Chapter 3 General Finite Element Framework
3.4. Contact search algorithm
3.4.2. Local contact search
The purpose of the local contact search algorithm is to find the point on the master surface closest to each slave node. In order to reduce the computational time associated to the contact detection, the local search is performed only within the set of master segments identified in the global contact search. However, the computational cost of the local search is typically higher than the one required by the global search procedure, because the local search is repeated in each iteration of every time step. The methods commonly adopted in the local search are the closest point projection algorithm [Hallquist 85], [Konyukhov 08], the pinball algorithm proposed by Belytschko and Neal [Belytschko 91] and the inside– outside algorithm suggested by Wang and Nakamachi [Wang 97]. For the closest point projection procedure, the minimum distance between each slave node and the master surface is calculated based on the normal projection of the node onto the surface. The Newton–Raphson method is typically used to solve this problem and find the contact point coordinates. On the other hand, the pinball algorithm is very efficient when combined with the penalty method, since it is based on simple checks that eliminate any iterative procedure. However, some inaccuracies concerning the real geometry of the contacting bodies are introduced, since the penetration between their surfaces is assumed as the interpenetration of two spherical balls. The inside–outside algorithm is based in the status of the projected point of the slave node along the mesh normal direction. Only two states are allowed, i.e. the projection point is located either inside or outside the master segment. This algorithm is fast, robust and does not requires any iterative procedure to perform the search (closed-form expression), when faceted finite elements are used in the contact surfaces description [Wang 97]. The method adopted in this study to perform the local search is the normal projection algorithm [Hallquist 85], [Konyukhov 08], since it is more accurate and adequate for smooth contact surfaces, which are the purpose of this dissertation.
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3.4.2.1. Closest point projection
The normal gap previously defined in (2.30) is strongly connected with the closest point projection used in the local contact detection procedure. The value of penetration is measured as the closest distance from the slave node x to the master surface s x , which m
is parameterized by the surface coordinates ξ( ,ξ ξ1 2), as shown in Figure 2.5. This leads
to the following minimization problem:
s m 1 2
( , ) min,
d x x ξ ξ (3.27)
which is frequently solved numerically using the Newton–Raphson method. Some difficulties arise in this procedure related to the uniqueness and existence of the closest point projection (see Figure 2.9), which are analysed by Konyukhov and Schweizerhof [Konyukhov 08]. These problems arise mainly when the contact surfaces are modelled by low order finite elements, leading to a piecewise bilinear representation of the surfaces [Hallquist 85], [Heegaard 96]. In fact, the smoothness of the master surface is necessary condition, although a not sufficient for the existence of the normal projection point [Yastrebov 13].
The aim of the closest point projection is to find, for each slave node, the point belonging to the contact master surface that is closest to the slave node, as shown in Figure 2.7. The coordinates of a generic slave node x can be correlated with a vector describing s
any point on the master surface x through the normal gap value, as given by the m
following equation:
proj 1 2 m 1 2 1 2 s
n n
( ,ξ ξ g, ) ( , )ξ ξ g ( , )ξ ξ ,
F x n x (3.28)
where the unit normal vector of the master surface is defined in (2.28) and the value of the normal gap function (2.30) is the third coordinate of the surface coordinate system [Konyukhov 05]. The solution of the minimization problem (3.27) is obtained from the solution of Fproj0 and provides the coordinates of the contact point, while assuring that the vector connecting this point with the slave node is collinear with the normal vector (see
Figure 2.7 and Eq. (3.28)). The Newton–Raphson method is used for solving the nonlinear system of equations, which can be summarized as follows:
1 proj proj 1 ( ) ( ), i i i i s s F s F s (3.29) where 1 2 T n , , i ξ ξ g i
s contains the solution vector at iteration i . Note that the solution yields simultaneously the normal gap gn and the local coordinates of the contact point on the master segment ξ(ξ ξ1, 2) [Heege 96]. In case of contact with a rigid master surface,
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Section 3.4.1.1). On the other hand, in case of contact between deformable bodies, the midpoint of the master contact segment is the initial guess selected for the iterative procedure. The maximum number of iterations allowed is limited to 10 in order to reduce the computational cost. Nevertheless, the normal projection expressed by Eq. (3.28) is applied to each of the ten master segments selected in the global contact search to determine the correct segment. In case of multiple solutions the algorithm selects the segment with minimum normal distance [Oliveira 03].
In order to employ the Newton–Raphson method to solve (3.28), it is necessary to determine the Jacobian matrix of the system of equations, at any point, which is defined as follows:
proj proj proj
proj 1 1 n , , , g ξ ξ F F F F (3.30)
where covariant basis vectors m( )
α
τ ξ defined in (2.27) and the gradient of the normal vector, with respect to the local coordinates, are involved in the formulation of the Jacobian matrix. The derivatives of the unit normal vector can be calculated directly using the Weingarten formula [Heege 96], [Konyukhov 05]:
1 2 1 2 m 1 2 ( , ) βγ( , ) ( , ), , , 1,2, αβ γ α b ξ ξ m ξ ξ ξ ξ α β γ ξ n τ (3.31)
where b denotes the covariant components of the symmetric curvature tensor and αβ m βγ
expresses the contravariant components of the metric tensor, both defined in (2.38). The full properties of the master surface are defined by two fundamental tensors: the metrics tensor (first fundamental tensor) and the curvature tensor (second fundamental tensor) [Konyukhov 08].
The finite element approximation of the master surface leads to a non-smooth surface representation between finite elements, which leads to greater difficulties to solve large slip contact problems [Pietrzak 97]. This problem arises mostly in the bilinear parametric representations, but also in high order finite elements. In fact, this situation leads several difficulties in finding the projection of the slave node on the master segment. Each master segment presents its “normal projection” zone, as shown in Figure 3.19, where the slave node can have at least one projection onto the master surface [Yastrebov 13]. However, sometimes the assembly of the “normal projection” zones does not fill the neighbouring space completely, creating deadzones where no normal projection exists. If several projections are found (see Figure 3.19), the projection point with minimum normal gap is selected to create a contact element [Heege 96]. Nevertheless, when no projection is found, as shown in Figure 3.19, serious numerical problems may arise [Zavarise 09b]. Two types of blind spots can be distinguished: internal and external. Slave nodes situated in external
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101 blind spots are not detected before they penetrate under the master surface, as shown in
Figure 3.19 (a). On the other hand, in the presence of internal blind spots the contact is predicted correctly (before penetration), but if the slave node penetrates under the master surface it is not detected, as shown in Figure 3.19 (b).
(a) (b)
Figure 3.19. Example of a slave node near a sharp corner/valley: (a) convex master surface; (b) concave master surface.
In brief, the finite element discretization of the master surface leads to discontinuities in its normal and tangential vector fields, which cause serious convergence difficulties in the solution of contact problems involving large sliding. The approaches adopted to overcome the above mentioned type of problems (blind spots) are typically divided in two groups: (i) smoothing the master surface (ii) extension of the master segments parameterization outside the standard domain. In the present study the first approach is adopted, avoiding the discontinuities introduced by the discretization procedure and providing a smooth description of the master contact surface. The necessary surface interpolation method is analysed in Chapter 4.
Although the frictionless contact problem only requires the evaluation of the normal gap function (2.30) and surface normal direction to determine the contact force, the frictional contact formulation also needs the tangential relative sliding (2.35) to calculate the frictional (tangential) force. This is due to the nature of the friction force that is path- dependent, thus requiring an incremental update procedure. The change of the closest point projection describes the tangential sliding between the contacting surfaces. In the case of an incremental solution of quasi-static frictional contact problems, the tangential relative sliding velocity (2.35) can be replaced directly by the corresponding tangential slip increment [Heege 96]. In order to build the sliding path, the coordinates of the solution point at the last converged configuration are stored and used in the current time step as input parameters. The history variables adopted in the mapping are the global coordinates of the solution point m 1 2
(ξ ξ, )
x instead the local coordinates 1 2
(ξ ξ, ) , avoiding the problems related with the sliding of a slave node over several master segments [Krstulović-
master slave
master slave
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Opara 02], [Wriggers 01]. Therefore, the tangential slip increment of a slave node is evaluated using the knowledge of its location relative to the master surface in the previous time step.
The coordinates of the closest point projection in the current time step are defined by
1 m 1 1 1 2
( , )
n n n
ξ ξ
x , while in the last converged configuration are denoted as m 1 2
( , ).
n n n
ξ ξ x
Note that the position of the master surface is updated in case of rigid surfaces and it can move and deform in case of deformable master body. The global coordinates of the closest point in the last converged configuration mapped into the current time step are written using the tilde symbol m 1 2
( , )
n n n
ξ ξ
x . The basic idea is to store the local coordinates of the projection point obtained in the last time step to evaluate its global coordinates in the current time step. Thus, the slip increment n1g is simply defined as the vector connecting
the solution point in the last converged configuration mapped into the current time step and the slave node in the current time step, which can be defined as follows:
1 1 s m( 1, 2),
n g n x nx nξ nξ (3.32)
where the position of the slave node tends to the master surface during the interactive procedure in order to eliminate the penetration, thus converging to 1 m 1 1 1 2
( , )
n n n
ξ ξ
x .
However, in case of curved contact surfaces, the slip increment vector is not lying in the tangential plane of the master surface at the current solution point. Hence, the tangential slip increment vector within the current time step is defined as:
1 1 1 1 1
t ( ) ,
n n n n n
g g g n n (3.33)
where n1n denotes the unit normal vector of the master surface at the closest point,
evaluated in the current time step. Since the normal vector is calculated through the projection point algorithm (2.28) and it is updated in each equilibrium iteration, the local frame system (normal vector and tangent plane) changes within the iterative procedure. Note that the tangential slip increment vector does not converges to zero at the solution, as happen for the normal gap function. In fact, tangential slip vector provides the direction of the friction force required for the contact slip status.
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(a) (b)
Figure 3.20. Definition of the slip increment vector for: (a) slave node in contact in the previous time step; (b) slave node not in contact in the previous time step.
The major problems in the definition of the tangential slip increment vector arise when a slave node comes into contact during the actual time step [Heege 96]. Figure 3.20 presents a scheme illustrating the definition of the slip increment vector for the two possible contact status (contact and gap) at the previous time step. If the slave node is in contact at the previous time step, the slip vector can be determined easily from the slip increment of the slave node relatively to the master surface (Figure 3.20 (a)). On the other hand, when the slave node comes into contact during the current time step (Figure 3.20 (b)), the slip increment vector is defined using the normal projection of the slave node obtained in the previous equilibrium step. This procedure was proposed by [Heege 96] for the contact of a deformable body with a rigid surface and it is also adopted for contact between deformable bodies involving large sliding [Wriggers 01].