The development of numerical methods for frictional contact problems is one of the most challenging task in the field of computational sciences. It combines several fields of research, including optimization theory to deal with arising inequality constraints, the development of appropriate discretization schemes in space and time, and tribology for the physical modeling of the surface interaction.
The numerical model to be developed can be subdivided into different tasks, using constitutive laws to describe the physical nature of the surface interaction and, on the other hand, kinematic relations to embed these laws into the generally nonlinear finite element method. We omit any small deformation considerations and focus on the large deformation frictional contact problem. For a comprehensive survey of contemporary methods, we refer to the textbooks of Laursen  and Wriggers .
Within this paper, we focus on the development of a new method for the contact kinematics. The governing constitutive laws have been extensively investigated by several authors (see among others He and Curnier  and Laursen and Oancea ). We apply the well-known Coulomb's law, using the established analogy between Coulomb friction and nonassociative plasticity (see for example ). However, the proposed approach is capable to embed all kinds of constitutive friction laws.
Our approach follows the lines of the work of Laursen [1, 6], referred to herein as the direct approach . Therefore, in contrast to earlier works (see for example Wriggers and Simo  and Parisch ), continuum mechanical arguments are used to derive the underlying variational formulation.
In contrast to previous works dealing with large deformation frictional contact, we propose a new augmentation technique for the description of the frictional kinematics. The augmentation technique relies on the introduction of additional variables representing the local convective coordinates. Note that the terminus augmentation is widely used for different methods, including the augmented Lagrangian update algorithm. Within our newly proposed method for frictional contact, we refer to augmentation techniques as used in the context of multibody systems . The additional variables are connected to the original ones by introducing additional algebraic constraints that are enforced by means of Lagrange multipliers. When compared with the traditional direct approach, the augmentation technique significantly simplifies the whole formulation.
After the discretization in space has been performed, the resulting differential-algebraic equations can be reformulated to reduce the size of the algebraic system to be eventually solved. That is, applying a size-reduction procedure within the discrete setting essentially recovers the size of the original system. It is worth noting that the size-reduction process does not affect the algorithmic conservation properties of the proposed discretization in space and time. In particular, our discretization approach inherits the conservation laws for linear and angular momentum from the underlying continuous formulation.
The development of structure-preserving time-stepping schemes for large deformation contact problems has been subject of extensive research, see Laursen and Chawla [11, 12], Armero and Petöcz [13, 14], and Hesch and Betsch [15, 16]. Similar to these works, the spatial discretization of the contact surface used in the present work is based on the node-to-surface (NTS) method. The extension to more sophisticated models using mortar-based formulations [17-22] will be dealt with in a subsequent work.
The article is organized as follows. The fundamental equations of the underlying problem in strong and weak form are outlined in Section 2. In this connection, the well-established description of the frictional kinematics is summarized, and the newly proposed augmentation technique is introduced. The spatial discretization and the NTS method together with the specific formulation of the augmentation technique is given in Section 3. In Section 4, we apply a suitable time integration scheme and verify algorithmic conservation of linear and angular momentum. Representative numerical examples are presented in Section 5. Eventually, conclusions are drawn in Section 6.
2 GOVERNING EQUATIONS
We consider continuum bodies written in their reference configurations for the large deformation problem at hand. To characterize the deformation, we assume the existence of a mapping , , such that we can introduce the deformation gradient as follows:
Note that remains nonsingular and invertible throughout the considered time interval , that is, . The material behavior is governed by the strain energy function where , denotes the right Cauchy–Green tensor, and we define the first Piola–Kirchhoff stress tensor as follows:
Eventually, the balance of linear momentum reads
where denotes the body force per reference volume and the inertia term using the reference density . The boundaries are subdivided into the Dirichlet boundary , the Neumann boundary , and the contact boundary (Figure 1). We require that these boundaries do not overlap; hence, they satisfy
Moreover, appropriate boundary conditions are given by
where denotes the unit outward normal. We assume that the bodies are in contact within the considered time interval and restrict our consideration on a two body contact problem neglecting self-contact for simplicity of exposition. Additionally, we provide the initial conditions
finalizing the strong form of the problem. Next, we rewrite the system in weak form to obtain the virtual work of the whole system. To this end, we define the solution space
and the space of test functions
where the Sobolev space consists of square-integrable functions and square-integrable first derivatives thereof. The virtual work contribution of each body reads
for and . Here, denotes the second Piola–Kirchhoff stress tensor. Now, the principle of virtual work for the two-body contact problem under consideration can be written as . Taking into account the balance of linear momentum across the contact interface
the contact contribution to the virtual work can be summarized in the expression
In the last statement, contains the collection of the mappings , i = 1,2 (similarly for ).
2.1 Contact formulation
We assume that a point , on the surface is in contact with the opposing master surface and define the projection
where is the closest point to . The master surface itself can be viewed as a 2D manifold parametrized by the convective coordinates , . Thus, the projection can be characterized by the relationships
where are calculated from (14). We further introduce tangent vectors
where denotes the derivative with respect to . Note that the vectors are directed tangentially along the coordinate curves at (Figure 2). Associated dual vectors are defined by
where is the inverse of the metric . Next, we introduce the gap function
where denotes the unit outward normal to at , defined as
Note that the tangent vectors along with the normal vector are covariant base vectors. Furthermore, the normal vector is assumed to be directed tangentially along the coordinate line (Figure 2). The variation of the gap function can now be written in the form
As usual, we decompose the contact traction in (13) into the normal and the tangential part
and require that . For the normal component, the Karush–Kuhn–Tucker conditions
have to hold, whereas the vector lies in the tangent space of the master surface and can be resolved via
The corresponding frictional constitutive law to define the tractions will be dealt with in the appendix.
Summarizing, the contact contribution to the virtual work can now be written in the form
The last statement depends crucially on the variation of the convective coordinates on which we will focus next.
2.2 Frictional kinematics
Next, we focus on the variation of the convective coordinates to complete the contact formulation given in (25). In particular, we outline the most common approach, referred to as the direct approach in the following  and present subsequently a new augmentation technique for the description of the frictional kinematics.
The convective coordinates can be obtained from the solution of the minimum distance problem (14). Correspondingly, the orthogonality condition
has to be valid. Computing the time derivative of the last equation yields
Using the unit length of the normal vector, that is, together with , we can rearrange the terms in (27) and obtain the rate of change of the convective coordinates
where denotes the inverse of and is the curvature of the surface. Replacing the velocity by the variation yields
Assuming that is valid at the contact interface, the variation of boils down to
Accordingly, the virtual work expression (25) can be recast in the form
where relation (21) has been used. The majority of previous works dealing with large deformation frictional contact problems rely on (31) . Note that statement (31) holds true if (29) is used instead of (30), because the additional terms to be considered only redefine the tractions in tangential direction.
2.3 Coordinate augmentation technique
Following the arguments in Hesch & Betsch , we extend a specific coordinate augmentation technique to frictional contact problems. This technique relies on the introduction of additional coordinates that represent the convective coordinates . To link the new coordinates to the original ones, we introduce two constraint functions
and require that . Similar to definition (17) for the tangent vectors, in (32), for . Analogous to the definition of the gap function (19), we introduce
where again follows from (20) by replacing with . The contact contribution to the virtual work can now be determined along the lines of the direct approach. Accordingly, similar to (31), we obtained
where . It is important to realize that the augmented coordinates are to be viewed as primary variables on an equal footing with the original variables . Consequently, the newly proposed augmentation technique strongly affects the discretization in space and time. Indeed, it will be shown in the sequel that the newly proposed augmentation technique simplifies the implementation significantly when compared with the direct approach.
3 SPATIAL DISCRETIZATION
We apply a suitable spatial discretization process to the bodies under consideration and to the contact constraints. More precisely, displacement-based finite elements are used for the bodies in contact, subdividing each body into a finite number of elements via
The polynomial approximations of the solution and the test space are written as
where represents the nodal position at point and the corresponding variation at point . Furthermore, are global trilinear Lagrangian shape functions. The semi-discrete version of the principle of virtual work can now be written in the form
Note that the nodal position vectors and have been collected in the vector (similarly for ). The last term in (37) denotes the discrete version of the contact virtual work, which will be dealt with in the following section. We can rewrite (37) as follows
where represents the nodal mass contributions, the internal nodal forces, and the external forces. Throughout the paper, we assume that the internal and external forces are associated with a potential energy function
The last two terms in (38) represent the normal contact and frictional forces, which will be derived in the next section.
3.1 Node-to-surface element
Similar to the approximations of the solution and the test space, we define the following approximations at the contact boundaries
where denote bilinear shape functions at the corresponding node , representing the set of all nodes on the contact interface. Using the direct approach, we have to compute the convected coordinates internally within each NTS element (Figure 3) by solving
for the convective coordinates using a Newton–Raphson iteration. The discrete nodal gap function (Figure 3) reads
using the set of nodes . Employing the discrete nodal gap function, we can define the constraint function in normal direction
along with the Lagrange multipliers that can be viewed as discrete counterpart of the normal traction .
The corresponding tangential tractions are dealt with in Appendix A for the case of Coulomb's law. Note that we incorporate the frictional response using a widely used regularization method based on a penalty parameter. Other types of enforcement of the frictional response are possible [24-26], because the proposed augmentation technique remains unaffected by, for example, the existence of additional Lagrange multipliers. The chosen approach is used for the sake of clarity in the presentation.
Similar to the kinematic relationship (29), the variation of the convective coordinates in the discrete setting reads
If we assume that the gap is zero, we obtain
and the corresponding discrete virtual contact work for a single NTS element reads
With regard to (38), we rearrange the frictional contributions using a single vector . Furthermore, we collect all normal constraints in a single vector and assemble the associated Lagrange multipliers in the vector . Then, the semi-discrete equations of motion read
where denotes the consistent mass matrix.
3.2 Coordinate augmentation technique
Next, we apply the coordinate augmentation technique described in Section 2.3 to the NTS element. In contrast to the direct approach, we calculate the convective coordinates on a global level, that is, we do not solve the algebraic system of equations in (41) internally but enforce them as additional constraints
Here, we make use of a vector for each NTS element, representing the convective coordinates . The associated Lagrange multipliers are given by . In addition to that, the constraints in normal direction are given by
As before, we collect all data in global vectors, that is, we collect all augmented coordinates in a single vector , where denotes the number of all convective coordinates. Furthermore, the augmented constraints (48) are arranged in a single vector and the associated Lagrange multipliers in a single vector . The semi-discrete equations of motion can now be written as follows
where and . Furthermore, combines the frictional tractions in a single vector.
To implement the newly proposed method in an efficient way, we eliminate the additional Lagrange multipliers using the algebraic condition . For a single NTS element, this condition reads
where represents the tangential tractions of the corresponding NTS element. The Lagrange multipliers can now be calculated analytically as follows
Accordingly, on the level of each NTS element, the Lagrange multipliers associated with the augmented coordinates can be expressed in terms of the extended set of coordinates and the contact traction in normal direction. Using (52) for each NTS element, the vector of Lagrange multipliers can be eliminated from the semi-discrete equations of motion (50). Accordingly, we arrive at
where the block diagonal matrix consists of the local projection matrix
for each NTS element. Note that is valid at the solution point, and we obtain the simplified system
The last set of equations defines the residual , which we have to solve with respect to , and . This first reduction step can be written in matrix notation using the modified projection matrix
where denotes the number of degrees of freedom of the configuration , the number of constraints , and the number of augmented coordinates . Premultiplication of (50) by the projection matrix in (56) yields (55).
In a second step, we eliminate the augmented coordinates within the Newton–Raphson iteration
that is used to solve (55). Here, and denotes the derivative of with respect to and , respectively. Next, we extract the equations for the augmented constraints of a single NTS element from (57)
and solve this last equation with respect to , such that we obtain
The last reduction step can also be written in matrix notation using
It is important to remark that the whole reduction procedure can be carried out on element level for each single NTS-element, because is block diagonal. The convective coordinates can be recovered using (59). The consistent linearization can now be carried out in two different ways:
As shown in (57), we have to linearize with respect to the configuration and the augmented coordinates . The involved constraints (48) and (49) are at most quadratic in the configuration and in the augmented coordinates, thus the only terms of higher order to be derived depending on the used constitutive law (this derivative is always necessary) and the inverse matrix , that is, we have to linearize .
In , we have used the projection matrix to obtain a new residual, which we have to linearize to obtain the matrix in (57). Alternatively, we can premultiply the full linearized original system (50) by and obtain
where terms labeled by the upper index represent the contributions arising from and . Next, we remove and the corresponding columns from the system, because we solve directly for using (52). The second reduction step follows as before, now avoiding the linearization of . Note that we take again advantage of its block diagonal structure, such that all steps can be carried for each contact element.
The linearization is extremely simplified, compared with traditional methods, where we need to calculate the linearization of the variation of the convective coordinates (cf. Laursen )
Remark: Although we use Lagrange multipliers to enforce the normal constraints, we can also apply an augmented Lagrangian method to calculate the exact values of .
3.3 Conservation properties
The conservation properties of the underlying mechanical system are well known, so we concentrate on the contact contributions. Reconsider the virtual work contributions of a single contact element
where we make use of (46) along with (42) and (44). The conservation of linear momentum may be verified by substituting , where is arbitrary and constant into the global virtual work of the constraint forces
using the direct approach in (44). To verify conservation of angular momentum, we substitute and obtain
Note that the simplified variation (45) conserves angular momentum only if the normal gap is equal zero.
Finally, we verify the conservation properties of the augmented system in (50). The corresponding contact virtual work reads
where we have taken () into account. Insertion of into the augmented system yields
whereas insertion of yields
Because the constraints are frame indifferent with respect to rigid body motions of the form
where is a constant vector , we can show that for each NTS element, the relation
is valid. Substituting , , and subsequent derivation with respect to yields
Thus, (68) holds for arbitrary and linear momentum is conserved. Substituting and , where is a skew-symmetric matrix, associated with the axial vector so that for any , we end up with
Thus, (69) holds for arbitrary and angular momentum is conserved for the semi-discrete system. The same statements are true for the reduced system, because the algebraic reformulation does not change the general characteristics of the system.
4 TEMPORAL DISCRETIZATION
The semi-discrete equations of motion (47) and (50) have to be discretized in time. Consider a time interval subdivided into increments and assume the state at time to be known. Now, for a typical time step , the full discrete version of (47) reads
Here, denotes the discrete gradient of the strain energy function . The discrete version of the frictional kinematics used in is related to the definition of the convective coordinates
Note that we deal with the adjoint discrete traction using a local evolution scheme in Appendix B. The time-discrete version of the augmented system in (50) reads
where , , and . As already mentioned, consists of the tractions , see (50). Accordingly, has to be evaluated as shown in Appendix B. Following the arguments outlined in the previous section, we create a local projection matrix as follows
and obtain for the reduced system
The second reduction step follows immediately from (60) using the discretized projection matrix in (77) evaluated at time
The full discrete system to be solved in each Newton–Raphson iteration now reads
where consists of the residual contributions in , and and denotes the consistent linearization of with respect to and , respectively. In a final step, we recover the augmented coordinates by solving
for each NTS element.
It is obvious that the linearization is extremely simplified compared with traditional schemes. Furthermore, the proposed scheme is more consistent because it ensures the exact fulfillment of the orthogonality conditions at each time node within the chosen mid-point type scheme.
4.1 Conservation properties
As before, we focus on the contact contribution and begin with the verification of the conservation of linear momentum. To this end, we substitute , into the weak form of the contact contributions
which confirms that the constraints do not affect linear momentum conservation. Following the arguments in (66), we substitute and obtain
which confirms that the constraints do not affect angular momentum conservation as well.
At last, we verify the conservation properties of the full discrete system in (76) and substitute
whereas we obtain
if we substitute . Once again frame indifference of the vector of constraints against rigid body motions is crucial for the fulfillment of the conservation laws. Proceeding along the lines of Section 3.3, we can easily verify that
where . Substituting , , yields
Analogous to the semi-discrete system, linear momentum is algorithmically conserved. Substituting and yields
Thus, angular momentum is algorithmically conserved for the full-discrete system. Note that the last statement is also true for the reduced system, because the algebraic reformulation of the system does not change any properties of the underlying formulation.
5 NUMERICAL RESULTS
In this section, we evaluate the accuracy and performance of the newly proposed method and compare the results to well-known and established methods. To solve the arising nonlinear system of equations, a Newton–Raphson solution procedure has been implemented.
5.1 Contact of two elements
To evaluate the properties of the algorithms under consideration, we investigate a simple nonlinear three-dimensional example, which is constructed such that we obtain reproducible results. In particular, we consider two 3D elements, using trilinear shape functions. A compressible Neo–Hookean material is used with associated strain energy density function
where and , are Lamé parameters corresponding to a Young's modulus of and a Poisson's ratio of . The reference density is given by and the coefficient of friction . The initial position of the 16 nodes are given in Table 1 together with the initial velocity (Figure 4). Because of the initial configuration, the tangent vectors of the master surface are not orthonormal.
Table 1. Nodal positions and initial velocity.
[ − 0.5, − 1, 2.1]
[0, 0.1, − 0.04]
[ − 1, − 1, 1]
[0 0 0]
[ − 0.5, 0, 2.1]
[0, 0.1, − 0.04]
[ − 1.5, 1.5, 1]
[0 0 0]
[ − 0.5, − 1, 1.1]
[0, 0.1, − 0.04]
[ − 1, − 1, 0]
[0 0 0]
[ − 0.5, 0, 1.1]
[0, 0.1, − 0.04]
[ − 1.5, 1.5, 0]
[0 0 0]
[ 0.5, − 1, 2.1]
[0, 0.1, − 0.04]
[1, − 1, 1]
[0 0 0]
[ 0.5, 0, 2.1]
[0, 0.1, − 0.04]
[1.2, 1, 1]
[0 0 0]
[ 0.5, − 1, 1.1]
[0, 0.1, − 0.04]
[1, − 1, 0]
[0 0 0]
[ 0.5, 0, 1.1]
[0, 0.1, − 0.04]
[1.2, 1, 0]
[0 0 0]
Both elements are flying free in space, that is, no boundary conditions are prescribed. First, we show results using an implicit Euler backward algorithm. Therefore, the z-position of node 8 of the upper block, which is directly in contact with the lower block, is plotted over time in Figures 5 and 6. In particular, Figure 5 shows the results for different time step sizes of the newly proposed algorithm using both reduction steps, whereas Figure 6 shows the results of the conventional direct approach. As expected, the Euler backward algorithm damps oscillations for larger time step sizes. In Figure 7, we show a comparison of the augmented system, the reduced system, and the direct approach for a time step size of . The results coincide extremely well for the used implicit Euler backward algorithm. Thus, the advantages of the new approach considered here relies on the simplified structure of the contact element.
Figure 8 shows additionally the results of the proposed new algorithm using a mid-point type evaluation, as proposed in Section 4. Using the mid-point type evaluation, we obtain even for large time step sizes reliable results, for example, for , using only 100 time steps for the whole simulation. Note that Lagrange multipliers have been used to enforce the constraints in normal direction throughout all shown examples.
Finally, total energy and total angular momentum are plotted in Figures 9 and 10 for the proposed scheme using the mid-point type evaluation. As can be seen, total energy is conserved after the frictional impact. Furthermore, total angular momentum is also conserved. Although not shown here, linear momentum is also conserved algorithmically.
5.2 Two tori impact problem
In this example, we consider an impact problem of two tori to demonstrate that the proposed algorithm is suitable for large systems. Initial values and the material properties have been taken from Yang and Laursen . The initial configuration is displayed in Figure 11. The inner and outer radius of the tori are 52 and 100; the wall thickness of each hollow torus is 4.5. Both tori are subdivided into 8024 elements, using a Neo–Hookean hyperelastic material with and . The initial densities are and the homogeneous, initial velocity of the left torus is given by . A time step size of 0.01 has been used throughout the whole simulation. The deformation at different time steps is shown in Figure 12.
The evolution of the total energy is shown in Figure 13, whereas the three components of angular momentum are shown in Figure 14. As expected, total energy decreases because of the frictional behavior. Because we used the proposed mid-point type evaluation of the system, angular momentum is conserved.
Finally, Figure 15 shows the deformation at for different friction coefficients, using and . The deformation changes significantly because large sliding effects are directly correlated with the friction coefficient.
A novel formulation for the frictional kinematics has been developed in the framework of large deformation contact. This new method rests on an augmentation technique, which substantially simplifies the underlying expressions at the cost of an enlarged global system of algebraic equations to be solved. To remedy this drawback, a size-reduction procedure has been proposed on the basis of the elimination of the Lagrange multipliers associated with the augmented coordinates. In a second step, the size of the system has been further reduced to the original size. In this connection, a new analytical representation of the projection of the convective coordinates to the configuration space has been established.
The new approach is much more simple compared with traditional methods. As shown, the internal Newton iteration to determine the actual convective coordinates has been removed, and the residual and the tangent contributions of the contact element are significantly simplified. We have demonstrated the usability of the proposed augmentation technique for large deformation problems. Because the size-reduction of the system relies on an analytical reformulation, the underlying conservation laws are not affected. Thus, we could ensure algorithmic conservation of linear and angular momentum. This provides enhanced numerical stability of the method for large simulations.
APPENDIX A: CONSTITUTIVE EVOLUTION EQUATIONS
Many researchers have investigated various constitutive laws used to describe the tangential tractions. We omit a further investigation and focus on a standard dry friction Coulomb law to complete the set of equations used for the numerical examples. On the basis of this specific formulation, we state that
The tangential displacement in the case of slip follows from where denotes the consistency parameter, which depends on (A.1), and we can write
In analogy to plasticity, we rewrite the last statement as follows
and regularize the equation for the tangential velocity using a penalty method
(Figure A.1). Note that the components in tangential direction read
where we have made use of .
APPENDIX B: LOCAL TIME STEPPING SCHEME FOR THE FRICTIONAL EVOLUTION EQUATIONS
Here, we apply a one-step integration scheme of the local evolution equations (A.5) following the arguments in the work of Armero and Petöcz . Consequently, the approximation of the tractions can be written as follows
where controls the corresponding time stepping scheme and should be chosen consistent with the global time stepping scheme. Taking the inequality conditions (A.3) into account, we obtain
Note that is represented by a Lagrange multiplier, constant within the time step. To implement (B.2), we apply a return mapping scheme and start by considering the stick case, that is,
which defines our trial state. Depending on the condition , slip occurs, and we obtain
by comparing (B.2) and (B.3). After short calculations using the relation , the consistency parameter in the case of slip is determined by
Thus, the final contribution in the case of slip reads
To summarize, the return mapping scheme can be written as follows
which completes the used definition for the tractions.
Support for this research was provided by the Deutsche Forschungsgemeinschaft (DFG) under grant HE 5943/1-1 and HE 5943/3-1. This support is gratefully acknowledged.