarily convex), and they can have a polygonal/polyhedral shape with straight edges (if d = 2) or planar faces (if d = 3).
This setting in particular allows for meshes with hanging nodes. The mesh sequence ( T
h)
h>0is assumed to be shape-regular in the sense of [22]. In a nutshell, each mesh T
hadmits a matching simplicial submesh
=
hhaving locally equivalent length scales to those of T
h, and the mesh sequence ( =
h)
h>0is shape-regular in the usual sense of Ciarlet. The meshsize is denoted h := max
T∈Thh
T, with h
Tthe diameter of the cell T and n
Tdenotes the unit outward normal to T . Discrete trace and inverse inequalities in the usual form are available on shape-regular polyhedral mesh sequences (see, e.g., [21]).
A closed subset F of Ω is called a mesh face if it is a subset with nonempty relative interior of some affine hyperplane H
Fand if (i) either there are two distinct mesh cells T
1, T
2∈ T
hso that F = ∂T
1∩ ∂T
2∩ H
F(and F is called an interface) (ii) or there is one mesh cell T
1∈ T
hso that F = ∂T
1∩ Γ ∩ H
F(and F is called a boundary face). Remark that this definition allows for hanging nodes, since a mesh cell can have various faces that belong to the same hyperplane. The mesh faces are collected in the set F
hwhich is further partitioned into the subset of interfaces F
hiand the subset of boundary faces F
hb. We assume that the meshes are compatible with the boundary partition ∂Ω = Γ
D∪ Γ
N∪ Γ
C, which leads to the partition of the boundary faces as F
hb= F
hb,D∪ F
hb,N∪ F
hb,C(with obvious notation).
Let k ≥ 1 be the polynomial degree. For all T ∈ T
h, let F
∂Tbe the collection of the mesh faces that are subsets of ∂T, let F
∂Ti:= F
∂T∩ F
hi, F
∂Tb,C:= F
∂T∩ F
hb,C, and we use a similar notation for F
∂Tb,Dand F
∂Tb,N. We set F
∂T\:= F
∂Ti∪ F
∂Tb,D∪ F
∂Tb,Nfor the collection of all the faces composing ∂T except those located on Γ
C. The local HHO discrete space is
U b
kT:= P
k(T ; R
d) × P
k/k+1( F
∂T; R
d),
where P
k(T ; R
d) is composed of the restrictions to T of d-variate polynomials of total degree at most k, and P
k/k+1( F
∂T; R
d) := P
k( F
∂T\; R
d) × P
k+1( F
∂Tb,C; R
d),
where P
k( F
∂T\; R
d) and P
k+1( F
∂Tb,C; R
d) are composed of the restrictions to F
∂T\and F
∂Tb,C, respectively, of piecewise (d − 1)-variate polynomials of total degree at most k and (k + 1), respectively. A generic element in U b
kTis a pair b v
T:= (v
T, v
∂T), where v
T∈ P
k(T ; R
d) is the cell component and v
∂T∈ P
k/k+1( F
∂T; R
d) is the face component. The degrees of freedom are illustrated in Fig. 2, where a dot indicates one degree of freedom (which is not necessarily computed as a point evaluation but can be for instance the moments against a selected set of basis functions defined on the cell or the face, see [17]).
(a) Pentagonal cell with no contact face (F∂T\ =F∂T)
(b) Pentagonal cell with a con- tact face in red (F∂T\ F∂T)
Figure 2: Face (black or red) and cell (gray) degrees of freedom in U b
kTfor k = 1 and d = 2 (each dot represents a degree of freedom which is not necessarily a point evaluation).
Remark 3.1 (Degree on contact faces). We use a polynomial order (k + 1) on the faces located on the
contact boundary (it is also possible to use this order on all the boundary faces). This choice is motivated by
the error analysis, where it will be shown that it allows one to recover error bounds with optimal convergence
order. Moreover this choice increases only marginally the computational cost with respect to using the same
order k for all the faces.
3.2 Local HHO operators
The first key ingredient in the devising of the HHO method is a local symmetric strain reconstruction in each mesh cell T ∈ T
h. Following [2, 7, 4], we define the local discrete symmetric gradient operator E : U b
kT→ P
k(T; R
dsym×d) such that, for all b v
T∈ U b
kT, E( b v
T) ∈ P
k(T ; R
dsym×d) solves the following local problem:
For all τ ∈ P
k(T ; R
dsym×d),
(E( b v
T), τ)
T= − (v
T, ∇· τ )
T+ (v
∂T, τ · n
T)
∂T. (11) The local problem (11) mimics an integration by parts at the discrete level. Moreover, the local discrete divergence operator D : U b
kT→ P
k(T ; R) is simply defined by taking the trace of the discrete symmetric gradient: For all v b
T∈ U b
kT,
D( b v
T) := trace(E( b v
T)). (12)
The second key ingredient is a local stabilization operator S : U b
kT→ P
k/k+1(∂T ; R
d) used to penalize in a least-squares sense the difference between the face unknown v
∂Tand the trace of the cell unknown v
T|∂T. Let Π
k/k+1∂Tand Π
kTbe the L
2-orthogonal projections onto P
k/k+1( F
∂T; R
d) and P
k(T ; R
d), respectively.
Then we set, for all b v
T∈ U b
kT,
S( b v
T) := Π
k/k+1∂Tv
∂T− R( b v
T)
|∂T− Π
kTv
T− R( b v
T)
|∂T
. (13)
Here R : U b
kT→ P
k+1(T ; R
d) is a local displacement reconstruction operator such that, for all b v
T∈ U b
kT, R( b v
T) ∈ P
k+1(T ; R
d) solves the following local problem: For all w ∈ P
k+1(T ; R
d),
( ε (R( b v
T)), ε (w))
T=( ε (v
T), ε (w))
T+ (v
∂T− v
T|∂T, ε (w) · n
T)
∂T. (14) The reconstructed displacement is uniquely defined by prescribing additionally that R
T
R( b v
T)dT = R
T
v
TdT and R
T
∇
ss(R( b v
T))dT= R
∂T 1
2
(v
∂T⊗ n
T− n
T⊗ v
∂T)d∂T , where ∇
ss(v) :=
12(∇v − ∇v
T) is the skew- symmetric part of the gradient (see [22]). Comparing with (11), one readily sees that ε(R( v b
T)) is the L
2-orthogonal projection of E( b v
T) onto ε(P
k+1(T; R
d)).
We use the above operators to mimic locally the exact local bilinear form a defined in (6) by means of the following local bilinear form defined on U b
kT× U b
kT(compare with (6)):
b a
T( b v
T, w b
T) :=2µ(E( b v
T), E( w b
T))
T+ λ(D( b v
T), D( w b
T))
T+ 2µh
−T1(S( b v
T), S( w b
T))
∂T. (15) The stabilization term is weighted (as in the linear case) by the Lam´e coefficient µ.
3.3 Global discrete problem
For simplicity we employ the Nitsche technique only on the subset Γ
Cwhere the nonlinear frictional contact conditions are enforced, whereas we resort to a strong enforcement of the homogeneous Dirichlet condition on the subset Γ
D. The global discrete space for the HHO-Nitsche method is
U b
kh:= P
k( T
h; R
d) ×
P
k( F
hi∪ F
hb,D∪ F
hb,N; R
d) × P
k+1( F
hb,C; R
d)
, (16)
leading to the notation v b
h:= (v
T)
T∈Th, (v
F)
F∈Fhfor a generic element v b
h∈ U b
kh. For all T ∈ T
h, we denote by b v
T:= (v
T, (v
F)
F∈F∂T) ∈ U b
kTthe local components of b v
hattached to the mesh cell T and the faces composing ∂T , and for any mesh face F ∈ F
h, we denote by v
Fthe component of b v
hattached to the face F. We enforce strongly the homogeneous Dirichlet condition on Γ
Dby considering the subspace
U b
kh,0:= {b v
h∈ U b
kh| v
F= 0 ∀ F ∈ F
hb,D} .
The HHO-Nitsche method uses a symmetry parameter θ ∈ {− 1, 0, 1 } and two penalty parameters γ
n> 0
and γ
t> 0 to enforce weakly the contact and friction conditions on Γ
C, respectively. Choosing θ := 1
leads to a symmetric formulation with a variational structure, choosing θ := 0 is interesting to simplify the
implementation by avoiding some terms in the formulation, and choosing θ := − 1 allows one to improve on
the stability of the method by exploiting its skew-symmetry and making it more robust in the incompressible
limit. It is convenient to define the subset T
hCas the collection of the mesh cells having at least one boundary face on Γ
Cand to set ∂T
C:= ∂T ∩ Γ
Cfor all T ∈ T
hC. The subset T
hNis defined similarly, and we set
∂T
N:= ∂T ∩ Γ
Nfor all T ∈ T
hN. We consider the following discrete HHO-Nitsche problem:
( Find u b
h∈ U b
kh,0such that
b b
h( b u
h; w b
h) = ` b
h( w b
h) ∀ w b
h∈ U b
kh,0, (17) where the global discrete semilinear form b b
hand the global discrete linear form ` b
hare defined as follows:
b b
h( b v
h; w b
h) := X
T∈Th
b a
T( b v
T, w b
T)
− X
T∈ThC
θ h
Tγ
n(σ
n( b v
T), σ
n( w b
T))
∂TC− X
T∈ThC
θ h
Tγ
t(σ
t( b v
T), σ
t( w b
T))
∂TC+ X
T∈ThC
h
Tγ
n([τ
n( b v
T)] , (τ
n+ (θ − 1)σ
n)( w b
T))
∂TC+ X
T∈ThC
h
Tγ
t([τ
t( v b
T)]
s, (τ
t+ (θ − 1)σ
t)( w b
T))
∂TC, and
` b
h( w b
h) := X
T∈Th
(f , w
T)
T+ X
T∈ThN
(g
N, w
∂T)
∂TN. Here, with a slight abuse of notation, we have written
σ( w b
T) := 2µE( w b
T) + λD( w b
T)I
d∈ P
k(T; R
dsym×d), (18) together with the decomposition σ( w b
T) · n
T:= σ
n( w b
T)n
T+ σ
t( w b
T), and we have introduced the linear operators (again with a slight abuse of notation)
τ
n( w b
T) := σ
n( w b
T) − γ
nh
Tw
∂T ,n, τ
t( w b
T) := σ
t( w b
T) − γ
th
Tw
∂T ,t,
together with the decomposition w
∂T:= w
∂T ,nn
T+ w
∂T ,tfor the face polynomials. Note that in the definition of τ
nand τ
t, the penalty parameters are rescaled by h
−T1in each mesh cell in T
hC. Proposition 2.2 still holds true with this rescaling of the penalty parameters since the only requirement there is that γ
nand γ
tbe positive.
Remark 3.2 (Comparison with FEM-Nitsche). In the FEM-Nitsche method devised in [11], one restricts the setting to simplicial meshes and considers the usual H
1-conforming finite element space V
hcomposed of continuous functions that are piecewise polynomials of degree at most k ≥ 1 in each mesh cell. The discrete problem is formulated in the subspace V
h,0explicitly enforcing the Dirichlet condition on Γ
Dand involves the global discrete semilinear form
b
h(v
h; w
h) := a(v
h, w
h) − (
θ˜γσ
n(v
h), σ
n(w
h))
ΓC− (
θ˜γσ
t(v
h), σ
t(w
h))
ΓC+ (
γ1˜[τ
n(v
h)] , (τ
n+ (θ − 1)σ
n)(w
h))
ΓC+ (
γ1˜[τ
t(v
h)]
s, (τ
t+ (θ − 1)σ
t)(w
h))
ΓC,
as well as the linear form `
h(w
h) := `(w
h), where a and ` are the same as for the continuous problem, the notation τ
n, σ
n, τ
t, and σ
tis that employed for the exact solution, the symmetry parameter θ is taken again in {− 1, 0, 1 } , and the penalty parameter ˜ γ is a piecewise constant function on Γ
Csuch that ˜ γ
|∂TC= h
−1Tγ with γ > 0 for all T ∈ T
hC. Note that there is only one penalty parameter in [11] since the analysis there did not consider the scaling with respect to the Lam´e parameters.
Remark 3.3 (Comparison with [10]). There are various differences compared to the HHO-Nitsche method
devised in [10] for the scalar Signorini problem. Here we address the vector-valued case and include Tresca
friction. Moreover we use the face polynomials in the definition of the operators τ
nand τ
twhich corresponds
to the face version considered in [10]. However we employ here a higher polynomial degree on those faces
located on Γ
C.
4 Stability and error analysis
In this section we perform the stability and error analysis of the above HHO-Nitsche discretization of the frictional contact problem. We first collect some useful analysis tools. Then we establish a stability property and infer the well-posedness of the nonlinear discrete problem (17). The stability property is valid if the penalty parameters γ
nand γ
tare bounded from below, this bound being 0 for the skew-symmetric variant θ = − 1. Then we slightly tighten these minimal values to derive an error estimate bounding the energy error and the error on the nonlinear boundary condition on Γ
Cand featuring optimal decay rates of order (k + 1) for smooth solutions. The following properties of projections onto a convex set will be useful:
([x] − [y] )(x − y) ≥ ([x] − [y] )
2≥ 0, ∀ x, y ∈ R, (19) ([x]
s− [y]
s) · (x − y) ≥ | [x]
s− [y]
s|
2≥ 0, ∀ x, y ∈ R
d. (20) We use the symbol C to denote a generic constant whose value can change at each occurrence as long as it is independent of the mesh size and the Lam´e parameters. The value of C can depend on the mesh regularity and the polynomial degree k ≥ 1. We abbreviate as a . b the inequality a ≤ Cb with positive real numbers a, b and a constant C > 0 as above and whose value can change at each occurrence.
4.1 Analysis tools for HHO operators
We equip the space U b
kTwith the following local discrete strain seminorm:
|b v
T|
21,T:= k ε (v
T) k
2T+ h
−1Tk v
∂T− v
T|∂Tk
2∂T, (21) where k·k
T(resp. k·k
∂Tand k·k
∂TC) is the L
2-norm on T (resp. on ∂T and ∂T
C). Notice that |b v
T|
1,T= 0 implies that v
Tis a rigid-body motion and that v
∂Tis the trace of v
Ton ∂T . The following local stability and boundedness properties of the strain reconstructon and stabilization operators are established as in [22, Lemma 4].
Lemma 4.1 (Boundedness and stability). Let E be defined by (11) and S by (13). There are 0 < α
[<
α
]< + ∞ such that, for all T ∈ T
h, all h > 0, and all v b
T∈ U b
kT, we have α
[|b v
T|
1,T≤
k E( b v
T) k
2T+ h
−1Tk S( b v
T) k
2∂T 12≤ α
]|b v
T|
1,T. (22) The key operator in the HHO error analysis is the local interpolation operator I b
kT: H
1(T; R
d) → U b
kTsuch that
I b
kT(v) := (Π
kT(v), Π
k/k+1∂T(v
|∂T)) ∈ U b
kT, (23) for all v ∈ H
1(T; R
d) and all T ∈ T
h. The global version I b
kh: V
D→ U b
kh,0is defined locally by setting the local HHO components of I b
kh(v) to ( I b
kh(v))
T:= I b
kT(v
|T) ∈ U b
kT, for all v ∈ V
Dand all T ∈ T
h. This definition makes sense since the functions in V
Ddo not jump across the mesh interfaces and vanish at the boundary faces located on Γ
D. The HHO interpolation operator allows one to obtain important local commuting properties satisfied by the reconstruction operators (see, e.g., [22, Proposition 3]), namely we have
E(b I
kT(v)) = Π
kT(ε(v)), D(b I
kT(v)) = Π
kT( ∇· v), (24) for all v ∈ H
1(T ; R
d), all T ∈ T
h, and all h > 0, where Π
kTis the L
2-orthogonal projection onto P
k(T ; R
dsym×d) and Π
kTthat onto P
k(T ; R ). For all v ∈ H
1+ν(Ω; R
d), ν >
12, and all w b
h∈ U b
kh,0, let us set (the reason for the notation will become clear in the proof of Theorem 4.6 below)
T
01,1(v, w b
h) := X
T∈Th
− ( ∇· σ(v), w
T)
T+ b a
T(b I
kT(v), w b
T)
+ X
T∈ThN∪ThC
(σ
n(v), w
∂T)
∂TN∪∂TC, (25)
where we recall that w b
T= (w
T, w
∂T) are the local components of the test function w b
h∈ U b
kh,0attached to the mesh cell T ∈ T
h. For a function z ∈ H
ν(T ; R ), ν >
12, we employ the notation
k z k
2],T:= k z k
2T+ h
Tk z k
2∂T,
and the same notation for tensor-valued functions.
Lemma 4.2 (Consistency for linear elasticity) . Let T
01,1(v, w b
h) be defined in (25) for all v ∈ H
1+ν(Ω; R
d) , ν >
12, and all w b
h∈ U b
kh,0. The following holds true:
|T
01,1(v, w b
h) |
2. A
1(v)
X
T∈Th
2µ | w b
T|
21,T, (26)
with the interpolation error
A
1(v) := X
T∈Th
1 2µ
(2µ)
2k ε (v) − Π
kT( ε (v)) k
2],T+ (2µ)
2k ε (v − Π
k+1T(v)) k
2T+ λ
2k∇· v − Π
kT( ∇· v) k
2],T. (27)
Proof. The proof essentially follows from [22, Theorem 8] and is only sketched here. Integrating by parts the term ( ∇· σ(v), w
T)
Tfor all T ∈ T
h, using (1) together with the definitions (11)-(12) for E( w b
T) and D( w b
T), respectively, re-arranging the terms, and setting η
T:= ε(v) − E(b I
kT(v)) = ε(v) − Π
kT(ε(v)) and ζ
T:= ∇· v − D( I b
kT(v)) = ∇· v − Π
kT( ∇· v) (owing to the commuting properties (24)) leads to (details are skipped for brevity)
T
01,1(v, w b
h) = X
T∈Th
2µ(η
T, ε(w
T))
T+ 2µ(η
T· n
T, w
∂T− w
T)
∂T− 2µh
−1T(S(b I
kT(v)), S( w b
T))
∂T+ λ(ζ
T, ∇· w
T)
T+ λ(ζ
Tn
T, w
∂T− w
T)
∂T.
Invoking the Cauchy–Schwarz inequality and the upper bound from Lemma 4.1 to estimate h
−T12k S( w b
T) k
∂T, we infer that
| T
01,1(v, w b
h) |
2.
X
T∈Th
1 2µ
(2µ)
2k η
Tk
2],T+ (2µ)
2h
−T1k S(b I
kT(v)) k
2∂T+ λ
2k ζ
Tk
2],T×
X
T∈Th
2µ | w b
T|
21,T.
Finally combining the ideas used in [22, Eqs. (20)&(35)] with a local multiplicative trace inequality and a local Korn inequality, we infer that
h
−T1k S(b I
kT(v)) k
2∂T. k ε(v − R(b I
kT(v))) k
2T≤ k ε(v − Π
k+1T(v)) k
2T, which completes the proof of (26).
4.2 Stability and well-posedness
Let us first establish an important monotonicity property of the semilinear form b b
hunder the assumption that the penalty parameters γ
nand γ
tare large enough. The lower bound on these parameters involves the constant C
dtfrom the following discrete trace inequality:
k v
hk
∂TC≤ C
dth
−T12k v
hk
T, (28) for all T ∈ T
hC, all h > 0, and all v
h∈ P
k(T ; R
q), q ∈ { 1, d } . We equip the global HHO space U b
kh,0with the norm
kb v
hk
2µ,λ:= X
T∈Th
2µ |b v
T|
21,T+ λ k D( b v
T) k
2T.
That k·k
µ,λdefines a norm on U b
kh,0follows from the usual arguments since face components are null on all
the faces in F
hb,Dand this set is nonempty by assumption.
Lemma 4.3 (Monotonicity) . Assume that the penalty parameters are such that
min(κ
−1γ
n, 2γ
t) ≥ 3(θ + 1)
2C
dt2µ, (29) recalling that κ := max(1,
2µλ). Then the semilinear form b b
his monotone and we have for all b v
h, w b
h∈ U b
kh,0,
b b
h( b v
h; v b
h− w b
h) − b b
h( w b
h; b v
h− w b
h) ≥ 1
3 min(1, α
2[) kb v
h− w b
hk
2µ,λ. (30) Proof. Let b v
h, w b
h∈ U b
kh,0and set b z
h:= b v
h− w b
h. Recalling the definition of the HHO-Nitsche semilinear form b b
hand exploiting the positivity of the local HHO bilinear form b a
T, we infer that
X
T∈Th
2µ k E ( b z
T) k
2T+ h
−1Tk S( b z
T) k
2∂T+ λ k D( b z
T) k
2T≤ T
1+ T
2,n+ T
2,t− T
3,n− T
3,t, (31)
where
T
1:= b b
h( b v
h; b z
h) − b b
h( w b
h; b z
h), T
2,n:= X
T∈ThC
θ h
Tγ
nk σ
n( b z
T) k
2∂TC, T
2,t:= X
T∈ThC
θ h
Tγ
tk σ
t( z b
T) k
2∂TC, and
T
3,n:= X
T∈ThC
h
Tγ
n([τ
n( b v
T)] − [τ
n( w b
T)] , τ
n( z b
T))
∂TC+ X
T∈ThC
(θ − 1) h
Tγ
n([τ
n( v b
T)] − [τ
n( w b
T)] , σ
n( z b
T))
∂TCT
3,t:= X
T∈ThC
h
Tγ
t([τ
t( b v
T)]
s− [τ
t( w b
T)]
s, τ
t( z b
T))
∂TC+ X
T∈ThC
(θ − 1) h
Tγ
t([τ
t( b v
T)]
s− [τ
t( w b
T)]
s, σ
t( z b
T))
∂TC.
Let us consider T
2,n− T
3,n. Setting δ
T:= [τ
n( b v
T)] − [τ
n( w b
T)] , we infer that T
2,n− T
3,n≤ X
T∈ThC
θ h
Tγ
nk σ
n( z b
T) k
2∂TC− k δ
Tk
2∂TC− (θ − 1) h
Tγ
n(δ
T, σ
n( z b
T))
∂TC≤ X
T∈ThC
1
4 (θ + 1)
2h
Tγ
nk σ
n( b z
T) k
2∂TC≤ X
T∈ThC
1
4 (θ + 1)
2C
dt2γ
nk σ
n( b z
T) k
2T,
where we used (19) in the first bound, Young’s inequality and the fact that θ +
14(θ − 1)
2=
14(θ + 1)
2in the second bound, and the discrete trace inequality (28) in the third bound. Recalling the definition (18) of the discrete HHO stress and using the triangle and Young’s inequalities, we infer that
T
2,n− T
3,n≤ X
T∈ThC
1
2 (θ + 1)
2C
dt2γ
n(2µ)
2k E( b z
T) k
2T+ λ
2k D( z b
T) k
2T≤ X
T∈ThC
(θ + 1)
2C
dt2γ
nµκ ×
2µ k E ( b z
T) k
2T+ λ k D( z b
T) k
2T.
The reasoning to bound T
2,t− T
3,tis similar except that σ
t( b z
T) has only off-diagonal contributions and is therefore independent of λ. We infer that
T
2,t− T
3,t≤ X
T∈ThC
1
2 (θ + 1)
2C
dt2γ
tµ × 2µ k E( b z
T) k
2T.
Combining the above bounds and using the condition (29) leads to T
2,n+ T
2,t− T
3,n− T
3,t≤ X
T∈Th
2 3
2µ k E ( b z
T) k
2T+ λ k D( b z
T) k
2T.
Recalling (31) leads to
T
1≥ X
T∈Th
1 3
2µ k E( b z
T) k
2T+ h
−T1k S( b z
T) k
2∂T+ λ k D( b z
T) k
2T.
Using the definition of T
1, the lower bound from Lemma 4.1, and the definition of the norm k·k
µ,λconcludes the proof.
Using the argument from [8, Corollary 15, p. 126] (see [13] for the application to FEM-Nitsche), we infer from Lemma 4.3 the following well-posedness result.
Corollary 4.4 (Well-posedness). The discrete problem (17) is well-posed.
Remark 4.5 (Lower bound (29)). The monotonicity result stated in Lemma 4.3 is robust in the incom- pressible limit provided that the condition (29) does not imply that the penalty parameters γ
nand γ
tneed to be chosen large when the ratio κ is large (remember that this ratio becomes large in the incompressible limit). Robustness happens in the two following situations: 1) for the skew-symmetric variant θ = − 1, for which the penalty parameters γ
nand γ
tneed only to be positive real numbers (instead, for θ ∈ { 0, 1 } , this property is lost for γ
nwhich needs to scale as µκ); 2) for bilateral contact and any value of θ, since only the parameter γ
tis used and it remains independent of κ.
4.3 Error analysis
This section contains our main theoretical results on the convergence of the HHO-Nitsche method for the frictional contact problem.
Theorem 4.6 (Error estimate). Let ∈ (0, 1]. Recall that κ := max(1,
2µλ). Assume that the penalty parameters are such that
min(κ
−1γ
n, 2γ
t) ≥ 3 (θ + 1)
2+ (4 + (θ − 1)
2)
C
dt2µ. (32)
Assume that the exact solution satisfies u ∈ H
1+ν(Ω; R
d) , ν >
12. Let u b
hbe the discrete solution of (17) with local components u b
Tfor all T ∈ T
h. Then we have
X
T∈Th
2µ k ε(u) − E( b u
T) k
2T+ λ k∇· u − D( b u
T) k
2T+
2(1 + ) X
T∈ThC
h
Tγ
nk [τ
n(u)] − [τ
n( u b
T)] k
2∂TC+ h
Tγ
tk [τ
t(u)]
s− [τ
t( b u
T)]
sk
2∂TC. A
1(u) + A
2(u), (33)
with A
1(u) defined in (27). Moreover A
2(u) is given by A
2(u) := X
T∈ThC
2
h
Tγ
nk δσ
n,Tk
2∂TC+ γ
nh
Tk δu
n,Tk
2∂TC+ h
Tγ
tk δσ
t,Tk
2∂TC+ γ
th
Tk δu
t,Tk
2∂TC, (34)
with δσ
n,T:= σ
n(u) − σ
n(b I
kT(u)), δσ
t,T:= σ
t(u) − σ
t(b I
kT(u)), the HHO interpolation operator is defined
in (23), and δu
n,Tand δu
t,Tare the normal and tangential components on Γ
Cof δu
T:= u
|∂T− Π
k+1∂T(u
|∂T).
Proof. Let us set z b
h:= b u
h− I b
kh(u), where I b
kh: V
D→ U b
kh,0is the global HHO interpolation operator defined in Section 4.1. The same manipulations as in (31) lead to
X
T∈Th
2µ k E( b z
T) k
2T+ h
−T1k S( b z
T) k
2∂T+ λ k D( b z
T) k
2T≤ T
1+ T
2,n+ T
2,t− T
3,n− T
3,t,
where the terms on the right-hand side are defined as above by setting b v
h:= u b
hand w b
h:= I b
kh(u). We use the fact that b u
his the discrete solution to infer that T
1:= b b
h( b u
h; z b
h) − b b
h( I b
kh(u), b z
h) = b `
h( z b
h) − b b
h( I b
kh(u), z b
h).
Recalling the definition of b b
hand b `
h, we obtain T
1:= T
1,1+ T
1,2,n+ T
1,2,t− T
1,3,n− T
1,3,twith T
1,1:= X
T∈Th
(f , z
T)
T− b a
T(b I
kT(u), z b
T)
+ X
T∈ThN
(g
N, z
∂T)
∂TN,
T
1,2,n:= X
T∈ThC
θ h
Tγ
n(σ
n( I b
kT(u)), σ
n( b z
T))
∂TC,
T
1,3,n:= X
T∈ThC
h
Tγ
n([τ
n(b I
kT(u))] , (τ
n+ (θ − 1)σ
n)( b z
T))
∂TC,
and similar expressions for T
1,2,tand T
1,3,t. We use the identities σ
n( I b
kT(u)) = (σ
n( I b
kT(u)) − σ
n(u)) + σ
n(u) and [τ
n( I b
kT(u))] = ([τ
n(b I
kT(u))] − [τ
n(u)] ) + [τ
n(u)] in T
1,2,nand T
1,3,n, respectively, and obtain
T
1:= T
01,1(u, b z
h) − T
01,2,n− T
01,2,t+ T
01,3,n+ T
01,3,t, (35) where T
01,1(u, b z
h) is defined in (25) and (recall that δσ
n,T:= σ
n(u) − σ
n( I b
kT(u)))
T
01,2,n:= X
T∈ThC
θ h
Tγ
n(δσ
n,T, σ
n( z b
T))
∂TC,
T
01,3,n:= X
T∈ThC
h
Tγ
n([τ
n(u)] − [τ
n(b I
kT(u))] , (τ
n+ (θ − 1)σ
n)( z b
T))
∂TC,
and similar expressions for T
01,2,tand T
01,3,t. To make the term T
01,1(u, b z
h) appear in (35), we used that
∇· σ(u) + f = 0 in Ω, σ
n(u) = g
Non Γ
N, whereas on Γ
Cwe used that σ
n(u) = [τ
n(u)] ,
hγTn
(θσ
n( z b
T) − (τ
n+ (θ − 1)σ
n)( z b
T)) = z
∂T ,n, a similar identity for the tangential component, and that σ
n(u) · z
∂T= σ
n(u)z
∂T ,n+ σ
t(u) · z
∂T ,t. Combining T
2,nwith T
01,2,n, we infer that
T
02,n:= T
2,n− T
01,2,n= X
T∈ThC
θ h
Tγ
nk σ
n( b z
T) k
2∂TC− (δσ
n,T, σ
n( b z
T))
∂TC,
together with a similar expression for T
02,t:= T
2,t− T
01,2,t. Moreover, combining T
3,nwith T
01,3,n, we infer that
T
03,n:= − T
3,n+ T
01,3,n= X
T∈ThC
h
Tγ
n([τ
n(u)] − [τ
n( b u
T)] , (τ
n+ (θ − 1)σ
n)( z b
T))
∂TC,
together with a similar expression for T
03,t:= −T
3,t+ T
01,3,t. Putting everything together, we infer that X
T∈Th
2µ k E( z b
T) k
2T+ h
−T1k S( z b
T) k
2∂T+ λ k D( b z
T) k
2T≤ T
01,1(u, b z
h) + T
02,n+ T
02,t+ T
03,n+ T
03,t. (36) Let us now bound the terms ( T
02,n+ T
03,n) and ( T
02,t+ T
03,t). We only detail the bound on ( T
02,n+ T
03,n) since the reasoning is similar for ( T
02,t+ T
03,t). Recalling the above expression for T
02,nand using Young’s inequality (with β
1> 0) for the second term on the right-hand side, we infer that
T
02,n≤ X
T∈ThC
h
Tγ
n(θ +
β21) k σ
n( z b
T) k
2∂TC+
2βθ21
k δσ
n,Tk
2∂TC.
Turning to T
03,n, since b z
T= b u
T− I b
kT(u) and the operator τ
nis linear, we have
(τ
n+ (θ − 1)σ
n)( b z
T) = − (τ
n(u) − τ
n( b u
T)) + (τ
n(u) − τ
n(b I
kT(u))) + (θ − 1)σ
n( z b
T), so that we can re-arrange the terms composing T
03,nas follows:
T
03,n= X
T∈ThC
h
Tγ
n− ([τ
n(u)] − [τ
n( b u
T)] , τ
n(u) − τ
n( b u
T))
∂TC+ ([τ
n(u)] − [τ
n( b u
T)] , τ
n(u) − τ
n( I b
kT(u)))
∂TC+ (θ − 1)([τ
n(u)] − [τ
n( b u
T)] , σ
n( b z
T))
∂TC.
Using (19) for the first term on the right-hand side and letting ω
n,T:= [τ
n(u)] − [τ
n( b u
T)] and δτ
n,T:=
τ
n(u) − τ
n( I b
kT(u)), we infer that T
03,n≤ X
T∈ThC
h
Tγ
n− k ω
n,Tk
2∂TC+ (ω
n,T, δτ
n,T)
∂TC+ (θ − 1)(ω
n,T, σ
n( z b
T))
∂TC.
Using Young’s inequality to bound the second and the third terms on the right-hand side (with β
2> 0 and β
3> 0), we infer that
T
03,n≤ X
T∈ThC
h
Tγ
n( − 1 +
β22+
|θ2β−1|3
) k ω
n,Tk
2∂TC+
2β12
k δτ
n,Tk
2∂TC+
|θ−21|β3k σ
n( z b
T) k
2∂TC.
Putting the bounds on T
02,nand T
03,ntogether leads to T
02,n+ T
03,n≤ X
T∈ThC
h
Tγ
n− ρ
1k ω
n,Tk
2∂TC+ ρ
2k σ
n( z b
T) k
2∂TC+
2βθ21
k δσ
n,Tk
2∂TC+
2β12
k δτ
n,Tk
2∂TC,
with
ρ
1:= 1 −
β22−
|θ2β−1|3, ρ
2:= θ +
β21+
|θ−1|2β3.
Let ∈ (0, 1] and let us choose β
1:= 2, β
2:=
1+, and β
3:=
|θ−1|2(1+). Then we have ρ
1=
2(1+)and ρ
2=
(θ+1)4 2+ (1 +
(θ−41)2), as well as
2βθ21=
θ42≤
41≤
1and
2β12=
1+2≤
1. Using the above bound on T
02,n+ T
03,ntogether with a similar bound on T
02,t+ T
03,tin (36), we infer that
X
T∈Th
2µ k E( b z
T) k
2T+ h
−T1k S( b z
T) k
2∂T+ λ k D( b z
T) k
2T(37)
+ ρ
1X
T∈ThC
h
Tγ
nk ω
n,Tk
2∂TC+ h
Tγ
tk ω
t,Tk
2∂TC≤ T
01,1(u, z b
h) + X
T∈ThC
ρ
2h
Tγ
nk σ
n( z b
T) k
2∂TC+ h
Tγ
tk σ
t( z b
T) k
2∂TC+ A
02(u),
with ω
t,T:= [τ
t(u)]
s− [τ
t( b u
T)]
sand A
02(u) := X
T∈ThC
