Node-to-Segment

Despite the drawbacks associated with the Node-to-Segment (NTS) discretization technique, it is the most widely used technique for large deformation contact problems due to its simplicity and flexibility [Zavarise 09b]. For these reasons, it is often implemented in commercial finite element codes being also the contact discretization technique adopted throughout this dissertation. The first step of the NTS discretization comprises the selection of one contacting surface as slave and the other one as master, leading to an asymmetry in the contact problem because the contact surfaces are treated differently. The impenetrability conditions are enforced only in a discrete number of points on the contact slave surface, preventing the slave nodes from penetrating on the contact master surface [Hallquist 85]. However, the master nodes are allowed to penetrate into the slave surface, as depicted in

Figure 3.9 (a). Each contact element (not structural) is composed by a slave node and the closest segment (element edge/facet) on the master surface (see Figure 3.9 (b)), which is selected through the orthogonal projection of the slave node. Nevertheless, particularly when the NTS approach is applied with low order finite elements [Crisfield 00], the identification of the master segment is either ambiguous or impossible, which may result in slow convergence or even in divergence of the numerical solution. Some strategies have been specially developed to deal with such problems in 2D frictionless contact problems [Zavarise 09b].

(a) (b)

Figure 3.9. Node-to-Segment contact discretization: (a) penetration conditions unchecked in master nodes; (b) contact element composed by a slave node and a master

segment.

3.3.1.1. Selection of master and slave surfaces

The proper selection of master and slave surfaces is fundamental to the success of the NTS contact discretization. After identifying the pair of surfaces that will interact, one surface is assigned as master and the other one as slave, leading to an asymmetric contact treatment, as shown in Figure 3.9 (a). The selection should be carried out taking into account that the slave nodes cannot penetrate into the master surfaces, but the master nodes can

master

slave slave node

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penetrate the slave surface. Hence, the main guidelines for choosing the master and slave surfaces are listed below:

 The contact surface that presents the coarse mesh should be the master surface, while the surface with the fine mesh is the slave surface;

 When the stiffness between the contacting bodies is different, the contact surface of the stiffer body should be the master surface and the other should be assigned as slave surface;

 When the contact occurs between a deformable body and a rigid obstacle, the surfaces of the rigid obstacle must be specified as master surfaces;

 In the case of contact between a convex surface with a flat or concave surface, the master surface should be the flat/concave surface;

 If one body slides over another with a contact surface considerably larger, the larger surface should be the master surface, in order to minimize the contact status changes.

In order to highlight the importance of the master/slave surface selection, the contact patch test example is presented. The original patch test [Taylor 91] is modified in this example, where two elastic cubes with identical geometry (each edge with 10 mm) and the same material properties ( E100 MPa and ν0.3 ) are pressed against each other considering frictionless contact. The finite element discretization of each cube using 8-node hexahedral elements is depicted in Figure 3.10 (a), showing that the meshes do not coincide at the contact interface. The bottom surface of the lower cube is constrained against vertical displacements and the four lateral surfaces of the cubes are constrained against displacements in its normal direction. The uniform pressure is imposed by applying a vertical displacement of 1 mm on the top surface of the upper cube.

The distribution of the vertical stress component in the cubes is depicted in Figure 3.10 (b), employing the NTS contact discretization with the upper cube (finer mesh) defined as master. The obtained results do not satisfy the contact patch test due to the non-conforming meshes at the contact interface. Since the finer mesh is assigned as master (incorrect choice), the penetration of some master nodes into the slave cube can be considered excessive (see

Figure 3.10 (b)), which leads to high deviations in the predicted contact stress, i.e. inaccurate transmission of constant normal stresses between two contacting surfaces. On the other hand, by exchanging the master and slave surfaces definition, the resulting distribution of vertical stress is shown in Figure 3.10 (c). The noise in the contact stress is reduced considerably when the coarse mesh is assigned as master surface, leading to a contact surface approximately flat after loading. Although the mesh refinement of the slave surface improves the accuracy, only matching meshes at the contact interface (NTN contact discretization) allows the complete elimination of the inaccurate transmission of stresses.

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(a) (b) (c)

Figure 3.10. Distribution of the vertical stress component for different choices of the master and slave surfaces in the NTS contact discretization: (a) finite element mesh of contacting cubes; (b) upper cube defined as master and lower as slave; (c) upper cube

defined as slave and lower as master.

The value of the vertical stress component (compressive) and the corresponding contact force applied on the upper cube can be exactly calculated performing the patch test using a conforming mesh at the contact interface. In this case, the obtained vertical stress is uniform in both cubes being the compressive contact stress 6.9 MPa and the contact force 690.0 N. Nevertheless, when the cube discretized with the finer mesh is wrongly defined as master (Figure 3.10 (b)), the predicted contact force is 646.9 N, which is a consequence of the local penetration of the cubes. On the other hand, the proper selection of the master surface (Figure 3.10 (c)) leads to a contact force value of 689.7 N, which is very close to the exact value. Recently, a modification of the NTS discretization has been proposed by Zavarise and De Lorenzis [Zavarise 09a], which passes the contact patch test in 2D frictionless contact problems using the penalty method to enforce the contact constrains. The basic idea of this algorithm is to create two virtual slave nodes located at the quarter points of each slave segment, improving the contact contribution to the stiffness matrix and to the internal force vector.

3.3.1.2. Two-pass contact

The master-slave formulation used in the NTS discretization is inherently asymmetric (see Figure 3.9 (a)), which is not in accordance with the physical observation of contact problems. Besides, as pointed by Taylor and Papadopoulos [Taylor 91], the single-pass NTS algorithm does not satisfy the contact patch test, which in some circumstances can yield unsatisfactory results, as shown in Figure 3.10. Therefore, the two-pass contact (also called [MPa]

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symmetric contact) approach was developed to try to overcome this problem. The main idea of the two-pass approach is the definition of the each contact surface as master and slave simultaneously, performing a double definition of the contact pair, exchanging the master and slave surfaces. This approach precludes penetration of the slave nodes into the master segments (first pass), while the master nodes are restricted from penetrating on the slave segments, when the master and slave surfaces are exchanged in the second pass. Therefore, this strategy allows eliminating the geometric asymmetry by reversing the role of master and slave surfaces and repeating the same process performed in the single-pass algorithm. Since the number of contact elements created is higher, the two-pass NTS algorithm is less efficient than the single-pass contact approach.

The NTS discretization associated with the two-pass contact algorithm passes the contact patch test in 2D and in some 3D mesh configurations (with sufficient symmetry) for low order finite elements [Taylor 91]. Nevertheless, this discretization technique based on the two-pass contact algorithm has the recognised deficiency of locking, due to the over- constrained system of equations [Puso 04a]. Indeed, if any two nodes on both contact surfaces have identical locations, the corresponding contact constraint is created in duplicate during the two-pass algorithm, which results in a rank deficient matrix (linearly dependent rows and columns) [El-Abbasi 01]. This situation can lead to some numerical difficulties, such as singularities and zero pivots, which can be avoided applying a search algorithm to detect and remove these duplications. On the other hand, the smoothing of the master surface can alleviate the locking problems, due to the continuous change of the surface normal vector [Puso 04b].

In order to evaluate the robustness of the adopted direct solver (Intel MKL DSS), briefly described in Section 3.2.1.1, a simple contact problem comprising 2 finite elements is performed using the two-pass NTS algorithm. The same geometry, material properties and the boundary conditions of the last example are considered (Figure 3.10), but each cube is discretized with a single element, leading to matching meshes at the contact interface. Thus, the selection of the master and slave surfaces is completely arbitrary. The augmented Lagrange method, described in Section 2.3.3, is applied for handling the inequality constraints due to the contact, using a penalty parameter value of 60. The single-pass NTS algorithm leads to a problem involving 12 dof’s, 8 representing the nodal displacements (only vertical components) and 4 for the nodal contact forces, which are evaluated only in the slave surface. The square matrix of the linear system of equations (3.19) arising in the first iteration is given by:

General Finite Element Framework 83 254.5 60 59.2 59.2 0 0 23.6 0 1 0 0 0 60 254.5 0 0 59.2 59.2 0 23.6 1 0 0 0 59.2 0 254.5 23.6 60 0 59.2 0 0 1 0 0 59.2 0 23.6 254.5 0 60 59.2 0 0 0 1 0 0 59.2 60 0 254.5 23.6 0 59.2 0 1 0 0 0 59.2 0 60 23.6 254.5 0 59.2 0 0 1 0 23.6 0 59.2 59.2 0 0 254.5 60 0 0 0 1 0 23.6             A , 0 0 59.2 59.2 60 254.5 0 0 0 1 60 60 0 0 0 0 0 0 0 0 0 0 0 0 60 0 60 0 0 0 0 0 0 0 0 0 0 60 0 60 0 0 0 0 0 0 0 0 0 0 0 0 60 60 0 0 0 0                          _     _        _    (3.23)

where the first 8 rows represent the displacements and the last 4 denote contact forces. The obtained matrix is full rank, i.e. all rows are linearly independent. Besides, the determinant of the matrix presented in (3.23) is 2.03e+17 and its condition number is 1.796e+3. Therefore, obtaining the solution for the linear system of equations (3.19) with the matrix (3.23) should be stable and accurate, whatever the numerical method adopted.

On the other hand, the application of the two-pass NTS algorithm in the same example leads to a problem with 16 dof’s, 8 representing to the nodal displacements and 8 denoting the nodal contact forces (contact forces evaluated on both slave and master surfaces). The final stiffness matrix for the first iteration is given by:

314.5 120 59.2 59.2 0 0 23.6 0 1 1 0 0 0 0 0 0 120 314.5 0 0 59.2 59.2 0 23.6 1 1 0 0 0 0 0 0 59.2 0 314.5 23.6 120 0 59.2 0 0 0 1 0 1 0 0 0 59.2 0 23.6 314.5 0 120 59.2 0 0 0 0 1 0 1 0 0 0 59.2 120 0 314.5 23.6 0 59.2 0 0 1 0 1 0 0 0 0 59.2 0 120 23.6 314.5 0 59.2 0 0 0 1 0 1 0 0              A 23.6 0 59.2 59.2 0 0 314.5 120 0 0 0 0 0 0 1 1 0 23.6 0 0 59.2 59.2 120 314.5 0 0 0 0 0 0 1 1 60 60 0 0 0 0 0 0 0 0 0 0 0 0 0 0 60 60 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 60 0 60 0 0 0 0 0 0 0 0 0 0 0 0 0 0 60 0 60 0 0 0 0 0 0 0 0 0 0 0 0 60 0 60 0 0 0 0 0 0 0 0 0 0 0 0 0 0 60 0 60 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 60 60 0 0 0 0 0 0 0 0 0            , 0 0 0 0 0 60 60 0 0 0 0 0 0 0 0                                                      (3.24)

where the last 8 rows are related with the nodal contact forces. Since the two-pass algorithm is applied with conforming meshes at the contact interface, some rows are linearly dependent, as observed in matrix presented in (3.24). Indeed, the rank of this matrix is 12 (rank deficient matrix), where the four pairs of linearly dependent rows are 9-10, 11-13, 12-

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14 and 15-16. Thus, the condition number is infinite and the matrix determinant is zero (the matrix is not invertible). Therefore, the linear system (3.19) has either no solution or an infinite number of solutions. Nevertheless, the usage of the direct solver from the Intel MKL allows to solve this system of equations without visible numerical problems, providing the solution obtained with the single-pass NTS algorithm (correct solution). This indicates the robustness of the adopted solver, which overcomes efficiently the principal drawback associated with the two-pass NTS algorithm, i.e. locking behaviour due to the over- constraint. Since every node of the contact interface acts as slave node in the two-pass algorithm, static variables are evaluated on both contacting surfaces, which can make the interpretation of some results (nodal contact forces) difficult.

(a) (b)

Figure 3.11. Distribution of the vertical stress component in the contact patch test using: (a) the single-pass NTS algorithm with the lower cube as master; (b) the two-pass NTS

algorithm.

The comparison between the single-pass and the two-pass NTS contact algorithm is presented in Figure 3.11 for the contact patch test, with the discretization employed in the previous section. For the single-pass NTS contact algorithm, the cube with the coarse mesh is defined as master, while the selection of the master and slave surfaces in the two-pass algorithm is arbitrary due to its exchange in the second pass. The distribution of the vertical stress component obtained with the single-pass algorithm is presented in Figure 3.11 (a), which are the same results shown in Figure 3.10 (c) using a different range of values. In fact, the single-pass algorithm does not satisfies the contact patch test. On the other hand, the two-pass contact algorithm exactly transmits the constant normal stresses between the contacting surfaces, as shown in Figure 3.11 (b), thus solving the contact patch test. The patch test is passed since the expected contact surface is horizontal (flat). In case of refined

[MPa]

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85 finite element meshes on both contact surfaces, the improvements obtained with the two- pass contact algorithm are insignificant while the required computational cost increases.

The NTS contact discretisation enables to use non-conforming meshes at the contact interface, allowing to take into account large sliding of the surfaces during the deformation process [Hallquist 85]. However, the two-pass contact algorithm does not satisfy the contact patch test for some contact surface mesh configurations [Puso 04b]. The patch test example previously presented is repeated using an unstructured mesh at the contact interface (non- conforming). The contact surface mesh of the upper and lower cube is shown in Figure 3.12 (a) and (b), respectively. The distribution of the vertical stress component obtained with the single-pass algorithm is depicted in Figure 3.12 (c), where the noise in the contact stress is within a range slightly higher than the one obtained with the structured mesh (see Figure 3.11). Although the contact stress clearly converges to the exact value using the two-pass NTS contact algorithm, it does not passes the contact patch test using this interface discretization, as illustrated in Figure 3.12 (c). The vertical component of stress (compressive) ranges between 6.84 and 6.96 MPa and the obtained contact force is 690.0 N, which coincides with the exact value.

(a)

(b) (c) (d)

Figure 3.12. Distribution of the vertical stress component in the contact patch test adopting an unstructured mesh: (a) upper surface mesh; (b) lower surface mesh; (c) single-pass NTS discretization defining the lower cube as master; (d) two-pass NTS

discretization.

In specific situations the identification of the contact pairs can be a big challenge, such as in self-contact problems, where the identification in advance of the individual contacting areas is very difficult (or impossible) due to large folding of the body. Moreover, the distinction between master and slave surfaces is not clear. In these cases is convenient to [MPa]

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apply the two-pass NTS contact algorithm because the classification of the master and slave surfaces is arbitrary. The two-pass algorithm also is particularly appropriate for problems in which both contact surfaces present very coarse meshes, since the contact constraints are enforced in more locations than in the single-pass contact algorithm.