Partitioned Modeling of Coupled Problems and Applications

Lecture 1: Formulation and Solution Algorithms

K.C. ParkCenter for Aerospace StructuresUniversity of Colorado at BoulderBoulder, CO 80309

Research Collaborations with: Carlos A. Felippa, Roger Ohayon, Udayan Gumaste, Yongju Lee, Rert Rebel, Xue Yue, K.F. Alvin, Greg Reich, Bertrand Herry, Yonghwa Park, Hiraku Sakamoto, Yasuhiro Morita,Yasuyki Miyazawa, Jose Gonzales, Damijan Markovic,Luis Solano

Lecture given at University of Sevilla (Wednesday, 31 May 2006)

• External acoustic-structure interactions• Membrane Space Structures• Partitioned Analysis (Today’s Lecture Topic)• Contact-friction problems• Coupled physics problem• Health monitoring of structural systems• MEMS (Micro-Electro-Mechanical Systems)

Current Research Activities (K. C. Park)

Acoustic-structure interactions Moonseok Park(KAIST) Youn-sik Park (KAIST) Roger Ohayon(CNAM) Carlos Felippa(CU-Boulder)

Membrane Space Structures Hiraku Sakamoto(CU and MIT) Yasu Miyazaki (Nihon University) Sebastian Kreissl (Tech. Univ. Minich)

Contact-friction problems Gert Rebel (Goodyear Tire) Yasu Miyazaki (Nihon University, Japan) Jose Gonzales (Sevilla, Spain) Luis Solano (Sevilla, Spain) Carlos Felippa (CU-Boulder)

Coupled Physics Carlos Felippa(CU-Boulder) Roger Ohayon (CNAM, Paris) Denis Caillerie (Domaine Universitaire, Grenoble)

Health monitoring of structural systems Ken Alvin (Sandia National Laboratories) Haru Namba (Shimizu Corp, Japan) Greg Reich (Wright Aeronautical Lab, US Air Force) Xue Yue (Mathlab, Boston)

MEMS Yonghwa Park(Samsung) Gyeongho Kim (KAIST) Timothy Straube(NASA/Johnson Center)

FEM and Related Kendall Pierson (Sandia National Laboratories) Damijan Markovic(ENS Cachan, France)

Contributors:Carlos Felippa, Roger Ohayon: Variational Principles

Yonghwa Park,Damijian Markovic: Flexibility-Based CMS Techniques

Euill Jung, Younsik Park: Optimization

Jose Gonzales, Yasu Miyazaki, Luis Solano,Gert Rebel, Xue Yue: Interface (contact) Problems

Hiraku Sakamoto, Yosu Morita: Vibrational Control

YongHwa Park, Michael Ross, Tim Straube,GyeongHo Kim: Multiphysics, MEMS, Soil-Structure-Dam

Origin of the Method of Classical Lagrange Multipliers

Joseph Lagrange developed his method for deriving the equilibrium equation for a body or a system using

(1)

Lagrange’s next task:

For a system involving two or more bodies, he wanted toemploy the procedure:

δL = ∑ δLj = ∑ (δTj - δUj ), j=1,2, …, nbodies.(2)

where Tj and Uj are the kinetic and potential energy of the j-thbody in completely free states.

Then he realized that he needed to prescribe the conditions ofconstraints on how the bodies are connected among themselves.

Thus, if one expresses such conditions of constraints as

δck (q1, q2, …, qn) = 0, k=1,2, …, mcconditions , (3)

Lagrange’s introduction of the undetermined coefficients:

The virtual work due to the conditions of constraints are givenas

δπ = ∑λk δck (4)

where λk is called the undetermined coefficients.

Remark:Since {δck=0, k =1,2, …, nconstraints}, the virtual work ofconstraints, δπ, contributes nothing to the assembled (or total)system, that is,

δπ = 0 (5)

Formulation of the total system is obtained by the stationary value of the modified Lagrangian:

δL = ∑δLj + δπ = ∑ (δTj - δUj ) + ∑λk δck = 0 (6)

In the present talk, Lagrange’s undetermined coefficients associated with the virtual work pertaining to the conditionsof constraints as formulated in (5) is designated as classical Lagrange’s multipliers.

Let us now examine the properties of classical Lagrange’s multipliers as part of the motivations of the present talk.

The method of classical Lagrange multipliers yields a unique constraint condition when involving only two interfaces:

(7)

Example: the use of all three constraints leads to one redundant constraints, causing singularity.

However, the method of classical Lagrange multipliers yields either non-unique or redundant constraint conditions when involving more than two interfaces:

(8)

Example: the use of two of the three constraints leads to non-singular constraint equations, but they are not unique; in fact, there are three possible non-singular constraint pairs.

Classical Lagrange multipliers - cont’d

(9)

What method does the finite element method utilizewhen assembling elements and/or partitioning the assembled structure ?

Answer: The FEM assembly and partitioning do not utilizethe method of classical Lagrange multipliers!

What does the FEM assembly and partitioning utilize then?

Answer: It utilizes the method of localized Lagrange multipliers.

Are you interested in listening to the rest of the talk?

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FEM assembly connectivity relation in every FEM code:

Partitioning the assembled 4 rod-element system involvesfour constraint conditions as shown below:

Observe all four partitioned nodes refer to the one global node u5.These four constraint equations are unique and rank-sufficient.

Classical λ-Method connects directly from one substructural interface node to that of another interface node:

Note that one has to choose three rank-sufficient conditions fromamong six possible conditions.

Comparison of Localized vs. Classical λ-Methods

For two interfaces, the number of multipliers with the localizedλ-method is twice that of the classical λ-method. Does that meanthe computational cost of the localized λ-method would be doublethat of the classical λ-method ?

Answer: Essentially, the computational cost of the two methodsare equivalent. Here’s why:

Partitioning of two coupled domains

We will focus on two issues: Non-Matching Interface Grids, and Heterogeneous Interfaces.

Advantages of localized Lagrange multipliers? - continued

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.

(A mixed interpolation!)

Mixed formulations can lead to mixed results!

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

Recap: What is a localized Lagrange multiplier?

Modeling of Multiply Connected Truss Elements

Observe that node 27 consists of 8 elements. Model update for node 27 must consider all of them.

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

If conventional partitioning were employed,

If localized partitioning were used,

Assembled

Classical Partitioning

Localized Partitioning

Classification of Partitioning

Split!

Split!

Loca

lizat

ion

Fram

e Fe

atur

es

Why increase the number of Lagrange multipliers?

Excerpts from what William Hamilton read at RoyalSociety of London in 1834:

``While science is advancing in one direction by theimprovement of physical laws, it may advance inanother direction also by the invention ofmathematical methods.’’ …

``This difficulty is therefore at least transferred fromthe integration of many equations of one class to theintegration of two of another.

f

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:

The localized modeling, as in Hamilton’s canonical equations, also increases the interface unknowns from N to 2N multipliers. Now the question is:

Is the theory of localized modeling just an intellectual pleasure or does it offer also a practical facility?

Localized Lagrange Multipliers

Localized Lagrange Multipliers on: How it is felt!

The Earth touches meAnd I touch the EarthThat is how Man can walk -- Navajo Night Chant

(Conceptual Reasoning for Heterogeneity Treatment)

Outline of Presentation:Why Do We Need Partitioning?

What is the Method of Localized Lagrange Multipliers?

Interface Treatment via Frames

Treatment of Localized vs. Global Variables

Applications (to be given 01 June 2006):Flexibility-Based Component Mode SynthesisOptimization of Vibration Problems

. . .Contact-Impact ProblemsLocalized Vibrational Control

. . .

What does localized partitioning offer?

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)

Algorithm for Determination of Frame Nodes

Step 1: Compute the interface loads that correspond to a constant stress stateof the subdomains.

Interface loads (not scaled) that correspond to a constant stress state

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.

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.

Linear and Quadratic Matched Interface

Nonmatching-Node Interface with Linear and QuadraticDiscretizations

Outline of Presentation:Why Do We Need Partitioning?

What is the Method of Localized Lagrange Multipliers?

Interface Treatment via Frames

Treatment of Localized vs. Global Variables

Applications (To be given 01 July 2006):Flexibility-Based Component Mode SynthesisOptimization of Vibration Problems

. . .Contact-Impact ProblemsLocalized Vibrational Control

. . .

Localized Lagrange multipliers (LLM) - formulation

Euler-Lagrange equation•Local dynamics

•Global-local compatibility

•Global equilibrium

Partitioned Four-Variable Equations of Motion

Step 1: Partition the assembled global system:

,

Step 2: Form Partitioned Functional:

Step 3: Decompose u(j) into deformation and rigid-body modes:

1st & 2nd equations: substructural equilibrium equations;

3rd and 4-th equations: localized interface constraints;

Bottom equation: Interface Newton’s 3rd law

Localized Lagrange multipliers (LLM) - displacement decompositions

zero energy modes (RBM)

deformation modes {,

Reason : substructures being floating objects

K singular, K-1 non-existent

u = K-1 f is replaced by d = K+ f

generalized inverse

And, α is computed via self-equilibrium considerations.

Partitioned system to solve

global

local

Summary of today’s lecture:

• The origin of localized Lagrange’s multipliers;

• Review of classical Lagrange’s multipliers;

• Formalism of the method of localized Lagrange’s multipliers;

• Derivation of the partitioned formulation via a mixed variational principle, and a proof that the present interface treatment leads to displacement-like method;

• A proof that the present interface formulation is unique and variationally complete;

• Application to non-matching interfaces;

• Procedure for the discretization of the interface frames.