1 Residual Vectors & Error Estimation in Substructure based Model Reduction - A PPLICATION TO WIND TURBINE ENGINEERING - MSc. Presentation Bas Nortier

2 80 m Introduction Trends in wind industry Increase in size of wind turbines `Going offshore` More wind offshore Decrease wind energy costs Decrease wind turbine costs 25 m

3 Introduction Cost reduction through optimisation Costs reduction cycle Turbine design Dynamic behaviour Structural dynamic analysis Design changes

4 Introduction Create accurate reduced model Reduced model `Simplify` the model Increase computational efficiency Approximation of dynamic behaviour Dynamic behaviour is influenced by excitation Current reduction methods Do not take excitation into account “Investigate and implement the modal truncation augmentation (MTA) method into the current structural dynamic tools"

5 Introduction Create accurate reduced assembly “Investigate error estimation techniques for accuracy determination and refinement strategies" Unreduced assembly Reduced assembly Which component models? Needs exact solution Accuracy Efficiency Refinement Comparison Level of reduction blades tower

6 Content Introduction MTA method Application to an offshore support structure Error estimation Application to an offshore wind turbine Conclusions & Recommendations MTA – Application – EE – Application – Conclusions

7 MTA method What is it? Extension of current reduction methods Taking excitation into account Create improved reduced model Model reduction ¼ Standard `Standard` modes MTA – Application – EE – Application – Conclusions Extended Force dependent modes `Standard` modes ¼

8 Blades Tower Application to an offshore support structure Model description Model 5-MW offshore turbine Jacket support structure Excited by waves Jacket MTA – Application – EE – Application – Conclusions

9 Application to an offshore support structure Experiment description Goal Create reduced jacket model Use standard and extended reduction methods 4 wave loads; low, medium, high, freak waves One model for each wave type One combined model MTA – Application – EE – Application – Conclusions

10 Application to an offshore support structure Results Low Medium High Freak Improved Similar Extended Standard Combined MTA – Application – EE – Application – Conclusions

11 Content Introduction MTA method Application to an offshore support structure Error estimation Application to an offshore wind turbine Conclusions & Recommendations MTA – Application – EE – Application – Conclusions

12 Error estimation Why, and what Estimate error Without knowing exact response Conservative  Upper bound Determine refinement of components Blades Jacket Tower Hub Exact error Estimated error Unreduced assembly Reduced assembly Refinement Comparison Time Displacement Exact Approximation Exact error Estimated error Error MTA – Application – EE – Application – Conclusions

13 Error estimation How does it work? Error results in global residual force Split global residual Conservative scaling per component MTA – Application – EE – Application – Conclusions Residual M Ä ~ u + K ~ u = f + r Accelerations Displacements Force M Ä u + K u = f  Interface  …  Tower r = r ( 0 )... r ( n ) Exact error Scaling Component residual jk e kj 2 · n P s = 0 1 ¸ ( s ) ° ° r ( s ) ° ° 2 ·

14 Error estimation Type of errors Errors for various situations Global eigenmode & eigenfrequency Accurate range Single eigensolution MTA – Application – EE – Application – Conclusions ::: ) Eigenmode + Eigenfrequency = Eigensolution Reduced assembly

15 Error estimation Iteration loop 1. Reduce model 2. Approximate response  Global residual 3. Domain contribution Tolerance? 4. Refinement strategy Optimal reduced model No Yes MTA – Application – EE – Application – Conclusions

16 Application to an offshore wind turbine Model description Same turbine model Rotor nacelle assembly (RNA) Tower Jacket Interface MTA – Application – EE – Application – Conclusions

17 Application to an offshore wind turbine Experiment description MTA – Application – EE – Application – Conclusions Create a reduced assembly Optimal component refinement Upper bounds Error on 10 th eigenfrequency Error on 10 th eigenmode

18 Application to an offshore wind turbine Results global eigensolution Component error MTA – Application – EE – Application – Conclusions Eigenfrequency Eigenmode

19 Conclusions & Recommendations Conclusions MTA method Able to produce more accurate reduced models Implemented in dynamic analysis tools Error estimation Can determine accuracy Used for refinement strategy “Investigate and implement the MTA method into the current structural dynamic tools" “Investigate error estimation techniques for accuracy determination and refinement strategies" MTA – Application – EE – Application – Conclusions

20 Conclusions & Recommendations Recommendations MTA method Generalise excitation Error estimation Create a practical tool Range of eigensolutions Combining the best of both worlds Error estimation & MTA method Component residuals  force dependent modes MTA – Application – EE – Application – Conclusions

21 Thank you for your attention

22 Backup slides Dynamic substructuring tools BHawC Global turbine model used for multiple simulations Simulation take half an hour Hundreds of simulation have to be run Dynamic Substructuring tools Counterpart of BHawC Input large FE models Use reduction methods to reduce large models Create superelements for input in BHawC 3 Tools Preparation Tool Assembly Tool Postprocessing Tool

23 Backup slides Reduction method Craig-Bampton Static constraint modes Fixed interface modes Dual Craig-Bampton Free vibration modes Rigid body modes Residual attachment modes … Both extended using MTA method

24 Backup slides MTA method Force dependent modes Based on external loading Based on interface loading Number equals interface DoF Can become limiting  Interface reduction Interface reduction Interface displacements (substructure and assembly) Interface forces Rigid interface displacements Effective modal mass Post-selection

25 Backup slides MTA method 2 Efficient computation Using Lanczos algorithm Postprocessing Lanczos iterations Separate step using (Block) Lanczos Frequency shift Create MTA vectors for specific frequency Creates a dynamic stiffness matrix Additional costs

26 Backup slides Force analysis using POD method Wave loads are time varying Need to obtain time-invariant force vectors Proper orthogonal decomposition (POD) method Used to obtain spatial force vector from time varying load data Proper orthogonal modes (POM); force shapes Proper orthogonal values (POVs); energy captured by POMs

27 Backup slides Extended results dynamic response High waves Freak waves Medium wavesLow waves

28 Backup slides Error estimation; which type Error estimation A priori  knowing error in advance A posteriori  computing error in hindsight (iteratively) Compatible with Craig-Bampton reduction Assembly Transformation Reduction

29 Backup slides Error estimation; different errors

30 Backup slides Error estimation; refinement schemes 2 Refinement schemes Selecting largest contributors (part of largest component error) Normal distribution Divide number of available DoF accordingly

31 Backup slides Uncoupling of component models u i = ª C ; i u b + ^ u i Compatible with Craig-Bampton reduction method System description Uncoupled component models Component models  Domains ^ M ¤ bb L ( 1 ) b T M ( 1 ) b i ¢¢¢ L ( n ) b T M ( n ) b i M ( 1 ) i b L ( 1 ) b M ( 1 ) ii M ( n ) i b L ( n ) b 0 M ( n ) ii Ä u b Ä u ( 1 ) i... Ä u ( n ) i ^ K ¤ bb L ( 1 ) b T K ( 1 ) b i ¢¢¢ L ( n ) b T K ( n ) b i K ( 1 ) i b L ( 1 ) b K ( 1 ) ii K ( n ) i b L ( n ) b 0 K ( n ) ii u b u ( 1 ) i... u ( n ) i = f b f ( 1 ) i... f ( n ) i