fe_cyclicb

SDT-rotor Contents Functions

fe_cyclicb

Purpose

Support for advanced cyclic symmetry computations.

Description

fe_cyclicb groups advanced commands used to build and manipulate reduced order models of single symmetric structures and their assemblies. For more details on the associated theory you can refer to [8].

Rotor Construction

DiskFromSector

This command builds a disk/rotor model from (a) physical sector model(s). Shafts can be generated by combining multiple calls to disk from sector. Once, disks are combine, their connection trough rim models is described in section 3.1.6.

Command DiskFromSector also handles the construction of the cyclic sector models. Cyclic symmetry information can be already given in the sector model (calls to fe_cyclic('build') done beforehand) or done in the command. In the later case, an optional epsl tol can be declared so that it is propagated to the subsequent call to fe_cyclic('build epsl tol',...), where tol is the desired tolerance for left-right interface node matching.

The example below demonstrates the capability of the function for two disks with 7 and 10 blades respectively.

 cf=demo_cyclic('buildstep0');
 sector1=cf.Stack{'disk1'};
 sector2=cf.Stack{'disk2'};

 % build disk1 from sector1
 model=fe_cyclicb('DiskFromSector epsl 1e-6',[],sector1);
 % build disk2 from sector2 and append to disk1
 model=fe_cyclicb('DiskFromSector epsl 1e-6',model,sector2);
 fe_cyclicb('DisplayFirst',model) % Avoids full display for large models

In cases when cf already contains one sector per disk, the shaft model can be created in a single operation with the command fe_cyclicb('diskfromsector',[],cf,{'sel_disk1','sel_disk2',...}); where sel_diski selects the sector model of disk i. The example below illustrate this by putting the two sector models into a single one prior to the rotor assembly.

 cf=demo_cyclic('buildstep0'); % See sdtweb demo_cyclic('Step0')
 sectors=cf.Stack{'disk1'}; % Build a model with two sectors
 sectors=feutil('addtest',sectors,cf.Stack{'disk2'});
 sectors.Stack={};
 cf.model=sectors;

 % build rotor from sectors and auto display
 fe_cyclicb('DiskFromSector epsl 1e-3',cf,{'group1:2','group3:4'});

During the build process, sectors are automatically renumbered so that node numbers are left interface, interior, right interface (in order matching that of the left interface). The renumbering can be forced with the -renumber option. This allows to have nodal overlap between the superelements of two adjacent sectors. The command then adds a mpc,diski_end multiple point constraint to account for the fact that the disk is closed circumferentially.

Mesh

Meshing utilities See fe_cyclicb Mesh.

ConnectionRing

ConnectionRing builds a "ring connection" where the structure is fixed axially and radially on a set of nodes and first point only in tangential direction

Display

Display commands group tools to build mesh views specific to disk assemblies.

[def,ENER]=fe_cyclicb('Displaysel',cf,def,Sel,'enerkdens') is used to recover mono-harmonic solutions on a partial selection For details on monoharmonic solutions see section 3.5. Examples can be found in section 3.5.2, section 3.5.3, ...
Sel is a cell array specifying how each stage is displayed. In the example from section 3.5.1, one uses
```
  Sel={'','EltName SE';    % Keep only SE for display (no interstage rim)
       'disk1(1:2)','groupall'; % all elements from sectors 1:2
       'disk2(1:3)','groupall'};% all elements from sectors 1:3
```
See sdtweb fesuper#SeBuildSel for details on partial superelement display and more examples on the way to define Sel). The last command can be any valid fe_stress command. Without output argument the result is displayed.
```
 [cf,def]=demo_cyclic('buildstep1');
 def=fe_def('subdef',def,def.data(:,2)>=0); % remove fixed edge solutions
 fe_cyclicb('Display',cf,def);fecom('ColorDataEvalA');

 fe_cyclicb('DisplaySel',cf,def,cf.Stack('ViewMesh'));
 fecom('ColorDataEvalA');
```
During the command one defines SE.cGL0 corresponding to a rotation by one sector. And the SE.Alpha for the harmonic shift. fesuper('sedefinit -rot',cf); is then used to define a restitution by disk. The SeRestit then contains the def so that SeDefinit is performed for every restitution.
fe_cyclicb('Display',cf,def) defines a full disk selection within feplot. def=fe_cyclicb('Display',cf,def) is used to recover full motion from mono-harmonic solutions. For large models, restitution on the full shaft model may be very costly (remember that one vector for 1e6 DOF requires 7.6 MB) and DisplaySel is typically preferred.
fe_cyclicb('DisplayAllEdges',cf[,sel]) displays 2D cuts of the disks specified in the cell array sel, with a typical entry {'diski'} to display only disk i (default displays all disks). This cut is basically the projection of the right interface (or equivalently the left interface when meshes are compatible) to the plane θ=0. Such a view is particularly well suited to the definition of the inter-stage rim nodes in MeshAddRim as well as to the construction of viewing meshes in MeshRimLine2Patch. Note finally that this view keeps the elements in their original groups.
```
 cf=demo_cyclic('buildstep0');
 fe_cyclicb('DisplayAllEdges',cf);
 fecom('colordatag');
```
fe_cyclicb('DisplayFirst',cf[,sel]) provides a simple command to display the first sector of each stage (cf can be replaced by a model resulting from fe_cyclicb DiskFromSector). A selection can also be specified to restrict the view to a subset of stages.
```
 cf=demo_cyclic('buildstep0');
 fe_cyclicb('DisplayFirst',cf,{'disk2'});
```
fe_cyclicb('DisplaySkin',cf[,sel]) displays the outer enveloppe of the selected disks in sel with the inter-sector common surfaces removed.
```
 cf=demo_cyclic('buildstep0');
 fe_cyclicb('DisplaySkin',cf,{'disk1'});
 fecom('showpatch');set(cf.o(1),'FaceAlpha',.33);
```

fe_cyclicb('DisplayInterDisk',cf,nodes) displays the two ring surfaces used to define the inter-stage volumic interface.

 cf=demo_cyclic('buildstep0');
 
 % rim nodes
 n1=feutil('getnode NodeId',cf.mdl,[12 18 24 1127 1133 1139]');
 fe_cyclicb('DisplayInterDisk',cf,n1);

fe_cyclicb('DisplaySymmetrySurface',cf) displays the nodes in a cyclic symmetric condition as colored surfaces.

Mono-harmonic solutions

These commands apply to sector models used to compute mono-harmonic eigensolutions.

ShaftEig,ShaftTEig,ShaftSolve

These commands compute mono-harmonic solutions with specified Fourier harmonics (classical cyclic solution for single stage models). For a tutorial on generating the proper models, see section 3.1.6. For the associated theory, please refer to [9].

The calling format is def=fe_cyclicb('shaft Teig delta_list',model);. ShatTeig accepts multiple diameters in the delta_list and packages individual calls that to ShatTeig. For a disk example section 3.5.2.

The main command options are

Diameter -1 is used to ask for the computation of fixed edge modes.
The eigenvalue options 'info','EigOpt' should be set in the model stack.
-ReAssemble forces reassembly rather than reuse of disk matrices that may have been precomputed and saved in the sector superelements.
-thermal takes thermal loading into account. Thermal state should be stored as a case entry called DofSet,ThermalState. See example in section 3.5.1.
for computations at prestressed state curve,StaticState should be defined.
-all xxx
-FixTan is used to enforce no tangential motion of one interface node. This is used for static analysis of freely rotating rotors (example in section 3.5.1).
-NoN stores the imaginary component of the eigenvector using a DOF shift by 50 (thus .51 is the imaginary x translation). This is necessary if further computations require complex fields (by considering different components there is no difficulty storing the spatial Fourier transform of a complex field).
model.Dbfile when the field is defined to a proper file name, intermediate matrices above the preference getpref('SDT','OutOfCoreBufferSize',100) (in MB) are stored in the database file which uses the standard HDF5 based .mat format (MATLAB >= 7.3).
-nlstep tol is used to compute the large non-linear large transformation problem with a tolerance δ q/q<tol. See an example in section 3.5.1.
-soft if present include centrifugal softening effects
For static computations, the centrifugal load is rebuilt at each step using model=fe_cyclic('loadcentrifugal',model); for the rim and sectors.

-batch option is used to compute eigenvectors with multiple diameter in a single run. For large models, this can take many hours so that intermediate file saves are used to allow restarts.

One then typically expects to have set a cf.mdl.Dbfile='file_DB.mat' to allow memory off-loading during the computation.

 cf=demo_cyclic('testrotor 7 -blade -cf 2');
 root=fullfile(sdtdef('tempdir'),'Disk_7_Batch')
 %setpref('SdtRuntime','ExecLocal',1) % May be needed
 % cf.mdl.Dbfile=[root '_DB.mat']; % out of core matrices

 fe_cyclicb('shaft Teig 0 1 5 -batch',[root '.mat']);
 % multiple Disk_7_batch_diam.mat files are generated

 % Now reload pointers to selected solutions
 RO=struct('Fmax',8000,'diams',[0 1 5]);
 d1=fe_cyclicb('DefList',root,RO);

ShaftSeAssemble

fe_cyclicb('ShaftSeAssemble -force',cf.mdl,fname) is used to assemble superelement matrices of each of the disk* superelements. If a curve,StaticState is defined in the model stack, fe_cyclic assemble is used, otherwise SE=fe_mknl(SE,'NoT') is called.

If the -force option is omitted an attempt to reload the variable Stack_SE_diski from the file is first made and assembly is only performed if that variable does not contain the matrices.

If a mdl.Dbfile field is defined, the argument fname may be omitted.

-reset xxx.

Deflist,Def ...

Cyclic symmetry results can be stored in three main forms

the basic form, valid for real vectors only, stores real and imaginary components for the spatial fourier transform as real and imaginary components at a single DOF
the long vector form uses additionnal DOFs shifted by 50. Starting from the basic form one would have d1.DOF=[d1.DOF;d1.DOF+.5]; d1.def=[real(d1.def);imag(d1.def)]
The test for usage of this format is that the last dof is above .05 rem(d2.DOF(end),1)>.5.
the double vector form uses the nominal DOFs of the first sector but store the real and imaginary parts as consecutive vectors.
d1.def=[real(d1.def(:,1));imag(d1.def(:,1)]; d1.data=[d1.data(1,:);NaN NaN]

DefDouble, DefLong, DefBasic commands allow transformations between formats while handling out-of-core files properly.

When reading results, def=fe_cyclicb('DefList','root'); reads all root*.mat files and combines the vectors into a single deformation set in the double vector format. Selection of diameters and frequency range during the read process is peformed using

RO=struct('Fmax',8000,'diams',[0 1 5 18]);
d1=fe_cyclicb('DefList','root',RO);

Specific cases require to sort the output vectors according to the list of diameters specified in the .diams field, especially when one wants to put fixed interfaces solutions first for reduction purposes. The command to use becomes DefListSortDiam.

Full rotor SE model

ShaftPrep

fe_cyclicb('ShaftPrep',cf,def); generates reduced sector super-elements.

Each bladed sector is divided into two regions. A first super-element is attached to the elements with the left interface nodes, it is called the inter-sector super-element. A second one is attached to the remaining elements to form the sector super-element.

A reduced kinematic subspace of the sector super-element (with the definition above) is built from def, disk by disk.

Vectors are first sorted with respect to their contribution to the considered disk if the -svdtruncate option is used.

Then, they are sorted according to their contribution to subsets of physical DOF of the initial sector. If one specifies -mseq 0 (default call), these subsets are

DOFs within either the inter-sector interface (right interface) or the inter-stage interface(s),
remaining DOFs (left interface and interior DOF).

If -mseq 1 is enforced, these subsets are

DOFs common to the inter-stage and inter-sector interface(s),
DOFs within the inter-sector interface (right interface),
DOFs within the inter-stage interfaces,
remaining DOFs (left interface and interior DOFs).

Both these sortings make the subsets of vectors linearly independent from each other. They require that fixed edge solutions are stored at the beginning in def.

The following step is to make the vectors linearly independent within each set. Vectors in sets (1.), (2.) and (4.), when applicable, are processed with an Iterative Maximum Sequence Algorithm ([10]). Vectors in set (3.), when applicable, are processed with a Singular Value Decomposition.

See Ref. [5] for details.

-handle option controls whether the resulting bases are stored in memory or on the disk.

-norestit suppresses the explicit construction of the Restit variable, normally stored in cf.mdl.Stack.

Once the sector superelements have been generated, the disk model is assembled using the subsequent fesuper('fassemble',cf) call which generates the reduced disk model in cf.Stack{'SE','MVR'}.

A compact example is provided below. A fully developed example can be found in demo_cyclic ShaftMulti.

 [cf,def]=demo_cyclic('buildstep1');
 fe_cyclicb('ShaftPrep -svdtruncate -mseq1 -norestit',cf,def);
 fesuper('fassemble',cf);

ShaftLoad,ShaftSELoad

fe_cyclicb('ShaftLoadMulti',cf,data); generates a reduced mono-harmonic load for each disk specified in data. It is a cell array whose a typical line is {'diski',def_i,delta_i} where for each disk i, the shape def_i and harmonic coefficient delta_i are specified.

fe_cyclicb('ShaftSELoad',cf,def); generates a reduced load from its physical counterpart. In practice, it is used for very specific loading cases, e.g. single DOF load or random load.

Developed examples are presented in demo_cyclic ShaftMulti.

% define model
cf=demo_cyclic('buildstep4');

% build the excitation on first all sectors, with specified
% diameters 3 on disk1 and 4 on disk 2
data={'disk1',cf.Stack{'disk1'}.Stack{'Enforce_mode_7'},3;
      'disk2',cf.Stack{'disk2'}.Stack{'Enforce_mode_7'},4
      'disk1l',struct('def',1,'DOF',cf.Stack{'disk1l'}.Elt(2,1)+.01),3;
      'disk2l',struct('def',1,'DOF',cf.Stack{'disk2l'}.Elt(2,1)+.01),4};
def34=fe_cyclicb('ShaftLoadMulti',cf,data); 
def34.def=sum(def34.def,2); % one column in def34.def per row in data
def34.data=[0 0];
fe_cyclicb('fourier 1 -red -cf 1',cf,def34); % checkout shape

% now keep the same shapes but force delta=0 on both disks
data(:,3)={0};
def00=fe_cyclicb('ShaftLoadMulti',cf,data);
def00.def=sum(def00.def,2); % one column in def00.def per row in data
def00.data=[0 0];
fe_cyclicb('fourier 1 -red -cf 1',cf,def00); % checkout shape

% now try a random load
r1=fesuper('fnode',cf.mdl);
r1=[r1(:,1)'+.01;r1(:,1)'+.02;r1(:,1)'+.03];
defrnd=struct('DOF',r1(:),'def',[]);
defrnd.def=randn(length(defrnd.DOF),1);
defrnd=fe_cyclicb('ShaftSELoad',cf,defrnd);
defrnd.data=[0 0];
fe_cyclicb('fourier 1 -red -cf 1',cf,defrnd); % checkout shape

% now consider a physical load on first sector of disk 1
r1=fesuper('fnode',cf.mdl);
r1=[r1(:,1)'+.01;r1(:,1)'+.02;r1(:,1)'+.03];
defsp=struct('DOF',r1(:),'def',[]);
defsp.def=fe_c(defsp.DOF,1+[.01;.02;.03])'*randn(3,1);
defsp=fe_cyclicb('ShaftSELoad',cf,defsp);
defsp.data=[0 0];
fe_cyclicb('fourier 1 -red -cf 1',cf,defsp); % retrieve a Dirac's comb

ShaftFRF [, D, MS, -rest]

This command allows to build the Frequency Response Functions of a rotor model, either full or reduced. A load and a set of observation DOF have to be defined and added to the model with fe_case. The frequency range is stored in the stack as a 'info','Freq' entry.

The general call is

 
 xF=fe_cyclicb('ShaftFRFD',disk,lossfac)
 xF=fe_cyclicb('ShaftFRFD -rest',disk,lossfac,cf,sel)
 xF=fe_cyclicb('ShaftFRFMS',disk,def,damp)
 xF=fe_cyclicb('ShaftFRFMS -rest',disk,def,damp,cf,sel)

The command FRFD assembles the matrices of the model then uses them to compute the response. An optional loss factor can be specified.

The command FRFMS synthetizes the response from a set of modeshapes. A damping ratio for all modes can be specified.

The option -rest restores the response computed on the reduced model to a given selection of physical DOF. Without selection, the response is restored to the whole physical DOF set. This option must be disabled when dealing with a full rotor model.

The example developed in demo_cyclic ShaftMulti builds the sythetized response of a reduced rotor model to a random excitation.

% define model
[cf,load,def]=demo_cyclic('buildstep5');

% Compute the response to the random excitation
mdl=fe_case(cf.Stack('mvr'),'dofload','Load',load);
mdl=stack_set(mdl,'info','Freq',linspace(900,2000,2201));
xF=fe_cyclicb('shaft frfms',mdl,def,.001); 

% select where to restore (upper blade corner)
Sel1={'','eltname SE';'disk1','selface & withnode{NodeId 154}'};
Sel1=fesuper('SeBuildSel -initrot',cf,Sel1);
Sel2={'','eltname SE';'disk2','selface & withnode{NodeId 154}'};
Sel2=fesuper('SeBuildSel -initrot',cf,Sel2);
% ... and do restore
xF1=fesuper('SeDef',Sel1.cna{1},xF);
xF2=fesuper('SeDef',Sel2.cna{1},xF);

% plot responses
ci=iiplot(3);
XF=iicom(ci,'curveXF');
XF('Disk1')=struct('w',xF.data,'xf',xF1.def','dof',xF1.DOF);
XF('Disk2')=struct('w',xF.data,'xf',xF2.def','dof',xF2.DOF);
iicom('subMagPha')
iicom(ci,'IIxOnly',{'Disk1','Disk2'});
ii_plp(def.data);

Fourier [,ind], ... [-phys, -mono, -red, -egy, -egyfrac, -sortbyd]

This command allows to perform a 3D Fourier analysis of given modeshapes. The maximum norm of each harmonic is plotted against the harmonic coefficient. The plot is different when dealing with a single modeshape or a set of modeshapes.

Accepted options are

ind is an optional selection of deformations. See also the alternate fe_def SubDef.
-phys, -mono and -red are used to distinguish between an analysis of physical modeshapes (full 3D), mono-harmonic modeshapes (the DFT step is omitted) and generalized modeshapes (reduced multi-harmonic model). In all cases, the user has to check that the physical or reduced models are geometrically periodic, i.e. that DOF come in repetitive groups, except for -mono where the concept of mono-harmonic modeshapes assumes that structures are periodic.
-egy and -egyfrac provides means for energy-based computations. Option -egy displays the fraction of energy in each existing harmonic within each disk so that the total amount of energy within each disk is 1. Option -egyfrac displays the fraction of energy in each existing harmonic within each disk so that that total amount of energy within the rotor is 1. The default displays these quantities disk by disk (from top to bottom) and for each disk, all the possible harmonics are displayed (from bottom to top), as depicted in the figure below.
-sortbyd groups these quantities first by harmonics (from bottom to top) and then by disk (from top to bottom), with the appropriate number of zeros for non present harmonics (typically when δ>N/2 for a given disk), as displayed in the figure below.

[cf,def]=demo_cyclic('buildstep1'); % sdtweb demo_cyclic('step1')

Curve=fe_cyclicb('fourier 1:50 -mono -egyfrac',cf,...
           fe_def('subdef',def,def.data(:,2)~=-1));
'xxx'%fe_cyclicb('fourier -mono -egyfrac -cf 3',Curve);

%  sdtweb demo_cyclic('step4') multi-harmonic analysis
[cf,def]=demo_cyclic('buildstep4'); %
fe_cyclicb('fourier 7:25 -red -egyfrac -cf 11',cf,def);
fe_cyclicb('fourier 7:25 -red -egyfrac -sortbyd -cf 13',cf,def);