Technical Application Note: Efficiency Features Added for CMS, Virtual Fluid Mass, vma.v707, TAN 4349
M. A. Gockel, December 20, 1999
References:
TAN 4009, MODAL DAMPING IN COMPONENT MODES, M. A. Gockel, Revised April 12, 1999
1. INTRODUCTION
A new path in modal dynamic analysis is provided for Version 70.7 in alter package vma.v707. This path provides a more efficient method of computing virtual fluid mass (VM) effects, and several efficiency features that can reduce solution time significantly for all methods of modal analysis.
This work has been delivered by parts in the past in several breadboard DMAP alters. A new DMAP alter that combines all of these features is described here. Although this work is defined under the heading of VM, the other capabilities provided will be of more interest to most of the user community. VM is merely the straw the broke the camel’s back with regard to the need for better integration and efficiency for the many techniques used in CMS. The modal damping for CMS is now integrated in V70.7. The efficiency improvements described here are now being integrated into the next major release of MSC/NASTRAN.
2. INPUT
CMS Modal Damping
PARAM, SESDAMP,YES For this value any SDAMP Case Control Commands apply to all superelements including the residual structure. For the default value of 'NO', they apply to the residual structures only, as is the case for systems prior to V70.7. This will be the user interface for some time to come, for both the built-in v70.7 capability and when used with this alter package.
VM Efficiency Improvements
PARAM, VMOPT, [integer] allows the values of 0 and 1.
VMOPT=0 (default) The modes of the structure only are first computed without VM effects to obtain Ritz vectors to use in a generalized coordinate solution, similar in concept to the older Generalized Dynamic Reduction method. The modes are re-computed in the generalized basis after the VM effects are added. This method generally costs slightly more than when production V70.7 is not set, but the accuracy of the solution is much better. When an o-set exists (usually because of the presence of ASETi-type entries, or for superelement reduction) the VM effects are added to the component modes (o-set level). When there is not an o-set the VM effects are added in the residual structure Phase II operations (a-set level), after the structure-mass-only eigensolution. The rigid body mass of the VM is output automatically, computed about the location of the grid point listed on PARAM, GRDPNT.
VMOPT=1 This path is unchanged except for minor efficiency improvements. The virtual mass effects are added before any constraints are processed (g-set level). No approximations are made in the solution. This is the most expensive option in terms of computer resource requirements and computation time. It is practical for only relatively small-size models, or when the VM is in one superelement only, with most elements in the superelement wetted.
PARAM, CMSMSG, YES. Prints many intermediate matrices. Useful for understanding small pilot models. Produces excessive output for moderate- through large-size models.
3. OUTPUT
Most output is unchanged. Exceptions are described here.
When PARAM, GRDPNT is present, the structural weight is added to the VM effects for PARAM, VMOPT, 1, as is true prior to V70.7, for the traditional weight summary output. However, for the default value of PARAM, VMOPT, 0 the structural weight and VM weight are output separately. The rigid body mass of the VM is always printed with a matrix format in place of the format normally used for the PARAM, GRDPNT option.
4. EXAMPLES
Problems useful for demonstration purposes include vm00.dat, vm01.dat
These are classic verification problems, based on a single element test. Both models are identical except that the default value of param, vmopt is used in vm00 (approximate solution) and the value of 1 is used in vm01(exact solution). Both problems give identical answers with and without the alter package.
5. LIMITATIONS
All methods, new and old, do not include VM effects in spc or mpc forces except for vmopt=1.
For vmopt=0, if any wetted elements are in a superelement, all wetted elements selected by the same MFLUID case control command must also be interior to that superelement. If any are not a blanket warning is given, and results may be of poor quality.
6. THEORY
The number of matrix triple multiplications of the form PHIoz*Koo*PHIoz are reduced by eliminating the old dynamic reduction modules, which were designed to compute approximated eigenvectors. Although the old GDR user inputs are still honored, they call the more modern Lanczos eigensolution which produces exact eigensolutions, where the triple product above is known a priori to be a diagonal matrix, ignoring truncation effects. The method for discarding redundant Ritz vectors produced by the residual vector capability has been improved in reliability and efficiency, including a second re-orthogonalization of the Ritz vectors. Other general cleanup work should result in lower cost analysis for most modal dynamic analysis. PARAM, MODEFF, is set with a default value of 'YES'. It is a vestigial effect from development, and should not be changed from its default value.
An omitted set (o-set) occurs when OMITi entries, or their complement the ASETi-type entries are present. They are also always present for superelement reduction, which uses identical equations for solution. The following discussion therefore applies to both o-set processing and superelement processing. This process is given the generic name Component Mode Synthesis (CMS) processing. When all options of CMS reduction are requested, the eigenvectors including the omitted set are computed first, followed by shape functions derived from static analysis. This collection of static and dynamic shape functions are further reduced and refined to eliminate generalized functions that may not represent independent generalized DOFs. Special effects such as Virtual Fluid Mass (VM) are added at an appropriate place for a balance of efficiency and accuracy. After modes including the o-set DOFs are complete, the remaining terms are included in the a-set, and the combined effects of the total model are computed.
The equations below are given in their most concise form. In the DMAP used to implement these equations an expanded form using the partitions of the matrix are used when this order of operations provides improved efficiency. Comments in the DMAP describe the relation between a single equation in this theory section and the expanded set of equations that replace it in the DMAP implementation.
The m-set and s-set are eliminated from the g-set equations conventionally to provide the f-set equations of motion. At the time these operations start the matrix of static shapes due to unit motion of the t-set [Got] are also available. The equations for frequency response are:
[Kff +i*omega*Bff –omega**2*Mff] uf = Pf
Similar equations are used for transient response.
The a-set is partitioned from the f-set to produce the o-set, generally much larger in size than the a-set. The o-set coefficients are reduced to a-set coefficients by combinations of static and dynamic reduction methods. The component modes of the v-set, a set that includes the o-set, are computed in the CMPMODE SubDMAP. Several matrix subscripts are used locally in this SubDMAP. The letters of these subscripts are not synonymous with members of the USET table, although some of their names are the same as USET member names for matrices that pass through SubDMAP calling sequences. The matrices using the local subscripts are also local to this SubDMAP. Their subscript names are preceded by N. For example, there is a matrix in the SubDMAP named PHIaz, where “a” refers to a member of the uset set table (the analysis set, or a-set), and z is the current retained set if columns. There are local parameters of the form N* that give the current size of the set.
NzNumber of component modes computed.
NpNumber of static loading functions selected by the user, then retained.
NyNumber of combined shape functions for output.
NcNumber of combined shape functions plus Got, for linear dependence checks
All four numbers are dynamic in the SubDMAP, as shape functions are discarded at several steps to delete those that do not describe linearly independent shapes. The discarding process requires some judgement for sorting out shapes that may contain independent data from those that are merely numerical noise. This process is controlled by parameters that the user may set. The default values for these parameters are selected to provide stable equations for solution at some risk that independent vectors may be discarded.
The f-set variables are partitioned into the v-set and its complement with respect to the f-set, symbolized as the x-set here. In terms of a progression towards the mutually exclusive sets,
uo
uf = [ ]
ua
uq
ua = [ur]
ub
uc
uo
uv = [ur]
uc
ub
ux = [ ]
uq
The real eigenvectors of the v-set are computed,
[Kvv – lambdaz*Mvv]*PHIvz
where Nz is the number of eigenvectors computed. Some of these shapes may not be linearly independent with respect to Got. These are removed by the following equation:
PHI1oz = PHIoz - Gorc*PHIrcz
The PHI* quantities on the right side are partitions of PHIvz, and Gorc contains the subset of columns of Got associated with its c- and r-subsets. The action of this reduction may make some column of PHI1oz merely numerical noise. This is particularly true of rigid body modes, and sometimes includes other modes. The columns that appear to be numerical noise are discarded, resulting in the final component modes PHIoz. CMPMODE returns the dynamic shape functions PHIoz and the corresponding roots of the eigensolution in the diagonal matrix RUTZ.
The residual vector capability is performed in the RESVEC SubDMAP, along with the virtual fluid mass calculation and various cleanup functions. This produces the improved eigensolution matrices including residual vector effects, some further discarding of generalized shapes that are not linearly independent with respect to other shapes, and a final re-orthogonalization. Static loads are assembled for computing static shape functions for use as residual vectors. They include Pg, the loads selected by the user with LOAD entries in subcase commands, Pr for inertial loads computed from rigid body accelerations, and P6, for unit loads applied at points selected by USET, U6 entries. These loads are reduced to o-set size, and appended to form Ppo,
Ppo = [Pgo | Pro | P6o]
The loads are swept against the mass matrix to identify loads that are not linearly independent with respect to one another,
P2po = -Moo*P1po + P1po
P2po is then renamed to Ppo. Columns that are numerical noise are deleted, and Np is reset when forming Ppo. The static shapes u1po are solved from the equation
Koo*upo = Ppo
These shapes are appended to the boundary interface modes Got and the dynamic shapes PHIoz to form the complete set of generalized shape functions PHIoc,
PHIoc = [Got | PHIoz | uop]
The generalized mass matrix Mcc is computed,
Mcc = PHIoc’*Moo*PHIoc
Mcc is decomposed into its factors Lcc and Dcc, where Lcc is lower triangular and Dcc is diagonal. Rows and columns where diag(M1cc)/Dcc are larger than a large number are regarded as rows that are linearly dependent on prior rows. They are discarded to form Myy and PHIoy, and Ny is reset. The generalized stiffness is then calculated,
Kyy = PHIoy*Koo*PHIoy,
and the improved eigensolutions in the generalized “y” basis are computed,
[Kyy –lambday*Myy]*PHIyy = 0
These vectors are used to take linear combinations of the physical basis shape functions to produce the final physical basis eigensolution, including residual vector effects,
PHI = PHIoy*PHIyy
DMAP IMPLEMENTATION
New SubDMAPs
VMGEN Generates the components of the VM matrix in the MGEN module (Gge, Mee, etc.) and sets control parameters.
VMSOLV Accepts the reduced structural mass Mxx, and generates the VM mass matrix Mvxx in the same basis, then outputs their sum, as well as other data blocks needed for functions such as data recovery. Great care is made with the order of operations to minimize computation costs. The exact same results as the older MGEN1 SubDMAP are produced except for numerical truncation effects. It is called at either the g-set size (VMOPT=1) or at the a-set size (VMOPT=0). The presence or absence of omitted points also influences this decision, as described by comments in VMGEN, the SubDMAP where the VMSET parameter is set to control this decision.
MAKVE1C, FBSX Utilities for VMSOLV
Modified SubDMAPs
SEMG. The components of the VM matrix are always computed at this step, even when the VM is inserted in a downstream SubDMAP. This allows reset of a new parameter VMSET when the user selects VM but neglects to input ELIST entries. VMSET is set to 'none' in this instance. It is a global NDDL parameter, and is transferred to many other places. This organization could also facilitate computing fluid matrices in Phase 0 to allow VM to span superelements. (See Section 8 for details).
PHASE1B Some forms of modal damping are added here.
SEMRM The main branch between the new and old methods is made here. The new method calls new SubDMAPs based on old SubDMAPs with similar names.
SEMR3VM This is a rewrite of the old SubDMAP with the same root name. Changes made downstream allow elimination of most triple products of the form PHI'*M*PHI at the v- and o-set levels, that is, for most DOFs of the models. The logic for expanding or contracting the generalized coordinate matrices to be of q-set size is concentrated in one location. Reforms in the modules described below allow elimination of 20% of the dmap lines in this rewrite. It calls the two revised SubDMAPs described next.
CMPMDVM The CMPMODE SubDMAP , Version 70.6, performs the following operations:
The number of generalized coordinates needed to meet a maximum frequency of interest (fmax) is determined by the DYCNTRL module, based on user input on selected DYNRED entries.
The factor of the v-set matrix is found by decomposition.
Approximate eigenvectors are found by the DYNREDU module.
A generalized stiffness and mass matrix are determined from the approximate eigenvectors.
An auto-omit operation is performed when a tridiagonal (non-Lanczos) eigensolution is requested.
An eigensolution is found in the reduced basis.
An improved solution is found for the v-set vectors from the generalized solution and the approximate v-set eigenvectors.
The r- and c-set components are swept out to obtain o-set eigenvectors
CMPMODE has been partially modernized for v70.7. A more modern form CMPMDVM is also provided in a parallel path when modern virtual mass is requested (PARAM, VMOPT, 0). Efficient CMS processing is always provided with the alter package. The modified steps are:
The number of generalized coordinates needed for GDR is determined in the same way.
Steps b-f above, which use the DYNREDU module, are replaced by an exact eigensolution using the Lanczos method, the hard-wired selection for this path. This eliminates the need for computing approximate eigenvectors, and for the auto-omit step. (Note that the auto-omit operation is retained for the a-set eigensolution in the MODERS SubDMAP.) All existing GDR input files will function without changes. They will provide better answers.
The o-set sweeping reduction is done as before, with the deletion of the first Nr eigenvectors added, a step moved from the RESVEC SubDMAP in the interests of simpler DMAP communication.
An additional step is used to remove eigenvectors that are only numerical noise after the sweeping operation is completed. These noise eigenvectors sometimes occur when the entire boundary is in the c-set.
f. RSVCVM The RESVEC SubDMAP, Version 70.6, performs the following operations:
Nr modes are discarded, where Nr is the number of DOFs listed on SUPORTi-type entries.
Eigenvector components from the CMPMODE SubDMAP that were calculated at v-set size are reduced to o-set size.
Shapes due to user-defined loads (optional) are calculated by static analysis.
The eigenvector and static shapes are normalized to unit generalized mass.
The two sets of shapes are checked against themselves and each other for linear independence. Static shapes with little independent content are discarded.
The static shapes are used to provide improved eigensolutions (o-set path) by a second eigensolution, or to provide additional quasi-eigenvectors (a-set path) by appending (hopefully) uncoupled DOFs for the static shape generalized coordinates to the modal generalized coordinates.
RESVEC was left unchanged for v70.7. A more modern form RSVCVM is also provided in a parallel path when using the alter package.