Consistency – patch tests

In order to demonstrate and validate the consistency of numerical methods or element formu-lations, patch tests are the most common choice, see e.g. Irons [126] and Taylor et al. [267].

In the context of mortar methods for computational contact mechanics, these tests are utilized to demonstrate the ability of the methods to represent a constant stress state across the active contact interface. First, this is investigated for a tied surface contact setting in 4.9.1.1 in order to validate the consistency of the shape function modification in Section 4.7. Second, a frictionless contact setting is analyzed with the combined point, line and surface contact algorithm in Sec-tion 4.9.1.2 to demonstrate the consistency of the combined contact framework of point, line and surface contact. Finally, a line contact scenario is considered where the active contact interface reduces to a curve and the algorithm is tested with regard to represent a constant stress state across this curve.

4.9.1.1. Mortar mesh tying with boundary modification

The first patch test investigated in this thesis is a 3D cuboid, which consists of six differently discretized subdomains connected by the mortar mesh tying scheme, see left part of Figure 4.17.

It is well-known that the employed mortar method with its variationally consistent interpolation of the interface traction via discrete Lagrange multipliers naturally guarantees for a satisfaction of classical patch tests. Nevertheless, this example is challenging, because it inevitably leads to crossing mesh tying interfaces, which require special treatment of the Lagrange multipliers at so-called crosspoints and crosslines in order to achieve a properly stated problem. There-fore, the discrete Lagrange multipliers at the nodes attached to crosspoints and crosslines are removed and the shape functions of neighbored Lagrange multipliers are modified according to Section 4.7. Since the matrices arising for mortar mesh tying schemes are strongly related to the matrices from mortar based computational contact mechanics, this example can be in-terpreted as validation for the shape function modification. In order to test the shape function modification for all commonly employed first-order and second-order finite elements, 4-node and 10-node tetrahedral elements and 8-node, 20-node and 27-node hexehadral elements are em-ployed, see again the left part of Figure 4.17. The cuboids dimensions in x-, y- and z-direction are3×3×8and the employed material model is of Saint-Venant-Kirchhoff type with Young’s modulus E = 22500 and Poisson’s ratio ν = 0.0. It is completely fixed at its lower surface and subjected to a load p = 1000in positive z-direction at its upper surface. The resulting dis-placement state and the Cauchy stresses are visualized in Figure 4.17. Here, the consistent shape function modification at crosspoints and crosslines allows for an exact representation of the con-stant stress state within the cuboid and consequently of the linear distribution of the displacement field. This demonstrates the consistency of the employed shape function modification regardless of which element type is employed.

Figure 4.17: 3D patch test for mortar meshtying with boundary modification of Lagrange mul-tiplier shape functions: Finite element mesh (left), displacement solution (middle) and Cauchy stress (right).

4.9. Numerical examples

Figure 4.18: Result for patch test with combined non-smooth contact formulation: displace-ment (left) and Cauchy stresses (right). The figure is taken from Farah et al. [71].

4.9.1.2. Mortar surface-to-surface contact

The next example is a simple patch test for a surface contact scenario, which is investigated to show the ability of the proposed method to represent a constant stress state across non-matching discretizations at the contact interface. It is well-known that mortar contact formulations are able to successfully pass this test setup, whereas classical node-to-segment formulations would fail, see El-Abbasi and Bathe [66] and Taylor and Papadopoulos [266]. However, the method that has been introduced in this contribution modifies the mortar contact formulation at vertices and edges of the contact boundary, and thus the patch test has to be revisited to demonstrate that these modification have no negative influence on the solution accuracy as compared with pure surface contact. The test setup consists of a large block with dimensions10×10×4and a small block with dimensions5×5×4. The larger block is completely supported at its lower surface and its upper surface acts as master contact side. The smaller block lies on top of the larger one and acts as slave body. The employed finite element meshes are shown in Figure 4.18. The nodes attached to vertices carry point Lagrange multipliers, the nodes on edges carry line Lagrange multipliers and all other slave nodes are subject to surface Lagrange multipliers. The upper surface of the slave body and the non-contact part of the upper surface of the lower body are loaded with the constant pressurep=−1.0in Z-direction. The employed material model for both bodies is based on a compressible Neo-Hookean material law with Young’s modulusE = 1000 and Poisson’s ratio ν = 0.0. In addition, frictionless contact is assumed for the simulation. The resulting displacements and Cauchy stresses are shown in Figure 4.18. It can be seen that the contact patch test requirements are perfectly fulfilled, i.e. the test is passed to machine precision. In addition, the resulting Lagrange multiplier values are visualized in the left part of Figure 4.19. Here, only the four Lagrange multiplier vectors of the inner surface nodes have noteworthy non-zero values. This is due to the surface Lagrange multipliers being able to represent the constant stress state within the contact interface and thus are able to completely fulfill the contact constraints.

Consequently, the point and the line Lagrange multipliers do not significantly contribute to the contact virtual work. Instead, their contact status can be described as limit case where the gap values are zero but no noteworthy non-zero Lagrange multiplier values occur. Numerically, this could lead to problems due to an arbitrarily changing contact status of the vertex nodes and the edge nodes for this example, while the constraint residual as well as the structural residual

Figure 4.19: Result for patch test with combined non-smooth contact formulation: Lagrange multiplier vectors (left) and scaled interface tractions (right). The figure is taken from Farah et al. [71].

converge perfectly. Therefore, convergence behavior of the Lagrange multiplier increment and the gap function are tracked and changes in the active set are ignored as convergence criterion when both quantities simultaneously approach zero.

However, the Lagrange multiplier solution in the left part of Figure 4.19 cannot be interpreted as interface traction since the shape function modification in (4.74) has been applied to the surface Lagrange multiplier shape functions. Taking into account the post-processing procedure explained in Section 4.8.4, a representative solution for the contact traction can be derived, which is visualized in the right part of Figure 4.19. There, the expected constant stress state at each slave node can be observed.

4.9.1.3. Mortar Edge-to-Surface contact

The next example is introduced to demonstrate the ability of the proposed contact algorithm to represent a constant stress state for edge-contact situations, i.e. it can be interpreted as an edge-to-surface contact patch test. The example consists of a rigid plate that is completely fixed and an elastic cube. The edge length of the cube is l = 2 and its material model is of Neo-Hookean type with Young’s modulusE = 22.5·10⁵ and Poisson’s ratioν = 0.0. It is rotated by45^◦twice around two different axes, such that its contact edge equals the diagonal of the fixed plate. The cube acts as slave body and the plate as master body, respectively. During the entire simulation, inertia effects and damping are neglected. The initial distance between the bodies isd = 2.29·10⁻²and the cube is pressed against the plate with a total prescribed displacement at its upper surfaces of d_max = 0.2. This displacement boundary condition is applied within12 quasi-static load steps. This setup is calculated with three different contact algorithms. First, the proposed algorithm with its combination of point, line and surface Lagrange multipliers. Second, with a classical mortar contact algorithm, and finally with a classical node-to-segment formu-lation. The resulting displacement solutions are shown in Figure 4.20. Here, the left part shows the solution for the proposed contact algorithm, which successfully enforces the non-penetration conditions and leads to a physically correct displacement state. The right part shows a solution computed with a classical node-to-segment algorithm, which also shows a reasonable displace-ment state. The classical mortar formulation in the middle of Figure 4.20 obviously produces a

4.9. Numerical examples

Figure 4.20: Displacement solution for the patch test for edge contact: combined formula-tion (left), classical mortar contact (middle) and classical node-to-segment algo-rithm (right). The figure is taken from Farah et al. [71].

Figure 4.21: Contact tractions for the patch test for edge contact: combined formulation (left), classical mortar contact (middle) and classical node-to-segment algorithm (right).

The figure is taken from Farah et al. [71].

large penetration and the contact is only detected very late. This is due to the surface weighted gap function which inherently arises for the classical mortar formulation, see Popp et al. [212].

In Figure 4.21, the interface tractions are visualized. From this, it can be further deduced that the proposed algorithm with its line Lagrange multipliers perfectly passes the patch test by pro-ducing a constant stress state, which is to be expected for this test setup. In order to compare the results, the visualized stress state is based on a post-processing procedure that considers element dimensions of the setup. The only discrete Lagrange multiplier with a non-zero value is the line Lagrange multiplier at the middle node of the contacting edge. The vertex Lagrange multipliers again exhibit the limit case, where the gap functions are zero but no noteworthy non-zero value for the point Lagrange multiplier arises. Consequently, the entire set of contact constraints are consistently enforced with only one discrete line Lagrange multiplier. In contrast, the classical mortar algorithm produces smaller stresses, since the predicted penetration is far from the phys-ically meaningful state of being zero. Finally, the node-to-segment algorithm, while yielding

Figure 4.22: Parallel edge-to-edge contact: initial setting (left) and deformed state with contact tractions (right). The figure is taken from Farah et al. [71].

plausible results from a qualitative point of view, is not able to produce a constant stress state, which is, of course, a well-known deficiency of this type of contact discretization.

4.9.1.4. Mortar parallel edge-to-edge contact

The next example is a parallel edge-to-edge contact situation of two elastic cubes. Here, the robustness of the proposed numerical evaluation of the line contact algorithm shall be demon-strated. The material model for both cubes is identical to the elastic cube from the edge-contact example in Section 4.9.1.3. Both cubes are rotated by 45^◦ around their individual X-axis such that their edges are perfectly parallel, see Figure 4.22. The cubes have identical dimensions of1×1×1, and tri-linear hexahedral elements are employed for the spatial discretization. The finite element meshes are non-matching at the contacting edges as visualized in Figure 4.22. The upper block is defined as slave side and the lower body represents the master side. The lower cube is supported at its lower surfaces and the upper cube is subjected to a prescribed motion at its upper surfaces. Their initial distance is d = 0.083 and the total prescribed displacement in negative Z-direction isd_max= 0.166, which is enforced within50quasi-static load steps. The re-sulting contact tractions and the deformed meshes are again shown in Figure 4.22. The proposed contact algorithm yields perfectly identical contact tractions at all slave nodes. This is due to the highly accurate segment-based integration scheme explained in Section 4.6.2. Furthermore, the introduced definition for the nodal normal field leads to a perfect edge-to-edge contact scenario, which can again be interpreted as a special kind of (edge-to-edge) contact patch test. It should be pointed out, however, that the solution of this example represents an academic limit case and is therefore rather sensitive with respect to nodal normal definitions and other numerical evaluation procedures.

4.9. Numerical examples

Figure 4.23: Non-parallel edge-to-edge contact: initial setting (left) and deformed state with nor-mal contact force att= 5(right). The figure is taken from Farah et al. [71].