Krylov Model Order Reduction of a Thermal Subsea Model
A subsea hydrocarbon production system can undergo planned and unplanned shutdowns during the life of the field. The thermal FEA is used to simulate the cool down to verify the insulation design of the subsea equipment, but it is also used to derive an acceptable insulation design for the cold spots. The driving factors of subsea analyses require fast responding and accurate models of the equipment cool down. This paper presents cool down analysis carried out by a Krylov subspace reduction method, and compares this approach to the commonly used FEA solvers. The model considered represents a typical component of a subsea production system, a closed valve on a dead leg. The results from the Krylov reduction method exhibits the least error and requires the shortest computational time to reach the solution. These findings make the Krylov model order reduction method very suitable for the above mentioned subsea applications.
[1] T. Bechtold, E. B. Rudnyi and J. G. Korvink, "Automatic Generation of
Compact Electro-Thermal Models for Semiconductor Devices," Ieice
Transactions on Electronics, vol. 86, pp. 459-465, 2003.
[2] J. Vesely and M. Sulitka, "Machine Tool Virtual Model," in Proceedings
of the International Congress MATAR 2008, 2008.
[3] P. Kolar, M. Sulitka and M. Janota, "Simulation of dynamic properties
of a spindle and tool system coupled with a machine tool frame," The
International Journal of Advanced Manufacturing Technology, vol. 54,
pp. 11-20, 2011.
[4] B. M. Irons, "Structural Eigenvalue Problems: Elimination of Unwanted
Variables," AIAA Journal, vol. 3, no. 5, pp. 961-962, 1965.
[5] R. J. Guyan, "Reduction of stiffness and mass matrices," AIAA Journal,
vol. 3, pp. 380-380, feb 1965.
[6] L. B. Bushard, "On the value of Guyan Reduction in dynamic thermal
problems," Computers & Structures, vol. 13, no. 4, pp. 525-531, 1981.
[7] M. Bampton and J. R. Craig, "Coupling of substructures for dynamic
analyses.," AIAA Journal, vol. 6, pp. 1313-1319, jul 1968.
[8] Y. Aoyama and G. Yagawa, "Component mode synthesis for large-scale
structural eigenanalysis," Computers & Structures, vol. 79, no. 6, pp.
605-615, 2001.
[9] D. Botto, S. Zucca, M. Gola, E. Troncarelli and G. Pasquero,
"Component modes synthesis applied to a thermal transient analysis of a
turbine disc," in Proceedings of the 3rd Worldwide Aerospace
Conference & Technology Showcase, Toulouse (FR), 2002.
[10] D. Botto, S. Zucca and M. M. Gola, "Reduced-Order Models for the
Calculation of Thermal Transients of Heat Conduction/Convection FE
Models," Journal of Thermal Stresses, vol. 30, no. 8, pp. 819-839, 2007.
[11] P. Nachtergaele, D. J. Rixen and A. M. Steenhoek, "Efficient weakly
coupled projection basis for the reduction of thermo-mechanical
models," J. Comput. Appl. Math., vol. 234, no. 7, pp. 2272-2278, aug
2010.
[12] Z. Bai and B.-S. Liao, "Towards an Optimal Substructuring Method for
Model Reduction," in Applied Parallel Computing. State of the Art in
Scientific Computing, vol. 3732, J. Dongarra, K. Madsen and J.
Waśniewski, Eds., Springer Berlin Heidelberg, 2006, pp. 276-285.
[13] J. O'Callahan, "A Procedure for an Improved Reduced System (IRS)
Model," in Proceedings of the 7th. IMAC, 1989.
[14] M. Friswell, S. Garvey and J. Penny, "Model reduction using dynamic
and iterated IRS techniques," Journal of Sound and Vibration, vol. 186,
no. 2, pp. 311-323, 1995.
[15] A. C. Antoulas, Approximation of Large-Scale Dynamical Systems
(Advances in Design and Control) (Advances in Design and Control),
Philadelphia, PA, USA: Society for Industrial and Applied Mathematics,
2005.
[16] A. C. Antoulas and D. C. Sorensen, "Approximation of large-scale
dynamical systems: An overview," Int. J. Appl. Math. Comput. Sci,
Vol.11, No.5, 1093-1121, 2001.
[17] T. Stykel, "Analysis and Numerical Solution of Generalized Lyapunov
Equations," Ph.D. thesis, Institut f├╝r Mathematik, Technische
Universität Berlin, 2002.
[18] E. Grimme. Krylov projection methods for model reduction. PhD thesis,
Univ. of Illinois at. Urbana-Champaign, USA, 1997.
[19] E. Grimme and K. Gallivan, Krylov Projection Methods for Rational
Interpolation, unpublished.
[20] Z. Bai, "Krylov subspace techniques for reduced-order modeling of
large-scale dynamical systems," Applied Numerical Mathematics, vol.
43, pp. 9-44, 2002.
[21] A. Odabasioglu, M. Celik and L. T. Pileggi, "PRIMA: passive reducedorder
interconnect macromodeling algorithm," in Proceedings of the
1997 IEEE/ACM international conference on Computer-aided design,
Washington, DC, USA, 1997.
[22] Z. Bai, K. Meerbergen and Y. Su, "Arnoldi Methods for Structure-
Preserving Dimension Reduction of Second-Order Dynamical Systems,"
in Dimension Reduction of Large-Scale Systems, vol. 45, P. Benner, D.
Sorensen and V. Mehrmann, Eds., Springer Berlin Heidelberg, 2005, pp.
173-189.
[23] T. Bechtold, E. B. Rudnyi and J. G. Korvink, Fast Simulation of Electro-
Thermal MEMS: Efficient Dynamic Compact Models, Springer, 2007.
[24] I. M. Elfadel and D. D. Ling, "A block rational Arnoldi algorithm for
multipoint passive model-order reduction of multiport RLC networks,"
in Proceedings of the 1997 IEEE/ACM international conference on
Computer-aided design, Washington, DC, USA, 1997.
[25] L. H. Feng, E. B. Rudnyi, J. G. Korvink, C. Bohm and T. Hauck,
"Compact Electro-thermal Model of Semiconductor Device with
Nonlinear Convection Coefficient," in In: Thermal, Mechanical and
Multi-Physics Simulation and Experiments in Micro-Electronics and
Micro-Systems. Proceedings of EuroSimE 2005.
[26] E. Zukowski, J. Wilde, E.-B. Rudnyi and J.-G. Korvink, "Model
reduction for thermo-mechanical simulation of packages," THERMINIC
2005, 11th International Workshop on Thermal Investigations of ICs and
Systems, 27 - 30 September 2005, Belgirate, Lake Maggiore, Italy, p.
134 - 138.
[27] R. S. Puri, D. Morrey, A. J. Bell, J. F. Durodola, E. B. Rudnyi and J. G.
Korvink, "Reduced order fully coupled structural-acoustic analysis via
implicit moment matching," Applied Mathematical Modelling, vol. 33,
no. 11, pp. 4097-4119, 2009.
[28] J. S. Han, E. B. Rudnyi and J. G. Korvink, "Efficient optimization of
transient dynamic problems in MEMS devices using model order
reduction," Journal of Micromechanics and Microengineering, vol. 15,
no. 4, p. 822, 2005.
[29] J. Han, "Efficient frequency response and its direct sensitivity analyses
for large-size finite element models using Krylov subspace-based model
order reduction," Journal of Mechanical Science and Technology, vol.
26, pp. 1115-1126, 2012.
[30] U. Baur, C. Beattie, P. Benner and S. Gugercin, "Interpolatory
Projection Methods for Parameterized Model Reduction," SIAM J. Sci.
Comput., vol. 33, no. 5, pp. 2489-2518, oct 2011.
[31] P. Koutsovasilis and M. Beitelschmidt, "Comparison of model reduction
techniques for large mechanical systems," Multibody System Dynamics,
vol. 20, pp. 111-128, 2008.
[32] W. Witteveen, "Comparison of CMS, Krylov and Balanced Truncation
Based Model Reduction from a Mechanical Application Engineer-s
Perspective," in Topics in Experimental Dynamics Substructuring and
Wind Turbine Dynamics, Volume 2, vol. 27, R. Mayes, D. Rixen, D.
Griffith, D. De Klerk, S. Chauhan, S. Voormeeren and M. Allen, Eds.,
Springer New York, 2012, pp. 319-331.
[33] F. Incropera, D. DeWitt, T. Bergman and A. Lavine, Fundamentals of
Heat and Mass Transfer 5th Edition, Wiley, 2001.
[34] R. Lewis, P. Nithiarasu and K. Seetharamu, Fundamentals of the Finite
Element Method for Heat and Fluid Flow, John Wiley & Sons, 2004.
[35] T. Hughes, The Finite Element Method: Linear Static and Dynamic
Finite Element Analysis, Prentice-Hall, 1987.
[36] H. Langtangen, Computational Partial Differential Equations: Numerical
Methods and Diffapck, Springer-Verlag, 2003.
[37] "ANSYS release 14, Help system, Theory reference".
[38] T. Davis, Direct Methods for Sparse Linear Systems, Society for
Industrial and Applied Mathematics, 2006.
[39] A. George and W. H. Liu, "The evolution of the minimum degree
ordering algorithm," SIAM Rev., vol. 31, no. 1, pp. 1-19, mar 1989.
[40] G. Karypis and V. Kumar, "A Fast and High Quality Multilevel Scheme
for Partitioning Irregular Graphs," SIAM J. Sci. Comput., vol. 20, no. 1,
pp. 359-392, dec 1998.
[41] Y. Saad, Iterative Methods for Sparse Linear Systems, 2nd ed.,
Philadelphia, PA, USA: Society for Industrial and Applied Mathematics,
2003.
[42] S. Gugercin, A. Antoulas and C. Beattie, "H2 Model Reduction for
Large-Scale Linear Dynamical Systems," SIAM Journal on Matrix
Analysis and Applications, vol. 30, no. 2, pp. 609-638, 2008.
[43] L. F. L. Feng, D. Koziol, E. Rudnyi and J. Korvink, "Model order
reduction for scanning electrochemical microscope: the treatment of
nonzero initial condition," CORD Conference Proceedings, pp. 1236-
1233, oct 2004.
[44] G. Flagg, C. Beattie and S. Gugercin, "Convergence of the Iterative
Rational Krylov Algorithm," Systems & Control Letters, Volume 61,
Issue 6, June 2012, Pages 688-691.
[45] E. F. Yetkin and D. H., "Parallel implementation of iterative rational
Krylov methods for model order reduction," in Proceeding of the
ICSCCW 2009.
[46] C. Beattie and S. Gugercin, "Inexact Solves in Krylov-based Model
Reduction," in 45th IEEE Conference on Decision and Control, 2006.
[47] P. Benner, L. Feng and E. B. Rudnyi, "Using the Superposition Property
for Model Reduction of Linear Systems with a Large Number of Inputs,"
in Proceedings of the 18th International Symposium on Mathematical
Theory of Networks and Systems (MTNS2008), 2008.
[1] T. Bechtold, E. B. Rudnyi and J. G. Korvink, "Automatic Generation of
Compact Electro-Thermal Models for Semiconductor Devices," Ieice
Transactions on Electronics, vol. 86, pp. 459-465, 2003.
[2] J. Vesely and M. Sulitka, "Machine Tool Virtual Model," in Proceedings
of the International Congress MATAR 2008, 2008.
[3] P. Kolar, M. Sulitka and M. Janota, "Simulation of dynamic properties
of a spindle and tool system coupled with a machine tool frame," The
International Journal of Advanced Manufacturing Technology, vol. 54,
pp. 11-20, 2011.
[4] B. M. Irons, "Structural Eigenvalue Problems: Elimination of Unwanted
Variables," AIAA Journal, vol. 3, no. 5, pp. 961-962, 1965.
[5] R. J. Guyan, "Reduction of stiffness and mass matrices," AIAA Journal,
vol. 3, pp. 380-380, feb 1965.
[6] L. B. Bushard, "On the value of Guyan Reduction in dynamic thermal
problems," Computers & Structures, vol. 13, no. 4, pp. 525-531, 1981.
[7] M. Bampton and J. R. Craig, "Coupling of substructures for dynamic
analyses.," AIAA Journal, vol. 6, pp. 1313-1319, jul 1968.
[8] Y. Aoyama and G. Yagawa, "Component mode synthesis for large-scale
structural eigenanalysis," Computers & Structures, vol. 79, no. 6, pp.
605-615, 2001.
[9] D. Botto, S. Zucca, M. Gola, E. Troncarelli and G. Pasquero,
"Component modes synthesis applied to a thermal transient analysis of a
turbine disc," in Proceedings of the 3rd Worldwide Aerospace
Conference & Technology Showcase, Toulouse (FR), 2002.
[10] D. Botto, S. Zucca and M. M. Gola, "Reduced-Order Models for the
Calculation of Thermal Transients of Heat Conduction/Convection FE
Models," Journal of Thermal Stresses, vol. 30, no. 8, pp. 819-839, 2007.
[11] P. Nachtergaele, D. J. Rixen and A. M. Steenhoek, "Efficient weakly
coupled projection basis for the reduction of thermo-mechanical
models," J. Comput. Appl. Math., vol. 234, no. 7, pp. 2272-2278, aug
2010.
[12] Z. Bai and B.-S. Liao, "Towards an Optimal Substructuring Method for
Model Reduction," in Applied Parallel Computing. State of the Art in
Scientific Computing, vol. 3732, J. Dongarra, K. Madsen and J.
Waśniewski, Eds., Springer Berlin Heidelberg, 2006, pp. 276-285.
[13] J. O'Callahan, "A Procedure for an Improved Reduced System (IRS)
Model," in Proceedings of the 7th. IMAC, 1989.
[14] M. Friswell, S. Garvey and J. Penny, "Model reduction using dynamic
and iterated IRS techniques," Journal of Sound and Vibration, vol. 186,
no. 2, pp. 311-323, 1995.
[15] A. C. Antoulas, Approximation of Large-Scale Dynamical Systems
(Advances in Design and Control) (Advances in Design and Control),
Philadelphia, PA, USA: Society for Industrial and Applied Mathematics,
2005.
[16] A. C. Antoulas and D. C. Sorensen, "Approximation of large-scale
dynamical systems: An overview," Int. J. Appl. Math. Comput. Sci,
Vol.11, No.5, 1093-1121, 2001.
[17] T. Stykel, "Analysis and Numerical Solution of Generalized Lyapunov
Equations," Ph.D. thesis, Institut f├╝r Mathematik, Technische
Universität Berlin, 2002.
[18] E. Grimme. Krylov projection methods for model reduction. PhD thesis,
Univ. of Illinois at. Urbana-Champaign, USA, 1997.
[19] E. Grimme and K. Gallivan, Krylov Projection Methods for Rational
Interpolation, unpublished.
[20] Z. Bai, "Krylov subspace techniques for reduced-order modeling of
large-scale dynamical systems," Applied Numerical Mathematics, vol.
43, pp. 9-44, 2002.
[21] A. Odabasioglu, M. Celik and L. T. Pileggi, "PRIMA: passive reducedorder
interconnect macromodeling algorithm," in Proceedings of the
1997 IEEE/ACM international conference on Computer-aided design,
Washington, DC, USA, 1997.
[22] Z. Bai, K. Meerbergen and Y. Su, "Arnoldi Methods for Structure-
Preserving Dimension Reduction of Second-Order Dynamical Systems,"
in Dimension Reduction of Large-Scale Systems, vol. 45, P. Benner, D.
Sorensen and V. Mehrmann, Eds., Springer Berlin Heidelberg, 2005, pp.
173-189.
[23] T. Bechtold, E. B. Rudnyi and J. G. Korvink, Fast Simulation of Electro-
Thermal MEMS: Efficient Dynamic Compact Models, Springer, 2007.
[24] I. M. Elfadel and D. D. Ling, "A block rational Arnoldi algorithm for
multipoint passive model-order reduction of multiport RLC networks,"
in Proceedings of the 1997 IEEE/ACM international conference on
Computer-aided design, Washington, DC, USA, 1997.
[25] L. H. Feng, E. B. Rudnyi, J. G. Korvink, C. Bohm and T. Hauck,
"Compact Electro-thermal Model of Semiconductor Device with
Nonlinear Convection Coefficient," in In: Thermal, Mechanical and
Multi-Physics Simulation and Experiments in Micro-Electronics and
Micro-Systems. Proceedings of EuroSimE 2005.
[26] E. Zukowski, J. Wilde, E.-B. Rudnyi and J.-G. Korvink, "Model
reduction for thermo-mechanical simulation of packages," THERMINIC
2005, 11th International Workshop on Thermal Investigations of ICs and
Systems, 27 - 30 September 2005, Belgirate, Lake Maggiore, Italy, p.
134 - 138.
[27] R. S. Puri, D. Morrey, A. J. Bell, J. F. Durodola, E. B. Rudnyi and J. G.
Korvink, "Reduced order fully coupled structural-acoustic analysis via
implicit moment matching," Applied Mathematical Modelling, vol. 33,
no. 11, pp. 4097-4119, 2009.
[28] J. S. Han, E. B. Rudnyi and J. G. Korvink, "Efficient optimization of
transient dynamic problems in MEMS devices using model order
reduction," Journal of Micromechanics and Microengineering, vol. 15,
no. 4, p. 822, 2005.
[29] J. Han, "Efficient frequency response and its direct sensitivity analyses
for large-size finite element models using Krylov subspace-based model
order reduction," Journal of Mechanical Science and Technology, vol.
26, pp. 1115-1126, 2012.
[30] U. Baur, C. Beattie, P. Benner and S. Gugercin, "Interpolatory
Projection Methods for Parameterized Model Reduction," SIAM J. Sci.
Comput., vol. 33, no. 5, pp. 2489-2518, oct 2011.
[31] P. Koutsovasilis and M. Beitelschmidt, "Comparison of model reduction
techniques for large mechanical systems," Multibody System Dynamics,
vol. 20, pp. 111-128, 2008.
[32] W. Witteveen, "Comparison of CMS, Krylov and Balanced Truncation
Based Model Reduction from a Mechanical Application Engineer-s
Perspective," in Topics in Experimental Dynamics Substructuring and
Wind Turbine Dynamics, Volume 2, vol. 27, R. Mayes, D. Rixen, D.
Griffith, D. De Klerk, S. Chauhan, S. Voormeeren and M. Allen, Eds.,
Springer New York, 2012, pp. 319-331.
[33] F. Incropera, D. DeWitt, T. Bergman and A. Lavine, Fundamentals of
Heat and Mass Transfer 5th Edition, Wiley, 2001.
[34] R. Lewis, P. Nithiarasu and K. Seetharamu, Fundamentals of the Finite
Element Method for Heat and Fluid Flow, John Wiley & Sons, 2004.
[35] T. Hughes, The Finite Element Method: Linear Static and Dynamic
Finite Element Analysis, Prentice-Hall, 1987.
[36] H. Langtangen, Computational Partial Differential Equations: Numerical
Methods and Diffapck, Springer-Verlag, 2003.
[37] "ANSYS release 14, Help system, Theory reference".
[38] T. Davis, Direct Methods for Sparse Linear Systems, Society for
Industrial and Applied Mathematics, 2006.
[39] A. George and W. H. Liu, "The evolution of the minimum degree
ordering algorithm," SIAM Rev., vol. 31, no. 1, pp. 1-19, mar 1989.
[40] G. Karypis and V. Kumar, "A Fast and High Quality Multilevel Scheme
for Partitioning Irregular Graphs," SIAM J. Sci. Comput., vol. 20, no. 1,
pp. 359-392, dec 1998.
[41] Y. Saad, Iterative Methods for Sparse Linear Systems, 2nd ed.,
Philadelphia, PA, USA: Society for Industrial and Applied Mathematics,
2003.
[42] S. Gugercin, A. Antoulas and C. Beattie, "H2 Model Reduction for
Large-Scale Linear Dynamical Systems," SIAM Journal on Matrix
Analysis and Applications, vol. 30, no. 2, pp. 609-638, 2008.
[43] L. F. L. Feng, D. Koziol, E. Rudnyi and J. Korvink, "Model order
reduction for scanning electrochemical microscope: the treatment of
nonzero initial condition," CORD Conference Proceedings, pp. 1236-
1233, oct 2004.
[44] G. Flagg, C. Beattie and S. Gugercin, "Convergence of the Iterative
Rational Krylov Algorithm," Systems & Control Letters, Volume 61,
Issue 6, June 2012, Pages 688-691.
[45] E. F. Yetkin and D. H., "Parallel implementation of iterative rational
Krylov methods for model order reduction," in Proceeding of the
ICSCCW 2009.
[46] C. Beattie and S. Gugercin, "Inexact Solves in Krylov-based Model
Reduction," in 45th IEEE Conference on Decision and Control, 2006.
[47] P. Benner, L. Feng and E. B. Rudnyi, "Using the Superposition Property
for Model Reduction of Linear Systems with a Large Number of Inputs,"
in Proceedings of the 18th International Symposium on Mathematical
Theory of Networks and Systems (MTNS2008), 2008.
@article{"International Journal of Mechanical, Industrial and Aerospace Sciences:51289", author = "J. Šindler and A. Suleng and T. Jelstad Olsen and P. Bárta", title = "Krylov Model Order Reduction of a Thermal Subsea Model", abstract = "A subsea hydrocarbon production system can undergo planned and unplanned shutdowns during the life of the field. The thermal FEA is used to simulate the cool down to verify the insulation design of the subsea equipment, but it is also used to derive an acceptable insulation design for the cold spots. The driving factors of subsea analyses require fast responding and accurate models of the equipment cool down. This paper presents cool down analysis carried out by a Krylov subspace reduction method, and compares this approach to the commonly used FEA solvers. The model considered represents a typical component of a subsea production system, a closed valve on a dead leg. The results from the Krylov reduction method exhibits the least error and requires the shortest computational time to reach the solution. These findings make the Krylov model order reduction method very suitable for the above mentioned subsea applications.
", keywords = "Model order reduction, Krylov subspace, subsea production system, finite element.", volume = "7", number = "5", pages = "790-8", }
