On some aspects for contact with rigid surfaces: surface-to-rigid surface and curves-to-rigid surface algorithms Alexander Konyukhov a,∗, Karl Schweizerhof a a Karlsruhe
Institute of Technology (KIT), Institute of Mechanics, Kaiserstrasse 12, D-76128, Karlsruhe, Germany
Abstract Special algorithms allowing a simplified description of contact between deformable body and rigid surfaces are developed based on the geometrically exact covariant description of contact. A special attention is given to various geometric combinations where the contact can be represented as a) contact between surfaces and b) contact between a curve and a surface. For contact between surfaces, leading to the Segment-To-Analytical Surface (STAS) approach, two algorithms can be distinguished based on the selection of a coordinate system for the Closest Point Projection (CPP) procedure: a) Rigid Surface is a “Slave” surface and b) Rigid Surface is a “Master” surface. A special combination of both contact kinematics for the Surface-To-Surface and for the Curve-To-Curve approaches is employed for the contact between a curve and a surface leading to the Curve-To-Rigid Surface (CTRS) approach. The last algorithm is verified with the well known Euler formula for the rope-cylinder interaction as well as with a new derived generalization into 3D a spiral rope on a cylinder. The developed algorithms can be straightforwardly implemented within an iso-geometric approach as well as within the conventional finite elements where rigid surfaces are given by CAD patches. Any type of elements can be employed for the contacting deformable surface/curve because the algorithms are formulated in a covariant form. Key words: covariant description, rigid surface, segment-to-analytical surface, curve-to-surface contact, beam-to-surface, rope on surface, geometrically exact contact
∗ Corresponding author. Email addresses:
[email protected] (Alexander Konyukhov),
[email protected] (Karl Schweizerhof). URLs: http://www.ifm.uni-karlsruhe.de/seite 203.php (Alexander Konyukhov), http://www.ifm.uni-karlsruhe.de (Karl Schweizerhof).
Preprint submitted to Computer Methods in Applied Mechanics and EngineeringAugust 25, 2014
Contents 1
Introduction
3
2
Kinematics of Segment (Deformable)-To-Analytical (Rigid) Surfaces Contact (STAS) – Two Strategies
6
2.1
STAS: rigid surface is a “slave” surface approach
8
2.2
On the solvability of the Closest Point Projection procedure for arbitrary surface
9
2.3
Order of contact interaction for surface-to-surface contact
10
2.4
STAS: rigid surface is a “master” surface approach
12
2.5
Computation of Contact Measures
16
3
Weak Formulation and Further Linearization of the STAS Contact
17
3.1
Computation of contact forces
18
3.2
Linearization of the Weak Form for the Surface-To-Surface Contact
20
4
Kinematics of the Curve-To-Rigid-Surface (CTRS) interaction
22
4.1
Order of contact interaction for curve-to-surface contact
23
4.2
Contact measures and their rates for CTRS contact
23
5
Weak Formulation for Curve-To-Rigid Surface (CTRS) Contact
27
5.1
Curve-To-Rigid Surface Contact – Transformation of Contact Forces
27
5.2
Linearization of the Weak Form for Curve-To-Rigid Surface Contact
28
6
Numerical examples
30
6.1
Covering a saddle – case without friction
30
6.2
Covering a saddle – case with friction – sticking and sliding areas
31
6.3
Euler-Eytelwein Problem – a rope on a circular cylinder
35
6.4
Rope in form of a spiral line (helix) around a cylinder
39
7
Conclusion
41
A
Rigid surface is a ”slave” approach: simplified CPP procedure
41
2
A.1 Contact with a rigid surface of revolution B
1
Closed form solutions for CPP procedure: rigid surface is a ”master” approach
41
43
B.1 Contact with a rigid sphere
43
B.2 Contact with a rigid cylinder
44
B.3 Contact with a rigid torus
45
B.4 Contact with a rigid cone
47
References
48
Introduction
Statements of contact problems between deformable and rigid bodies historically appeared as Signorini problems. A simple iteration method for a nonlinear setup within Finite Element Analysis has been reported among the first methods to solve the problem, see Kikuchi and Oden [10]. Within the finite element methods, in general, contact with rigid bodies is regarded as rather particular case of contact between deformable bodies. The major steps in this development can be summarized as follows. A special consistent linearization for the Newton iterative scheme to solve problem numerically has been proposed in Wriggers and Simo [47]. A gap between bodies has been introduced via the closest point projection procedure and involved into linearization. A linearization procedure via convective coordinates has been used for 3D problems in Laursen and Simo [21]. The approach requiring linearization became standard in contact mechanics. Since, linear finite elements were the most popular choice, some difficulties concerning computation of the normal vectors on element boundaries are reported, see discussion and further references in the monographs of Wriggers [46] and Laursen [20]. As a possible remedy of this problem smoothing techniques have been proposed. Again earlier proposals were dealing with smoothing only a rigid surface, see Schweizerhof and Hallquist [37] and in Heege and Alart [7]. Then various aspects of smoothing techniques for master contact surfaces for both non-frictional and frictional cases have been extensively studied in Pietrzak and Curnier [32], Padmanabhan and Laursen [30], Puso and Laursen [34] Krstulovic-Opara and Wriggers [19], Stadler, Holzapfel and Korelc [40] and others. A special group of Mortar methods effectively using averaging of normals is reported to be numerically effective. This group can be subdivided into a group of Mortar methods with penalty enforcement of contact constraints and into a group of Mortar methods with the Lagrange multiplier enforcements. Within the first group special 3
integration techniques based on projection of the overlapping contact areas are discussed in Laursen and McDevitt [29], Puso and Laursen [35], [36]; special integration techniques based on complex Gauss formulas are discussed in Fischer and Wriggers [4]. This approach is also known as Segment-To-Segment algorithm, see Zavarise and Wriggers [49], for which integration with subdivisions and influence of the order of integration has been discussed in Harnau, Konyukhov and Schweizerhof [6]. The dual Lagrange multiplier method initiated by Wohlmuth in [45] has become the major key development within the group of Mortar methods with the Lagrange multiplier enforcements. Various aspects have been developed within a series of publications: a direct linearization procedure in Popp et.al. [33], an application of energy conservative integration scheme in dynamics in Hesch and Betsch [8]. Additional aspects are devoted to application of high-order finite elements and iso-geometric finite element methods for modeling of contact. For small displacement problems an hp-version of the finite element method has been discussed in Paszelt et.al. [31]. Special contact layer finite elements involving the exact geometry of contacting bodies are discussed in Konyukhov and Schweizerhof [15]. Oscillations of normal contact stresses for the contact zone inside the elements have been reported as a main difficulty. Various techniques within high-order finite element schemes are developed in Franke et.al. [5] to solve this problem. Recently, isogeometric techniques combined with the Mortar method are found to be effective to solve this problem see Temizer, Wriggers and Hughes [43], [44] and using special points to locate the contact in Matzen, Cichosz and Bischoff [28]. Though, contact between rigid bodies and deformable surfaces can be represented as a particular case of the general case of deformable bodies, a particular geometric technique based on the selection of special coordinate systems can be developed to simplify these cases. A special consideration is particularly necessary for contact between beams or ropes and rigid surfaces. Only a few publications are devoted to a beam-to-surface contact algorithm: a special problem of mooring cables has been considered in Souza de Cursi [39] as a 2D contact problem of Signorini type; Maker and Laursen [27] have developed the one-dimensional slide-line algorithm. It is also necessary to overview algorithms especially dealing with beamto-beam contact for a part of the current work dealing with the curve-tosurface contact algorithm: Wriggers and Zavarise [48] have proposed beam-tobeam non-frictional contact algorithm for straight beams with circular crosssections, extended then into frictional problems in Zavarise and Wriggers [50]. A generalization of this algorithm for the case of rectangular cross-sections is given in Litewka and Wriggers [26] and [24] for both frictional and nonfrictional cases respectively. In order to deal with curvilinear beams, Litewka in [22] and in [23] has worked with the direct derivation of the obtained functionals via mathematical software. The theoretical basis for a curve-to-curve algorithm, including existence and uniqueness results of the closest point pro4
jection procedure, description of kinematics and linearization in an arbitrary curvilinear Serret-Frenet coordinate system is developed in Konyukhov and Schweizerhof [17] and applications with iso-geometric beam elements can be found in [18]. The parallel tangent case has been recognized as a case without an unique solution and with the consequential difficulties to solve it. Durville [1] has been working with discrete contact algorithms to overcome this problem and Litewka proposed a multiple-point contact element [25]. In this contribution we discuss a very special case of contact – contact between rigid surfaces and deformable objects in a geometrically exact fashion. The main aspect is the consideration of all parameters in covariant fashion independently on the selection of the coordinate system pointed out in [12] and recently published in the monograph [16]. The main steps of the geometrically exact approach can be summarized as follows: • The existence and uniqueness analysis of the Closest Point Projection (CPP) procedure allows to split the full contact analysis into the analysis with the following contact pairs: surface-to-surface, curve-to-surface, point- tosurface, curve-to-curve, point-to-curve, point-to-point (joint). Then the solution of the corresponding CPP procedure exists and is unique for those pairs. • The various associated CPP procedures give rise to local curvilinear coordinate systems: Gaussian surface coordinates system for the surface-tosurface and point-to-surface contact pairs; Serret-Frenet coordinate system for curve-to-curve and point-to-curve contact pairs. • All contact measures are naturally defined in those coordinate systems – penetration and measures of tangential and other interaction. • All necessary operations, including formulation of the weak forms, its linearization, necessary constitutive relations and return-mapping algorithms for contact forces etc. are formulated in the covariant form in those coordinate systems. This leads to an a-priori independent-of-discretization structure of the algorithm. • The type of discretization is chosen (conventional finite elements, high-order finite elements, isogeometric finite elements) in the last step of the numerical modeling in order to derive the discretized form for the corresponding contact pairs. Various techniques are possible at this stage including the standard Node-To-Segment discretization as well as more general Mortar methods. According to this scheme, first, a general procedure based on the selection of a coordinate system is described – this leads to the Segment-To-Analytical Surface (STAS) approach. Two cases for such a selection can be distinguished leading to different approaches to compute the penetration: a) Rigid Surface 5
is a “Slave” Surface and b) Rigid Surface is a “Master”. First, the cases with available analytical solutions are summarized and discussed. Then, the same approach is followed to obtain another special case such as a Curve-To-Rigid Surface (CTRS) approach. The latter requires a special combination of both contact kinematics for the Surface-To-Surface and for the Curve-To-Curve approaches. The Curve-To-Rigid Surface contact approach is finally studied numerically for the known Euler-Eytelwein problem for contact between a cylinder and a rope. The rigid surface can be modeled in various ways: (1) A set of constrained (fixed) nodes within conventional finite elements. (2) A set of NURBS patches (as typical for the CAD modeling) or in some cases via analytical functions. The geometric features of the rigid surface are not fully captured within the first approach, because, the surface is either represented as a set of fixed nodes, or inherits the geometry of the finite element approximation. All geometrical features are fully captured only within the second approach, and the rigidity of surfaces can be easily incorporated. Namely, this makes the second approach advantageous in many applications such as deep drawing and forming simulation etc. in which many surfaces can be modeled directly as rigid surface patches instead of modeling with various solid or shell finite elements. The current development can be, therefore, used straightforwardly with conventional finite element discretizations for deformable bodies, as well as with isogeometric finite element techniques.
2
Kinematics of Segment (Deformable)-To-Analytical (Rigid) Surfaces Contact (STAS) – Two Strategies
First, we consider a convention which is used in the geometrically exact theory [16] concerning the selection of master and slave side of the contacting bodies. Considering two interacting bodies, we are free to choose a “master” and a “slave” body at the beginning. We are calling the “master” bodies one of the contacting body on the surface of which (resp. master surface) we are setting up a local coordinate system. The local coordinate coordinate system follows all geometrical features of the surfaces – namely, this is e.g. a surface coordinate system in the case of a surface-to-surface pair, or a curve SerretFrenet coordinate system in the case of a surface edge etc. The other contacting body is then called “slave” body. On the surface of the slave body in the direction of the normal to the “master” surface we consider a slave point S, which is e.g. an integration point or a nodal point. Two bodies are coming into contact, if a slave point of the second surface penetrates into the master 6
surface, where penetration is defined as the shortest distance between the two surfaces of the contacting bodies. The kinematics for surface-to-surface (STS) contact interaction is described in the local coordinate system based on the the closest point projection (CPP) procedure onto a surface. This projection allows to define a coordinate system as follows: rs (ξ 1, ξ 2 , ξ 3 ) = ρ(ξ 1, ξ 2 ) + ξ 3 n(ξ 1 , ξ 2 ).
(1)
The vector rs is a vector for the “slave” point, ρ is a parametrization of the “master” surface, n is a normal to the surface. The slave point is taken to be either a node from the finite element mesh for the well-known Node-ToSegment algorithm or an integration point for arbitrary approximations within the Segment-To-Segment or Surface-To-Surface algorithms. If one of both contacting bodies is rigid and is described by e.g. a set of NURBS patches globally, then a stand-alone technique can be applied to describe this specific type of contact. Two such strategies can be distinguished if one of the contacting bodies is rigid and can be analytically parametrized (directly by known analytical functions, or by a set of suitable NURB splines in a CAD system). These strategies depend on the selection of the master or slave part in a surface coordinate system leading to various techniques to compute the penetration: • the rigid surface is a “slave” surface; • the rigid surface is a “master” surface. In the first case the standard CPP procedure [14] is not used – one has to develop a separate algorithm to define the penetration point. In both cases with a simple geometry it is even possible to obtain a closed form solution for the CPP procedure. Remark 1 According to the definition of master and slave parts, we distinguish two situations for the contact pair: (1) selection of the coordinate system for computation of the penetration, thus leading to two strategies; (2) selection of the coordinate system for further formulation of the weak form (including computation of contact parameters such as contact forces etc.) and its linearization. Remark 2 During the modeling of the contact with rigid surfaces, the weak form for contact (contact integral) and the forthcoming linearization should be computed in the coordinate system of the deformable body independently of the method which has been used for computation of the penetration.
7
2.1 STAS: rigid surface is a “slave” surface approach If the rigid surface described globally analytically is a “slave” surface then we can write the following coordinate system: rs (α1 , α2 ) = ρ(ξ 1 , ξ 2 ) + ξ 3 n(ξ 1 , ξ 2 ).
(2)
The rigid surface rs is then parametrized by internal Gaussian coordinates α1 , α2 . A “master” deformable segment from the finite element mesh is parametrized as ρ(ξ 1 , ξ 2 ), n is the normal to the master segment. In other words, a point rs of the rigid surface is observed in the local coordinate system of the contact master element. The standard closest point projection procedure now turns into the determination of the surface point defined by equation (2). Taking the integration points (ξI1 , ξI2 ) on the “master” deformable segment we are searching the corresponding internal coordinates α1 , α2 of the rigid surface as well as the penetration ξ 3 – in other words, the point on the rigid surface in which the normal n of the “master” segment is pointing. Computation is provided then by the Newton method. For this algorithm we define a function F(α1 , α2 , ξ 3 ) with the components given in eqn. (2)
F(α1 , α2 , ξ 3 ) = rs − ρ − ξ 3n =
xs1 xs2
3
− x1 − ξ n1 − x2 − ξ
3
n2
with xi = xi (ξ 1 , ξ 2). (3)
xs3 − x3 − ξ 3 n3
Its derivative with respect to the coordinates (α1 , α2 , ξ 3) is:
∂xs1 ∂α1 ∂x ′ F = s21 ∂α ∂x s3 ∂α1
∂xs1 ∂α2 ∂xs2 ∂α2 ∂xs3 ∂α2
−n1 −n2
−n3
xs1,1
xs1,2
xs3,1
xs3,2
= xs2,1
xs2,2
−n1
. −n2 −n3
(4)
Then, the Newton iteration procedure gives as follows for iteration step n:
1 ∆αn
∆αn = ∆αn2 = −(F′ )−1 n Fn , ∆ξn3
8
(5)
αn+1 = αn + ∆αn .
2.2 On the solvability of the Closest Point Projection procedure for arbitrary surface The solvability analysis of the Closest Point Projection (CPP) procedure for arbitrary surfaces eqn. (2) is a rather complicated algorithmic task, however, the theoretical basis of this is well understood and is described in a series of publications [14], [16], [17]. The solvability analysis is based on the analysis of the surface geometry and leads to the building of the proximity domain surrounding the surface continuously. Any slave point form this proximity domain is projected uniquely onto the master surface. The full solvability analysis for the CPP procedure can be summarized as follows: • for C2-continuous points of a surface it is possible to construct the 3D proximity domains, and so far from both sides of the surface – convex as well as concave – in which the solution of the CPP procedure exists and is unique. The finite proximity domains can be constructed based on values of corresponding principal curvatures of the surface. • for C1-continuous (but not C2-continuous) points on a surface it is possible to construct the continuous proximity domain. • if C1-continuity of the surface is no longer preserved – e.g. in the case of an edge of a surface – the analysis should be continued with the CPP procedure onto the curve representing the edge. • for C2-continuous points of a curve it is possible to construct the 3D proximity domains surrounding the curve in which the solution of the CPP procedure exists and is unique. The finite proximity domains can be constructed based on the main curvature of the curve. • for C1-continuous (but not C2-continuous) points of a curve the continuity of the proximity domain is preserved. • if C1-continuity of the curve is no longer preserved – as in the case of angular points (vertexes) of a curve – the analysis is finalized with the construction of the proximity domain based on the CPP procedure for the point. • the resulting global proximity domain for the arbitrary surface (including edges and angular points (vertexes)) is constructed as Boolean sum of proximity domains for surfaces, curves and edges. • the full analysis allows to define the zones with possible multiplicity of solutions for the CPP procedure – they are identified as Boolean overlapping of 3D proximity domains leading to surfaces as well as Boolean overlapping of bounding surfaces for 3D proximity domains leading to curves. • more careful analysis may be required for particular points of the surface (e.g. flat points) and curves in order to construct the proximity domains more precisely. 9
Applying this fundamental result of the solvability of the CPP procedure in eqn. (2), one can ensure a unique solution, if the “slave” point rs is positioned in the global proximity domain of the “master” surface ρ(ξ 1 , ξ 2 ).
2.3 Order of contact interaction for surface-to-surface contact If the solvability of CPP procedure is concerned with the analysis of the geometry of the master surface (vector ρ(ξ 1 , ξ 2) in eqn. (2)), then the order of contact interaction is concerned with the geometry of the slave surface in eqn. (2). In order to classify contact we will exploit the definition of contact between two manifolds: two manifolds have a contact of order k if they have the same value at a point S and also the same derivatives there, up to order k. The order of contact depends on the structure of the slave surface. For at least continuous surfaces ρ(ξ 1 , ξ 2) and rs (α1 , α2 ) we can define the following: (1) Observing the slave surface rs (α1 , α2 ) in the coordinate system of the master surface in eqn. (2), zero order contact is defined if ξ 3 = 0. With this condition only the intersection of two surfaces is considered geometrically. It is known that it results in a set of isolated points, a set of curves, a set of overlapping surfaces, or any combination of these cases in 3D, and to a line, or a point on the tangent plane to the surface ρ(ξ 1 , ξ 2) at ξ 3 = 0. (2) Considering C1-continuity for both ρ(ξ 1 , ξ 2 ) and rs (α1 , α2) first order contact at point S, or tangent contact, is defined if two tangent planes at this point are coinciding. (3) Considering C2-continuity for both ρ(ξ 1 , ξ 2 ) and rs (α1 , α2) second order contact at point S, or osculating contact, is defined if for both surfaces the principal directions and the principal curvatures are coinciding. Remark 3 One can see that for C1-continuous surfaces the enforcement of the shortest distance between the master surface and slave point leads to first order contact at point S, or tangent contact. This is an analog of the requirement for the derivatives to be zero for the min/max point in Cartesian space. Several cases are rather special within the rigid surface is a “slave” approach e.g.: • contact with a rigid plane, possessing a closed form solution for the pene10
tration; • contact with a surface of revolution.
2.3.1 Contact with a rigid plane The simplest example for the rigid surface as a “slave” strategy is “contact with a rigid plane” allowing a closed form solution for the penetration. Consider a rigid “slave” plane given in analytical form as: (r − r0 ) · N = 0,
(6)
where r0 is any point on the plane, N is the normal vector for the plane and r is a vector with Cartesian coordinates {x, y, z}. Assuming now that from one
C
master n
ρ
N
ξ3
Z rs Y
S r0 slave
X
Figure 1. Contact with a rigid plane given analytically
side the slave vector rs belongs to the plane see eqn. (6) and from another side it is observed from the master segment, i.e. it is satisfying eqn. (2), we can write the following system combining equations (6) and (2): (rs
− r0 ) · N = 0 rs = ρ(ξ 1 , ξ 2) + ξ 3 n(ξ 1 , ξ 2 ).
(7)
Just inserting rs from the second equation into the first one, we can compute the value for the distance ξ 3 , and therefore for the penetration as: 11
1 2 (ρ(ξi.p. , ξi.p. ) − r0 ) · N 1 2 n(ξi.p. , ξi.p. )·N The distance can be always computed if the denominator is not zero
ξ3 = −
1 2 n(ξi.p. , ξi.p. ) · N 6= 0,
(8)
(9)
which means the slave plane must not be orthogonal to the tangent plane of the master segment. 1 2 For further computations, as a vector ρ(ξi.p. , ξi.p. ) is given at integration points 1 2 1 2 ξi.p. , ξi.p. of the master segment (deformable), the normals n(ξi.p. , ξi.p. ) are also computed at these points.
Another example allowing simplification of the CPP procedure is a surface of revolution, see A. 2.4 STAS: rigid surface is a “master” surface approach Preserving the variables ξ 1 , ξ 2 only for the finite element approximations for the case rigid surface is a “master” surface we can write the following coordinate system: rs (ξ 1 , ξ 2) = ρ(α1 , α2) + p n(α1, α2 ). (10) 1 2 Now, an integration point rs (ξ , ξ ) from the deformable “slave” finite element segment is found in the direction of the normal n(α1 , α2 ) to the rigid “master” surface ρ(α1 , α2 ). This distance – denoted as p – plays the role of the penetration. It is important to note that the distance p between the master surface and slave point is not coinciding with the penetration ξ 3 measured from the finite element contact segment, because, the normals from the master and the slave, in general, are not parallel. The situation is illustrated for the contact between the rigid sphere and a “slave” segment, see Figure B.1 in Appendix B. However, when two bodies are close to contact then the normals are almost parallel, in fact, they are enforced to be parallel by the contact algorithm. This leads to the possibility to use the current approach for contact mechanics. In this case, the weak form (normal and tangent vectors etc.) is computed in the coordinate system of the “slave” surface, see Remark 2. The corresponding penetration is taken as ξ 3 = p. Newton’s method is exploited in this approach in order to solve eqn. (10) defining then a point with the coordinates α1 , α2 on the rigid surface and the distance p between this surface and a selected integration point on the “slave” segment rs (ξ 1 , ξ 2). In this case an integration point rs (ξ 1 , ξ 2) is projected onto the rigid “master” surface, or in other words, the CPP procedure to rigid surfaces is involved. In due course, the fundamental result of existence and 12
uniqueness of the CPP procedure presented in [14] is fully applicable now based on the geometry of rigid surfaces. Remark 4 This approach allows to work with arbitrary finite element approximations (including isogeometric FE techniques) for deformable bodies as well as for rigid surfaces.
2.4.1 Contact with an arbitrary rigid surface – an example with a hyperboloid As an example of an arbitrary approximated surface within the rigid surface is a “master” approach we consider a hyperboloid z = axy, see Fig. 2, written via Gaussian coordinates as:
1
2
ρ(α , α ) =
α
1
α2 aα1 α2
(11)
Now, we have to work in the coordinate system of the “master” in eqn. (10) defined following the CPP procedure on this surface. Thus, the CPP procedure for an arbitrary point of the deformable body rs (ξ 1 , ξ 2) = {x, y, z}T (in 1 2 computations an integration point ξip , ξip ) is defined as the minimization of 1 2 the function F (α , α ): 1 F = (r − ρ)(r − ρ) −→ min 2
(12)
Following the solution scheme discussed in [14], we define, first, geometrical parameters of the hyperboloid: • tangent vectors (first derivatives)
1 ∂ρ ρ1 = = 0 1 ∂α aα2
0 ∂ρ ρ2 = = 1 2 ∂α aα1
(13)
• metric tensor with covariant components aij = ρ1 · ρ2 :
2
1 + a
[aij ] =
2
2 2
2
(α )
1
aα α
2
aα α 2
2 1 2
1 + a (α )
• second derivatives of the surface vectors ρij 13
1
(14)
0 0 0 ∂2ρ ∂2ρ ∂2ρ 0. = ρ = , = ρ = , = ρ = 0 0 11 22 12 1 1 2 2 1 2 ∂α ∂α ∂α ∂α ∂α ∂α 0 0 a (15)
5
z
10 0
5 0
−5 10 −5
5 0
x
−5
−10
−10
y
Figure 2. Hyperboloid z = axy as an example of an arbitrary approximation within the rigid surface is a “master” approach
Then, the first and the second derivatives of the function F in eqn. (12) are computed as
F′ = −
ρ1 · (r − ρ)
ρ2 · (r − ρ)
= − 14
1
2
1
2
(x − α ) + aα (z − α α ) (y − α2 ) + aα1 (z − α1 α2 )
(16)
a11 − ρ11 · (r − ρ) a12 − ρ12 · (r − ρ)
F′′ =
a21 − ρ21 · (r − ρ) a22 − ρ22 · (r − ρ)
2 2 1 + (aα )
=
2a2 α1 α2 − az
a2 α1 α2 − a(z − aα1 α2 )
(17)
1 + (aα1 )2
in order to construct the Newton iterative process as
∆α(n) =
1 ∆α(n) ∆α2
(n)
′ = −(F′′ )−1 (n) F(n)
(18)
α(n+1) = α(n) + ∆α(n) , where the inverse of the second derivative is computed (F′′ )−1 =
1 2 1 + (aα )
1 det F′′ az − 2a2 α1 α2
az − 2a2 α1 α2 1 + (aα2 )2
with det F′′ = (1 + (aα1 )2 )(1 + (aα2 )2 ) − (2a2 α1 α2 − a)
(19)
For further analysis of this numerical example, initial values for the iterative process in eqn. (18) are supplied as corresponding coordinates of a slave point 1 2 rs (ξip , ξip ) α(0) =
α1 (0)
α2
=
(0)
x(ξ 1 , ξ 2 ) ip
ip
y(ξ 1 , ξ 2 ) ip
.
(20)
ip
Once, the computation is finished i.e. the coordinates of the projection α1 , α2 are computed, the penetration into the hyperbolic surface is computed directly as: n
ξ 3 = (rs − ρ) · n = x − α1 , y − α2 , z − aα1 α2 2
1
1
2
oT
·n
= aα (α − x) + aα (α − y) + z − aα1 α2 ,
(21)
where the normal vector n to the hyperboloid is defined as
n=
2 −aα
ρ1 × ρ2 1 −aα1 . =q 1 2 1 2 |ρ1 × ρ2 | (aα ) + (aα ) + 1 1
15
(22)
Remark 5 All parameters are computed for an integration point belonging to a finite element (or iso-geometric finite element) of the deformable body. Algorithmically it requires only one additional subroutine describing the presence of the single rigid surface. Namely, the strategy “one rigid surface element for all elements of the deformable body” allows to simplify modeling of contact with rigid surfaces in comparison with the alternative strategy – “use surface-to-surface contact algorithm and apply boundary conditions to make one surface to be rigid”. In order to construct an effective contact algorithm, the global searching algorithm should be modified including the special analysis of the projection domains based on the existence and uniqueness of the CPP procedure, see [14]. 2.5 Computation of Contact Measures Contact measures are naturally appearing after consideration of the relative velocity vector in the corresponding surface basis during contact, i.e. on the tangent plane either with ξ 3 = 0, or p = 0, see the iscussion in [12]. The following pair is obtained: • a relative velocity on the master surface obtained after derivation of eqn. (2) and substituting ξ 3 = 0 vs (α1 , α2 ) − v(ξ 1 , ξ 2) = ξ˙i ρi + ξ˙3 nξ (ξ 1 , ξ 2 ) (23) • a relative velocity on the slave surface obtained after derivation of eqn. (10) and substituting p = 0 (vector ρ(ξ 1 , ξ 2) is used now only for the deformed body) v(ξ 1, ξ 2 ) − vs (α1 , α2 ) = α˙ j rj + pn ˙ α (α1 , α2 ).
(24)
Consider now eqns. (23-24) for a pair of points from master and slave surfaces in contact and assuming first order contact (tangent contact) see Remark 3 – tangent vectors from both the deformable surface ρi and the rigid surface ri are laying in the same tangent plane, and normal vectors are satisfying nξ = −nα . A a sum of the two equations we obtain the following split: • for the tangential plane
ξ˙i ρi + α˙ j rj = 0
(25)
ξ˙3 nξ (ξ 1, ξ 2 ) + pn ˙ α (α1 , α2 ) = 0.
(26)
• for the normal direction
The measure for normal interaction ξ 3 = p is computed with regard to the approach used: either the rigid body as a ”slave”, or the rigid body as a ”master” as described before. 16
Measures for the tangential interaction in an incremental form are arising from the tangential convective velocities as ∆ξ i and ∆αi . The tangential measure is precisely computed on the master surface as a path passed by the projected slave point. For the rigid body as a ”master approach”, first, the measure ∆αj is computed as a difference between two projections at current (k) and previous (k − 1) load steps: j j ∆αj = α(k) − α(k−1) ;
(27)
and, then, ∆ξ i from eqn. (25) is computed in the basis ρ1 , ρ2 as ∆ξ i = −∆αj (rj · ρi ).
(28)
Other methods are discussed in detail in [13] and also in the monograph [16].
3
Weak Formulation and Further Linearization of the STAS Contact
There are some peculiarities in the formulation of the weak form for the surface-to-analytical segment (STAS) type contact. We consider contact tractions Ts , Tm acting on the infinitesimally small slave dss surface and master dsm surface respectively. δrs and ρ are corresponding variations of the displacement field. The virtual work δWc of the contact tractions, taking into
Figure 3. Contact tractions acting on the infinitesimally small master surface ds and slave surface dss .
17
account the equilibrium equation at the contact boundary, Ts dss = −Tm dsm , is obtained by the following surface integral expressed equivalently in both master and slave surfaces: δWc =
Z
ss
Ts · (δrs − δρ)dss =
Z
Tm · (δρ − δrs )dsm .
sm
(29)
Only integration over the slave surface is employed due to the special sequence of operations during the numerical solution: a) b) c) d)
setup of integration points on the slave surface; CPP procedure for all slave points to master surface; computation of normal and tangential measures, penetration check; computation of normal and tangential forces via a constitutive law in the case of regularization; return-mapping scheme.
Now we consider in detail the STAS approach with the rigid surface as a ”master” surface. Preserving now notation ρs (ξ 1, ξ 2 ) for the deformable slave surface and rm (α1 , α2 ) for the rigid master surface, we are expressing the integral in eqn. (29) in the metrics of the deformable surface δWc =
Z
ss
Ts · (δρs − δrm )dss
(30)
using the relative variation vector in analogy to eqn. (23) (a projection point is observed in the basis ρ1 , ρ2 ) δrm (α1 , α2 ) − δρ(ξ 1, ξ 2 ) = δξ i ρi + δξ 3 nξ ;
(31)
and using the expression of the contact traction vector Tslave in the basis ρ1 , ρ2 , nξ , see Fig. 3 Tslave ≡ T = Nnξ + T i ρi . (32) The final expression is δWc = −
Z
s
Nδξ 3 dss −
Z
s
Ti δξ i dss
(33)
The minus sign is expressing that the computation of both variations eqn. (31) and forces eqn. (32) are performed in the same metric of the deformable surface. 3.1 Computation of contact forces A penalty regularization is used to compute both normal N and tangent Ti forces. From the mathematical point of view the penalty method leads to approximately satisfied contact conditions and from a mechanical point of view it is a constitutive equation for contact interfaces. 18
The penalty condition for the normal force is formulated as 0
N=
if ξ 3 > 0 no contact
(34)
3 3 N ξ if ξ ≤ 0 penetration.
ǫ
Penalty condition is enforced point wisely, namely, the active set of Gauss points used for further computation of contact integral is formed by a set of penetrating points. This leads to detection of the contact zone by Gauss point, which will be illustrated in the numerical example Sect. 6.2. 3.1.1 Tangential contact conditions. Return-Mapping Scheme. Additional constitutive equations are necessary for the tangential contact tractions Tj . Traditionally, the Coulomb friction law is taken to define the transfer between the sticking and sliding situations. In order to compute the real tangential traction, i.e. in accordance with the Coulomb friction law, the return-mapping algorithm is employed, see [20], [46], [38].
3.1.1.1 Trial step. The final result will be computed in the basis of the (k+1) requires deformable surface ρ1 , ρ2 , nξ . Computation of trial forces (T tr )i at the load step (k + 1) the increments ∆ξ j expressed via the increments ∆αi in eqns. (27-28) N
(k+1)
= ǫN ξ
(k+1)
(T tr )i
3,(k+1)
(k)
= Ti
(k)
− ǫT ∆ξ j aij = Ti
tr (k+1) Φtr | (k+1) := |T(k+1) | − µ|N
|Ttr (k+1) | =
q
(k+1)
(T tr )i
(k+1) ij a
(T tr )j
where, • • • •
+ ǫT ∆αj (rj
· ρi )
,
ǫN is a normal penalty parameter; ǫT is a tangential penalty parameter; (k) Ti is the real force from the previous converged load step (k); ρi , i = 1, 2 are basis vectors of the ”slave” deformable surface; 19
(35)
• aij = (ρi · ρj ) are covariant components of the metric tensor; • aij are contravariant components of the metric tensor, matrix of which is computed as inverse of the matrix with covariant components; • ri , i = 1, 2 are basis vectors of the ”master” rigid surface; • µ is the coefficient of Coulomb’s friction.
3.1.1.2 Return mapping scheme The stick-slide condition is checked (k+1) within the return mapping process and, finally, the real forces Ti are computed (k+1)
Ti
=
(k+1)
(T tr )i
(k+1) (T tr )i (k+1) | µ|N |Ttr |
if Φtr (k+1) ≤ 0 (stick)
.
(36)
if Φtr (k+1) > 0 (slide)
(k+1)
An iterative solution of e.g. Newton’s type is performed to solve the global equilibrium equations on the current load step (k + 1).
3.2 Linearization of the Weak Form for the Surface-To-Surface Contact The final form of the weak form in eqn. (33), written in the basis of the deformed surface (see Remark 2), allows to use directly the structure of the linearized weak form obtained in a covariant form for the surface-to-surface contact (first developed in [12]). We just recall these expressions here Linearization of the normal contact part δWcN gives us the following expression: L(δWcN ) = =
−
Z |
s
N
Z
|s
ǫN H(−ξ 3)(δrs − δρ) · (n ⊗ n)(vs − v)ds {z
}
Lm N
(37a)
δρ,j · aij (n ⊗ ρi )(vs − v) + (δrs − δρ) · aij (ρj ⊗ n)v,i ds {z
LrN
−
Z |
s
N (δrs − δρ) · hij (ρi ⊗ ρj )(vs − v)ds . {z
LcN
20
}
}
(37b)
(37c)
Linearization of the tangential contact part δWcT – case of sticking gives us the following expression used if ”sticking” is identified in the returnmapping scheme eqn. (36): L(δWcT ) = −εT −
Z
s
Z
s
(δrs − δρ)aij ρi ⊗ ρj (vs − v)ds
(38a)
Ti (δrs − δρ) ail ajk ρk ⊗ ρl vj + δρ,j aik ajl ρk ⊗ ρl (vs − v) ds (38b) +
Z
s
Ti hij (δrs − δρ) · ρj ⊗ n + n ⊗ ρj (vs − v)ds.
(38c)
Linearization of the tangential contact part δWcT – case of sliding gives us the following expression used if ”sliding” is identified in the returnmapping scheme eqn. (36):
L(δWcT ) = −
Z
s
− +
Z
s
−
Z
s
Z
s
!
ǫN µTi aij (δrs − δρ) ρj ⊗ n(vs − v) ds |T|
(39a)
!
ǫT µ|N|aij ρi ⊗ ρj (vs − v) ds (δrs − δρ) |T| !
ǫT µ|N|Ti Tj aik ajl (δrs − δρ) ρk ⊗ ρl (vs − v) ds |T|3
(39b) (39c)
µ|N|Ti (δrs − δρ) ail ajk ρk ⊗ ρl vj + δρ,j aik ajl ρk ⊗ ρl (vs − v) ds |T| (39d) ! Z µ|N|Ti ij + (39e) h (δrs − δρ) · ρj ⊗ n + n ⊗ ρj (vs − v) ds |T| s
Remark 6 Though, the linearized expression contains vectors from both deformable slave and rigid master surface, the final expression after the discretization will include the approximation matrix A for the deformed surface only ρ = Ax, see details of the finite element implementation of the STAS non-frictional algorithm in [9]. Furthermore, the global acceleration of the iterative solution based on application of the Large Penetration scheme and the analysis of the main part eqn. (37a), the “rotational” part (37b) and the “curvature” part (37c) is proposed.
21
4
Kinematics of the Curve-To-Rigid-Surface (CTRS) interaction
The kinematics of the Curve-To-Rigid-Surface (CTRS) contact interaction are based on the combination of both the Segment-To-Analytical-Surface (STAS) and the Curve-To-Curve (CTC) contact kinematics. The CTC contact kinematics are constructed, see [17], as follows: one of the curves is observed in the natural Serret-Frenet coordinate system of the other curve ρ2 (s1 , r, ϕ1 ) = ρ1 (s1 ) + re1 (s1 , ϕ1 ); with auxiliary vectors e1 = ν 1 cos ϕ1 + β 1 sin ϕ1 g1 = τ 1 × e1 .
1 ⇆ 2;
(40) (41) (42)
Here, the vector ρ2 is a vector describing a contact point of the second curve, ρ1 (s1 ) is a parametrization of the first curve; a unit vector describing the shortest distance e1 is written via the unit normal ν 1 and the bi-normal β 1 defining the Serret-Frenet coordinate system of the first curve. Eqn. (40) describes the motion of the contact point on the second curve in the coordinate system attached to the first curve. The description is symmetric with respect to the choice of the curve 1 ⇆ 2. Convective coordinates are used as measures: r – is mutual for both curves and is a measure for normal interaction; sI – for tangential interaction and ϕI – for rotational interaction for the I-th curve.
Figure 4. Curve-To-Curve contact kinematics. Definition of the Serret-Frenet coordinate system.
22
4.1 Order of contact interaction for curve-to-surface contact Before developing an algorithm, we introduce here the order of contact between a curve and a surface, in analogy to the Section 2.3 For at least a continuous surface ρ(ξ 1 , ξ 2 ) and a continuous curve rs (η) we can define the following: (1) Observing the curve rs (η) in the coordinate system of the master surface similar to eqn. (2) (as rs = ρ + ξ 3 n), zero order contact is defined if ξ 3 = 0. Geometrically with this condition only the intersection of the curve with the surface is considered. It is known that it results in a point (piercing). (2) Considering C1-continuity for both the curve rs (η) and the surface ρ(ξ 1 , ξ 2) first order contact at point S, or tangent contact, is defined if the tangent vector of the curve is laying in the tangent plane of the surface. Remark 7 One can see that for both C1-continuous curves and surfaces the enforcement of the shortest distance between the master surface and slave point leads to the first order contact at point S, or tangent contact.
4.2 Contact measures and their rates for CTRS contact For further discussion we are assuming that the requirements for the tangent contact can be fulfilled, see Remark 7. For the CTRS algorithm all parameters are formulated dually in the surface coordinate system (via Gaussian coordinates ξ 1 , ξ 2 ) and in the Serret-Frenet curve coordinate system (via coordinate ζ). The curve rs (η), see Fig. 5, is assumed to be deformable, thus, all parameters for the curve are considered as developed for the Curve-To-Curve contact. A rigid surface ρ(ξ 1 , ξ 2 ) is assumed to have an arbitrary analytical description, e.g. via NURBS surfaces. In order to describe contact between deformable curves and rigid surfaces the Segment-To-Analytical-Surface (STAS) algorithm is modified as follows: • A set of contact points (integration points) ηi is set on the curve segment AB: all kinematical parameters are considered then in the Serret-Frenet curve coordinate system τ , ν, β; • The contact point (one of the integration points) is projected onto the rigid surface. At each point all kinematical parameters are considered in the surface coordinate system ρ1 , ρ2 , n via convective coordinates ξ 1 , ξ 2 . The combination of both CTC and STAS strategies leads to the Curve-To23
Rigid (analytical) Surface contact algorithm and to the following definition of the coordinate system on the master rigid surface: rs (η) = ρ(ξ 1, ξ 2 ) + ξ 3 n(ξ 1 , ξ 2),
(43)
where rs (η) is defining an integration point positioned on the curve, or in the case of the beam contact on the mid-line of the curvilinear beam element, ρ(ξ 1 , ξ 2) is the parametrization of the rigid “master” surface. The integration point ρs (η) is found in the direction of the normal n(ξ 1 , ξ 2) to the rigid “master” surface. The shortest distance between integration points and the surface denoted as ξ 3 plays the role of a penetration. The Closest Point Projection (CPP) procedure as the projection of a point onto the rigid surface is necessary for this. In general, Newtons method is exploited to solve the CPP procedure defining then a point on the rigid surface and the penetration ξ 3 between this surface and the selected integration point S. A set of analytical solutions is possible for some surfaces as discussed above. Kinematical relations during the contact can be obtained dually considering the relative velocity of contact points during contact: • The relative velocity on the tangent plane from the surface point of view, i.e. in the surface coordinate system with ξ 3 = 0 in eqn. (1): vs − v = ξ˙3 n + ξ˙j ρi , i, j = 1, 2, (44) v being the velocity of the projection contact point on the rigid surface. • The relative velocity on the curve from the point of view on the curve, i.e. in the Serret-Frenet coordinate system at r = 0, attached to the deformed curve, see eqn. (40): v − vs = vτ τ + ve e + vg g.
(45)
Since, in this contact approach the point is fixed on the curve, we introduce here notations vτ , ve , vg for projections of the relative velocity vector on the curvilinear coordinate system τ, e, g instead of using the convective velocities s, ˙ r˙ and ϕ. ˙ Summing up both equations (44) and (45), we obtain: ξ˙j ρi + ξ˙3 n + vτ τ + ve e + vg g = 0.
(46)
The vector e, defined in eqn. (40), is re-represented in the surface coordinate system, see eqn. (1), at the contact krs − ρk = |ξ 3| = 0 on the tangent plane as ρ − rs e= = −n. (47) kρ − rs k From eqn. (46), we can define the following components of the relative velocity vector: 24
• normal relative velocity during contact ve = ξ˙3 = r˙ = (vs − v) · n = (v − vs ) · e
(48)
• pulling relative velocity vτ = −(ρi · τ )ξ˙i ,
i = 1, 2
(49)
i = 1, 2
(50)
• dragging relative velocity vg = −(ρi · g)ξ˙i ,
Remark 8 From eqn. (49) one can see, that during contact a tangent vector of the curve τ is laying in the tangent plane of the surface (because it can be expressed via ρi ). Remark 9 The dragging velocity vg is not zero, if the shortest distance becomes zero, r = 0 and can be represented as a limit vg = limr→0(r ϕ), ˙ i.e. the velocity represented by the rolling of the single wire on a surface, if the radius of the wire goes to zero. Since, the slave contact point rs is fixed in the curve coordinate system, all kinematical parameters for the CTRS contact should be defined, first, in the surface coordinate system ρ1 , ρ2 , n as follows: • Measure of normal interaction (in the case of a curve edge) ξ 3 = (rs − ρ) · n,
(51)
or in the case of a curvilinear beam ξ 3 = (rs − ρ) · n − R,
(52)
now in eqn. (52) R is a radius of the curvilinear beam (rope, cable etc.) as shown e.g. in Fig. 5. • Measure of tangential interaction (similar for both STS and STAS contact) in rate form as : ξ˙i = (vs − v) · ρj aij ,
(53)
Further computation of tangential measures e.g. in an incremental form ∆ξ i follows the STS algorithm: ∆ξ i = ξ (n+1), i − ξ (n), i ,
(54)
where ξ (n), i are projection point coordinates onto the master rigid surface at the load (time) step (n). 25
Figure 5. Kinematics of the Curve-To-Rigid Surface (CTRS) Contact – definition of both the Gaussian surface and the Serret-Frenet curve coordinate systems.
These measures will be used to compute the contact tractions (both normal and tangential ones) in the form of Surface-To-Surface contact. However, the weak form should be formulated in the coordinate system of the deformed curve, thus, in the form of Curve-To-Curve contact. In order to fulfill this, the contact tractions should be projected into the Serret-Frenet coordinate system, associated with a curve rs (η), see further Sect. 5.1. Combining the result for the relative velocity vector in eqn. (45) with equations for normal, pulling and dragging velocities in eqns. (48, 49, 50) we obtain the following variations: • variation of normal displacement δue = δξ 3 = δr = (δρ − δrs ) · e
(55)
• variation of pulling displacement δuτ = −(ρi · τ )δξ i
(56)
• variation of dragging displacement δug = −(ρi · g)δξ i
(57)
The relative displacement vector is then written in the curve coordinate system as: δρ − δrs = δuτ τ + δue e + δug g. (58)
26
5
Weak Formulation for Curve-To-Rigid Surface (CTRS) Contact
Considering the developed kinematics for the Curve-To-Rigid Surface interaction, we are formulating the weak form in the curve Serret-Frenet coordinate system as an integral along the curve l. It is formulated via parameters expressed either in the curve Serret-Frenet coordinate system τ , e, g, or via parameters defined in the Gaussian surface coordinate system ρi , n, thus using the projection of the force defined in Sect. 5.1, eqn. (67). Starting with the weak form expressed at the curve, we have: (1) via curve parameters δW =
Z
l
Rs · (δrs − δρ)dl
(59)
Z
(60)
= − (δue N + δuτ T + δug B)dl =
l
Z
l
(T (δrs − δρ) · τ + N(δrs − δρ) · e + B(δrs − δρ) · g) dl (61) (62)
(2) or via surface parameters δW = =
Z h Zl
l
i
N(δrs − δρ) · e + T (ρi · τ )δξ i + B(ρi · g)δξ i dl
[N(δrs − δρ) · e i
− (T j ρi · (δrs − δρ)) ρi · (τ ⊗ τ + g ⊗ g)ρj dl
(63)
(64)
For the linearization of the weak form, the result for CTC algorithm is available in [17] and can be used directly, in this case the weak form is written as set of eqns. (60-61).
5.1 Curve-To-Rigid Surface Contact – Transformation of Contact Forces Taking into account the kinematics for Curve-To-Rigid Surface (CTRS) contact, see Sect. 4, the contact traction vector RC is defined first in the local surface coordinate system at the point C, see Fig. 5, as RC = NS n + T i ρi .
(65)
The contravariant components of the trial tangent vector T i can be defined during further computations in the surface coordinate system similar to the 27
Surface-To-Surface algorithm as (n+1), i
Ttr
(n), i
= Treal − εT (ξ (n+1), i − ξ (n), i ).
(66)
As a slave point is observed on the rigid surface, the return-mapping scheme should be involved in order to compute the real components of the tangent (n), i vector Treal at the corresponding load (time) step in the surface coordinate system ρ1 , ρ2 , see eqns. (35-36). The curve is deformable, however, all parameters are initially computed on the rigid master surface. Thus, taking into account the equilibrium conditions at the contact point, we can compute the contact traction RS acting on the deformable curve and expanded by coordinate vectors τ , e, g, using the kinematics of the Curve-To-Curve interaction, see Fig. 5 for CTRS kinematics: RC + RS = 0 with RC = NS n + T i ρi RS = Ne + T τ + Bg.
(67)
During contact we have n = −e, therefore, we can construct the following relations to compute the contact force in the curve coordinate system: • normal force N = NS
(68)
T = −T i (ρi · τ )
(69)
B = −T i (ρi · g)
(70)
• pulling force • dragging force
Thus, the components N, T , B of the contact force RS in the curve coordinate system are just obtained by the projection of the contact force RC computed in the surface coordinate system.
5.2 Linearization of the Weak Form for Curve-To-Rigid Surface Contact The weak form for the Curve-To-Rigid Surface given in Sect. 5 is linearized in the curve Serret-Frenet coordinate system according to the results obtained for the Curve-To-Curve algorithm in [17]. Z
Ls,r,ϕ [δW ] = Ls,r,ϕ [{Rs · (δrs − δρ)}]dl = l
28
=
Z
l
Z
{Ls,r,ϕ [Rs ] · (δrs − δρ)}dl + {Rs · Ls,r,ϕ [(δrs − δρ)]}dl.
(71)
l
Here, rs is a vector of a ”slave” point of the curve, and ”ρ” is a vector of the point – projection on the surface. Two parts can be distinguished in the linearized expression in eqn. (71): a) the first part with linearization of the force Ls,r,ϕ [Rs ] and b) the second part with linearization of the variations Ls,r,ϕ [(δrs −δρ)]. The first part is computed differently for sticking and sliding cases, the second part remains the same, however, with the real force Rs computed via the return-mapping scheme. The sticking case identified, however, on the surface after the return-mapping scheme, is characterized by the following first part : Z
l
Z
=
l
{Ls,r,ϕ [R] · (δrs − δρ)}dl =
{ (δrs − δρ) · (ετ τ ⊗ τ + εr e ⊗ e + εg g ⊗ g) (vs − v)} dl.
(72)
The second part representing geometrical nonlinearity is obtained fully following the CTC algorithm, but now in the form of an integral along the curve: Z
l
{R · Ls,r,ϕ [(δrs − δρ)]}dl =
Z
= {(δrs − δρ) · [ ( T k cos ϕ (τ ⊗ e + e ⊗ τ ) + N k cos ϕ (τ ⊗ τ )+
(73a)
l
+B (k sin ϕ τ ⊗ τ ) ] (vs − v) }dl, (73b) where ϕ is an angle between the surface normal n and the curve normal ν cos ϕ = (n · ν).
(74)
The parts representing geometrical nonlinearity in eqns. (73a-73b) are used for both sticking and sliding cases. The constitutive parts in the case of sliding are depending on the model of friction and can be taken in analogy to CTC contact. In the case of isotropic Coulomb friction the following parts are added: Z
= − {δrs − δρ) · (εr µτ ⊗ e + εg µτ ⊗ g)(vs − v)}dl. l
(75)
The part for normal interaction is always present and is used for both sticking and sliding cases. It has the following form: Ls,r,ϕ [δW ] = +
Z
l
N {δrs − δρ) ·
(
Z
l
εr {δrs − δρ) · [e ⊗ e] (vs − v)dl+ )
1 k cos ϕ1 τ ⊗ τ − g ⊗ g (vs − v)dl. (1 − rk cos ϕ) r 29
(76a) (76b)
Here, both linearized parts for forces and for variations are taken into account. Similar as for STAS approach, discretization can be introduced only for the deformable curve (one rigid surface element for all curve elements) as rs = Ax,
(77)
where x is a vector of nodal (knots or control points knot for NURB spline approximations) coordinates for the curve. Exemplary, the part of the tangent matrix for normal interaction in eqns. (76a76b) after discretization via the matrix A in eqn. (77) is written in the following form N
K =
Z
L
+
Z
L
6
N AT
(
εr AT [e ⊗ e] Ads
(78a) )
1 k cos ϕ1 τ ⊗ τ − g ⊗ g Ads. (1 − rk cos ϕ) r
(78b)
Numerical examples
As a set of representative numerical examples we select: • For STAS contact: non-frictional and frictional contact of a suspended plate with a hyperbolical surface – covering a saddle; • For STAS contact: frictional contact of a suspended plate with a hyperbolical surface – covering a saddle; • For CTRS frictional contact: · Contact of a rope with a cylindrical surface – verification with the 2D Euler-Eytelwein Problem; · Contact of a spiral rope with a cylindrical surface – verification with a new 3D analytical solution, see in [11].
6.1 Covering a saddle – case without friction A square plate ABCD with size 20 × 20 and thickness h = 0.2 (dimension is consistent) is suspended at four springs attached at four corners above the rigid surface of the hyperboloid, see Fig. 6. The hyperboloid has the geometrical structure analyzed in Sect. 2.4.1 with a = 2.5 · 10−2 . A plate is modeled with 20 × 20 “solid-shell” finite elements allowing arbitrary non-linear deformations. Mechanical properties of the plate is E = 2.5 · 105 , ν = 0.3 with 30
the Saint-Venant linear elastic law. The stiffness of each spring is subjected to the nonlinearly weakening law during the loading process as c = 1000 e0.005t with t = 0, 1, 2, ..., 1000 is a loading parameter. The plate is loaded by the 1102, 5 uniform load with the final value qF = = 2.756F orce/Area units. (20 × 20) The final value of the load is applied incrementally as the distributed loading qF t with t = 0, 1, 2, ..., 1000 as a loading parameter. The increasing q(t) = 1000 loading process together with the weakening springs leads to the covering of a hyperboloidal saddle with a plate, thus forming a hyperbolic saddle surface from a plate. Contact is modeled via STAS approach discussed in Sect. 2.4 as rigid surface is a “master” approach with the following example of a hyperbolic surface in Sect. 2.4.1. The normal penalty parameter εN = 2.5 · 105 . Each STAS contact segment of the deformable plate is covered with 7 × 7 Gauss integration points to integrate the contact integral in eqn. (33). Active set includes only Gauss points with negative penetration in eqn. (21). Fig. 7 represents diagram of the loading process. q
B
A C D
Figure 6. Covering a saddle set-up: A loaded plate hanging on four springs above the hyperboloid.
6.2 Covering a saddle – case with friction – sticking and sliding areas
Exact contact area in the previous example can be studied by drawing a cloud of active Gauss points. We are repeating now this computation with the frictional STAS algorithm, thus, the return-mapping algorithm in eqn. (35) is performed at each coordinate of the hyperboloid αj , j = 1, 2 corresponding j at each active Gauss point ξip , j = 1, 2 of the plate. The tangential penalty 31
(a) t = 0 (b)
(c) t = 250 (d)
(e) t = 500 (f)
(g) t = 750 (h)
(i) t = 1000 (j) Figure 7. Covering a saddle. Vertical uniformly distributed loading. Front and isometric view at the plate during the loading process.
parameter during computation is ǫT = εN = 2.5 · 105 . The coefficient of friction is taken µ = 0.125. It should be noted that due to combination of vertical loading q and curvlinear structure of the hyperbolic, the contact area is split into sticking and sliding zones. It is even possible to make a very rough estimation of this zone analytically just considering a mass point in the equilibrium 32
at the hyperbolic surface under the gravity force mg; normal reaction N and friction force F with its limit Coulomb value µN. The solution of this problem is the equilibrium is possible if the following inequality is satisfied: µ > tan γ,
(79)
where γ is cosine between the normal of the surface and the z-axis. Determining the normal n from eqn. (22), we can transform eqn. (79) as follows
µ>
q
1 − n2z nz
=
q
(aα1 )2 + (aα1 )2 .
(80)
The boundary line of the sticking area is a circle (α1 )2 + (α1 )2 =
µ2 . a2
(81)
0.125 In the current computation the radius of this circle R = µa = 2.5·10 −2 = 5 gives the estimation of the radius for the circular sticking area. Fig. 8 represents the devolopemnt of the contact area and its split into sticking and sliding zones. The sticking Gauss points are depicted with green color (central area), sliding Gauss points are depicted with red color.
33
10 8 6 4
(a) t = 250
2 0 −2 −4 −6 −8 −10 −10
−8
−6
−4
−2
0
2
4
6
8
10
2
4
6
8
10
2
4
6
8
10
2
4
6
8
10
(b) 10 8 6 4
(c) t = 500
2 0 −2 −4 −6 −8 −10 −10
−8
−6
−4
−2
0
(d) 10 8 6 4
(e) t = 750
2 0 −2 −4 −6 −8 −10 −10
−8
−6
−4
−2
0
(f) 10 8 6 4
(g) t = 1000
2 0 −2 −4 −6 −8 −10 −10
−8
−6
−4
−2
0
(h) Figure 8. Covering a saddle. Vertical uniformly distributed loading. Development of the contact area during the loading process: 34 central area is sticking area, dark red area is sliding area.
6.3 Euler-Eytelwein Problem – a rope on a circular cylinder
Equilibrium under frictional forces of a rope wrapped around a cylinder is the next example to show the algorithm for frictional contact between curves and rigid surfaces. As is well known, this 2D solution gives then the ratio between the pulling forces as: T1 = expµϕ , (82) T2 where µ is a coefficient of friction, ϕ is the angle in radians formed by the first and last spots the rope touches the cylinder, see setup in Fig. 9. This case is very well known from many undergraduate books on mechanics as a belt friction formula, capstan formula, belt friction formula, or Euler-Eytelwein formula referring to the most earlier publications in [2], [3]. Two different
µ T1 T2
ϕ
Figure 9. Standard formulation of the Euler-Eytelwein problem as a rope around the cylinder.
finite element models are used here to simulate a rope behavior: (1) C1-spline isogeometric finite beam element; (2) solid-beam finite element, see in [41], [42]. Both finite elements have been presented in comparative analysis for curveto-curve contact in [18], [42]. 35
In the first example an angle ϕ = 180o is kept constant, but the coefficient of friction is varied. The inextensible rope behavior is modeled within the finite beam element model as follows: bending stiffnesss are taken in both directions EI = 2.1 · 104 with a assumption of the circular crosssection, the shear stiffnesss are reduced in 104 times (compared to the standard shear stiffnesss for the circular cross-section) and are taken in both directions GA = 0.29; then stiffness along the line EA = 2.1 · 104 and finally torsional stiffness GI = 2.1 · 104 , both remain as for the circular cross-section. The finite element model includes 10 spline elements along the arc with the angle π and then additional elements are taken just for better transportation of forces, see Fig. 10. The inextensible rope behavior is modeled within µ
ϕ R
T Figure 10. Finite element model for the Euler-Eutelwein problem. Case of ϕ = 180o for contact. Additional straight elements are supplied outside of contact.
the solid-beam finite element as follows: Saint-Venant orthotropic law inherited with the elliptic geometry of the solid-beam is taken: Elastic moduli in all three directions (along mid-line, radial and circumferential) are taken as E = 2.1 · 104 , shear moduli are reduced in 104 times compared to isotropic case and are taken as G = 0.81, and all Poisson ratios are taken as ν = 0.3. Curve-To-Rigid Surface (CTRS) contact approach with penalty formulation is used here. Both normal and tangential penalty parameters are taken as εN = εT = 2.1 · 104. Fig. 11 shows the comparison with the analytical solution for a large variation of the coefficient of friction. 36
2D Euler−Eytelwein Problem. Variation of friction coefficient 25
Analytical Solution Solid−Beam FE Spline Beam Element
20
T/T0
15
10
5
0 0
0.2
0.4
0.6
0.8
Friction Coefficient
Figure 11. Comparison with analytical solution for the large variation of coefficient of friction with constant angle ϕ = 180o . Curve-To-Rigid Surface contact approach. Two types of FE to model a rope: C1-spline beam finite element and solid-beam finite element.
An analogous numerical experiment is performed for the variation of the angle: 0 < ϕ < 2π, see Fig. 12. Now the coefficient of friction is kept constant µ = 0.3. A rather good correlation with the analytical solution in eqn. (82) is obtained for both spline and solid beam finite elements together with the implemented Curve-To-Rigid Surface contact approach.
37
1
Euler−Eytelwein 2D Variation of angle 7
Analytical Solution Solid Beam FE Spline Beam Element
6
T/T0
5
4
3
2
1 0
1
2
3 4 Angle in Radians
5
6
Figure 12. Comparison with analytical solution during variation of angle 0 < ϕ < 2π. Coefficient of friction is kept constant. Curve-To-Rigid Surface contact approach. Two types of FE to model a rope: C1-spline beam finite element and solid-beam finite element.
38
6.4 Rope in form of a spiral line (helix) around a cylinder Though, the solution of the Euler-Eytelwein problem in 2D formulation considered in Section 6.3 is mostly used for the rope-cylinder interactions, in practical applications one can observe a pitch between coils, thus leading to a spiral line or helix wrapped over the cylinder, see Fig. 13. The solution of this problem is discussed recently in [11] and has the following form: ϕs
T = T0 e
R2
µR
H + 2π
2
,
(83)
where R – radius of cylinder, ϕ – an angle in cylindrical coordinate system, H – pitch and µ – coefficient of friction. Equation (83) gives us 3D-generalization of the Euler-Eytelwein formula considering pitch. The standard 2D case is obviously recovered with H = 0.
Figure 13. Spiral line (helix) with a pitch H. Geometry and finite element model with solid-beam FE.
In the verification example the polar angle for the arc is kept constant ϕ = 180o and the pitch H is varied. The inextensible rope is modeled with solid-beam finite element as in example in Sect. 6.3. The finite element model includes 32 solid-beam elements along the arc (coordinate 0 ≤ ϕ ≤ 180o) The Curve-To-Rigid Surface (CTRS) contact approach with penalty formulation is used here. Both normal and tangential penalty parameters are taken 39
as εN = εT = 2.1 · 104 . The number of Gauss points to check contact is 3 along the mid-line of the solid-beam finite element. The coefficient of friction is set to µ = 0.1. Fig. 14 shows the comparison with the analytical solution in eqn. (83) for a variation of the pitch. One can see quite a good correlation with 3D analytical solution, while 2D standard solution is independent of the pitch H and shown for the comparison.
Solid−Beam FE Analytical Solution 3D Euler Solution 2D
1.375
1.37
T/T0
1.365
1.36
1.355
1.35
0
0.5
1 Height
1.5
Figure 14. Comparison with the analytical solution for the variation of pitch H. Polar angle for the arc is ϕ = 180o . Curve-To-Rigid Surface contact approach. 2D solution is independent of H.
40
2
7
Conclusion
Special algorithms allowing a simplified description of contact with rigid surfaces are developed in the current contribution. Kinematically two algorithms can be distinguished based on the selection of a coordinate system for the Closest Point Projection (CPP) procedure: a) Rigid surface is a “slave” surface and b) Rigid surface is a “master”. Geometrically two cases are considered: a) contact between surfaces leading to the Segment-To-Analytical Surface (STAS) approach and b) contact between curves (beams, ropes) and surfaces leading to the Curve-To-Rigid Surface (CTRS) approach. A special combination of both contact kinematics for the Surface-To-Surface and for the Curve-ToCurve approaches is employed to build CTRS algorithm. This algorithm is verified with the well known Euler formula for the rope-cylinder interaction as well as with a new derived generalization into 3D – a spiral rope on a cylinder. The developed algorithms can be straightforwardly implemented into an isogeometric approach as well as into conventional finite elements, where rigid surfaces are given by CAD patches. Any type of elements can be employed for the deformable surface/curve because the algorithms are formulated in a covariant form.
A
A.1
Rigid surface is a ”slave” approach: simplified CPP procedure
Contact with a rigid surface of revolution
For a surface of revolution, given by an analytical function, or described by NURBS, a matrix form solution for the coordinate increments in eqn. (5) can be directly developed. In the simplest case f (r) can be a plane curve uniquely projected onto the r axis, see Fig. A.1. The revolution of the curve about the axis OZ gives a surface of revolution. In a Cartesian coordinate system it can be written as
rs (r, φ) =
xs ys
zs
=
r cos φ r sin φ .
f (r)
Then the iteration vector ∆αn in eqn. (5) gets the following form: 41
(A.1)
Z
slave f(r) S
ξ3
O
n
r
φ
Y
C
master
X Figure A.1. The surface of revolution
∆rn
(A.2a)
∆αn = ∆φn where
∆ξn3
1 · ((x3 − f (r))(n1 cos φ + n2 sin φ) D + n3 (r − x1 cos φ − x2 sin φ)) 1 · (f (r) − x3 − rf ′(r))(n1 sin φ − n2 cos φ)+ ∆φn = Dr ∆rn =
+ f ′ (r)(n1 x2 − n2 x1 ) + n3 (x1 sin φ − x2 cos φ) ∆ξn3 =
1 · f ′ (r)(x1 cos φ + x2 sin φ − r) + f (r) − x3 Dr
(A.2b)
(A.2c)
(A.2d)
+ ξ 3 [f ′ (r)(n1 cos φ + n2 sin φ) − n3 ]
and with the determinant D = − n3 + f ′ (r)(n1 cos φ + n2 sin φ).
(A.2e)
Remark 10 Since, the parametrization of the master segment is considered arbitrary (the segment in Figures is shown intentionally curved), the corresponding contact approach works with any kind of approximation for finite elements including iso-geometric finite elements.
42
B
Closed form solutions for CPP procedure: rigid surface is a ”master” approach
Several cases of the surfaces - a sphere, a cylinder, a torus and a cone – possessing closed form solutions for the penetration (of the, in general, nonlinear equation (10)) have to be noted within this approach and are described in detail in Appendix B. • • • •
contact contact contact contact
with with with with
a a a a
rigid rigid rigid rigid
sphere; cylinder; torus; cone.
B.1 Contact with a rigid sphere The computation of the penetration for contact with a rigid sphere is the most trivial case for the rigid surface is a “master” strategy, because the absolute value of a vector, and therefore, a distance in the Cartesian coordinate system is defined in a form of an equation for a sphere. However, we formally start with the definition of a coordinate system assigned to the spherical surface: rs = RC + ρsph + p nsph ,
(B.1)
where RC is the center of the sphere, ρsph is a vector to the surface of the sphere with radius R, satisfying kρsphk = R. Since ρsph is parallel to the unit normal nsph of the sphere surface, see Fig. B.1, we can rewrite eqn. (B.1) as rs − RC = R nsph + p nsph,
(B.2)
Taking then the absolute value we obtain krs − RC k = |R + p|,
(B.3)
and the distance p as
p=
krs
− RC k − R – for an outward normal
R − kr
(B.4) s
− RC k – for an inward normal .
The first part in equation (B.4) is describing the simple geometrical fact AS = CS − CA in Fig. B.1 as a positive distance from the sphere into the outward direction of the sphere. A continuum body is the interior of the sphere in this case. The second equation describes a positive distance into the inward 43
direction of the sphere. In this case the continuum is the exterior of the sphere.
n ξ3 S
ρ sph
Z RC
A nsph
C
O
p slave
Y X master Figure B.1. Contact with a rigid sphere given by an analytical equation
B.2 Contact with a rigid cylinder Another example of a closed form solution within the rigid surface is a “master” strategy is the distance between a cylinder and a point, see Fig. B.2. The cylinder with a radius R can be given in the following form: ρ(ϕ, z) = RC + z ez + R eϕ (ϕ) .
(B.5)
where RC is an arbitrary point on the central axis of the cylinder CCz , ez is a unit vector of the central axis, eϕ (ϕ) is a unit vector in the radial direction of the polar coordinate system in the orthogonal plane, see Fig. B.2. The unit coordinate vectors are orthogonal (eϕ (ϕ) · ez ) = 0. Noting that the vector eϕ is normal to the cylinder, the slave point S is observed in the coordinate system assigned to the cylindrical surface as: rs = ρ(ϕ, z) + p eϕ (ϕ),
(B.6)
Substituting ρ in eqn. (B.6) by eqn. (B.5) we obtain: RC − rs + z ez + (R + p) eϕ (ϕ) = 0.
(B.7)
First, a coordinate z is defined after taking the scalar product with ez : z = −(RC − rs ) · ez ; 44
(B.8)
ξ
Cz eϕ
z
ez
C Z
ϕ
A p
3
n S
rs
RC
O
Y
X
Figure B.2. Contact with a rigid cylinder given by an analytical equation
afterward the absolute value in eqn. (B.7) is taken as kRC − rs + z ez k = |R + p|,
(B.9)
then the distance p is defined as
p=
kRC
− rs − (RC − rs ) · ez ez k − R – for an outward normal
R − kR
C − rs − (RC − rs ) · ez ez k – for an inward normal . (B.10) The distance p is defined to be positive in both outward and inward direction.
B.3 Contact with a rigid torus Consider a torus as a result of rotating a circle given in the XOZ-plane along the OZ-axis by increasing the angular coordinate ψ in the XOY -plane, see Fig. B.3. The torus can be given in the following form: ρ(ϕ, ψ) = R eR (ψ) + r eϕ (ϕ), 45
(B.11)
where eR and eϕ are radial unit vectors for the polar coordinate system given by two orthogonal planes. First, we define a scalar product eR ·ρ. This product −→ defines a projection of the vector OA = ρ = (x, y, z) on the plane XOY , therefore, q
(B.12)
kρ − R eR k2 = kr eϕ k2 .
(B.13)
ρ · eR =
x2 + y 2.
This product allows to get rid of eϕ in eqn. (B.11), by taking an absolute value:
Transformation of the scalar products leads to a torus equation in the following form kρk2 + R keR k2 − 2R (ρ · eR ) = kr eϕ k; or written by coordinates: q
x2 + y 2 + z 2 + R2 − 2R x2 + y 2 = r 2 .
(B.14)
Now we can proceed with the observation of the slave point rs = (xs , ys , zs )
Z O
r C
R eR
Y
X
ξ
A
ϕ
eϕ
rs
3
n S
ψ
p
Figure B.3. Contact with a rigid torus given by an analytical equation
at the distance p in the toroidal coordinate system, remembering that eϕ is a normal vector to the torus surface rs = R eR (ψ) + (r + p) eϕ (ϕ).
(B.15)
Similar transformations are leading to the following distance:
p=
r 2 xs r
−
q
+ ys2 + zs2 + R2 − 2R x2s + ys2 − r
r
q
x2s + ys2 + zs2 + R2 − 2R x2s + ys2
– for an outward normal – for an inward normal .
(B.16) The distance p is defined to be positive in both outward and inward directions. 46
B.4 Contact with a rigid cone A cone can be considered also within the rigid surface is a “master” strategy. The cone, see Fig. B.4, has the OZ-rotation axis and is defined as: ρ(r, ϕ) = r er (ϕ) + r tan α ez ,
(B.17)
where er and ez are unit vectors of the cylindrical coordinate system. The outward normal vector n is simply defined from the geometry of a triangle, see Fig. B.4 n = sin α er − cos α ez . (B.18) The slave point at the distance p from the cone surface is written first in the cone coordinate system and then in the cylindrical coordinate system as: rs = ρ(r, ϕ) + pncone = r tan α ez + r er + p(sin α er − cos α ez ) = (r tan α − p cos α) ez + (r + p sin α) er .
(B.19)
The definition of the distance p is tremendously simplified as we observe that the vectors ρ and n are orthogonal ρ · n = 0,
(B.20)
p = (rs · n) = sin α (er · rs ) − cos α (ez · rs ).
(B.21)
then from eqn. (B.19) we have
It is more convenient to consider rs in a cylindrical coordinate system in order to compute the scalar product in eqn. (B.21) as rs =
q
x2s + ys2 er (ϕ) + zs ez ,
(B.22)
then the penetration in eqn. (B.21) becomes: p=
q
x2s + ys2 sin α − zs cos α.
(B.23)
Remark 11 The rigid surface is a “master” strategy does not give the answer to the solvability of the contact algorithm, though the penetration is always uniquely defined for the cone. This criterion is appearing from the geometry analysis of the contact segment. Thus, considering contact with a rigid cone described by the rigid surface is a “slave” strategy as a surface of revolution in Sect. A.1 it is necessary for the solvability that the determinant in eqn. (A.2e) is not zero: D 6= 0. Taking f ′ (r) = tan α, the determinant is transformed as D=
sin α(n1 cos φ + n2 sin φ) − n3 cos φ . cos α 47
(B.24)
C
z r
A
master Z
ncone
ez Y
O X
α ϕ
p rs
er
n S slave
B Figure B.4. Contact with a rigid “master” cone surface
Using then the definition of the normal for the cone in eqn. (B.18), the determinant can be written as D=
nξ · ncone 6= 0. cos α
(B.25)
Now, the criterion of the solvability is clear – a segment normal nξ should not be orthogonal to the cone normal ncone . The case with cos α = 0 leading to α = π/2 represents the cone being degenerated into a plane and is also not accepted.
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