CDH/AMLS Users’ Guide

By Mladen Chargin

CDH AG Advanced Engineering Am Marktplatz 6 79336 Herbolzheim Germany Phone: +49 (0) 7643-9377-26 Fax: +49 (0) 7643-9377-29 E-Mail: david.bella@cdh-ag.com URL: http://www.cdh-ag.com CDH Aktiengesellschaft Vorstand: Dr Leo W. Dunne Aufsichtsratsvorsitzender: Dr Theodor Seitz Sitz und Registergericht: Augsburg, HRB 19468

November 2008

Table of Contents

1. Introduction.......................................................................................................... 3

2. Background.......................................................................................................... 3

3. EIGRL Specification............................................................................................ 4

4. Accuracy Issues ................................................................................................... 4

5. Accuracy Issues (solid models) ............................................................................ 6

6. Enforced Motion .................................................................................................. 8

7. Use of CDH/AMLS for Component Modes Synthesis (CMS) Calculation............ 10

8. Structural Damping Definition ............................................................................. 10

9. Modal/Grid/Panel Participation Factor Calculation............................................... 11

10. Residual Vector Issues ....................................................................................... 11

11. CDH/AMLS SMP Parallel ................................................................................. 13

12. I/O Enhancement for IBM and SGI Computers .................................................. 13

13. Use of MFLUID................................................................................................. 14

14. Rigid Body, Massless Mechanism Modes, and Singular Stiffness....................... 16

References: ................................................................................................................. 19

Appendix A: Enforced Motion for Modal Frequency Response................................ 20

Appendix B: CDH/AMLS Parameter List ................................................................ 22

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1. Introduction

This document will describe the software product CDH/AMLS (Automated Multi-Level Substructuring), its interaction with Nastran, and the most effective way to use it in a variety of analysis situations. CDH/AMLS supports all the modal solution in Nastran, sol103, 110, 111, 112 (normal modes, modal complex eigenvalue, modal frequency response, modal transient response) as well as optimization, sol200. A number of references listed at the end of this document may be of interest to some users. They may be obtained by request to mladen.chargin@cdh-ag.com. Another purpose of this document is to describe some of the parameters that may be specified by the user. For a typical response analysis, the user needs to be concerned about very few details. However, there are some specific cases that one must be aware of, because the solution depends on a proper understanding of these cases. Any discussion of CDH/AMLS capability relates to version 3.2.r159 (even though some, but not all, were included in earlier versions) or later. The only necessary requirement is that the CDH/AMLS Delivery Data Base be used. CDH provides the customers with the appropriate DB for MSC (2001, 2004, 2005,, 2007, 2008, MD) and NX (5, 6) Nastran versions for various computers. CDH/AMLS is a very efficient approximation method for analyzing vibration of large finite element models of structures. This fact is evident from the speed with which CDH/AMLS has been adopted by nearly the entire international automobile industry within the past several years.

2. Background

CDH/AMLS begins by dividing a finite element model into two substructures, then dividing each of these into its own substructures, and recursively continuing until thousands of substructures have been defined in a “tree” hierarchy. Then modes of substructures, whose natural frequencies are below a certain cutoff value, are computed and the model is transformed from the finite element representation to one in terms of substructure modes using the Craig-Bampton method. Modes of the overall structure are approximated in terms of the substructure modes. Finally, the response of the overall structure is represented in terms of these approximate global modes, and exact static response vectors if the residual flexibility technique is used.

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There are only two required items that the user must provide in the Nastran input file, an include statement in the FMS Nastran section and an EIGRL data in the Bulk Data deck. There are other user options which will be discussed in subsequent sections. The include statement is:

include ‘/full/path/name/amls_assign’

The file, amls_assign, is part of the AMLS delivery package. EIGRL requirement is discussed in the next section. In order to facilitate its adoption as a production analysis tool, CDH/AMLS has been implemented in a form that is as transparent to users as possible and requires very little training for productive use. However, some explanation of the issues related to various aspects of CDH/AMLS, such as accuracy, will be helpful for users and will be discussed in Sections 4 and 5.

3. EIGRL Specification

The only requirement for using AMLS (besides the use of the AMLS Delivery Data Base) is that one must define an EIGRL in the Bulk Data deck and request it in the Case Control deck. The reason for this is that AMLS needs to know the cutoff frequency below which it will calculate the eigenvalues and eigenvectors. The necessary value is the F2 entry on the EIGRL. Only specifying the ND value (number of desired roots) is unacceptable and CDH/AMLS will issue a fatal message.

4. Accuracy Issues

The accuracy of the CDH/AMLS approximation depends mainly on the cutoff frequency used for the substructure modes calculation. This cutoff frequency is determined as the product of the cutoff frequency for the global modes (set as the EIGRL upper frequency limit F2) and the Bulk Data parameter ss2gcr (“substructure-to-global cutoff ratio”).

param, ss2gcr, 5.0 [default]

ss2gcr’s default value of 5.0 can be changed by adding the line param, ss2gcr, 7.5, for example, to the Bulk Data file, to increase accuracy. The default value has been found to give excellent accuracy in typical structures (such as car bodies) composed of mostly shell elements and where the structure behaves as a shell structure. A later section will

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describe the appropriate value of solid structures. Increasing ss2gcr to improve accuracy only modestly increases the cost of analysis. The lowest-frequency global modes are approximated very accurately by AMLS, but the highest-frequency global modes are less accurate because the truncation of substructure modes has the greatest effect at the highest frequencies. The accuracy of the global modes whose natural frequencies are within the range of excitation frequencies is more important to the accuracy of the computed frequency response than the accuracy of the global modes above the highest excitation frequency. Typically, global modes are computed to a cutoff frequency fifty percent higher than the highest excitation frequency when residual flexibility is used. Then the highest computed global modes are far enough from resonance that the effect of the small error in their natural frequencies is difficult to detect in the frequency response functions. If only transfer mobilities are needed, keep in mind that residual flexibility does not improve the accuracy of transfer mobilities nearly as much as it improves the accuracy of point mobilities. CDH/AMLS makes it possible to analyze complex structures at much higher frequencies than were feasible with previous methods. An important point to remember, however, is that there are so many modes involved in the response at higher frequencies that individual modes do not have as much influence on the shape of the frequency response functions as at lower frequencies. CDH/AMLS produces Rayleigh-Ritz approximations of the natural frequencies and modes, so the approximate natural frequencies are always greater than or equal to exact ones. Because natural frequencies are very closely spaced at higher frequencies, Rayleigh-Ritz approximations of the exact modes tend to be combinations of the modes that are very close in frequency. This makes it difficult to match approximate global modes with exact global modes. The use of MAC (Modal Assurance Criteria) to compare the “exact” and AMLS mode shapes is discouraged because it leads to very erroneous results. A much better technique using SVD (Singular Value Decomposition) is a more appropriate method of comparing these two sets of mode shapes. Nevertheless, the computed frequency response can be very accurate even if individual modes do not agree well with the exact modes, because the modes combine to form the frequency response. It is more important to approximate well the subspace of global modes in a particular frequency range than to approximate the individual modes very

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accurately. Another way in which CDH/AMLS differs from the Lanczos eigensolution method is that with the Lanczos method, if the modes in a narrow frequency range are needed, it is more economical to solve for only the modes in that range and not the modes below that range in frequency. In contrast, with AMLS, very little is saved by not calculating the modes whose natural frequencies are below the lower frequency limit (the EIGRL frequency limit F1) of the frequency range of interest. Therefore, it is generally more advisable to set the lower frequency limit to zero (or not specify it at all, i.e., leave it blank). Because of the importance of residual flexibility to frequency response accuracy, CDH/AMLS computes exact static responses for residual flexibility calculations. If the structure is unrestrained, it is not necessary when using AMLS to supply Nastran “support” information (in fact it should not be used), because it is ignored by AMLS. It should be noted that AMLS takes the applied loads, residual loads, and requested response points into consideration when defining the substructure tree to improve accuracy. As a result, some slight differences may be observed between results from different CDH/AMLS analyses if there are differences in the definitions of applied loads or the selection of response points. One can also improve the accuracy by simply increasing the value of F2 on the EIGRL. Thus, if one uses a rule-of-thumb of requiring modes 1.5 times the highest excitation frequency, simply use a value of 1.75, for example. This will only slightly increase the CDH/AMLS CPU time as will be seen in the next section. However, this approach will also increase the calculation time of the modal frequency response, especially if the modal equations are fully coupled.

5. Accuracy Issues (solid models)

Much of the above discussion is relevant to the analysis of any type of structure. However, the default value of ss2gcr is based on the use of CDH/AMLS for modal analysis of shell type structures. However, when analyzing structures consisting primarily of solid elements and behaving in a true 3D sense (non-zero stress through the thickness) the value of ss2gcr of 5.0 does not provide sufficient engineering accuracy. Thus, one needs to increase the magnitude of ss2gcr and a minimum value of 10.0 is recommended. As can be seen by the following example of a modal analysis of an engine block,

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increasing the value of ss2gcr has a small influence on the overall solution time. This is owing to the fact that in Craig-Bampton method, the substructure modes consist of two distinct sets of shapes. The “constraint modes” are obtained by a static analysis and the “dynamic modes” are obtained by an eigenvalue analysis of each substructure. Note that the “constraint modes” must be calculated irrespective of the excitation frequency of interest. In true 3D structures, as compared to 2½D structures, the calculation of “constraint modes” dominates the solution time, thus it is does not make much difference if the value of ss2gcr is 5, 10 or 15. However, the result accuracy improves with the larger value of ss2gcr. At some value one reaches a point of diminishing returns, thus a good compromise value should be between 10 and15.

SS2GCR ROOTS ELAPSED(min)

15 270 296:41 10 268 287:49 7 266 283:33 5 261 281:20 7 854 296:221 7 1387 307:412

Lanczos 271 363:47

Table 1. Comparison of Elapsed Times for Different SS2GCR Values

1Fmax=15810 (3x max freq of excitation) 2Fmax=21080 (4x max freq of excitation) As was already mentioned, an alternative to increasing the ss2gcr value is to simply increase the F2 value on the EIGRL. The following example illustrates the computational cost associated with the increase in the value for F2. All jobs run on an IBM P5 1.9GHz, 4CPUs, and 4GB memory. Problem size: 2348611 Grids, 6.89M DOFs Engine Block Model.

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Modes Below (Hz)

Number of Modes

AMLS Elapsed (min)

Total Elapsed (min)

2500 237 226:32 248:59 5000 672 236:49 259:33 10000 1981 268:05 291:35 20000 5704 423:15 448:55

Lanczos-2500 239 430:28 Lanczos-5000 680 943:05

Table 2. Comparison of Elapsed Times

for Different Cutoff Frequencies Clearly, by increasing the number of modes from 237 to 5704 (a factor of ~24) the elapsed time increases less than a factor of two. Similarly, for modes below 2500 Hz, CDH/AMLS calculates only two modes less than Lanczos and for modes below 5000 Hz, eight modes less. Also note that more than doubling the number of modes from 237 to 672 increased the elapsed time only ten minutes. It was simply impossible to calculate modes at the other two cutoff frequencies (10000 Hz and 20000 Hz) with Lanczos because the computation time would have been beyond any reasonable wait time.

6. Enforced Motion

In Nastran, there are a number of ways of specifying enforced motion. The original schemes from the beginning days of Nastran were the Large Mass/Spring methods and the Lagrange Multiplier method. Large Mass (LM) method was the preferred choice. However, in the early 1990’s CDH discovered that the Large Mass method, in combination with Lanczos (the preferred eigenvalue extraction technique), lead to unreliable dynamic response results. Consequently, CDH developed an exact technique that was eventually incorporated into both MSC and NX Nastran. This method required an additional decomposition of the stiffness matrix, a FBS (Forward-Backward Substitution) with the number of right-hand-sides equal to the number of enforced DOFs, as well as a number of matrix multiplies

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(see Eqs. (4) and (7) in Appendix A). The algorithmic details can be seen in Appendix A. However, since the advent of CDH/AMLS it became very obvious that the use of LM for enforced motion is again the preferred choice owing to the fact that it has absolutely zero additional computational cost. The reason that the LM technique failed is that Lanczos was used to extract the modes. In Lanczos the default block size is seven. This number was derived based on the expected multiplicity of eigenvalues, six being a typical number because of six rigid body modes. However, when using the LM method, the multiplicity of eigenvalues may be relatively large, e.g., 40 or more. Thus, when using Lanczos in Nastran, the results may be unpredictable. CDH/AMLS does not use the Lanczos method; rather it uses the Householder method whose accuracy is not sensitive to the multiplicity of eigenvalues. This document will not describe the LM method because it is already well documented in various Nastran publications, such as Dynamics Handbook. The only remaining question concerns the size of the large mass used in LM method. In the past, it was recommended that the value be between 103-106 times the total mass of the structure. This value took into consideration the fact that Lanczos was being used. When using CDH/AMLS, the recommended value is anything greater than 106 (no problems were encountered even when using a value of 1010). Another item of interest when using the LM technique occurs when the user specifies or requests residual vectors by defining (param, resvec, yes) or resvec = yes (MSC V2004+). By default, the residual vectors are obtained from the applied loads, which in the case of enforced motion, refers to all the loads on the large masses. If one simply accepts the default, then one would obtain two sets of almost identical residual vectors. The first set would correspond to the zero frequency mode shapes for each large mass and the second set would be obtained from the static analysis subject to applied loads at the large masses. The duplicity of almost identical residual vector could cause numerical difficulties. To avoid this condition, a user defined Bulk Data parameter, lgmass, can be specified. The function of this parameter is to eliminate any loads on DOFs which have a mass larger than the value of lgmass during the calculation of residual vectors,

param, lgmass, 1.e+5 [default]

The user will be informed regarding which residual loads have been eliminated by a message in the f06 file such as:

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================================================ RESIDUAL VECTOR DOFS THAT WILL BE ELIMINATED BECAUSE THEY HAVE A LARGE MASS ================================================ MATRIX MSVEC

POINT VALUE POINT COLUMN 1 1001 T1 1.00000E+08 1001 T3

1504 T1 1.00000E+08 1504 T3

In the above example, the value of 1.000000E+08 represents the value of the large mass at the particular grid and DOF and since the default value of lgmass is 1.e+5, the loads on these DOFs have been removed during the residual vector calculation.

7. Use of CDH/AMLS for Component Modes Synthesis (CMS) Calculation

One of the user options that Nastran provides is the ability to perform modal analysis of a component, e.g., body-in-white, trim body, etc. Some of these components can have millions of DOFs and it can be very time consuming to calculate the required eigenvalues and eigenvectors using Lanczos. CDH/AMLS can be used for the eigenvalue calculation for this case as well. The only requirement is that the user define an additional parameter

param, amlscmp, yes

This parameter can be placed in the Bulk Data deck, in which case it will be applied to all the superelements including the residual structure. However, this may not be what the analyst desires. For example, some superelements may be very large and others very small. Likewise, one may wish to calculate all the modes of the residual structure in which case a Householder or Givens methods may be more appropriate. In this case, one would place a param, amlscmp, yes in each subcase for which one wishes to use CDH/AMLS.

8. Structural Damping Definition

Nastran has several ways of defining structural damping, i.e., ge specification on the MATi, CELAS2, PELAS, and PBUSH. The structural damping matrix after all the reductions (spc, mpc, …) has a name of K4AA in the Nastran DMAP. This matrix is reduced to modal coordinates by CDH/AMLS. This is accomplished simply by

k4hh = !TK4AA!

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This operation may be quite expensive, depending on the number of non-zero columns in K4AA. If one had a single MAT1 record in the Bulk Data and that particular MAT1 had a ge specification, then the K4AA matrix would look just like the structural stiffness matrix KAA and the above operation would consume significant CPU time. To avoid this situation the following approach is highly recommended. Find all the MAT1s that have the same value of ge or rather that set of MAT1s that provide structural damping for the largest number of DOFs with the same value of ge . Delete that value of ge (assume it is x.xx) from that set of MAT1s and specify a parameter,

param, g, x.xx

For any other value of ge specified on other MAT1s or CELAS2, PELAS, or PBUSH, replace their ge value by (ge – x.xx).

9. Modal/Grid/Panel Participation Factor Calculation

The following Bulk Data Deck parameter is no longer required when performing the above calculations. However, it is still available, hence its default value is ‘no’ as shown.

param, ahh, yes [default=no]

This parameter causes CDH/AMLS to return to Nastran all the eigenvector components that correspond to any structural DOF that is “connected” to the acoustic domain by the “area matrix”. Note that CDH/AMLS ordinarily does not do this since it reduces the “area matrix” to structural modal coordinates by the following operation, a = !

s

TA and

returns a to Nastran. Subsequently, Nastran generates the modal “area matrix” by post-multiplying a by the acoustic eigenvectors, i.e., ahh = a! f . If one requests panel participation factor calculation via the standard Nastran commands, AMLS will automatically return to Nastran all the eigenvector components corresponding to the structural degrees of freedom associated with the requested panels.

10. Residual Vector Issues

At present, there is only one limit imposed on the user by CDH/AMLS, and that is a limit on the number of DOFs that may be loaded by a single residual vector or the total number

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of residual vectors, which by default is 500. The default value was selected based on the years of experience with residual vectors and it exceeds the number in any practical application. If this number is exceeded then the user will get a message in the f06 file:

=================================================== FATAL MESSAGE: TOO MANY DOFS ARE IN THE RESVEC AMLS PUTS ALL RESVEC DOFS IN THE RESIDUAL STRUCTURE MOST PROBABLY LOADS HAVE BEEN APPLIED TO DEPENDENT DOF ON RBE3. MAKE REFGRID INDEPENDENT, IF POSSIBLE! NUMBER OF ALLOWABLE RESVEC DOFS = 500 NUMBER OF RESVEC DOFS = 666 ===================================================

As mentioned in the message above, owing for the need to calculate the static response for the residual loads exactly, CDH/AMLS places all the DOFs loaded by residual load in the last (residual) substructure. Since CDH/AMLS works primarily with very small substructures (approximately several thousand DOFs), a larger value than 500 for the residual load DOFs may cause some problems. If the user insists on having a number greater than 500, it can be changed, but only on request to CDH. The above message is usually caused by applying a load to a dependent DOF of an RBE3, which has over 500 independent DOFs. A simple UNIX script, rbex, provided in the CDH/AMLS delivery package, will read the messages in the .f06 file to help the user identify which of the RBE3 elements are connected to how many independent grid points. There are other reasons that all RBE3s should be written such that the REFID is an independent grid point. This means that each RBE3 should have a “um” continuation record that will indicate which of the independent DOFs will become dependent. Please refer to Remark 3 in the Nastran QRG for the rule governing this action. An alternative approach available in both MSC and NX Nastran is to use the AUTOMPC option. In either case, the user must provide additional input. It has been a well-accepted fact that adding any number of residual vectors will never cause bad results. Some vectors may not help much, but they should never generate a worse solution. That, of course, is true in a perfect numerical world with infinite precision. However, there have been a number of cases where adding some residual vectors indiscriminately has, in fact, generated a nonsense solution. Thus, one should always keep in mind the following statement: residual loads are a MUST for accurate point mobility response, and in general, do not help much the transfer mobility accuracy. Only the addition of more modes will improve the transfer mobility accuracy.

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11. CDH/AMLS SMP Parallel

CDH/AMLS was designed to work very efficiently with multiple CPUs in Shared Memory Parallel (SMP). To instruct CDH/AMLS to use multiple CPUs is very straightforward. All one does is specify parallel=ncpus on the Nastran JCL command line, e.g.,

nastran jid=job.dat mem=200 parallel=ncpus del=AMLS_2005 ….

This approach requires that the user have a valid license to run Nastran in an SMP mode. There may be users that purchase an SMP CDH/AMLS license, but do not have an SMP Nastran license. In this case, the above command would cause Nastran to fail with a FATAL message that the SMP capability is not enabled. To avoid this problem, the user may specify a Bulk Data parameter, amlsncpu, e.g.,

param, amlsncpu, 4

Thus, Nastran would use a singe CPU while CDH/AMLS would use four CPUs.

12. I/O Enhancement for IBM and SGI Computers

IBM and SGI each provide special I/O software packages that improve the I/O performance of any 3rd party software, such as CDH/AMLS. IBM and SGI refer to their respective packages as MIO and FFIO. AMLS is capable of using both of these packages. Essentially, both use some of the available computer memory for I/O buffering which can reduce the overall run time. (Note that this capability is built into most Linux operating systems, but it is not controllable by the user.) The amount of memory reserved by MIO and FFIO are controlled by Bulk Data parameters, mio and ffio, respectively.

param, mio, 2000. [default=1000.] param, ffio, 128 [default=0]

Note that the mio parameter must be specified as a real number, including a (.) whereas the ffio parameter is an integer. The units for mio are in MB and for ffio the reserved memory is ffio *16 MB. The ffio default is set to 0 because its effect is very dependent on the particular operating system environment. Thus, one can experiment with some different values to determine if ffio has a positive effect, i.e., reduction in solution elapsed time. Suggested values for ffio are numbers with power of 2, e.g., 128, 256…

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13. Use of MFLUID

MFLUID capability in Nastran is typically used to model fluids such as gasoline in a fuel tank. It has both compressible and incompressible options, though most of the time it is used with the incompressible option. In this case, MFLUID generates a so-called “added mass matrix” or Virtual Mass (VM) which is completely full and its size is equal to the number of wetted structure grid points times 3 (for 3 translational DOFs). Unfortunately, this completely full matrix for the fuel tank, for example, can dramatically increase the eigenvalue extraction time if one uses Lanczos. It will also increase the CDH/AMLS time significantly. There are some alternative options built into Nastran that are controlled by a parameter VMOPT. If VMOPT=1, then the virtual mass will be included in the mass matrix at the same time as all other mass elements, thus the component modes will reflect the virtual mass. If VMOPT=2 the modes of the structure or component without the fluid are computed first (“dry” modes). The fluid effects are added in the modal basis during the residual flexibility computation to produce the “wet” modes for the component. Both eigenvalue tables are printed, allowing comparison of the dry and wet modes. The wet modes are used in modal dynamic analysis. The cost savings result from the dense VM matrix being kept out when computing dry modes in the physical basis. Its presence can increase memory and computation times by an order of magnitude. However, based on the CDH experience, neither of the VMOPT values led to a satisfactory solution. If VMOPT=1 is used, the cost is excessive. In addition, if VMOPT=2 is used, the CPU time is reduced, but the solution is not very accurate. Consequently, CDH has developed a special technique, which is controlled by the parameter, mfluid, and an optional parameter, scalec. Their respective values are:

param, mfluid, [amls, modes, mfreq], (default=modes) param, scalec, 0.05 (default)

If the value of mfluid is amls then the solution is exactly the same as specifying VMOPT=1 in for standard Nastran operation. However, the major difference occurs when the value of mfluid is either modes or mfreq. In either of these two cases the VM matrix is modified by the parameter scalec. The function of this parameter is to temporarily eliminate a significant number of terms from the virtual mass (VM) matrix thus reducing its density, i.e., VM = VM1 + VM2. In the f06 file one will find the

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message:

=================================================== ACTUAL VIRTUAL MASS: COLUMN 1 (X,Y,Z) AMLS VIRTUAL MASS: COLUMN 2 (X,Y,Z) =================================================== MATRIX VMASS (GINO NAME 101 ) IS A DB PREC 2 COLUMN X 3 ROW RECTANG MATRIX. COLUMN 1 ROWS 1 THRU 3 ------------------------- ROW 1) 1.8523D-02 3.4813D-02 5.2897D-02 0COLUMN 2 ROWS 1 THRU 3 ------------------------- ROW 1) 1.7791D-02 3.6005D-02 4.5756D-02

The message above indicates the total VM in the X, Y, and Z directions and the actual VM1 used by CDH/AMLS after the modification by scalec. Clearly, the values are very similar in magnitude. The structure eigenvalues are determined by CDH/AMLS using the matrix VM1. Subsequently, the eigenvectors and eigenvalues returned from CDH/AMLS are modified by the so-called “modal correction method” if param, mfluid, modes. Otherwise, if param, mfluid, mfreq, the matrix !TVM2! is added as a direct input matrix (M2PP in Nastran syntax) to the modal response equations. The question one may ask is: Which value of parameter mfluid should one use? The first decision to be made concerns the kind of an analysis one wants to perform. If it is SOL111 then the following line of reasoning applies. Assume that some kind of damping exists, BJJ matrix (cdamp2, etc.) or the structural damping matrix, K4JJ, (caused by elemental damping, such as Ge on MAT1 data or Ge for PELAS). (Note, the presence of param, G, ALONE does not matter). The choice is now obvious. Set param, mfluid, mfreq. This means that the VM matrix from MFLUID will be added to the modal mass matrix. Note that in this case the modal mass matrix will be absolutely full, i.e., 100% dense, but that is not a problem because the modal equations are already fully coupled owing to the presence of BJJ or K4JJ. This is the cheapest solution by far and just as accurate as any other approach. However, if one is running SOL 111 and there is no damping as described above (modal equations are uncoupled) or if one is using SOL 103 then the choice becomes more problematic. Now one can specify either

param, mfluid, [amls, modes]

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If one chooses “modes” then CDH/AMLS will calculate the structural modes with VM1 (thus making AMLS very efficient, in other words, there is very little change in the AMLS run time with or without MFLUID) and after the eigenvectors are returned to Nastran, a further eigenanalysis will be performed (modal correction method) to modify the modes by adding VM2. This approach works well if the number of output DOFs is relatively small, <100000, irrespective of the number of modes that were calculated by AMLS. However, if one sets DISP=all, this could become very expensive if there are many modes, e.g., (> several thousand). If there are few modes (< 100) and DISP=all, then this approach would still be acceptable. Alternatively, if one has DISP = all and many modes (> several thousand), then the only choice is param, mfluid, amls. This means that the full VM matrix is passed on to CDH/AMLS. However, this will significantly increase the elapsed time for CDH/AMLS.

14. Rigid Body, Massless Mechanism Modes, and Singular Stiffness

For some problems, CDH/AMLS will issue several possible warning messages that are listed in the *.log file and which require some explanation. There are three possible warning messages. The first is:

**********************AMLS WARNING********************** Singularity found in the stiffness matrix ******************************************************** Grid point Solution to problem 4362 Apply SPC

This is the simplest to explain. It means that there is one or more (most likely one) DOF associated with grid point 4362 that has both a singular stiffness and mass and CDH/AMLS will automatically apply an SPC to that DOF (very similar to autospc in Nastran). There are occasions where Nastran does not manage to constrain all singular DOF. Typically, this will be a rotational DOF. The second warning message that may be in the log file is:

**********************AMLS WARNING********************** Singularity found in the stiffness matrix ******************************************************** Grid point Solution to problem 143548 Treat as a mechanism

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This message means that for some DOF of grid point 143548 (probably a rotational DOF) there exists a vector x such that xT Kx ! 0 , but xT Mx ! 0 , thus CDH/AMLS treats this DOF as a mechanism. There is no need for any user intervention; however, one should review the model near this grid point. The user should check if this is a mechanism one intended, such as a steering wheel mechanism or is a spurious mechanism that should not exist. The third possible warning message is:

**********************AMLS WARNING********************** Rigid body mode(s) found for a non-root substructure ******************************************************** One or more rigid body modes were found for a non-root substructure in the substructure tree.

This indicates that: - there is a portion of the structure that is free to rotate and/or translate relative to the rest of the structure (a mechanism), or

- the finite element model does not represent the structure, as it should.

For each rigid body mode, the eigenvalue and the grid point associated with the largest entry in the associated eigenvector are printed below in case this information is useful for addressing modeling deficiencies:

Eigenvalue Grid Point ------------------------------ -7.408036E-06 146435

Modes in non-root substructures may lead to an AMLS error. If an error related to these modes is found, try one of the following:

1. Try setting k6rot equal to 0., and snorm equal to 45.:

param, k6rot, 0. param, snorm, 45.

2. In the Case Control section, before any subcases, create a SET containing the above grid points and set DATAREC equal to the SET's setid. As an example, if the above grid points are 12345 and 23456, choose a SETid of, e.g., 98765, and insert these lines in Case Control before any subcases:

set 98765 = 12345, 23456 DATAREC = 98765

3. Contact mladen.chargin@cdh-ag.com for assistance with this problem.

In this particular case, neither the stiffness nor the mass matrices are singular at the grid point mentioned above. However, there exists an eigenvalue whose value is -7.408036E-06 and this particular mode is considered a “rigid body mode”. This fact will not cause an error in phase3 but it may later cause a problem in phase4. If it does, one would get a fatal message in the log file from phase4 that would indicate the following:

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***********************AMLS ERROR*********************** Mass matrix singularity associated with rigid body modes See preceding AMLS Error message. ********************************************************

Since this message would indicate that AMLS stopped with a fatal error, the user should follow the instructions in Steps 1 or 2 above. The most likely solution would be to use instruction in Step 2, which means that for the example above, the user would define the following two lines before all subcases in the Case Control deck:

set 666 = 146435 DATAREC = 666

If everything fails, then simply go to Step 3.

CDH/AMLS

19

References:

1. J.K.BENNIGHOF and C.K. Kim, “An adaptive multilevel substructuring

method for efficient modeling of complex structures”, Proceedings of the AIAA, 39TH SDM Conference, 33rd SDM Conference, 1631-1639, Dallas, TX, 1992.

2. J.K.BENNIGHOF and M.F. KAPLAN, “Frequency sweep analysis using multi-

level substructuring, global modes and iteration”, Proceedings of the AIAA, 39TH SDM Conference, Long Beach, CA 1998.

3. J.K. BENNIGHOF, M.F. KAPLAN, M.B. MULLER, M. KIM, “Meeting the NVH

computational challenge: automated multilevel substructuring”, Proceedings of the International Modal Analysis Conference XVIII , San Antonio, Texas, 909-915, 2000.

4. J.K. BENNIGHOF, M.F. KAPLAN, M. KIM, C.W. KIM, M.B. MULLER,

“Implementing automated multi-level substructuring in Nastran vibroacoustic analysis”, Proc. of the SAE Noise and Vibration Conference Traverse City, Michigan, SAE paper 2001-01-1405, 2001.

5. J.K. BENNIGHOF AND R.B.LEHOUCQ, “An automated multilevel

substructuring method for the eigenspace computation in linear elastodynamics”, SIAM Journal Sci. Comput., 25, 2084-2106, 2004.

6. H. Voss, “Automated Multi-Level Substructuring (AMLS), Technische

Universität Hamburg-Harburg, Lecture Presentation. 7. M. Bennur, “Superelement, Component Mode Synthesis, and Automated

Multilevel Substructuring for Rapid Vehicle Development”, 2008 SAE World Congress, Paper 08B – 11.

CDH/AMLS

20

Appendix A: Enforced Motion for Modal Frequency Response

After all the constraints are eliminated, the equation of motion is written in the Nastran terminology as

!"2M

aa[ ] ua{ }+ i" Baa[ ] ua{ }+ K

aa[ ] ua{ } = Pa{ } (1)

Equation (1) can be partitioned into two sets, c and b, where the c set is free to displace and the b set consists of all the degrees of freedom that have a prescribed motion, i.e., displacement, velocity, or acceleration. The partitioned equation is written as

!" 2M

ccM

cb

Mbc

Mbb

#

$%

&

'(uc

ub

)*+

,+

-.+

/++ i"

Bcc

Bcb

Bbc

Bbb

#

$%

&

'(uc

ub

)*+

,+

-.+

/++

Kcc

Kcb

Kbc

Kbb

#

$%

&

'(uc

ub

)*+

,+

-.+

/+=

Pc

Pb

)*+

,+

-.+

/+ (2)

Assume that ub{ } is a known function of frequency and u

c{ } is represented by the normal modes plus static shapes, i.e.,

uc{ } = !c[ ] qi{ }+ Gcb[ ] ub{ } (3)

Each column of Gcb[ ] represents a static deformation of the c set due to a unit

motion of a single degree of freedom in the b set. The Gcb[ ] matrix is obtained

from an extra Guyan reduction (static condensation) represented by the following equation

Kcc[ ] Gcb[ ] = ! K

cb[ ] (4)

The flexible modes are calculated from the c set, or

Mcc[ ] !c[ ] "2#$ %& = K

cc[ ] !c[ ] (5)

Substitute Eq. (3) into Eq. (2) and premultiply by !c[ ]T to obtain the equations for

the dynamic response in the following form:

!" 2

miqi + i"biqi + kiqi = #c[ ]T!Fc{ } (6)

where

!Fc{ } = P

c{ }+! 2M

ccGcb+M

cb[ ] ub{ }" i! BccGcb+ B

cb[ ] ub{ } (7)

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21

Since this is a harmonic analysis, the acceleration and velocity may be obtained from

!!ub{ } = !"

2ub{ }and

!ub{ } = i! u

b{ } (8)

CDH/AMLS

22

Appendix B: CDH/AMLS Parameter List

Parameter Default Value

Comment

SS2GCR 5.0 Scale factor for determining the cutoff frequency of the substructure modes. SS2GCR*cutoff frequency for the global modes (F2 on EIGRL entry)

LGMASS 1.0E+5 Used to eliminate any loads on DOFs that have a mass larger than the value of LGMASS during the calculation of residual vectors.

AMLSCMP YES Use CDH/AMLS for the eigenvalue calculation of superelement component modes. Set to NO for standard eigenvalue operation.

AHH NO Set to YES if panel participation output is requested. If set to NO (default), panel participation results will be incorrect.

AMLSNCPU 1 Set to the number of CPUs to be used for the CDH/AMLS calculations. Default set to the value of “parallel=ncpus” on the nastran submit command.

MIO 1000.0 Available computer memory for I/O buffering on IBM machines. The units for MIO are in MB.

FFIO 0 Available computer memory for I/O buffering on SGI machines. The reserved memory is FFIO*16 MB. FFIO is an integer parameter.

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23

MFLUID MODES Options are:

AMLS - Same as specifying VMOPT=1 in for standard Nastran operation, virtual mass will be included in the mass matrix at the same time as all other mass elements

MODES - Fluid virtual mass (VM) is split into two parts: VM= VM1 + VM2. VM1 is used in the CDH/AMLS modes calculation. After the eigenvectors and eigenvalues are returned from CDH/AMLS, they are modified by the “modal-correction method” to add VM2.

MFREQ - Fluid virtual mass (VM) is split into two parts: VM= VM1 + VM2. VM1 is used in the CDH/AMLS modes calculation. The matrix !TVM2! is added as a direct input matrix (M2PP in Nastran syntax) to the modal response equations.

SCALEC 0.05 Used with PARAM, MFLUID to split the fluid virtual mass (VM) into two parts: VM= VM1 + VM2