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Thermoelastic effect: finite element approach

Michele Bonaldi, Enrico Serra. Thermoelastic effect: finite element approach. South Kensington Campus, Imperial College. London, October 26th - 27th, 2006 . Presentation outline. We are interested in the simulation of dissipative

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Thermoelastic effect: finite element approach

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  1. MicheleBonaldi, Enrico Serra Thermoelastic effect: finite element approach South Kensington Campus, Imperial College London, October 26th - 27th, 2006

  2. Presentation outline • We are interested in the simulation of dissipative effects in solids which are joined together with direct Silicon Wafer Bondingprocess. • We developed Ansys tools for the solution of thermoelastic problems. • The problem of small flexural vibrations of a thin beam is analyzed for validation • The extension to detector test masses give rise to convergence problems South Kensington Campus, Imperial College London, October 26th - 27th, 2006

  3. The thermoelastic problemsolved with FEM Navier –Lamè -Stokes Eq. Thermal conduction Eq. Temperature definition By using a FEM discretization scheme based on v.p. Coupling matrices South Kensington Campus, Imperial College London, October 26th - 27th, 2006

  4. The thermoelastic damping computation Boundary Value Problem (solved in harmonic regime ) From the solution of the Boundary Value Problem BVP in case of harmonic force load South Kensington Campus, Imperial College London, October 26th - 27th, 2006

  5. Small flexural vibrations in a thin beam (benchmark) The cases under investigation are summarized in the table where two different materials are considered. Comparisons with the model developed by R. Liftshitz and M.L. Roukes Phys Rew B wasdone. The analysis performed are: - Modal analysis with TED - Harmonic analysis TED - Harmonic analysis TED and SD TED = Thermo-Elastic Damping SD = Structural Damping Problem’s domain Reid et al. Phys lett A Reference temp. Harmonic uniform pressure adiabatic conditions on the beam’s boundaries Mechanical and thermal properties are retrieved at MPDB http://www.jahm.com London, October 26th - 27th, 2006 South Kensington Campus, Imperial College

  6. Modal Analysis (Si) Modal analysis was performed for the computation of the cantilever’s resonant modes including only TED - The analysis is not limited to mode frequencies. Remark. The use of high aspect ratio elements is not critical up to 10^3 3D Beam’s FEM model Element Aspect ratio 73.26 682.07 Hz 38.737 Hz 243.18 Hz 454.45 Hz London, October 26th - 27th, 2006 South Kensington Campus, Imperial College

  7. Harmonic Analysis (Si) Temperature gradient Two Harmonic Analysis were performed for the computation of the loss angle in presence of TED. Harmonic analysis at 300 K varying the force frequency [500-200000 Hz] 2. Harmonic analysis at 681Hz (third flexural mode) varying the temperature 4.2<T<300 K The expected loss angle is evaluated by using theory developed by L-R South Kensington Campus, Imperial College London, October 26th - 27th, 2006

  8. Good agreement with theory (10 % agreement) 2) 1) 1) • Considerations: • FEM data accounts for all modes: L-R solution are smaller • Good solution convergence (linear scale) • Hysteretic damping simply sums up to TED • - The modelling of composite cantilever is also possible 3)

  9. Discussion on convergence When the test body’s dimensions are increased convergence problems appear ! We consider the case of beam with square cross-section (10 mm beam’s width – in this case L-R theory is approximate). Convergence is obtained for Silicon but not for Fused Silica. Loss angles are: Too small. We are approaching the numerical precision ? South Kensington Campus, Imperial College London, October 26th - 27th, 2006

  10. Conclusions • The tool for thermoelastic analysis works only in presence of thin structures • The application of the tool for the computation of dissipation effects in bonding (and coatings) problems is promising The developed tool present convergence problems in case of mass (like DUAL, mirrors) because of low dissipation rate) South Kensington Campus, Imperial College London, October 26th - 27th, 2006

  11. Future developments What we can do now: Develop Ansys tools for the simulation of dissipative effects in bonding layers (or coatings) on Silicon substrate Understand the convergence problem ! Possible solutions: - using adaptive mesh strategy in highly stressed regions - increasing the numerical precision by using arbitrary precision arithmetic modules in combination with a proprietary FEM tool (it must be realized) South Kensington Campus, Imperial College London, October 26th - 27th, 2006

  12. End of the presentation • Contacts: • Michele Bonaldi • Istituto di Fotonica e nanotecnologie CNR-ITC and INFN Trento, I-38050 Povo(Trento), Italy • Email: michele.bonaldi@science.unitn.it • Enrico Serra • Istituto Trentino di Cultura, ITC-irst Microsystem Division, 38050Povo(Trento),Italy • Email: eserra@itc.it

  13. What will I do • First of all understand if the convergence problem appears during the matrix assembly phase or when the algebraic system is solved: • External solvers with arbitrary precision can be used (this strategy is quite simple) • Development of a FEM tool (is much mode difficult and we need a small group of sw developers ) South Kensington Campus, Imperial College London, October 26th - 27th, 2006 • Understanding if the convergence problem • is originated • Utilizzo di solutori esterni a precisione arbitraria (abbastanza semplice) • Sviluppo di un fem dedicato (più difficile e necessità di un piccolo gruppo di sviluppatori)

  14. Ansys realease v10.0 • Offer the possibility to perform a Direct Coupled-Field Analysis for thermoelastic problems by solving mechanical and thermal equations at the same time - 2D and 3D elements with (ux,uy,uz,temp) dofs are available A second order version of the thermoelastic element is available South Kensington Campus, Imperial College London, October 26th - 27th, 2006

  15. Structural and thermoelastic losses Thermoelastic and structural losses can be considered at the same time. SD+TED For a very low quality factor the structural damping is dominant along all the temperature range. In case of high structural quality factor the thermoelastic effect is dominant for certain temperature range and it is combined with structural damping in another range. South Kensington Campus, Imperial College London, October 26th - 27th, 2006

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