Abstract. 1 Introduction

Transcription

1 Contact detection algorithm using Overhauser splines M. Ulbin, S. Ulaga and J. Masker University of Maribor, Faculty of Mechanical Engineering, Smetanova 77, 2000 Maribor, SLOVENIA uni-mb.si Abstract A contact detection algorithm using Overhauser spline was used, solving contact problems with finite element method. Contact area defined by objects boundaries could be approximated in different ways. It can be modelled using straight lines between nodes, finite element shape functions or interpolation functions over boundary nodes respectively. As line description gives a poor approximation of the boundary, shape functions are widely used to describe the analysed geometry. C inter-element continuity of the geometry description is provided by the isoparametric elements. As the 'smooth' and accurate geometry description is crucial to contact stress analysis, decoupling of the geometry from the polynomial shape functions and implementation of the Overhauser splines is suggested in the present work. Single parametric curve is used to model the contacting surface offering the C* continues description of the boundary enabling exact boundary condition imposition. On the other hand geometry description using splines provides the data required for actual contact area determination and exact contact size calculation. In presented contact algorithm redundant boundary definition is introduced to eliminate the problem of poor inter-element continuity of shape functions. Implementation of suggested approach as it is used in developed finite element code is presented. Benefits and drawbacks of proposed approach are discussed. The developed code is then used for solving contact problems in gears. 1 Introduction Contact detection e.g. identification of the region of contact is usually the first task in contact problem analysis. Contact detection algorithm depends on object boundary approximation. It could be approximated with straight lines between nodes, with finite element shape functions, some other approximation functions over boundary nodes or some other special technique Belytschko [1]. Using straight lines is too simple so usually finite element shape functions are used English [3]. Shape functions geometry definition is unambiguous only inside the element and not at corner nodes between elements.

2 102 Contact Mechanics III Therefore boundary is defined as discrete segments where each segment is defined with element shape function. Instead of the piecewise continues and non-smooth boundary presentation provided by the element shape function, boundary of the object can be described by unique, smooth and C* continues parametric spline. For presented contact determination algorithm boundaries of contacting surfaces were approximated using Overhauser splines Brewer [2] instead of finite element shape functions. Introducing redundant boundary definition eliminates problem of inter-element discontinuities of shape functions. 2 Overhauser spline The implementation of the Overhauser spline is simple, if applied to the discretised boundary. While most of other splines require definition of the control points and tangent vectors, Overhauser spline requires only the positions of the control points and spline is drawn exactly through all but first and last of the control points. General equation of the Overhauser spline is shown in Equation 1. P(«) =!>/«) u=[0,l) (1) s=l Overhauser spline segment is shown in Figure 1. Four points define each segment of the spline. Spline is defined only between p; and p*+i, which means that additional point, must be supplied at the start and at the end of the spline. Introducing the Overhauser spline is only possible when contact surface can be predicted and spare points must be available before and after contact surface. Nodes on surface are control points used for Overhauser spline definition. / u=0 Figure 1: Overhauser spline

3 Contact Mechanics HI 103 Overhauser spline is defined with four parametric functions and it is constrained with four control points: Pi Pi+i Pl+2. (2) L u 2 1 u u 2 (3) In Equation 2 and Equation 3 the parameter u has the value between 0 and 1 because the equations are valid only for one segment of the spline. 3 Contact detection algorithm For contact determination a suitable algorithm that uses spline representation of the contacting surfaces is required. In Figure 2 target boundary p is shown. Due to load increment the contactor point was deformed in that way that its relative position moves from point S to point Q. Line SQ is therefore the penetration line. First of all it must be decided whether the contactor point Q is inside the target body or not. When using finite element shape functions first task is identifying the segment of the target body where the contact occurs. This is done by a rough check of the co-ordinates of a potentially contacting point against the maximum and minimum co-ordinates of an imaginary envelope constructed from the co-ordinate range of a potentially contacting target elements surface nodes. If potentially contacting point is inside of the envelope the point is tested with respect to identified segment. When potentially contacting point is outside the envelope the point is no longer contact candidate. Beside cases when the point is outside the target body, hat may occur when penetration in iteration step is too big so that the point penetrates beyond envelope, while program identifies that point is outside the body. If the contact surface is defined with the Overhauser splines, then the test is simple and exact. First the shortest distance between a point Q and the spline p must be determined.

4 104 Contact Mechanics HI target nodes o. Figure 2: Contact detection The orientation of the normal vector to the spline that goes through the test point shows whether the test point has penetrated the target body or not. Therefore it is necessary that the sequence of control points of the spline is unique and such that the normal of the spline always points inside the target body. Shortest distance between test point Q and the spline p can be found by solving (p(v)-q)p'(u) = (4) In Equation 4 U has the values between 0 and the number of segments of the curve. U can be divided into integer part, which is represented by i in the Equation 2 and the remainder, which is represented by u in the Equation 3. Equation 4 can be solved using Newton-Raphson method Nakamura [4]:,=//_. + (5) where f(u) = (p(u) - q)p'(u) /"(!/)= p'(u)p"(u) + (p(u) - q)p~(u) (6) In Equation 6 first and second derivative of the curve p are: Pi-, (7)

5 Contact Mechanics HI 105 Fl(u) = Fl(u) = FS(«) = 9 2 ' -5w I" 2 1 (8) + 4w 4- "2" 2 /% and = [FT(u\ FT(u), FT(u), FT(u)] Pi Pi+l (9) 3%+ 2 9w -5 9w + 4 3w- 1 (10) The next problem is finding the intersection between penetration line SQ and surface p as seen in Figure 2. When usual approximations with finite element shape functions are used, the segment on which penetration line penetrates the boundary must be identified first. After that Newton-Raphson method is used to evaluate intersection. Intersection between Overhauser spline and any line can be found by inserting spline function into the line Equation 11: Ax(U)+Ey(U)+C=0 (11) where x(u) and y(u) are components of the vector p : P(U) = (12) which can be again solved with the Newton-Raphson method where functions/ (U) and/"(lo are now: (13)

6 106 Contact Mechanics HI By performing the above calculations it is possible to solve all boundary contact calculations. The determined penetration value, normals and tangents are then included into augmented matrix. Overhauser spline is extendible into Overhauser surface and can be used in similar fashion using Equation 14 instead of Equation 1. (14) Implementation of the Overhauser class makes contact calculation independent of the rest of the finite element code and require no changes to the main finite element code. Finite element program stays fully operational and implementing the contact problem does not change any other function. The same program can now be used for any analysis and in addition offers the possibility of solving contact problems. 4 Contact size evaluation Accurate geometry description of contacting surfaces is extremely important in contact problem analysis, but due to its particular formulation it is often in conflict with the conventional isoparametric finite element approach. In some technical applications (such as gears, bearings, orthopaedic prostheses) the knowledge of the exact size of the contacting area is an important issue. Using isoparametric finite element codes the element size restricts the determination accuracy of the actual size of the contact area in equilibrium. While the element shape functions give only a piecewise continuity along the boundary, parametric splines provide C* continuity and therefore accurate boundary description, which leads to exact determination of the contact size and position of the contact area. The proposed algorithm is capable of the contact size determination for different shapes of the contact surfaces and for single as well as multiple contact. In the first step of the algorithm the intervals between the control points of the Overhauser 's spline are detected, where the sign of the contact changes (no contact => contact or contact => no contact). If the contact is detected, the exact points of intersection between contactor and target splines are determined. Next the length of single contacting intervals is evaluated by integration along the boundary between the calculated intersection points Nakamura [4]. The procedure is repeated for every load increment and one can follow the contact development as the load increases.

7 Contact Mechanics HI 707 Spline intersection calculation Total contact size Figure 3: Contact size determination In Figure 3 a simple float chart is presented. Contact area is calculated in every load step giving the total of contact area at the end. 5 Conclusions Using spline for approximation of the contact area certainly results in better and robust algorithms for contact detection. But it introduces redundant data, which requires more careful data handling. Apart form that spline approximation is applicable only to the contact problems where contact surfaces are known in advance or can be predicted. Developed computer program for contact problem analysis was used for contact problems in gears, where approximate contact surfaces are obvious. Exact contact area is then determined with above algorithms. After that contact problem is solved using Lagrange multiplier method and finite element method.

8 108 Contact Mechanics HI Index Finite element method, contact problems, spline functions, contact detection. References [1] Belytscho, T., Neal, M. O. Contact-Impact by the Pinball Algorithm with Penalty and Lagrangian Methods, International Journal for numerical methods in Engineering, 1991, Vol. 31, [2] Brewer, J., Anderson, D. C. Visual interaction with Overhauser curves and surfaces, Computer Graphics, Vol. 11, No. 2, 1977, pp [3] English, G. R. Lagrange Multiplier Method for Contact and Friction: Implementation and Theory, PhD thesis, University of Liverpool, Department of Mechanical Engineering, [4] Nakamura, S. Applied Numerical Methods in C, Prentice-Hall International, Inc., 1993.