Partitioned Modeling of Coupled Problems and Applications: New Interpretations of Lagrange Multipliers

Transcription

1 Partitioned Modeling of Coupled Problems and Applications: New Interpretations of Lagrange Multipliers Carlos A. Felippa and K. C. Park Center for Aerospace Structures University of Colorado at Boulder Boulder, CO WCCM-VII, Los Angeles, July 2006.

2 Partitioned modeling of coupled systems is to to transform a tightly coupled model into a loosely coupled system (pleasing to your eyes?):

3 Classification of Coupling in Multiphysics Tight Coupling: Through differing physical phenomena occupying the same space; Via hierarchical and/or multi-level variables; Loose Coupling: Through constitutive relations(?); Along spatial interfaces - Today s topic

4 Coupled physics Roger Ohayon (CNAM, Paris), Denis Caillerie (Grenoble), Michael Ross(CU) Contact-friction problems Gert Rebel (Goodyear Tire), Yasu Miyazaki (Nihon University, Japan) Jose Gonzales (Sevilla, Spain), Luis Solano (Sevilla) Reduced-order modeling and FEM-related work Damijan Markovic(ENS Cachan), Yonghwa Park (Samsung), Kendall Pierson (Sandia National Lab) Membranous space structures Hiraku Sakamoto(CU and MIT), Yasu Miyazaki (Nihon University) Sebastian Kreissl (Tech. Univ. Minich) Health monitoring of structural systems Ken Alvin (Sandia National Lab), Haru Namba (Shimizu Corp, Japan) Greg Reich (Wright Aeronautical Lab), Xue Yue (Mathlab) MEMS Yonghwa Park (Samsung), Gyeongho Kim (KAIST), Timothy Straube(NASA/Johnson) Structural optimization E.I. Jung (KAIST), Youn-sik Park (KAIST)

5 Outline of Presentation: A précis of the method of classical Lagrange multipliers(the CLM method) Can we localize Lagrange s multipliers? Properties of the method of localized Lagrange multipliers(llm) Interface treatment via the LLM frames Treatment of interface heterogeneities Applications: Inverse problems for damage identification Contact-impact problems Localized vibration control strategy Reduced-order model synthesis Optimization of vibration problems Multi-physics simulation BEM-FEM coupling

6 La méthode de quantités indéterminées and Lagrange s philosophy on mechanics: On ne trouvera point de Figures dans cet Ouvrage. Les methods que j y expose ne demandent ni constructions, ni raisonnemens géomtriques ou méchaniques, mis seulement des opérations algébriques, assujetties à une marche réguliere et uniforme... (One will find no pictures in this volume...) An old Asian saying: A picture is worth a thousand words ( ) may offer additional insight into the method of Lagrange multipliers...

7 Suppose that you are asked to model a Navajo dancing with the following night chant: The Earth touches me And I touch the Earth That is how Man can walk.

8 Formulation according to Lagrange s world of mechanics: Les methods que j y expose..., mis seulement des opérations algébriques... Done and what else is there to be discussed?

9 The method of classical Lagrange multipliers (CLM) yields a unique constraint condition when involving only two interfaces: (7)

10 However, the CLM method yields either non-unique or redundant constraint conditions when involving more than two interfaces: Example: the use of all three constraints leads to one redundant constraints, causing singularity. (8)

11 Classical Lagrange multipliers - cont d Example: the use of 2 of 3 constraints leads to non-singular constraint equations, but they are not unique; in fact, there are 3 possible non-singular constraint pairs from which one must choose one particular pairs. (9)

12 Classical λ-method (the CLM method) connects directly from one substructural interface node to that of another interface node: Note that one has to choose 3 rank-sufficient conditions from among 6 possible conditions.

13 Modeling of Multiply Connected Truss Elements for Inverse Problems Observe that node 27 consists of 8 elements. Model update for node 27 must consider all of them.

14 When model update or damage is detected in node 27, it could be all or some or just one of the eight elements.

15 If conventional partitioning that would befit the CLM method were employed,

16 What interface procedure does the finite element method utilize when assembling elements and/or partitioning the assembled structure? Answer: The FEM assembly and partitioning do not utilize the method of classical Lagrange multipliers! ============================================ ============================================ What does the FEM assembly and partitioning utilize then? Answer: It utilizes the method of localized Lagrange multipliers.

17 Outline of Presentation: A précis of the method of classical Lagrange multipliers(clm) Can we localize Lagrange s multipliers? Properties of the method of localized Lagrange multipliers(llm) Interface treatment via the LLM frames Treatment of interface heterogeneities Applications: Inverse problems for damage identification Contact-impact problems Localized vibration control strategy Reduced-order model synthesis Optimization of vibration problems Multi-physics simulation BEM-FEM coupling

18 Suppose the earth is humanized as Mother Earth, and express the level of touches figuratively by the size of a Greek letter: The Earth touches me, And I touch the Earth That s how man can walk.

19 Now you are asked to model a Navajo dancing with the following night chant: The Earth touches me And I touch the Earth That is how Man can walk. Which of the following two cartoons would lead to a good model? Global interface Localized interface

20 Classical Notion of Lagrange Multipliers f Partitioning node, f, is lost!

21 New Interpretation of Lagrange Multipliers f Partitioning node, f, is preserved!

22 Localization of Classical Lagrange Multipliers Assembled Classical Partitioning Split! Localized Partitioning Localization is achieved by introducing a frame node, f

23 Localization Process Split!

24 Localization Frame Features

25 Why increase the number of Lagrange multipliers? Excerpts from what William Hamilton read at Royal Society of London in 1834: ``While science is advancing in one direction by the improvement of physical laws, it may advance in another direction also by the invention of mathematical methods. ``This difficulty is therefore at least transferred from the integration of many equations of one class to the integration of two of another.

26 Why increase the number of Lagrange multipliers? -- cont d ``Even if it should be thought that no practical facility is gained, yet an intellectual pleasure may result from... Here, Hamilton refers to the canonical transformation of N equations of motion to 2N equations: f

27 Localized Lagrange Multipliers The localized modeling, as in Hamilton s canonical equations, also increases the interface unknowns from N to 2N multipliers for two interfaces. Now the question is: Is the localized modeling just an intellectual pleasure(exercise) or does it offer also a practical utility?

28 A visualization of FEM assembly and connectivity relation in every FEM code:

29 Partitioning the assembled 4 rod-element system involves four constraint conditions as shown below: Observe all four partitioned nodes refer to the one global node u 5. These four constraint equations are unique and rank-sufficient.

30 For two interfaces, the number of multipliers with the localized λ-method is twice that of the classical λ-method. Does that mean the computational cost of the localized λ-method would be double that of the classical λ-method? Answer: Essentially, the computational cost of the two methods are equivalent. Here s why:

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32 Outline of Presentation: A precis of the method of classical Lagrange multipliers(clm) Can we localize Lagrange s multipliers? Properties of the method of localized Lagrange multipliers(llm) Interface treatment via the LLM frames Treatment of interface heterogeneities Applications: Inverse problems for damage identification Contact-impact problems Localized vibration control strategy Reduced-order model synthesis Optimization of vibration problems Multi-physics simulation BEM-FEM coupling

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34 We will focus on two issues: Advantages of localized Lagrange multipliers? - continued (1) Non-Matching Interface Grids, and (2) Heterogeneous Interfaces. The continuum expressions of the two constraint functional indicate that all the interface variables have to be discretized, including the Lagrange multipliers, with the exception of matching node interfaces. Let us now examine the case of non-matching interfaces.

35 (A mixed interpolation!) Mixed formulations can lead to mixed results! Or live with the LBB hoops...

36 Localized λ-method, if co-located pairs of (u (j), λ (j) ) are used, transforms the mixed method into a displacement-like method, viz., into that of the interpolation of the frame displacement, u f.

37 Outline of Presentation: A precis of the method of classical Lagrange multipliers(clm) Can we localize Lagrange s multipliers? Properties of the method of localized Lagrange multipliers(llm) Interface treatment via the LLM frames Treatment of interface heterogeneities Applications: Inverse problems for damage identification Contact-impact problems Localized vibration control strategy Reduced-order model synthesis Optimization of vibration problems Multi-physics simulation BEM-FEM coupling

38 Modeling of Multiply Connected Truss Elements Observe that node 27 consists of 8 elements. Model update for node 27 must consider all of them.

39 When model update or damage is detected in node 27, it could be all or some or just one of the eight elements.

40 If conventional partitioning were employed,

41 If localized partitioning were used,

42 Outline of Presentation: A precis of the method of classical Lagrange multipliers(clm) Can we localize Lagrange s multipliers? Properties of the method of localized Lagrange multipliers(llm) Interface treatment via the LLM frames Treatment of interface heterogeneities Applications: Inverse problems for damage identification Contact-impact problems Localized vibration control strategy Reduced-order model synthesis Optimization of vibration problems Multi-physics simulation BEM-FEM coupling

43 Localization Frame Features

44 Essential Features of Present Interface Algorithm Determine the frame nodes via a localized discretization procedure, whereas in classical procedure one must interpolate the interface forces globally. Localized Global (classical)

45 Algorithm for Determination of Frame Nodes Step 1: Compute the interface loads that correspond to a constant stress state of the subdomains. Interface loads (not scaled) that correspond to a constant stress state

46 Algorithm for Determination of Frame Nodes -- cont'd Step 2: Map the interface forces (Lagrange multipliers) onto the frame. Mapping Interface loads onto the frame.

47 Algorithm for Determination of Frame Nodes -- cont'd Step 3: Compute the forces and moments along the frame at arbitrary frame locations. Computing Forces and Moments at a point on the frame.

48 Nonmatching-Node Interface with Linear and Quadratic Discretizations

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52 Outline of Presentation: A precis of the method of classical Lagrange multipliers(clm) Can we localize Lagrange s multipliers? Properties of the method of localized Lagrange multipliers(llm) Interface treatment via the LLM frames Treatment of interface heterogeneities Applications: Inverse problems for damage identification Contact-impact problems Localized vibration control strategy Reduced-order model synthesis Optimization of vibration problems Multi-physics simulation BEM-FEM coupling

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59 Outline of Presentation: A precis of the method of classical Lagrange multipliers(clm) Can we localize Lagrange s multipliers? Properties of the method of localized Lagrange multipliers(llm) Interface treatment via the LLM frames Treatment of interface heterogeneities Applications: Inverse problems for damage identification Contact-impact problems Localized vibration control strategy Reduced-order model synthesis Optimization of vibration problems Multi-physics simulation BEM-FEM coupling

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61 Localization Example for Engine Support Structure Problem

62 Damage indication based on global flexibility changes Damage indication based localized flexibility changes

63 Scale Model of Nuclear Containment Vessel

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66 Simplified Model of the Test Article

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68 Damage Indication based on global nodal stiffness changes Damage Indication based on localised flexibility changes

69 Outline of Presentation: A precis of the method of classical Lagrange multipliers(clm) Can we localize Lagrange s multipliers? Properties of the method of localized Lagrange multipliers(llm) Interface treatment via the LLM frames Treatment of interface heterogeneities Applications: Inverse problems for damage identification Contact-impact problems Localized vibration control strategy Reduced-order model synthesis Optimization of vibration problems Multi-physics simulation BEM-FEM coupling

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71 Three Dimensional Contact Patch Test Problem

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73 Outline of Presentation: A precis of the method of classical Lagrange multipliers(clm) Can we localize Lagrange s multipliers? Properties of the method of localized Lagrange multipliers(llm) Interface treatment via the LLM frames Treatment of interface heterogeneities Applications: Inverse problems for damage identification Contact-impact problems Localized vibration control strategy Reduced-order model synthesis Optimization of vibration problems Multi-physics simulation BEM-FEM coupling

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83 Outline of Presentation: A precis of the method of classical Lagrange multipliers(clm) Can we localize Lagrange s multipliers? Properties of the method of localized Lagrange multipliers(llm) Interface treatment via the LLM frames Treatment of interface heterogeneities Applications: Inverse problems for damage identification Contact-impact problems Localized vibration control strategy Reduced-order model synthesis Optimization of vibration problems Multi-physics simulation BEM-FEM coupling

84 Hybrid Localized/Global Active Vibration Control

85 Hybrid Localized/Global Active Vibration Control- cont d

86 Outline of Presentation: A precis of the method of classical Lagrange multipliers(clm) Can we localize Lagrange s multipliers? Properties of the method of localized Lagrange multipliers(llm) Interface treatment via the LLM frames Treatment of interface heterogeneities Applications: Inverse problems for damage identification Contact-impact problems Localized vibration control strategy Reduced-order model synthesis Optimization of vibration problems Multi-physics simulation BEM-FEM coupling

87 Design Optimization via Structural Modification

88 Outline of Presentation: A precis of the method of classical Lagrange multipliers(clm) Can we localize Lagrange s multipliers? Properties of the method of localized Lagrange multipliers(llm) Interface treatment via the LLM frames Treatment of interface heterogeneities Applications: Inverse problems for damage identification Contact-impact problems Localized vibration control strategy Reduced-order model synthesis Optimization of vibration problems Multi-physics simulation BEM-FEM coupling

89 Acoustic Fluid-Structure Interactions (PhD Thesis by Michael Ross) Model Problem

90 Gravity Dam FRF Output DOF Input DOF Output DOF Input DOF ormalized Vibration Magnitude Normalized Vibration Magnitude Alone Assembled Normalized Vibration Magnitude Compared with rmalized Vibration Magnitude

91 Outline of Presentation: A precis of the method of classical Lagrange multipliers(clm) Can we localize Lagrange s multipliers? Properties of the method of localized Lagrange multipliers(llm) Interface treatment via the LLM frames Treatment of interface heterogeneities Applications: Inverse problems for damage identification Contact-impact problems Localized vibration control strategy Reduced-order model synthesis Optimization of vibration problems Multi-physics simulation BEM-FEM coupling

92 FEM-Boundary Integral Coupling via Non-matching Interfaces

93 Basic Theory Felippa, C. A. and Park, K. C., Synthesis Tools for Structural Dynamics and Partitioned Analysis of Coupled Systems, Multi-physics and Multi-scale Computer Models in Non-linear Analysis and Optimal Design of Engineering Structures Under Extreme Conditions (NATO ARW PST.ARW980268), ed. A. Ibrahimbegovic and B. Brank, University of Ljubliana, 2004, K. C. Park, C. A. Felippa and G. Rebel, (2002), A Simple Algorithm for Localized Construction of Nonmatching Structural Interfaces, International Journal of Numerical Methods in Engineering, 2002; 53: Felippa, C. A. and Park, K. C., The construction of free-free flexibility matrices for multilevel structural analysis, Computer Methods in Applied Mechanics and Engineering, 191(19-20) (2002) K. C. Park, C. A. Felippa and G. Rebel, (2001), Interfacing Nonmatching FEM Meshes: The Zero Moment Rule, in: Trends in Computational Structural Mechanics, ed. by W. A. Wahl, K.-U. Bletzinger and K. Schweizerhof, CIMNE, Barcelona, Spain, 2001, p Park, K. C. and Felippa, C. A., A Variatioanl Principle for the Formulation of Partitioned Structural Systems, International Journal of Numerical Methods in Engineering, vol. 47, 2000, Felippa, C. A., Park, K. C. and Justino, M.R., The Construction of Free-Free Flexibility Matrices as Generalized Stiffness Inverses, Computers & Structures, vol.68 (1998), Park, K. C. and Felippa, C. A., A Variational Framework for Solution Method Developments in Structural Mechanics, Journal of Applied Mechanics, March 1998, Vol. 65/1, C. A. Felippa and K. C. Park, A direct flexibility method, Computer Methods in Applied Mechanics and Engineering, 149 (1997)

94 Parallel Computing Gumaste, Udayan and Park, K. C. (2000), Interfacing an explicit nonlinear finite element code with an implicit parallel solution algorithm, to be presented at the International Congress on Computational Engineering Sciences, August 5-8, 2000, Los Angeles, CA. Gumaste, Udayan, Park, K. C. and Alvin, K. F., A Family of Implicit Partitioned Time Integration Algorithms for Parallel Analysis of Heterogeneous Structural Systems, Computational Mechanics: an International Journal, 24 (2000) 6, Park, K. C., Gumaste, Udayan, and Felippa, C. A., A Localized Version of the Method of Lagrange Multipliers and its Applications, Computational Mechanics: an International Journal, 24 (2000) 6, Park, K. C., Justino, M. R, Jr. and Felippa, C. A., An Algebraically Partitioned FETI Method for Parallel Structural Analysis: Algorithm Description, International Journal of Numerical Methods in Engineering, 40, (1997). Justino, M. R, Jr., Park, K. C. and Felippa, C. A., An Algebraically Partitioned FETI Method for Parallel Structural Analysis:Implementation and Numerical Performance Evaluation, International Journal of Numerical Methods in Engineering, 40, (1997).

95 System Identification and Inverse Problems Xue Yue and K. C. Park (2002), Modeling of Joints and Interfaces, in : Modeling and Simulation-Based Life Cycle Engineering, K. Chong, S. Saigal, S. Thynell and H. Morgan (des.), Spon Press, London, pp Reich, G.W., Park, K. C. and Namba, H. (2001), Health Monitoring of a Reinforced Concrete ContainmentVessel by Localized Methods, Proc. of the Third International Workshop on Structural Health Monitoring, Technomic Publishing Company, Inc., 2001 Reich, G.W. and Park, K. C. (2001), A Theory for Strain-Based Structural System Identification, in: Journal of Applied Mechanics, 68(4), Reich, G.W. and Park, K. C., On the Use of Substructural Transmission Zeros for Structural Health Monitoring, AIAA Journal, Vol. 38, No. 6, 2000, Alvin, K. F. and Park, K. C., Extraction of Substructural Flexibilities from Global Frequencies and Mode Shapes, AIAA Journal, vol. 37, no.11, 1999, p Park, K. C., Reich, G. W. and Alvin, K. F. Structural Damage Detection Using Localized Flexibilities, Journal of Intelligent Material Systems and Structures, Vol. 9, No. 11, 1998, pp Park, K. C. and Felippa, C. A., A Flexibility-Based Inverse Algorithm for Identification of Structural Joint Properties, to appear in ASME Symposium on Computational Methods on Inverse Problems, November 1998, Anaheim, CA. Reich, G. W. and Park, K. C., Structural Health Monitoring via Structural Localization, Proc AIAA SDM Conference, Paper No. AIAA , April , Long Beach, CA. Park, K. C., Reich, G. W. and K. F. Alvin, Damage Detection Using Localized Flexibilities, in : Structural Health Monitoring, Current Status and Perspectives, ed. F-K Chang, Technomic Pub., 1997,

96 Coupled Problems Park, K. C., Felippa, C. A. and Ohayon, R., Reduced-Order Partitioned Modeling of Coupled Systems: Formulation and Computational Algorithms, Multi-physics and Multi-scale Computer Models in Non-linear Analysis and Optimal Design of Engineering Structures Under Extreme Conditions (NATO ARW PST.ARW980268), ed. A. Ibrahimbegovic and B. Brank, University of Ljubliana, 2004, Park, K. C., Felippa, C. A. and Ohayon, R., Partitioned Formulation of Internal Fluid-Structure Interaction Problems via Localized Lagrange Multipliers, Computer Methods in Applied Mechanics and Engineering, 190(24-25), 2001, Park, K.C., Felippa, C. A. and Ohayon, R. (2001), Localized Formulation of Multibody Systems, in: Computational Aspects of Nonlinear Systems with Large Rigid Body Motion (ed. J. Ambrosio and M. Kleiber), NATO Science Series, IOS Press, p Contact-Impact Problems Y. Miyazaki and K. C. Park, A formulation of conserving impact system based on localized Lagrange multipliers, to appear in International Journal of Numerical Methods in Engineering, G. Rebel, K. C. Park and C. A. Felippa (2002), A Contact Formulation Based on Localised Lagrange Multipliers: Formulation and Application to Two-dimensional Problems, International Journal of Numerical methods in Engineering, 2002; 54: G. Rebel and K. C. Park, Application of the Localised Lagrange Multiplier Method to a 3D Contact Patch Test Proc AIAA SDM Conference, Paper No. AIAA , April 2002, Denver, CO.

97 Vibration Control H. Sakamoto, K. C. Park, and Y. Miyazaki, Distributed and localized active vibration isolation in membrane structures, submitted to Journal of Spacecraft and Rockets, Hiraku Sakamoto, K.C. Park and Yasuyuki Miyazaki, Distributed Localized Vibration Control of Membrane StructuresUsingPiezoelectricActuators, PaperNo. AIAA , Proc. the 46th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference (SDM), April 2005, Austin, TX. Park, K. C., Kim, N. I., and Reich, G.W., A Theory of LocalizedVibration Control via PartitionedLQRSynthesis, Paper No , Proc Smart Structures and Materials Conference: Mathematics and Control in Smart Structures, Newport Beach, CA, March 6-9, Reduced-Order Modeling D. Markovic and K. C. Park, Reduction of substructural interface degrees of freedom in flexibility-based component mode synthesis, submitted to International Journal of Numerical Methods in Engineering, K. C. Park, Partitioned formulation with localized Lagrange multipliers and its applications, in: Structural Dynamics (Eurodyn 2005), Millpress, Roterdam, 2005, pp D. Markovic and K. C. Park, Reduction of Interface Degrees of Freedom in Flexibility-Based Component Mode Synthesis, Proc. 5th EUROMECH Nonlinear Dynamics Conference, Eindhoven, The Netherlands, August 7-12, 2005, pp Park K. C. and Park, Yong Hwa, Partitioned Component Mode Synthesis via A Flexibility Approach, AIAA Journal, 2004, vol.42, no.6,

98 BEM-BEM and BEM-FEM Modeling J. A. Gonz alez, K. C. Park and C. A. Felippa, FEM and BEM coupling in elastostatics using localized Lagrange multipliers, submitted to International Journal of Numerical Methods in Engineering, J. A. Gonz alez, K. C. Park and C. A. Felippa, Partitioned formulation of frictional contact problems, to appear in Comm. Num. Meth. Engr., MISC Topics Eui-Il Jung, Youn-Sik Park and K. C. Park, Structural Dynamics Modification via Reorientation of Modification Elements, Finite Element Analysis and Design, 42(1),2005, Park, Y.H and Park, K. C., Anchor Loss Evaluation of MEMS Resonators - I: Energy Loss Mechanism through Substrate Wave Propagation, Journal of Microelectromechanical Systems, Vol. 13, No. 2, 2004, Park, Y.H and Park, K. C., Anchor Loss Evaluation of MEMS Resonators - II: Coupled Substrate-REsonator Simulation and Validation, Journal of Microelectromechanical Systems, Vol. 13, No. 2, 2004, Park, K. C., Partitioned Solution of Reduced Integrated Finite Element Equations, Computers & Structures, 74 (2000)

99 Conclusions: The underlying concept of the method of localized Lagrange multipliers (the LLM method) and several applications are presented. Its construction is unique for multiply constrained cases; If the interface nodal variables and localized multipliers are colocated, the interface functional is transformed into a displacement functional, hence bypassing the mixed formulation challenge. It offers a natural framework for heterogeneity regularization.

100 Conclusions - continued: It should noted that the coupling via the LLM method has been applied so far to spatial interfaces, or loosely coupled systems. In order to apply the LLM methods to treat strongly coupled (e.g., field coupling, constitutive, multi-level, etc.) problems, the coupling phenomena have to be transformed into a loosely coupled problems. We are presently engaged to develop transformation procedures that can recast strong couplings into loose couplings. (Your participation is welcome!)

101 Fin!