Nastran 2012 Superelements Ug

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MSC Nastran 2012

Superelements Users Guide


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Disclaimer

MSC.Software Corporation reserves the right to make changes in specifications and other information containedin this document without prior notice.

The concepts, methods, and examples presented in this text are for illustrative and educational purposes only,

and are not intended to be exhaustive or to apply to any particular engineering problem or design. MSC.SoftwareCorporation assumes no liability or responsibility to any person or company for direct or indirect damages resultingfrom the use of any information contained herein.

User Documentation: Copyright 2011 MSC.Software Corporation. Printed in U.S.A. All Rights Reserved.This notice shall be marked on any reproduction of this documentation, in whole or in part. Any reproduction ordistribution of this document, in whole or in part, without the prior written consent of MSC.Software Corporation is

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copy of the METIS product documentation is included with this installation. Please see "A Fast and High QualityMultilevel Scheme for Partitioning Irregular Graphs". George Karypis and Vipin Kumar. SIAM J ournal on Scientific

Computing, Vol. 20, No. 1, pp. 359-392, 1999.

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Contents


1 Introduction and Fundamentals

Introduction to this Guide 2

Why Use Superelements? 4Reduced Cost 4Quicker Turnaround 4Reduced Risk 4Large Problem Capabilities 4Partitioned Input and Output 5Security 5Automation with High Performance Computing 5

Fundamentals of Superelement Analysis 7

Partitioned Solutions 10

Key Concepts in Superelement Partitions 10Static Condensation Process 11

Manual Solut ion of a Small Superelement Example 16

Conventional Analysis 17

Superelement Analysis 19

List of Superelement Enhancements Released Since Version 69 34

2 How to Define a Superelement

Introduction 40

Superelement vs. Residual 41

Three Types of Superelements 43

Defining List Superelements 44

List Superelement Definition with the SESET Entry 44List Superelement Definition with the GRID Entry 45List Superelement Definition on Element ID the SEELT entry 46

Interior versus Exterior Points 47Superelement Partitioning 54


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Defining PART Superelements 71

Defining Parts 71

The Bulk Data Section Using PARTs 71Format of the Input File When PARTs are Used 72Connecting PARTs to Other PARTs 78

Defining and Attaching External Superelements 94

Discussion of 2-step vs. 3-step methods 94Definition of External Points 102Creating External Superelements with EXTSEOUT 105Using EXTSEOUT (2-Step) External Superelements 114Using PARAM,EXTOUT (3-Step) External Superelements 126

The Superelement Map SEMAP 138

Contents of the Superelement MAP List Superelement 138Contents of the Superelement MAP PART Superelement 141

3 Single Level Superelement Analysis

Introduction 150

Baseline Static Example using Patran 153

Single-Level Analysis Using List Superelements 166

Static Example using Patran. (flyswatter) 166

Single-Level Analysis Using PART Superelements 196

Static Example using Patran (flyswatter) 196

Single-Level Analysis Using External Superelements 204

Static Example Using Patran (flyswatter) 204

Comparison of Methods 223

4 Loads, Constraints, Case Control , and Parameters in Static Analysis

Introduction 228

Mechanical Loads in Static Analysis 229

List Superelements 229PART Superelements 234External Superelements 234

Thermal Loads in Static Analysis 235List Superelements 235


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ii iCONTENTS

PART Superelements 241External Superelements 241

Boundary Conditions in Static Analysis 242Single Point Constraints (SPCs) 242Grid Point Singularity Processing 243Multipoint Constraints (MPCs) and Rigid Elements 244

Case Control 260

The SUPER Command 261Condensed Case Control 263

Expanded Case Control 265Output Control 269

Parameter Controls 274


Special Considerations 276SNORM for PART or External Superelements 276

5 Inertia Relief Analysis Using Superelements

Introduction 282

The Concept of Inertia Relief 283

Interface for Inertia Relief Using Superelements 284

Manual Definition of Reference Points 284Automatic Definition of Reference Points 286

Inertia Relief Examples (Freedom) 287

Baseline Residual Example 288List Superelement Example 292

PART Superelement Example 295External Superelement Example 297Comparison of results 297

6 Multiple Loading in Static Analysis

Introduction 300

Internal Case Control Partitioning 301Use of Load Sequences 301


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Examples for Multiple Loading Conditions (cantilever plate) 303

Baseline Residual Example 306

List Superelements 307PART Superelements 308External Superelements 310Comparison of Results 313

7 Multi-Level Superelement Analysis

Introduction 316

Comparison of Single- and Multi-Level Superelements 317

User Interface 327


Example Multi-Level Superelement Solved Manually 331

Example Multi -Level Superelement (Freedom) 339

Baseline Residual Solution 340List Superelement Solution 341PART Superelement Solution 345External Superelement Solution 349Comparison of Results 369

8 Output Description and Control in Static Superelement Analysis

Introduction 372

Diagnostic/Connection Output 373

Superelement Definition Tables 373Controlling Diagnostic Output with PARAMs SEMAP, SEMAPOPT, SEMAPPRT

375Part Superelement Diagnostic Output 387

Sorted Bulk Data 387Boundary Grid Search Output 388

List (SESET) Superelement Diagnostic Output 391

Sorted Bulk Data 391Boundary Grid Search Output 392

Visualizing Model with OUTPUT(PLOT) 394


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vCONTENTS

Result Output 395

List Superelements 396

PART Superelements 397External Superelements 399Controlling Results Output with SEDR 405

9 Introduction to Dynamic Analysis Using Superelements

Introduction 408

Description of Dynamic Reduction Process 410Static Condensation 410Dynamic Reduction Component Modes Synthesis 413

Illustrative Example 450

Accuracy Improvements with CMS 450

Nastran Set Definitions The USET Table 458

10 Input and Output for Dynamic Reduction

Introduction 464

Case Control for Dynamic Reduction 465

Single Level Dynamic Reduction 468


Multi-Level Dynamic Reduction 514

Defining the Superelement Tree 514Multi-Level Superelement Component Modes Synthesis Connection 517Multi-Level Modal Reduction Example (fly swatter) 518

11 Dynamic Loading on Superelements

Introduction 542

Direct Reference to EXCITEID 550

Indi rect Reference to EXCITEDID: LOADSET / LSEQ Method 554

Superelement Damping 559


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Modal Transient Illus trative Example 560

Non-Superelement Solution 560

List Superelement LOADSET / LSEQ Reference to EXCITEDID 574List Superelement Direct Reference to EXCITEDID 578PART Superelement LOADSET / LSEQ Reference to EXCITEDID 581PART Superelement Direct Reference to EXCITEDID 587

Frequency Response Illustrative Example 593

Non-Superelement Solution 593List Superelement LOADSET / LSEQ Reference to EXCITEDID 606List Superelement Direct Reference to EXCITEDID 609PART Superelement LOADSET / LSEQ Reference to EXCITEDID 612PART Superelement Direct Reference to EXCITEDID 618

External Superelement Dynamic Loading 624

Residual Vectors 624Applying a Dynamic Load an External Superelement 624Applying the Time History to the External Loading 627Combining External Superelement Dynamic Loads with Residual Dynamic Loads

629External Superelements and Damping 632

12 External Superelement Examples

Introduction 634

Connections 635Automatic Connections 635Manual Connections 636Potential Conflicts with SPC/MPC dof 639

Static Examples 642

Static Examples Using 2-Step Method (EXTSEOUT) 642Static Examples Using 3-Step Method 650

Modal Examples 660Modal Examples Using 2-Step Method (EXTSEOUT) 660Modal Examples Using 3-Step Method 668

Transient Response Examples 676

Transient Response Examples Using 2-Step Method (EXTSEOUT) 676Transient Response Examples Using 3-Step Method 683

Frequency Response Examples 695Frequency Response Examples Using 2-Step Method (EXTSEOUT) 695


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vi iCONTENTS

Frequency Response Examples Using 3-Step Method 702

13 Practical Image Superelements

Introduction 716

List Superelements 718

Using CSUPER 718Example Using CSUPER 720

PART Superelements 734

Using SEBULK and SELOC to image a PART 734Using SEMPLN to define a mirror plane 737Example Using SEBULK, SELOC, and SEMPLN 737

External Superelements 749

Using SECONCT to Attach and External Superelement 749Example using SECONCT 751

Multiple Image Example for Electronic Components 756Baseline Solution (Full Model) 756CSUPER Image Superelement Solution 757PART Superelement Image Solution 762External Superelement Image Solution 766Comparison of Results 774

14 Preparing Adams Flexible Bodies

Introduction 778

Creating a Flexible Body with ADMSMNF 779

Simple Linkage Example 781

Addi tional Reference Material 790

15 Design Sensit ivity and Optimization with Superelements

Introduction 792

16 Superelements in Aeroelastic ity

Introduction 794

Example of Swept Wing 796


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List Superelement 799PART Superelement 801External Superelement 805

17 Considerations in Nonlinear and Buckling Analysis

Introduction 812

Superelement Limitations in Nonlinear and Buckling Analysis 813

Buckling Example Showing Poor Approximation for Superelements 815

Applying Loads on Upstream Superelements in Nonlinear Stat ics 824

Superelement Loading in SOL 106 824Superelement Loading in SOL 400 830

Practical Buckling Example Isolating and Individual Panel 833

Superelements in Heat Transfer 840

18 Random Vibration with Superelements

Introduction 846

Enforced Motion (SPCD) Examples 851

Baseline Model, Cantilever Plate 851List Superelement, Cantilever Plate 863PART Superelement, Cantilever Plate 866External Superelement 869

Enforced Motion (Large Mass) Examples 882

Electronics Board Example 884

Baseline 884List Superelement 887Part Superelement 892

External Superelement 897Comparison of Results 913

19 Output with XYPLOT

Introduction 916

Superelement Plotting with PLOT 917

PART Superelement Example 920CSUPER Example 926


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ixCONTENTS

XYPLOT Commands 930

Illustrative Example: Modal Transient 934

Illustrative Example: Modal Frequency Response 944Illustrative Example: Random Vibration 950

APPENDICES

Glossary 2

Boundary Element 2Branch Element 2Collector Superelement 2

Component Modes Synthesis 2Constraint Modes 2External Superelement 2Fixed Boundary Solution 3Free Boundary Solution 3List Superelement 3Load Sequence 3Mixed Boundary Solution 3

Multilevel Superelement Tree 3Mutually Exclusive Set 3Main Bulk Data Section 4PART Superelement 4Phase 1 Processing 4Phase 2 Processing 4Phase 3 Processing 4Qualifier 4

SEP1 5SEP1X 5Single Level Superelement Tree 5Processing Order 5

S l t U G id
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Chapter 1: Introduction and Fundamentals

1 Introduction and Fundamentals Introduction to this Guide

Why Use Superelements?

Fundamentals of Superelement Analysis

Partitioned Solutions

Manual Solution of a Small Superelement Example

List of Superelement Enhancements Released Since Version 69

Superelements Users Guide2


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Superelements User s Guide

Introduction to this Guide

2

Introduction to this GuideIn finite element analysis, demand for computer resources has always exceeded existing capabilities. Inthe early days of computers, when engineers were solving 3 x 3 problems by hand, computers were ableto handle problems as large as 11 x 11. Once engineers discovered this ability, the size of engineeringproblems quickly grew to exceed the capacity of the existing systems. This process has repeated itselftime and time again. Today modern computers are capable of solving problems involving more than100,000,000 equations with 100,000,000 unknowns, which is still not enough to satisfy the needs ofmany engineers as more detail is added to finite element models and higher fidelity solutions arerequired.

The limits on hardware resources, combined with budget restrictions (large runs and stochastic variationscan be time-consuming), limits the ability of engineers to solve large, complicated problems with highfidelity meshes. A solution to these problems (both hardware and time budget), can be achieved for manymodels by using superelements in MSC.Nastran.

By using superelements, the analyst can not only analyze larger models (including those which exceedthe capacity of your hardware), but he can also become more efficient in performing the analysis, thusallowing more analytical design cycles or iterations in the analysis. Another benefit of superelements

efficiency can be realized when models are subjected to probabilistic or stochastic analysis by varyingportions of the structure. In design optimization, the use of superelements has become automated to helpreduce the overall optimization costs through a process called Automated External SuperelementOptimization (AESO) which is briefly described in Chapter 15: Design Sensitivity and Optimizationwith Superelements, and more fully described in the Design Sensitivity and Optimization Users Guide.

The principle used in superelement analysis is often referred to as substructuring. That is, the model isdivided into a series of components, each of which is processed independently resulting in a set ofmatrices that are reduced to a boundary and describe the behavior of the component as seen by the restof the structure. Often these components are comprised of logical groupings of elements (an engine, awing, a fender, the exhaust system, etc.), hence the term superelement.

The reduced boundary matrices for the individual superelements are combined to form assemblymatrices which are referred to as the residual matrices. The residual matrices are solved using standardtechniques for calculating displacements (and velocities, accelerations for dynamic solutions). Theresidual solution is then imposed on the boundary of each superelement so that the data recovery(calculation of displacements, stresses, etc.) for the boundary can be combined with the data recovery for

the body loads on the superelement.

In static analysis the theory used in superelement processing is exact. In dynamics the reduction of thestiffness is exact, but approximations occur during the reduction of the mass and damping matrices. Thedynamic solution can be improved dramatically by augmenting the static reduction with additionaldynamic degrees of freedom in a method called component modal synthesis, which is described inChapter 10: Input and Output for Dynamic Reduction.

In buckling analysis, the reduction of the differential stiffness uses the same theory as the static reduction,

but doesnt provide an accurate solution compared to the non-reduced system. Unfortunately, there is noimprovement available as in dynamic analysis, however, a savvy user can use this to his advantage as

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3CHAPTER 1Introduction and Fundamentals

described in Practical Buckling Example Isolating and Individual Panel (Ch. 17). Superelements can be

used in nonlinear analysis, but the superelement is limited to a linear reduction in its initial orientation.

This Users Guide is intended to be tutorial in format. That is, the emphasis is on how to use

superelements, not on the theory of superelements. Sufficient theory is presented for those who wish tounderstand the operations. Hand-solved samples are included to help the user understand the operationsinvolved when superelements are used. Sample MSC.Nastran input files and selected output are also

presented at appropriate points for clarity. All of the example files used in this guide are also deliveredwith the standard MSC.Nastran delivery in the install_dir/doc/seug/chapter#/subjectsubdirectories.

This Users Guide presumes that the reader is experienced in finite element analysis and wants to addsuperelement technology to his repertoire of skills. The Guide is arranged so that an experienced finiteelement analyst can start at the beginning and read only the information applicable to the type of analysisdesired. Overall information on superelements is presented first, followed by information for staticanalysis, followed by dynamics and other features. It is recommended that the user read the first 3chapters for foundation as well as Chapter 4 because much of the information presented in the sectionon statics is applicable in subsequent chapters. However, an engineer should be able to read theapplicable sections without having to read unnecessary information.

Note: Even though the theory of static condensation is exact for static solutions, the numericconditioning of the structural matrices can affect the overall solution. If the superelementstiffness matrices are well conditioned, then there will be only miniscule differencesbetween a residual-only solution and a superelement solution.



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Why Use Superelements?Efficiency is the primary reason to use superelements. A finite element model is rarely analyzed only

once. Often the model is modified and re-analyzed time and time again. By analyzing only the part of thestructure which changes, the user can save significant time. Without using superelements, each analysiscan cost the price of a complete solution. Here is a partial list of the advantages of superelements:

Reduced Cost

Instead of solving the entire model each time, superelements offer the advantage of incremental

processing. On restarts this advantage is magnified by the need to process only the parts of the structuredirectly affected by the change. This means that if the user thinks ahead when defining superelements, itis possible to achieve performance improvements on the order of anywhere from 2 to 30 times faster thannon-superelement methods (or more).

Quicker Turnaround

Because superelements can be processed individually with less computer resources required than a

complete, non-superelement solution, it is often possible to submit individual superelement processingruns usingfastqueues (or on local workstations instead of servers), rather than waiting and running thecomplete problem at once using an overnight queue. As stochastic and Monte Carlo simulations arebecoming more popular to understand the robustness of a structural design, fast solutions are a must.

Reduced Risk

Processing a model without using superelements is an all-or-nothing proposition. If an error occurs, theentire model must be processed again once the error is corrected. When using superelements, eachsuperelement need be processed only once, unless a change requires reprocessing the superelement. If anerror occurs during processing, only the affected superelement and the residual structure (finalsuperelement to be processed) need be reprocessed. The superelements that did not have an error do notneed to be processed again until a change is made to those superelements.

Large Problem Capabili tiesAll computers have hardware limits. MSC.Nastran is designed so that problem size will not be limitedby the program. This means that limits on available disk space or memory are the only limitations thatshould be encountered by a user. When the size of a model becomes too large to be processed on acomputer without using superelements, the user can use multiple computer resources to process eachsuperelement, or process one superelement at a time on a single computer resource. The reduced matricesfor each superelement can be stored on separate drives and brought together for the residual solution.

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Introduction and Fundamentals

Then data recovery can be done on each superelement separately, if desired. This process frees file spaceand reduces disk usage and storage costs.

Partit ioned Input and Output

Because superelements can be processed individually, separate analysis groups or organizations canmodel individual parts of the structure and perform model checks without information from other groups.An excellent example is the International Space Station which has many contractors working on the

structure. Each contractor models his own components and sends either complete or reduced models toa system integrator, who assembles the models to represent the many possible configurations, performsanalysis for each configuration, and sends results back to the individual contractors for their use. The

partitioned output format used in superelements allows for segmented data recovery; i.e., data recoverycan be requested for only the desired segments of the structure.

Security

Many companies work on proprietary or secure projects. These may range from keeping a new designfrom the competition, to keeping material data proprietary, to working on a highly confidential defense

program. Even when working on open programs, there is a need to send a representation of the model toothers so that they may perform a coupled analysis of an assembly which incorporates the component.The use of external superelements allows users to send reduced boundary matrices that contain nogeometric information about the actual component-only mass, stiffness, damping and loads as seen at the

boundary. Upon receiving a set of reduced matrices in any format that can be read by MSC.Nastran, anengineer can define an external superelement using those matrices and attach the foreign structure to his

model.

Automation with High Performance Computing

As models become larger and users demand high fidelity system level responses, the discipline of HighPerformance Computing (HPC) has become a primary consideration for industries such as theautomotive industry. There are many techniques involved in HPC, including efficient use of CPU core

memory (i.e. L1, L2, L3 cache), shared memory parallel (SMP) schemes for enhanced matrix solutiontimes using multi-core CPU machines, and distributed memory parallel (DMP) schemes used to takeadvantage of compute clusters with many different machines linked with a high-speed intranet to split

Note: Prior to MSC.Nastran Version 2005.5, there was a memory limit of 2 Gigawords (8Gigabytes) because of a 32 bit integer address used by MSC.Nastran (i4). Beginning inMSC.Nastran Version 2005.5, this limitation was removed by providing the option to usea 64 bit integer address (i8), thus making the number of words available for memoryaddress = 2^64, which is, in effect, an unlimited memory address. The higher memory canbe made available by specifying MODE=i8 on the command line. (note: the modekeyword cannot be specified in an rc file as of MSC.Nastran v2010, so the user must

specify the mode keyword on the command line).



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and recombine solutions. While many of the schemes are purely mathematical (cache memory, SMP),the distributed computing utilizes superelement technology as its core method for splitting andrecombining the solution of a large model. There are several schemes available to the user, such asGeometric Domain Decomposition which automatically splits the model into superelements based onGRID connectivity, Matrix Domain Decomposition which automatically splits the model intosuperelements based on matrix characteristics, and Automated Component Modes Synthesis (ACMS)which automatically splits the model based on GRID or Matrix characteristics and adds a ComponentModes Synthesis calculation for improved dynamic characteristics. While distributed HPC is based onsuperelement technology, the discipline of HPC is beyond the scope of this document and will be coveredin another manual.

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Superelement analysis can be described as a form of substructuring. That is, a model can be divided intosuperelements by the user in such a way that MSC.Nastran will process each superelement independentlyof all other superelements. The processing of each superelement results in a reduced set of matrices(mass, damping, stiffness, and loading) that represent the properties of the superelement as seen at itsconnections to adjacent structures. Once all superelements have been processed, these reduced matricesare assembled in what is known as the residual structure, and the assembly solution is performed. Datarecovery for each superelement is performed by expanding the solution at the attachment points, usingthe same transformation that was used to perform the original reduction on the superelement.

Superelements can consist of physical data (elements and grid points) or can be defined as an image ofanother superelement or as an external superelement (a set of matrices from an external source to beattached to the model). Figure1-1 demonstrates a simplified illustration of condensing a structure to its

boundaries, solving a reduced system, and back-expanding the solution to obtain the data recovery forthe superelement. Example files are available to the user in the following installation directory and filenames: /doc/seug/chapter1/clamp-baseline.bdf, clamp-seset.bdf, and clamp-part-se.bdf.

Figure 1-1 Simplified Depiction of Superelement Reduction, Solution, and Data Recovery

The following figures illustrate the possible types of superelement. In Figure1-2, a model of a portionof a gear is shown. The physical model of one tooth can be represented as a superelement. This type iscalled a primary superelement-one where the actual geometry for the superelement is defined in the bulk

data. Other gear teeth, as shown in Figure1-2, are images of the first (primary) tooth. An imagesuperelement is a superelement that uses the geometry of another superelement to describe it forMSC.Nastran. These image superelements can save processing time in that they are able to re-use the



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reduced stiffness, mass, and damping matrices from their primary superelement, which reduces theamount of calculations needed. Full data recovery is available for image superelements. An imagesuperelement can be an identical image, as shown in Figure1-2, or a mirror image, as shown inFigure1-3. InFigure1-3,the right side of the plate is a mirror image copy of the primary. Please note thatimages can have their own unique loadings. Only the stiffness, mass and damping is identical to theprimary. Another type of superelement is the external superelement, where a part of the model isrepresented by using matrices from an outside source (the matrices can come from another MSC.Nastranrun). For these matrices no internal geometry information is available; only the grid points to which thematrices are attached are known. An external superelement is shown in Figure1-4. In this figure the finiteelement model is on the left and the external superelement is represented by the dashed lines on the right.

Figure 1-2 A Primary Superelement and Several Identical Images

Figure 1-3 A Primary Superelement and its Mirror Image Superelement



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Figure 1-4 An External Superelement

In static analysis the theory used in superelement processing is exact. In dynamics the reduction of thestiffness is exact, but approximations occur during the reduction of the mass and damping matrices. Thedynamic solution can be improved dramatically by augmenting the static reduction with additionaldynamic degrees of freedom in a method called component modal synthesis, which is described inChapter 10: Input and Output for Dynamic Reduction.



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Key Concepts in Superelement Partit ionsThere are several key concepts that must be understood in the superelement formulation and processing;these are:

The input is partitioned into a separate set for processing each superelement.When MSC.Nastran is processing the bulk data for a model, the input is partitioned into aseparate set for each superelement, based on user instructions. The input used to accomplish this

partitioning is discussed in Chapter 2: How to Define a Superelement.Once the bulk data is partitioned into separate sets, each superelement is processed individually.The degrees-of-freedom (DOFs) for each superelement are partitioned into sets in a manneridentical to that used in conventional analysis. That is, all DOFs for a superelement arecombined to create a G-set. Then MPCs and R-elements are used to define the M- and N-sets,etc. (see Constraint and Set Notation (Ch. 1) in theMSC Nastran Reference Manualfor acomplete description of MSC.Nastran sets). The only change in the definition of sets is thedefinition of exterior DOFs. For each superelement the exterior DOFs are defined as the A-set,

which can contain physical and modal degrees of freedom.

Boundary / Exterior DOFs are best described as those that are retained or kept forfurther analysis. A superelements exterior DOFs are best described as those that are retainedfor further analysis, or you can think of them as boundary or attachment DOFs, where thesuperelement connects to other superelements or the residual. Note that exterior DOF are notrequired to attach to any other DOF. Structural matrices are assembled for each superelement,and the matrices go through reduction processing until the only remaining terms are for the A-set

or attachment DOFs. These reduced matrices are used to represent the properties of thesuperelement when it is attached to the rest of the model.

Interior DOFs can be thought o f as those that are condensed out duringsuperelement processing. All DOFs of a superelement that are not exterior are calledinterior DOFs (the omitted or O-set). These are the DOFs that are condensed out of the matricesduring the reduction process. Using either static or dynamic reduction, the stiffness, mass,damping, and loading on these interior DOFs are transferred to the exterior DOFs.

Each superelement is processed individually. The reduction process is best illustrated

using the process known as static condensation. For illustration purposes, we will ignoreLagrange DOF in this discussion, since they are not compatible with superelement processing. Instatic condensation we will start with the superelement matrices after all MPCs, R-elements, and

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SPCs have been processed. The set of DOFs remaining at this point are in terms of the F-set(DOFs that are not constrained), which contains the O- and A-sets as subsets. Although the

interior DOFs include the M- and S-sets also, the interior DOFs in this guide will often be oftenreferred to as the O-set.

Static Condensation Process

The key to superelement processing is the reduction of the omitted DOF to the equivalent boundarymatrices as shown in the following figure:

Figure 1-5 Pictorial Example of Static Condensation

Note: Lagrange Rigid Element Processing. Typically, the dependent dof associated withRBEs are placed in the mr set, while the dependent dof associated with MPCs areplaced in the mp set; which collectively define the M-Set. However, if the user specifiesRIGID=LAGRAN in the case control, the dependent dof are carried into the ASET as partof the lm dof. Currently the MSC.Nastran processing does not handle the

RIGID=LAGRAN for superelements. Further discussion on this subject can be found inNastran Set Definitions The USET Table (Ch. 9).



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A comprehensive description of the equations used by MSC.Nastran can be found in Dynamic Reductionand Component Mode Synthesis in SubDMAP SEMR3 (Ch. 1) in theMSC Nastran Reference Manual.Amore simplified presentation of the static condensation theory is included here for completeness. It is

assumed that the reader has some familiarity with the MSC.Nastran set notation for more informationon set notation, please refer to Degree-of-Freedom Sets (Ch. 7) in theMSC Nastran Quick ReferenceGuide.

In this formulation we will start with the superelement matrices after all M-set (MPCs and Rigidelements), and S-Set (active SPCs and permanent constraints on grid entries) have been processed. Theset of DOFs remaining at this point are in terms of the F-set (dof that are not constrained), which containsthe O- and A-sets as subsets. Although, in general, the interior dof may also include the M- and S-sets

also, the interior dof in this guide will often be often referred to as the O-set.

The static equation for the F-set is

(1-1)

This equation may be expanded to show the A-set and O-set partitions as

(1-2)

where the bar over a term ( and ) indicates that the sub-matrix represents the associated matrix

of terms for that set before the reduction operation. In a static solution, the T-set is equivalent to the A-set and is defined as the retained physical dof. So, for a static solution the previous equation becomes:

(1-3)

If we look at the upper part of the equation, we obtain

(1-4)

Pre-multiplying both sides of the equation by produces

(1-5)

Kf f Uf Pf =

Koo Koa Koa

TKaa

Uo Ua

Po Pa

=

Kaa Pa

Koo Kot

Kot T

Kt t

Uo

Ut

Po

Pt

=

Koo Uo Kot Ut + Po =

Koo 1

Uo total

Koo 1

Kot Ut Koo 1

Po +=

http://../reference/reference/solseq.pdfhttp://../reference/reference/solseq.pdfhttp://../reference/reference/solseq.pdfhttp://../reference/reference/solseq.pdfhttp://../reference/qrg/qrg.pdfhttp://../reference/qrg/qrg.pdfhttp://../reference/qrg/qrg.pdfhttp://../reference/qrg/qrg.pdfhttp://../reference/qrg/qrg.pdfhttp://../reference/qrg/qrg.pdfhttp://../reference/reference/solseq.pdfhttp://../reference/reference/solseq.pdf


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We can break up the total solution into two parts: the Fixed Boundary Solution , , and the

Free Boundary Solution, . To simplify the equation we define the physical boundarytransformation matrix between the exterior and interior motion as .

(1-6)

Physically, the matrix represents the influence coefficients to the free boundary solution, also

referred to as the Constraint Modes. That is, each column of this matrix represents the motion of the

interior points when one boundary dof is moved one unit while the other boundary points are heldconstrained. Therefore, the transformation matrix has one column for each exterior (boundary) dof (theAset for the superelement), and the number of rows are equal to the number of interior dof (the O-set forthe superelement). The constraint modes are discussed further in Example of Constraint Modes (Ch. 9)which includes a graphic example.

When the constraints mode influence coefficients are multiplied by the boundary displacements of the

residual solution, the free boundary solution is obtained:

(1-7)

Where is the partition from the residual structure solution of the physical dof to the superelement

boundary dof.

In addition to the free boundary solution, the fixed boundary solution of the superelement must be

calculated in order to obtain the total solution for the superelement:

(1-8)

This matrix represents the static solution for the displacements of the superelement when the loads areapplied and the exterior points are held fixed. Based on these definitions, the total displacement of theinterior points can be defined as

(1-9)

Uo Uo f i xed

Uo f ree

Got

Got Koo 1

Kot =

Got

Note: Computational Cost of Constraint Modes

The calculation of the constraint modes more specifically is typically the highest

cost associated with a static superelement solution because of the cost of calculating the

matrix . Even in dynamic solutions the cost associated with the constraint modes

is often a significant cost of the overall solution.

Got

Koo 1

Uo free

Got Ut =

Ut

Uo f ixed

Uoo Koo

1Po = =

Uo total Uo free Uo f ixed+=



14


26/972

A physical representation of this equation demonstrates the concepts of fixed boundary solution and freeboundary solution for a cantilever beam example.

Figure 1-6 Pictorial representation of fixed boundary solution and free boundary solution

Continuing with the reduction theory we rewrite the equation for the lower part of Equation 1-3 as:

(1-10)

From which, we can obtain the reduced stiffness and boundary loading of the superelement:

(1-11)

(1-12)

The residual structure consists of all components of the model that were not assigned to any othersuperelement, plus the assembly of the reduced superelement matrices. Each superelement is processedin this manner, and its associated matrices are reduced to the exterior dofs. Once all superelements havebeen processed, the reduced matrices are assembled into a system matrix during the residual structureprocessing.

Thus, the total assembled stiffness matrix of the residual structure, , is represented by

Kot T

Got Ut Uoo + Kt t Ut + Pt =

Kt t Pt

Kt t Kot T

Got Kt t +=

Pt Got T

Po Pt +=

Kgg



27/972

(1-13)

The system or assembly solution is performed using the assembled matrices for the residual structure.Once the assembly solution is known, the boundary solution is found for each superelement. This

boundary solution is used to calculate the interior displacements for each superelement, then standarddata recovery is available for all superelements, including the residual structure. Any output that isavailable in standard (non-superelement) analysis is available in superelement analysis. The differenceis that the output is now partitioned by superelement.

Kgg Kj j Kaa +=



16


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Manual Solution of a Small Superelement ExampleThe following small problem is used to demonstrate how a static analysis is performed using

superelements. The solution to the problem is first shown using conventional analysis, then by usingsuperelements.

Figure 1-7 Small NON-Superelement Example

For this example we are looking only at motion along the axis of the points, thus the problem is simplifiedto contain only five DOFs. Note: this example is solved in MSC.Nastran and provided as part of thedocumentation. The conventional analysis model is: /doc/seug/chapter1/simple-conventional.bdf andthe superelement solution is: /doc/seug/chapter1/simple-superelement.bdf.

The output for the simple-conventional.bdf file is as follows. Note that this .f06 listing, and other listingsin this book may remove page headings and slightly re-arrange the format slightly to fit the page, so theactual .f06 output format may be slightly different than shown



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.

Listing 1-1 MSC.Nastran Output for the Simple Example w/o Superelements

Conventional Analysis

In conventional analysis the problem is formulated in matrix form, constraints are applied, and theresulting reduced problem is solved. The five-by-five stiffness matrix is as follows:

MSC NASTRAN J OB CREATED ON 19- MAR-11 AT 16: 09: 34 MARCH 19, 2011 MSC NASTRAN 7/ 15/ 10 PAGE 10

DEFAULT

0 BASELI NE MODEL SUBCASE 1

D I S P L A C E M E N T V E C T O R

POI NT I D. TYPE T1 T2 T3 R1 R2 R3

1 G 0. 0 0. 0 0. 0 0. 0 0. 0 0. 0

2 G 2. 500000E+00 0. 0 0. 0 0. 0 0. 0 0. 0

3 G 4. 000000E+00 0. 0 0. 0 0. 0 0. 0 0. 0

4 G 3. 500000E+00 0. 0 0. 0 0. 0 0. 0 0. 0

5 G 0. 0 0. 0 0. 0 0. 0 0. 0 0. 0

L O A D V E C T O R


2 G 1. 000000E+00 0. 0 0. 0 0. 0 0. 0 0. 03 G 2. 000000E+00 0. 0 0. 0 0. 0 0. 0 0. 0

4 G 3. 000000E+00 0. 0 0. 0 0. 0 0. 0 0. 0

F O R C E S O F S I N G L E - P O I N T C O N S T R A I N T


1 G - 2. 500000E+00 0. 0 0. 0 0. 0 0. 0 0. 0

5 G - 3. 500000E+00 0. 0 0. 0 0. 0 0. 0 0. 0

F O R C E S I N S C A L A R S P R I N G S ( C E L A S 1 )

ELEMENT FORCE ELEMENT FORCE ELEMENT FORCE ELEMENT FORCE

I D. I D. I D. I D.

1 - 2. 500000E+00 2 - 1. 500000E+00 3 5. 000000E- 01 4 3. 500000E+00



18


30/972

(1-14)

Each row (or column) in the above matrix represents the terms associated with one DOF in the model.The terms are in ascending order; that is, the first column represents DOF 1, and the last columnrepresents DOF 5. Replacing the springs by their numeric values, we have

(1-15)

We now apply the constraints to the problem. In finite element analysis, constraints are applied byremoving the associated rows and columns from the matrix; therefore, after applying constraints we havethe static equation for the constrained structure

(1-16)

or, in numeric terms

(1-17)

Solving the equations provides the solution for the free dof:

(1-18)

Kgg

K12 K12

K12 K12 K23+ K23

K23 K23 K34+ K34

K34 K34 K45+ K45

K45 K45

=

Kgg

1 1 0 0 0

1 2 1 0 0

0 1 2 1 0

0 0 1 2 1

0 0 0 1 1

=

U2

U3

U4

K12 K23+ K 23

K23 K23 K34+ K 34

K34 K45+

1P2

P3

P4

=

U2

U3

U4

2 1 0

1 2 1

0 1 2

11

2

3

=

U2

U3

U4

2.5

4.0

3.5

=



31/972

The total solution vector becomes:

(1-19)

The constraint forces are obtained by partitioning the G-set stiffness matrix and solution vector asfollows:

Equation 1-20 (1-20)

(1-21)

Element Data Recovery

For this example, we can calculate the element forces based upon:

(1-22)

The CELAS1 convention for calculating element force is (Refer to Eq. (3-64) in theMSC NastranReference Manual) :

(1-23)

(1-24)

The remainder of the elements forces can be calculated similarly.

(1-25)

(1-26)

(1-27)

Superelement AnalysisWe now formulate and solve the same problem using superelements, as shown in Figure1-8. Becausethe method of defining superelements has not been discussed yet, some of what follows has not been

Ug

U1

U2

U3

U4

U5

0.0

2.5

4.0

3.5

0.0

= =

Fs Ks f Uf =

F1

F5

1 0 0

0 0 1

2.5

4.0

3.5

2.5

3.5

= =

Felem Kelem Uelem +

F12 K12 U1 U2 =

F12 1.0 0.0 2.5 2.5 = =

F23 1.5 =

F34 0.5 =

F45 3.5 =



20


32/972

defined. However, as you read further, more of the information will become clear. First a flowchartshowing the order of processing used to perform superelement analysis is shown in Figure 1-9, Flowchartfor Superelmeent Processing.

Figure 1-8 Small Superelement Example

Figure 1-9 Flowchart for Superelmeent Processing



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Continuing with the superelement solution for our simple example, the definitions of the model shownin Figure1-8:

Superelement 1 (SEID = 1)

Grid points 1 and 2 are interior points. (These grid points are condensed out during the Phase1 operations for superelement 1.)

Elements and are interior or belong to superelement 1.

The constraint at grid point 1 is contained in superelement 1.

The load applied on grid point 2 is in superelement 1.

Note: Note to advanced users: in the MSC.Nastran SubDMAP listings, the parameter LPFLG isused to control entry and processing within a superelement loop. Also, several of theMALTER statements are strategically placed at the top and bottom of superelement loopsas follows:

Table 1-1 Strategic DMAP MALTER Locations Associated with Superelements

$MALTER: TOP OF PHASE 1 SUPERELEMENT LOOP AFTER PARAMETERS AND

$MALTER: QUALI FI ERS SET

$MALTER: AFTER SUPERELEMENT STI FFNESS, VI SCOUS DAMPI NG, MASS, AND

$MALTER:ELEMENT STRUCTURAL DAMPI NG GENERATI ON ( KJ J Z, BJ J X, MJ J X, K4J J )

$MALTER: AFTER TOTAL SUPERELEMENT STI FFNESS, VI SCOUS DAMPI NG, AND MASS

$MALTER: FORMULATED, STRUCTURAL + DI RECT I NPUT ( KJ J , BJ J , MJ J )

$MALTER: AFTER SUPERELEMENT LAD GENERATI ON (PJ )

$MALTER: AFTER UPSTREAM SUPERELEMENT MATRI X AND LOAD ASSEMBLY

$MALTER: ( KGG, BGG, MGG, K4GG, PG)

$MALTER: AFTER SUPERELEMENT MATRI X AND LOAD REDUCTI ON TO A- SET,

$MALTER: STATI C AND DYNAMI C ( KAA, MAA, BAA, K4AA, PA)

ENDI F $ EXTER $MALTER: BOTTOM OF PHASE 1 SUPERELEMENT LOOP

$MALTER: AFTER SUPERELEMENT DI SPLACEMENT RECOVERY ( UG)

$ SET THE FOLLOWI NG FOR DBCPATH

ENDI F $ MALTER: BOTTOM OF SUPERELEMENT DATA RECOVERY LOOP



22


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Grid point 3 is exterior to superelement 1. (After all reduction [Phase 1] is completed forsuperelement 1, all that remains is a set of matrices representing the superelement attached togrid point 3.)

Superelement 2 (SEID = 2)

Grid points 4 and 5 are interior to superelement 2.

Grid point 3 is exterior to superelement 2.

The load on grid point 4 is in superelement 2.

Elements and are interior to or belong to superelement 2.

The constraint on grid point 5 is contained in superelement 2.

Residual structure (R.S. OR SEID = 0)

Grid point 3 is interior to the residual structure.

There are no elements left to belong to the residual structure.

The load on grid point 3 is in the residual structure.

Superelements 1 and 2 are processed independently, then the reduced matrices are assembledat the residual.

Phase 1 Processing of Superelement 1

After the model is divided into superelements, the data for superelement 1 contains the followinginformation:

Figure 1-10 Simple Model, Superelement 1 Partition

Based on this model, is the exterior DOF and belongs to the A-set for superelement 1. Therefore, we

want to generate matrices for superelement 1, apply any constraints, and reduce the matrices to theexterior DOF. The G-set for this superelement consists of the DOFs associated with grid points 1, 2, and3. The following are the G-sized matrices:

u2



35/972

(1-28)

(1-29)

The boxed superscript 1 ( ) shown on the matrices indicates that they belong to superelement 1. Notice

that the force on grid point 3, , is not included in the Superelement Processing because the force is

applied to an exterior point, it is not included in the superelement, but is accounted for in the residualstructure. This fact is indicated in the matrix for the loading by placing a bar over the P3 term and

indicating that this represents only loading on grid point 3 associated with superelement 1.

Looking at the model, we see that grid point 1 is constrained. Because that grid point is interior to thesuperelement, the constraint is applied as a part of the processing for superelement 1. The resulting(reduced) stiffness matrix is

0 (1-30)

This matrix is now divided into interior (O-set) and exterior (A-set) DOFs, and a static condensation isperformed to reduce the matrices to the exterior DOFs.

First we compute the boundary transformation for superelement 1 becomes (recall equation(1-6)):

1 (1-31)

The physical meaning of this equation is that if Point 3 is moved +1.0 units, then Point 2 will move 0.5units. This is exactly as expected considering that Point 1 is constrained.

Now, we use the transformation to compute the reduced stiffness at the boundary:

(1-32)

Again, the results make sense because there are two springs in series, for which the equation is readilyavailable in text books or online services such as Wikipedia(http://en.wikipedia.org/wiki/Hooke%27s_law#Derivation) :

Kgg

1

K12 K12 0

K12

K12

K23

+ K23

0 K23 K23

1 1 0

1

2 1

0 1 1

= =

Pg 1

P1

P2

P3

1

0

1

0

= =

1

P3

1

Kf f 1 K12 K23+ K23

K23 K23

Koo Kot

Kto Kt t

2 1

1 1= = =

G0 t 1

Koo 1

Kot 1

K12 K22+------------------------- K23

11 1+------------- 0.5 = = = =

Got

Kt t 1

Kot T

Got Kt t + 1.0 0.5 1.0 + 0.5 = = =



24


36/972

(1-33)

Now we have to reduce the applied loadings to the boundary. After applying the constraint to the loadingmatrix, we have

(1-34)

Referring back to 1-12, the loads reduction to the boundary becomes:

(1-35)

Inspection reveals that this also makes sense. If grid points 1 and 3 are constrained, then of the loadwould be distributed to each point.

Phase 1 Processing of Superelement 2

After the model is divided into superelements, the data for superelement 1 contains the followinginformation:

Figure 1-11 Simple Model, Superelement 2 Partition

Degree of freedom is the exterior dof and belongs to the A-set for superelement 2. The reduction of

the stiffness and loads to the exterior dof follows. Since this is similar to superelement 1, only the criticalequations are shown.

Kequiv

K12K23

K12 K+ 23

------------------------- 1 11 1+------------- 0.5= = =

Pf 1 P2

P3

1

1.0

0.0

= =

Pt 1

P3 1

Got T

P2 P3 1

+ 0.5 1.0 0.0 + 0.5 = = = =

u3



37/972

(1-36)

(1-37)

Again, since grid point 3 is exterior to superelement 2, the load is not part of the load vector forsuperelement 2. Recall, forces on exterior points are not included in the superelement matrices.

The constraint will be applied, this time at dof 5, thus the boundary transformation will be calculated andapplied to the stiffness and loads matrices, resulting in the following:

(1-38)

(1-39)

(1-40)

The transformation and reduced matrices make sense. If grid point 3 is moved 1.0 unit, grid point 4 willmove 0.5 units. As before, the stiffness is two springs in series, resulting in a combined stiffness of 0.5,and the load of 3.0 units at grid point 4 gives a 1.5 unit reaction at point 3 if it is constrained.

Residual Structure Processing

The remaining dof, or in this case grid point 3, is defined as the residual structure.

Phase 1 Processing

The phase 1 matrices are generated for the residual structure, based on any elements or loads remaining,then the reduced matrices from the superelements are added at the appropriate dof.

Phase 2 Processing

After the combined (or assembled) matrix for the residual is formed, and constraints applicable to theremaining DOFs are applied and the residual structure problem is solved as part of phase 2 operations.

Phase 3 Processing

Kgg

2

K34 K 34 0

K34

K34

K+45

K45

0 K 45 K45

1 1 0

1

2 1

0 1 1

= =

Pg 2 P3

2

P4

P5

0

3

0

= =

Got

2

Koo

1

Kot K34

K34 K45+-------------------------1

1 1+------------- 0.5 = = = =

Kt t 2

Kot T

Got Kt t + 1.0 0.5 1.0 + 0.5 = = =

Pt 2

P3 2

Got T

P4 P3 2

+ 0.5 1.5 0.0 += = = =



26


38/972

Phase 3 represents the data recovery. In this case, since there is only one grid, the data recovery is trivial.In the general case, the data recovery for the residual will include the residual element stresses, strains,forces, etc.

Figure1-1,depicts how the superelements feed into the residual structure. The individual componentsthat are assembled to make up the residual are shown on the left. The resulting assembly model is shown

on the right, where the system stiffness is: . The residual structure for this model contains

no elements, only one grid point, the physical load on that point, and the reduced matrices from thesuperelements.

Figure 1-12 Simple Model, Residual Structure

Because all physical constraints have been applied at the superelement level, no reduction is performedat the residual level for this model. If there were a physical model for the residual, then it would also gothrough the application of constraints and a reduction to a final set of analysis matrices. Therefore, the

assembly matrix is the result of adding the superelement matrices together at grid point 3, or

(1-41)

(1-42)

(1-43)

Where the matrices and represent the reduced superelement stiffness matrices, and the

matrix represents the stiffness matrix resulting from any elements in the residual structure. In

this problem there are no elements in the residual structure; therefore, is null. Since there are

no SPCs or MPCs in the residual structure, there are not eliminations or reductions require, so

K K1 K2+=

Kgg 0

Kj j 0

i 1=n

Kt t i

+=

Kgg 0

0 Kt t 1

Kt t 2

+

+=

0 0.5 0.5 + + 1.0 ==

Kt t 1

Kt t 2

Kj j 0

Kj j 0



39/972

(1-44)

Similarly, the loading matrix is the physical loadings applied on the residual, plus the reducedsuperelement loads. Because grid point 3 was in the residual, its load was not applied to the upstreamsuperelements, so the 2.0 unit force on grid point 3 is finally included at this point.

(1-45)

(1-46)Now that the stiffness and loading matrices have been generated and reduced, we are ready to solve theresidual structure problem for the physical (T-set) displacements. This is referred to as the Phase 2solution:

(1-47)

(1-48)

Now that the residual solution vector is available, the data recovery can be performed. In this case, thereis no additional data recovery for the residual structure since there are no elements, SPC constraints, orMPC constraints. The data recovery will be performed for the superelements in the subsequent sections.Review of the MSC.Nastran output from file /doc/seug/chapter1/simple-superelement.bdf. confirms thesolution.

Listing 1-2 MSC.Nastran Output for the Residual Structure of the Simple Example with

Superelements

Detailed explanations of the output will be provided in subsequent chapters.

Kt t 0

Kgg 0

1.0 = =

Pt 0

Pj 0

Pj 0

Pt 1

Pt 2

+

+= =

Pt

0

2.0 0.5 1.5 +

+

4.0 = =

Ut Kt t 1

Pt =

U3 0 11--- 4.0 4.0 = =

1 MSC NASTRAN J OB CREATED ON 19- MAR-11 AT 16: 09: 34 MARCH 19, 2011 MSC NASTRAN 7/ 15/ 10 PAGE 22

DEFAULT SUPERELEMENT 0 , 1

0 SUPERELEMENT MODEL SUBCASE 1



3 G 4. 000000E+00 0. 0 0. 0 0. 0 0. 0 0. 0

L O A D V E C T O R


3 G 2.000000E+00 0. 0 0. 0 0. 0 0. 0 0. 0



28


40/972

Data Recovery for Superelement 1

The first part of the Phase 3 data recovery involves partitioning the residual solution to the superelementboundary. In this case, it is trivial since there is only one residual dof. The next step for superelement

data recovery is to calculate the solution vector for the interior dof. The interior solution has twocomponents:

Free boundary displacements

The free boundary displacements are based on the boundary solution of the residual structure (i.e. the

external dof are, in general, free dof in the residual). For this example, the only unknown becomes

based on the external dof boundary displacements (i.e. the T-set), or

Figure 1-13 Simple Example Superelement 1 Free Boundary Displacement DataRecovery

(1-49)

(1-50)

(1-51)

Fixed boundary displacements

The fixed boundary solution is the solution vector for the interior dof when the T-set is fixed.

u2t

u2

uot

seid

Got se id

Ut se id

=

u2t

1

Got 1

Ut 1

=

0.5 4.0 = 2.0 =

u2o



41/972

Figure 1-14 Simple Example Superelement 1 Fixed Boundary Displacement DataRecovery

(1-52)

(1-53)

(1-54)

Total interior solution:

The total interior solution is the summation of the free boundary solution and the fixed boundary solution

(1-55)

(1-56)

The F-set (free set) displacements

The solution vector for the degrees of freedom allowed to move is obtained by merging the T-set and O-Set:

(1-57)

uoo

seid

Koo 1 seid

Po seid

=

uoo

1

Koo 11

Po 1

=

11 1+

------------------ 1.0 = 0.5 =

u2

Uo seid

Uot

seid

Uoo

seid

+=

U2 1

2.0 0.5 + 2.5 = =

Uf

Uf seid Ut

Uo

seid

=



30


42/972

(1-58)

The G-set displacements

The total solution vector for all dof in the superelement is calculated by merging the SPC constraints andback-expansion of the MPC dependent dof. In this example, there are only SPCs, so the total solutionvector becomes:

(1-59)

(1-60)

Rearranging:

(1-61)

Constraint Forces

The constraint forces are obtained by partitioning the G-set stiffness matrix and solution vector asfollows:

(1-62)

(1-63)

Element Data Recovery

For this example, we can calculate the element forces based upon:

(1-64)

U2

U3

1

=2.5

4.0

=

Ug

Ug seid Uf

Us se id

=

U1

U2U3

1

0.0

2.54.0

=

Ug

1U1

U2

U3

0.0

2.54.0

= =

Fs Ks f Uf =

F1 1 02.5

4.0

2.5 = =

Felem Kelem Uelem =



43/972

(1-65)

(1-66)

Similarly:

(1-67)

The MSC.Nastran output for this superelement matches the hand calculations

F12 K12 U1 U2 =

F12

1.0 0.0 2.5 2.5 = =

F23 1.0 2.5 4.0 1.5 = =



32

:


44/972

:

Listing 1-3 MSC.Nastran output for Simple Example Superelement 1

Data Recovery for Superelement 2

Following the same procedure, the data recovery for superelement 2 produces the following results:

(1-68)

The constraint forces are:

(1-69)


DEFAULT SUPERELEMENT 1 , 10 SUPERELEMENT MODEL SUBCASE 1



1 G 0. 0 0. 0 0. 0 0. 0 0. 0 0. 0

2 G 2. 500000E+00 0. 0 0. 0 0. 0 0. 0 0. 0

3 G 4. 000000E+00 0. 0 0. 0 0. 0 0. 0 0. 0

L O A D V E C T O R


2 G 1.000000E+00 0. 0 0. 0 0. 0 0. 0 0. 0



1 G - 2. 500000E+00 0. 0 0. 0 0. 0 0. 0 0. 0



I D. I D. I D. I D.

1 - 2. 500000E+00 2 - 1. 500000E+00

Ug 2 U3

U4

U5

4.03.5

0.0

= =

F5 1 13.5

0.0

3.5 = =


And the element forces are:


45/972

And the element forces are:

(1-70)

(1-71)

The MSC.Nastran output for this superelement matches the hand calculations:

Listing 1-4 MSC.Nastran output for Simple Example Superelement 2

F34 1.0 3.5 4.0 0.5 = =

F45 1.0 0.0 3.5 3.5 = =


DEFAULT SUPERELEMENT 2 , 1

0 SUPERELEMENT MODEL SUBCASE 1



3 G 4. 000000E+00 0. 0 0. 0 0. 0 0. 0 0. 0

4 G 3. 500000E+00 0. 0 0. 0 0. 0 0. 0 0. 0

5 G 0. 0 0. 0 0. 0 0. 0 0. 0 0. 0

L O A D V E C T O R


4 G 3.000000E+00 0. 0 0. 0 0. 0 0. 0 0. 0



5 G - 3. 500000E+00 0. 0 0. 0 0. 0 0. 0 0. 0



I D. I D. I D. I D.

3 5. 000000E- 01 4 3. 500000E+00



34

List of Superelement Enhancements Released Since


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List of Superelement Enhancements Released SinceVersion 69

Superelement technology is constantly being advanced by the development staff of MSC.Nastran andNew versions of MSC.Nastran are released periodically. This guide has been prepared with minimalreference to the version number and is consistent with the MSC.Nastran 2010. The previous version ofthis guide was released in conjunction with the MSC Nastran 2001. There have been several significantreleases in the meantime and it is felt to be of historical interest to show how each of these releases addto the Superelement capability. It is also of great practical interest to those who are not using the mostrecent release to see which of the capabilities described in this Guide are not available to them. The majorenhancements since Version 69 are identified here. Discussions of each of these topics are included in

this Guide.

Version 69

Introduction of PART Superelements

Release Guide Contents, Chapter 5:

Automatic Attachment of Partitioned Superelements

Multilevel Superelements

Superelements in Dynamic Analysis

Restarts with Superelements

Image Superelements

MSC/NASTRAN Plotter

Parts Assembly with Manual Control

Upward Compatibility

MSC/PATRAN Interface

Definitions and Acronyms

Summary of New or Modified Bulk Data Entries

Sample Files

Version 69.1

External Superelements for SOL 101 and SOL 103 3 Step Method

PARAMs added: EXTOUT, EXTDROUT, EXTDR

Support for MATRIXDB, DMIGDB, DMIGOP2, DMIGPCH

Addition of EXTERNAL to SEBULK entry

EXTRN bulk data entry introduced

Version 70.0

Enhanced External Superelements

Better documentation on usage


Residual Vectors for improved dynamic


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Residual Vectors for improved dynamic

Superelement Mass Reduction performance improvement

Version 70.5

Splining Upstream Superelements in Aeroelastic Analysis

Use of External Superelements in Aeroelastic Analysis

Coupled Fluid-Structure Models with Interface DOFs in Superelements

PARAM, FLUIDSE

External SE Enhancements

Assembly of external in SOL 101 thru 159

Data Recovery support added for SOL 107 through 112

CSUPER-type support for external SE via EXTOUT PARAMs

PARAM,VMOPT,1 to support virtual mass at GSET level

Better component mode handling byu bypassing INREL module

Version 70.7

Distributed Parallel Linear Static Analysis (SOL 101)

Differential Stiffness for Upstream Superelements

External Superelement Data Recovery for SOL 146

Modal Damping for Upstream Superelements, PARAM,SESDAMP,YES

Add Modal Damping to Structural Damping for Superelements, PARAM,KDAMP,-1

Special Superelement Reserved for Fluid Elements, PARAM,FLUIDSE,seid Reduced Data Recovery Matrices, PARAM,MINIGOA

Automated QSET Generation, PARAM,NQSET,n

Automatic Removal of Unused QSET dof, PARAM,SMALLQ

Version 2001

Shell Normal Default, PARAM,SNORM

Multiple DMIG input

Force Resultant Output Enhancements

Simplified Static Loading Data in Dynamic Analysis (removal of LOADSET/LSEQrestrictions)

OMODES Case Control of Modal Output

OTIME Case Control of Temporal Output

WEIGHTCHECK Mass Summation Output Including Upstream Superelements

Parallel Processing Enhancements: Geometric Domain Decomposition



36

Automated Component Modes Synthesis (ACMS)


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Version 2004

Enhancements to External Superelements 2 Step Method EXTSEOUT Cast Control

BNDFIX/BNDFIX1, BNDFREE, BNDFREE1

Scale Factors on X2GG and X2PP Matrices

Support of Adjoint Sensitivity in Superelement Analysis

Design Responses Spanning Across Subcases or Superelements

Specify DELAY or DPHASE in Upstream Superelements Residual Vector Enhancements

Nonstructural Mass Support for Superelements (NSM, NSM1, NSMADD)

Modal Participation Factors for Fluid Superelement

Control of Superelement Differential Stiffness Calculation, PARAM,SEKD

Adams Flexible Body Support, ADMSMNF

Punch File Identification of Superelement ID

Version 2004R3 (2004.5)

ACMS with Matrix Domain Decomposition

ACMS compatible with Superelements, Including External Superelements

Version 2005

CSET Improvements, PARAM,MHRED Automatic QSET generation, PARAM,AUTOQSET

Version 2005R2 (2005.1)

External Superelement Support in SOL 600

ADMSMNF Support in SOL 600

USET Support for MATMOD Option 16

Support for Fixed Boundary Displacements in SOL 101 for EXTSEOUT Assembly Runs

Support for Data Recovery with EXTSEOUT(DMIGPCH) option

Version 2005R3 (2005.5) / MDR1

Efficiency Enhancements for MTRXIN handling of DMIG entries

BSETi/BNDFIXi and CSETi/BNDFREEi Entries can be commingled

Enhancements to EXTSEOUT ASMBULK AUTO and MANQ options


FSCOUP option to Store Fluid-Structure Coupling Matrix


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DMIGSFIX option to Define DMIG Matrix Names

Automatic QSET numbering scheme with PARAM AUTOQSET Automatic OTM output for PLOTEL elements

MATDB name equivalent to MATRIXDB

Version 2007 / MDR2

ACMS for External Superelement Reduction

Automatic External Superelement Optimization (AESO)

Version 2007.1 / MDR2.1

ACMS for External Superelement Reduction of Fluid

EXTSEOUT support for OUTPUT4 with MATRIXOP4

Merged Superelement Output, PARAM,FULLSEDR

Version 2008 / MDR3

Local Adaptive Mesh Refinement

Improved SMP throughput for Static Condensation

Version 2010

Support for Part Superelement Optimization

SOL 400 Support for Linear Solutions with Superelements



38


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Chapter2: How to Define a Superelement


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2 How to Define a Superelement Introduction

Superelement vs. Residual

Three Types of Superelements

Defining List Superelements

Defining PART Superelements Defining and Attaching External Superelements

The Superelement Map SEMAP


Introduction

40

Introduction


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Now that the basic concept of superelements has been explained, we can focus on how to define

superelements in MSC.Nastran. Superelements are defined using the Bulk Data Section of the input fileand controlled via File Management Section (FMS), Executive Control, PARAMeters, and Case Control.

There are three methods available for defining superelements:

List Superelement: Commonly referred to as SESET superelements

PART Superelement: Commonly referred to as BEGIN SUPER superelements

External Superelement: Commonly referred to as EXTSEOUT or EXTOUT or DMIGsuperelements

The purpose of this chapter is to describe the input required to define each type of superelement as wellas discuss some of the advantages and disadvantages of each method.

Note: BEGIN SUPER vs. BEGIN BULK Superelement Partit ion ing

MSC.Nastran maintains two distinct paths for superelement processing in the solutionsequences. When a BEGIN SUPER entry is present the program uses the more modern

SEP1X module to make the SEMAP table used to control partition of superelements. Whenthere is a BEGIN BULK entry but no BEGIN SUPER entries a parallel path using the olderSEP1 module is used instead. For a more detailed discussion of these two methods, pleaserefer to the Superelement Analysis (p. 394) in theMSC Nastran Reference Manual.

41CHAPTER 2How to Define a Superelement

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Each superelement in MSC.Nastran is identified by an integer identification known as the SEID. Each

SEID must be a unique positive integer (with the exception of the residual structure, which is known assuperelement 0). If no superelements are defined, the model is assumed to be a residual-structure-onlymodel, and a conventional (non-superelement) solution is performed. By default, all superelementsolution sequences perform a conventional solution if no superelements are defined. The superelementsolution sequences for MSC.Nastran are as follows:

Table 2-1 Solution Sequences that Support Superelements

SOL Number SOL Name Description

101 SESTATIC Statics with options:

Linear steady state heat transfer.

Alternate reduction.

Inertia relief.

103 SEMODES Normal modes.

105 SEBUCKL Buckling with options:

Static analysis.

Alternate reduction.

Inertia relief.

106 NLSTATIC Nonlinear or linear statics

107 SEDCEIG Direct complex modes108 SEDFREQ Direct frequency response

109 SEDTRAN Direct transient response

110 SEMCEIG Modal complex eigenvalues

111 SEMFREQ Modal frequency response

112 SEMTRAN Modal transient response

128 SENLRHM Nonlinear harmonic response129 NLTRAN Nonlinear or linear transient response

144 AESTAT Static Aeroelastic response

145 SEFLUTTER Aerodynamic flutter

146 SEAERO Aeroelastic response



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Table 2-1 Solution Sequences that Support Superelements



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153 NLSCSH Static structural and/or steady state heat transfer analysis with options:

Linear or nonlinear analysis.

159 NLTCSH Transient structural and/or transient heat transfer analysis with options:

Linear or nonlinear analysis.

200 DESOPT Design Optimization

400 NONLIN Nonlinear static and transient analysis



Three Types of Superelements


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As mentioned in the Introduction to this chapter, MSC.Nastran has 3 basic methods for defining

superelements:

List Superelements: as the name implies, List Superelements are defined by specifying a list(or set) in the main Bulk Data section of the input file. These are often referred to as SESETsuperelements because the most common way of defining them is with the SESET bulk dataentry. When superelements are defined using this approach, the model defined in this section ofthe input is cut apart into separate components (each component is a superelement). A good wayto describe this is to say that the program is using a cookie-cutter approach with the model,

taking a model and dividing it into superelement lists for processing. Many models of this typecould be run as a stand-alone, or one shot model without superelement processing. However,a savvy user who has repeated components can use the List Superelements to efficiently image(copy, repeat, mirror) parts of the model.

PART Superelements: PART superelements are defined by defining each superelement in itsown Partitioned Bulk Data section. These separate sections of the bulk data are self-contained inthat each section contains all geometry, elements, properties, constraints, parameters, andloading data for that component of the model. When PARTs are used the program works in a

manner similar to an assembly process. That is, a series of separate components are assembledinto the final finite element model, i.e. the residual structure.

External superelements: External Superelements are similar to PART superelements in manyrespects, except rather than solving the model in a single run, the superelement can be processedand output for use at a later time. There are many advantages of external superelements: 1) thereduced matrices are compact and can be added to another structure while maintaining fullfidelity of the component behavior on the system, 2) they can be easily re-used as many times asnecessary at a very low runtime cost, 3) they can protect design information (proprietarygeometry) and material information (composite layup), 4) key results can be monitored withoutthe need for full data recovery, 5) files can be easily shared and maintained across differentorganizations or design groups.

The three approaches can be used independently or together depending upon the application. In versionsprior to Version 69, only the list superelements were available. Input files from versions prior to Version69 of MSC.Nastran can be used in later versions, and any superelement input will be treated as before.Once PARTs are defined, the program uses a different set of rules to partition the Main Bulk Data Section

into superelements. The modern external superelements were first introduced with PARAMs in V69.1and enhanced to include convenient case control commands in V2004. A list of enhancements byversion can be found in List of Superelement Enhancements Released Since Version 69.



44

Defining List SuperelementsAs the name implies List Superelements are defined by specifying a list or set; but List Superelements


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As the name implies, List Superelements are defined by specifying a list or set; but List Superelementscan only be defined in the main Bulk Data section of the MSC.Nastran input file. The superelementprocessing partitions the model into separate sections based on a list of interior grid points and/orelements defined by the user. The Main Bulk Data Section is defined as the first bulk data input sectionwhich occurs after BEGIN BULK or BEGIN SUPER [=0].

List Superelements can be defined on the GRID entry, the SESET entry, or the SEELT entry. In addition,a superelement can be defined as a copy of another superelement (image superelement) or by usingmatrices from an external source (external superelement) by using the CSUPER entry. Refer toChapter 13: Practical Image Superelements for more details on image superelements.

Any grid points defined in the main bulk data that are not assigned to a superelement, using either aSESET, GRID, or a SEELT entry will automatically be assigned to the residual structure (SEID = 0).

For list superelements defined with BEGIN BULK: if SPOINTs or EPOINTs exist in the input stream,they are automatically and permanently assigned to the residual structure and cannot be reassigned to beinterior to a superelement. However, if the user specifies BEGIN SUPER, then SPOINTs and EPOINTscan be assigned to any superelement. The reason is that BEGIN BULK and BEGIN SUPER undergodifferent superelement processing as described in the Superelement Analysis (Ch. 1) in theMSC Nastran

Reference Manual

List Superelement Defin ition wi th the SESET Entry

The SESET entrycan be used to define the interior GRIDs associated to a Superelement. The usersimply defines the SESET, its SEID and the associated GRID list for the interior GRIDs.

Defines interior grid points for a superelement.

Format:

Example:

SESET Superelement Interior Point Definition

1 2 3 4 5 6 7 8 9 10

SESET SEID G1 G2 G3 G4 G5 G6 G7

SESET 5 2 17 24 25 165


Alternate Format and Example:
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The SESET entry takes precedence over the SEID field on the GRID entry defined below. SESETdefines grid and scalar points to be included as interior to a superelement. SESET may be used as the

primary means of defining superelements or it may be used in combination with SEELT entries whichdefine elements interior to a superelement. For additional comments on the SESET entry, please referto the SESET (Ch. ) in theMSC Nastran Quick Reference Guide.

There is no limit on the number of SESET entries that can be used to define a superelement, and theTHRU option on the SESET entry, can have open sets. That is, not all grid points in the range specifiedneed to exist. If a nonexistent grid point is referenced by an SESET entry, that part of the entry is ignored.

If BEGIN SUPER is used and SEELT is present, then SEELT will take precedence over both the SESETentry and GRID entry SEID field.

List Superelement Defini tion with the GRID EntryAn interior superelement GRID can also be defined on the GRID entryin theSEID field.

Defines the location of a geometric grid point, the directions of its displacement, and its permanentsingle-point constraints.

Format:

Example:

SESET SEID G1 THRU

G2

SESET 2 17 THRU 165

Field Contents

SEID Superelement identification number. Must be a primary superelement. (Integer > 0)

Gi Grid or scalar point identification numbers. (0 < Integer < 1000000; G1 < G2)

GRID Grid Point

1 2 3 4 5 6 7 8 9 10

GRID ID CP X1 X2 X3 CD PS SEID

GRID 2 3 1.0 -2.0 3.0 316



46

Field Contents
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*See the GRDSET entry for default options for the CP, CD, PS, and SEID fields.

List Superelement Defin ition on Element ID the SEELT entry

As an alternative to defining the superelement based on GRID ids, the superelement can be defined basedon element ids using the SEELT entry. If the main bulk data section is defined with BEGIN SUPER[=0]instead ofBEGIN BULK, then the SEELT entry can be used to define the elements which belongto each superelement. The advantage is that sometimes it is easier to define element ranges rather thanGRID ranges for large models. Refer to the note on superelement processing based on BEGIN SUPERvs. BEGIN BULK Superelement Partitioning.

Reassigns superelement boundary elements to an upstream superelement

Format:

Field Contents

lD Grid point identification number. (0 < Integer < 100,000,000, see Remark9.)

CP Identification number of coordinate system in which the location of the gridpoint is defined. (Integer > 0 or blank*)

X1, X2, X3 Location of the grid point in coordinate system CP. (Real; Default = 0.0)

CD Identification number of coordinate system in which the displacements,degrees- of-freedom, constraints, and solution vectors are defined at the gridpoint. (Integer > -1 or blank, see Remark3.)*

PS Permanent single-point constraints associated with the grid point. (Any ofthe Integers 1 through 6 with no embedded blanks, or blank*.)

SEID Superelement identification number. (Integer > 0; Default = 0)

Note: Note that a SESET entry will override the definition on the GRID entry. Also, theGRDSETentrycan be used to define the default SEID for all GRIDs in the main bulk datasection.

SEELT Superelement Boundary Element Reassignment

1 2 3 4 5 6 7 8 9 10

SEELT SEID EID1 EID2 EID3 EID4 EID5 EID6 EID7


Example:

SEELT 2 147 562 937


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Alternate Format and Example:

The SEELT entry can also be used to assign elements connected entir

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