1. Alessandro BertarelliTS department Seminar, 3 rd May 2006 EDMS Alessandro Dallocchio 1,2 Alessandro Bertarelli 1 1 TS department – Mechanical and Material Engineering Group CERN, Geneva 2 Mechanical Engineering Department– Politecnico di Torino, Italy FE Simulation of 450 GeV Injection Error Test on Collimator Workshop on Materials for Collimators and Beam Absorbers 3-5 September, 2007 CERN - Geneva

2. A. Dallocchio - A. BertarelliWorkshop on Materials for Collimators and Beam Absorbers Outline  Introduction  450 GeV Injection Error Test  Analytical Method  Finite Element Model  Simulation Results  Simulations vs Experimental Measurements  Conclusions

3. A. Dallocchio - A. BertarelliWorkshop on Materials for Collimators and Beam Absorbers Introduction Thermally Induced Vibrations   The interaction between high energy particle beams and solids provokes, on the hit structure, a non-uniform temperature distribution that gives birth to thermal stresses and deformations.   In case of short beam pulses, a very fast heating process occurs: mass inertia cannot be neglected   Dynamic thermo-mechanical phenomena on structures having direct interaction with particle beams (beam targets, collimators and windows) can be usually included in the so called Thermally Induced Vibrations

4. A. Dallocchio - A. BertarelliWorkshop on Materials for Collimators and Beam Absorbers 450 GeV Injection Error Test Robustness test in the SPS ring – TT40 extraction line 5 mm Maximum intensity: 3.2 10 13 protons at 450 GeV over 7.2 µs Several  s shots ranging from 1 to 5 mm at different beam intensity. The collimation bloc is a slender structure (namely a beam with rectangular cross-section). Thermally induced vibrations occur at every beam extraction Power Density distribution [W/m 3 ] from FLUKA team Typical collimator cross-section

5. A. Dallocchio - A. BertarelliWorkshop on Materials for Collimators and Beam Absorbers General Assumptions   Given that the time scale is of the order of nanosecond (1ns each bunch) heat conduction Fourier law is still holding. This should be true down to time scales of a few picoseconds.   Since stresses are well below the Elastic modulus, the hydrodynamic theory is not compulsory (no shock waves). It is possible to treat the problem following the rules of thermoelasticity: a FEM model with implicit scheme of integration can be used to simulate thermally induced vibrations.   A preliminary estimation is necessary in order to set the integration time step. Although the implicit method is intrinsically stable, the Courant principle of stability should be applied to avoid numerical damping (typical of implicit codes).

6. A. Dallocchio - A. BertarelliWorkshop on Materials for Collimators and Beam Absorbers Analytical Method This approach was developped to study CNGS beam targets; more details on the analytical method can be found on: A.Bertarelli, A. Dallocchio, T Kurtyka - Dynamic Response of Rapidly Heated Cylindrical Rods: Longitudinal and Flexural Behaviour - accepted for publication on the Jounal of Applied Mechanics Collimation bloc considered as a rectangular beam simply supported at the extremities TEMPERATURE DISTRIBUTION T(x,y,t) Min Max Flexural vibrations Thermal Bending Moment Natural Frequency and Modal Shape Dynamic Flexural Oscillation Temperature distribution x y z

7. A. Dallocchio - A. BertarelliWorkshop on Materials for Collimators and Beam Absorbers Simple predictions Once expected temperature increase is know, simple formulas may be applied to predict plasticization … Assuming that no expansion at all occurs, we can simply evaluate compressive strains and linear elastic stresses … … maximum stress is well above the proportional limit of OFE-Copper (~50 MPa), hence, after the thermal shock, plastic strains will be remaining. Area where residual plastic strains are expected Permanent Deflection Mechanism Compressive residual strains on the 3-mm thick plate are eccentric with respect to the neutral axis of the metal support, leading to a permanent deflection, away from the beam axis.

8. A. Dallocchio - A. BertarelliWorkshop on Materials for Collimators and Beam Absorbers 1 st load step - Thermal analysis - T(x,y,z,t) 2 nd load step – Structural dynamic analysis – Temperature distribution as a function of time applied as nodal load at different time steps 3 rd load step – Static analysis – Residual plastic deformation can be evaluated. Elasto-plastic analysis performed using multilinear kinematic hardening model for metallic components Finite Element Model 3D Thermo-Mechanical Fast-Transient Elasto-Plastic Analysis – (Implicit Integration Scheme) Cooling Pipes – Internal pressure 15bar SOLID 45 - 8node brick element with initial strain to simulate spring preload (3bar) 3D Contact elements with friction

9. A. Dallocchio - A. BertarelliWorkshop on Materials for Collimators and Beam Absorbers Finite Element Model 3D Thermo-Mechanical Elasto-Plastic Analysis – an Implicit Method 3-D linear orthotropic model for C-C composite jaw Temperature dependent material properties No damping is considered in the model Integration time step and mesh size have been carefully chosen on the base of the preliminary analytical estimation. Δt=0.1μs Deposited Heat Power (W/m 3 ) from FLUKA simulations (HGEN on volumes - 3D tables) Convection (12360W/m 2 /K) + inlet temp. (27ºC) Simply supported extremities Min Max

10. A. Dallocchio - A. BertarelliWorkshop on Materials for Collimators and Beam Absorbers Simulation Results A small amount of residual plastic deformation is found on cooling pipes Transverse residual displacement - 16μm 3D Thermo-Mechanical Elasto-Plastic Analysis – an Implicit Method Transverse residual displacement - 350μm Closely matches the measured value (300μm) 2004 2006

11. A. Dallocchio - A. BertarelliWorkshop on Materials for Collimators and Beam Absorbers Simulation Results 3D Thermo-Mechanical Elasto-Plastic Analysis – an Implicit Method Qualitative estimation performed via analytical method has been confirmed by FEM 1st frequency of flexural oscillation ~45Hz with an amplitude of 1.5mm No damping is considered in the FE model

12. A. Dallocchio - A. BertarelliWorkshop on Materials for Collimators and Beam Absorbers FEM vs LDV measurments Simulation results have been compared with measurements performed via Laser Doppler Vibrometer and a good agreement has been found Experimental data: Courtesy J. Lettry, R. Wilfinger and H. Richter Dynamic response is two times the static deflection (as predicted by the analytical model) Quasi-static deflection due to thermal bending moment.

13. A. Dallocchio - A. BertarelliWorkshop on Materials for Collimators and Beam Absorbers Conclusions   The Analytical method has proved very useful in anticipating and interpreting results of FEM simulations (frequencies, oscillation amplitude and modal shapes)   The problem can be solved in three steps: 1. 1. Transient thermal simulation allows to calculate temperature distribution as a function of time 2. 2. Dynamic structural analysis can be solved with nodal temperature T(x,y,z,t) applied as nodal load at different time steps. 3. 3. Static structural analysis: on a long time scale, temperature gradients vanish and the system is in static equilibrium between plastic residual stresses and elastic forces.   Simulation results are in good agreement with experimental measurements. Differences found in oscillation amplitudes may be explained by the approximations of the FE model (no damping, friction coefficient, material properties…) and energy distribution input (FLUKA uncertainties). Also, uncertainties may be present in LDV measurements (problems with calibration factor? ).   An implicit FE model can been successfully used to study thermally-induced vibrations on the short time scale and predict residual deformations on the long time-scale.