Author manuscript, published in "ECCM 2010, IV European Comparative study of the use of C 1 − continuous finite Conference on Computational Mechanics, Paris : France (2010)" hal-00657382, version 1 - 6 Jan 2012 elements and splines for contact problems with large slidings B. Magnain a , A. Batailly b , N. Chevaugeaon c , M. Legrand b , C. Pierre b (a) Institut PRISME (b) Structural dynamics and (c) G´ eM, UMR CNRS 6183 ENSI Bourges vibration laboratory Pˆ ole calculs et structures ´ McGill University Ecole Centrale de Nantes G G e e M M Institut de Recherche en G énie C ivil et M écanique
C 1 − continuous methods:Hermite and B-splines Introduction Large slidings cases Conclusion Problem description Problem description hal-00657382, version 1 - 6 Jan 2012 Contact problems are highly nonlinear and may lead to complex and inefficient simulations In particular, high sensitivity of contact simulations when large slidings occur on curved contact interfaces discontinuity of the orientation of the normal to the contact surface ( facetization ) non smooth approximation of the gap function mesh refinement may not be a solution when large models are involved May 19 th 2010 ECCM 2010: C 1 − continuous finite elements and splines for contact problems with large slidings 2 / 24
C 1 − continuous methods:Hermite and B-splines Introduction Large slidings cases Conclusion Practical manifestation Practical manifestation hal-00657382, version 1 - 6 Jan 2012 Rotor/stator interaction study may lead to the simulation of permanent contact between a blade and the surrounding casing rubbing modal interaction � y � x α = π 2 C ∀ α, g = 0 Contact detection results from discretization errors May 19 th 2010 ECCM 2010: C 1 − continuous finite elements and splines for contact problems with large slidings 3 / 24
C 1 − continuous methods:Hermite and B-splines Introduction Large slidings cases Conclusion Practical manifestation Practical manifestation hal-00657382, version 1 - 6 Jan 2012 Beam: Q4 (20 × 5) Ring: Q4 (120 × 8) contact effort (N) 0.3 0.2 0.1 y � 1215 0 π π 0 2 π 3 π 4 π 1029 10 2 10 10 10 α (rad) � x α = π displacement ( µ m) 2 12 8 4 0 -4 π π 0 2 π 3 π 4 π 10 2 10 10 10 α (rad) May 19 th 2010 ECCM 2010: C 1 − continuous finite elements and splines for contact problems with large slidings 4 / 24
C 1 − continuous methods:Hermite and B-splines Introduction Large slidings cases Conclusion Practical manifestation Layout hal-00657382, version 1 - 6 Jan 2012 Introduction 1 Problem description Practical manifestation C 1 − continuous methods:Hermite and B-splines 2 Description Gap inaccuracy Contact normal orientation approximation Contact detection Large slidings cases 3 validation case:cube and rings blade tip/casing Conclusion 4 May 19 th 2010 ECCM 2010: C 1 − continuous finite elements and splines for contact problems with large slidings 5 / 24
C 1 − continuous methods:Hermite and B-splines Introduction Large slidings cases Conclusion Description C 1 − continuous methods hal-00657382, version 1 - 6 Jan 2012 Objective: better representation of a curved contact surface Means: mortar elements superposition of linear elements and smoothing methods (B-splineS) integration of smoothing methods within the element (Hermite element) Focus of our study: compare B-spline and Hermite methods that are compatible with our solution algorithm. May 19 th 2010 ECCM 2010: C 1 − continuous finite elements and splines for contact problems with large slidings 6 / 24
C 1 − continuous methods:Hermite and B-splines Introduction Large slidings cases Conclusion Description B-splines hal-00657382, version 1 - 6 Jan 2012 Description: third degree polynomial basis interpolation splines: control points chosen in agreement with mesh Assets: Independent on the element type of the mesh can be easily extended to 3D cases Requirements: specific detection procedure with Newton-Raphson method higher computation time spline/contact line � n s � n n � n n+1 n + 1 n May 19 th 2010 ECCM 2010: C 1 − continuous finite elements and splines for contact problems with large slidings 7 / 24
C 1 − continuous methods:Hermite and B-splines Introduction Large slidings cases Conclusion Description Hermite finite element hal-00657382, version 1 - 6 Jan 2012 Description : 24 dof isoparametric element with cubic edge dof include coordinates of tangent vector to the edges Assets : continuity of normal vector orientation from an element to another automatic update of the contact surface with mesh deformation Requirements : specific mesh construction higher computation time 4 η ∂ y v 3 u i = { u i ,∂ u i ∂ x ,∂ u i ∂ y , v i ,∂ v i ∂ x ,∂ v i v 3 4 3 ∂ x v 3 ∂ y } ∂ y u 3 1 ∂ x u 3 ξ Y i = (1 , 2 , 3 , 4) . 1 2 2 u 3 X May 19 th 2010 ECCM 2010: C 1 − continuous finite elements and splines for contact problems with large slidings 8 / 24
C 1 − continuous methods:Hermite and B-splines Introduction Large slidings cases Conclusion Description C 1 − continuous methods: validation hal-00657382, version 1 - 6 Jan 2012 The pertinence of each strategy is assessed by: checking geometrical errors 1 ◮ gap inaccuracy ◮ contact normal orientation approximation validating contact detection 2 simulating large sliding contact cases 3 All the simulations presented are quasi-static simulations without friction using bi-potential method for contact management in total Lagrangian framework. May 19 th 2010 ECCM 2010: C 1 − continuous finite elements and splines for contact problems with large slidings 9 / 24
C 1 − continuous methods:Hermite and B-splines Introduction Large slidings cases Conclusion Gap inaccuracy Gap inaccuracy hal-00657382, version 1 - 6 Jan 2012 30 18 1.6 gap inaccuracy g (%) 16 gap inaccuracy g (%) gap inaccuracy g (%) 1.4 25 14 1.2 12 20 1 10 15 8 0.8 6 0.6 10 4 0.4 2 5 0.2 0 0 -1.5 0 -1 1 -1 1 -1 1 ξ ξ ξ (a) Linear discretization (b) Spline discretization (c) Hermite discretization Gap inaccuracy for η = 2 ; η = 4 ; η = 8 ; η = 14 and η = 20 , η being the number of elements over the 1 2 ring. ξ = 1 ξ = 1 ξ = − 1 ξ = − 1 η = 4 η = 8 May 19 th 2010 ECCM 2010: C 1 − continuous finite elements and splines for contact problems with large slidings 10 / 24
C 1 − continuous methods:Hermite and B-splines Introduction Large slidings cases Conclusion Contact normal orientation approximation Contact normal orientation approximation hal-00657382, version 1 - 6 Jan 2012 contact orientation error θ (rad) contact orientation error θ (rad) contact orientation error θ (rad) 0.8 0.6 0.036 0.5 0.6 0.027 0.4 0.4 0.018 0.3 0.2 0.009 0.2 0 0 0.1 -0.2 -0.009 0 -0.4 -0.018 -0.1 -0.6 -0.027 -0.2 -0.8 -0.3 -0.036 -1 1 -1 1 -1 1 ξ ξ ξ (d) Linear discretization (e) Spline discretization (f) Hermite discretization Contact orientation error for η = 2 ; η = 4 ; η = 8 ; η = 14 and η = 20 . ξ = 1 ξ = 1 ξ = − 1 ξ = − 1 η = 4 η = 8 May 19 th 2010 ECCM 2010: C 1 − continuous finite elements and splines for contact problems with large slidings 11 / 24
C 1 − continuous methods:Hermite and B-splines Introduction Large slidings cases Conclusion Contact normal orientation approximation Contact normal orientation approximation hal-00657382, version 1 - 6 Jan 2012 Both the addition of B-spline over linear elements and the use of Hermite elements allow for significant improvements of the gap inaccuracy and the contact normal orientation error . The choice of the mesh parameters for our study is driven by the following conditions: maximum gap error must be g max < 1% of the outer radius of the ring convergence must be observed for static simulations (sensitive bending of structures modeled with Q4 elements) Mesh parameters: 10 × 8 May 19 th 2010 ECCM 2010: C 1 − continuous finite elements and splines for contact problems with large slidings 12 / 24
C 1 − continuous methods:Hermite and B-splines Introduction Large slidings cases Conclusion Contact detection Contact detection hal-00657382, version 1 - 6 Jan 2012 Assessment of contact detection: beam in contact over a 1 2 ring, flexible or rigid. � F Load applied on the tip of the beam. y � maximum displacement is equal for the three � x discretization methods no penetration detected Good contact detection Y Z X May 19 th 2010 ECCM 2010: C 1 − continuous finite elements and splines for contact problems with large slidings 13 / 24
C 1 − continuous methods:Hermite and B-splines Introduction Large slidings cases Conclusion Contact detection Contact detection hal-00657382, version 1 - 6 Jan 2012 Assessment of contact detection: beam in contact over a 1 2 ring, flexible or rigid. � F y � (loading) � x � y � x 0 0.143 0.285 May 19 th 2010 ECCM 2010: C 1 − continuous finite elements and splines for contact problems with large slidings 14 / 24
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