In this tutorial, the capabilities of the VIBES Toolbox for Component Mode Synthesis are demonstrated. Component Mode Synthesis (CMS) is a dynamic substructuring method for modally reduced numerical models. This tutorial demonstrates this method by assembling a building from repeated substructures and assessing its dynamic behavior.
Start by loading the FE models of the substructures as vibes.MCKModel objects. Each model comprises a mass matrix, a stiffness matrix, the nodes of the FE elements and a DoF-mapping table. Two substructures are defined:
Both substructures comprise two stories and can be combined to create a taller and wider building. The models also come with a set of FE element definitions for the floors and walls, which are used to show the mode shapes as faces of the vibes.ui.Mesh objects.
% Load the substructure models
mckA = vibes.load(‘VIBES,Examples,Datasets’,’mat’,’Building’,’substructureA.MCKModel.mat’);
mckB = vibes.load(‘VIBES,Examples,Datasets’,’mat’,’Building’,’substructureB.MCKModel.mat’);
Plot a mesh of both objects in the 3D viewer to see their shape and orientation. Note that faces are obtained from the field Parameters.Faces in the objects.
V = vibes.figure3d(); V.clf();
% Set camera
Figure 3-9: Plotting the mesh of the building in the 3D viewer
Looking at the meshes, it appears that the models are oriented incorrectly. We can rotate both models around the corner of the building to place them horizontally using the vibes.MCKModel.rotate command.
% Rotate about the X-axis (roll) to place the buildings on the ground (XY-plane)
mckA.rotate([90 0 0]);
mckB.rotate([90 0 0]);
% Move the models in Y-direction
mckA.move([0 6 0]);
mckB.move([0 6 0]);
Plot the new models in the 3D viewer.
% Clear the 3D viewer
% Plot the rotated models
% Save camera
cam1 = V.Camera;
Figure 3-10: Rotating the model
Calling the vibes.MCKModel.move or vibes.MCKModel.rotate methods with an output variable will create a shifted or rotated duplicate of that model, leaving the original unchanged. We will use this syntax to create shifted copies of the two substructures and build the larger building.
% Add a side element to close the right side of the building
mckC = mckA.move([25 0 0]);
mckC.Name = ‘Substructure C’;
% Add another floor
mckD = mckA.move([0 0 6]);
mckD.Name = ‘Substructure D’;
mckE = mckB.move([0 0 6]);
mckE.Name = ‘Substructure E’;
mckF = mckA.move([25 0 6]);
mckF.Name = ‘Substructure F’;
Plot the additional models too to preview the whole assembly.
% Plot the complete building
cam2 = V.Camera;
Figure 3-11: Duplicating the model
In order to construct the complete building efficiently, we will perform a Hurty/Craig-Bampton model reduction on the two substructure models that make up the assembly. The H/CB method reduces the full-sized system matrices to reduced system matrices comprising boundary DoFs and fixed-interface vibration modes.
The coupling will take place at the boundary DoFs. To find these boundary DoFs, the complete set of DoFs is first passed through the vibes.DoF.uniqueMatch method, which applies the default DoF-matching logic to get the “primal” set of DoFs to retain after coupling. The DoFs that are not unique can be found by taking the set difference of all DoFs and the unique ones, yielding the boundary DoFs. Finally, by taking the “DoF-matched” intersection of all DoFs with these boundary DoFs, the full set of boundary DoFs is found.
% Get all substructure DoFs
uA = mckA.DoFs;
uB = mckB.DoFs;
uC = mckC.DoFs;
uD = mckD.DoFs;
uE = mckE.DoFs;
uF = mckF.DoFs;
% Stack to get the complete set of DoFs
u = [uA; uB; uC; uD; uE; uF];
Use vibes.DoF.uniqueMatch to find the DoFs that are unique. This applies DoF-matching on 42972 vibes.DoF objects, so may take a while.
% Find the unique set of DoFs
q = uniqueMatch(u);
% Get the (redundant) coupling DoFs by taking the set difference of u and q
u_2r = setdiff(u,q);
% Get the complete set of boundary DoFs for all substructures
u_2 = intersectMatch(u,u_2r,’SingleMatch’,false);
As the substructures were copied and then moved, each substructure has the same node numbers as its original. We can use this to our advantage by reducing only substructure A at those nodes that are on the boundaries of A, C, D and F, and do the same for substructure B at the boundaries of B and E.
% Get the node numbers of the boundary DoFs
nnr_2 = unique(u_2.getNodeNumbers());
% Get the boundary DoFs of A and B
uA_2 = uA.select(‘NodeNumber’,nnr_2);
uB_2 = uB.select(‘NodeNumber’,nnr_2);
% Get boundary nodes from the models
nodesA_2 = mckA.selectNodes(‘NodeNumber’,nnr_2);
nodesB_2 = mckB.selectNodes(‘NodeNumber’,nnr_2);
% Plot the boundary nodes
% Make the meshes transparent
Figure 3-12: Performing a Hurty/Craig-Bampton reduction
Reduce the full vibes.MCKModel objects of just A and B to vibes.CMSModel objects using the found boundary DoFs. We will be using a Hurty/Craig-Bampton reduction with 20 internal vibration modes for each substructure. Setting the KeepTransformation property to true allows to later on visualize the fixed-interface vibration modes of the Hurty/Craig-Bampton model. However, it is computationally less efficient to do so, as these matrices may become considerably large.
% Define a number of fixed interface vibration modes
n_r = 20;
% Reduce the MCK models
cmsA = mckA.reduce(‘HCB’,uA_2,’nComponentModes’,n_r,’KeepTransformation’,true);
cmsB = mckB.reduce(‘HCB’,uB_2,’nComponentModes’,n_r,’KeepTransformation’,true);
Use the vibes.CMSModel.spy method to show the structure of the reduced matrices. Notice the diagonal structure of the section of the matrices containing the fixed interface modes.
Figure 3-13: Structure of the reduced matrices
Let us now inspect the modal basis of the reduced CMS models. Plotting a mesh from a vibes.CMSModel object that has its reduction matrix stored in the ‘T’ field will automatically pass that information onto the mesh object. Press SPACE and CTRL-UP/DOWN to animate the fixed interface mode shapes.
% Clear 3D viewer
% You can define mesh colormaps to show color gradients in the faces
meshA = cmsA.plotMesh();
meshA.Animation.ColorMap = winter();
meshB = cmsB.plotMesh();
meshB.Animation.ColorMap = autumn();
% Activate the colormap for the faces
V.Camera = cam1;
% Animate mode
Figure 3-14: Mode shapes of the reduced model
To create the full building from the reduced models, we will duplicate and shift the newly created vibes.CMSModel objects to their respective positions, similar as before.
cmsC = cmsA.move([25 0 0]);
cmsC.Name = ‘Substructure C’;
cmsD = cmsA.move([0 0 6]);
cmsD.Name = ‘Substructure D’;
cmsE = cmsB.move([0 0 6]);
cmsE.Name = ‘Substructure E’;
cmsF = cmsA.move([25 0 6]);
cmsF.Name = ‘Substructure F’;
We can assemble the six reduced substructure models together on their boundary nodes. To prevent double node numbers from showing up when coupling the substructures, we will reset the node numbers for each substructure before coupling. We will set them with offsets of 2000 per substructure, and also reset the names of the nodes to represent the node number.
Also plot the other parts to show the whole assembly.
% Also plot the boundary nodes
nodes_2 = u_2r.getUniqueNodes();
V.Camera = cam2;
Figure 3-15: Duplicating the reduced model
The assembly of the substructures is performed by the vibes.mixin.PhysicalDomain.couple method, which automatically matches the set of corresponding DoFs. Note that this only assembles the system matrices of the substructures, but does not yet give access to the coupled-system response.
[cms,coupleOpts] = couple(cmsA,cmsB,cmsC,cmsD,cmsE,cmsF);
cms.Name = ‘Assembled system’;
The ‘couple’ method returns a vibes.operation.NumericalCouplingOperation object that allows you to inspect the coupling procedure. For instance, you can visualize the result of the automatic DoF-matching procedure in a matrix view.
Figure 3-16: Matrix view of the automatic DoF-matching
The faces of the coupled system are simply the faces of all substructures combined.
cms.Parameters.Faces = [cmsA.Parameters.Faces; cmsB.Parameters.Faces;….
Using the spy method on the coupled system shows the overlapping of the matrices at the boundary DoFs, while the fixed interface modes are simply concatenated.
Figure 3-17: Assembled system
Clear 3D viewer
% Plot a mesh of the assembled system
mesh = cms.plotMesh();
Figure 3-18: Mesh of the assembled system
Since the mesh is quite large, automatically scaling the mode shapes will give a distorted image of the actual displacement of the nodes. Instead, we will turn off auto-scaling and scale up every mode by the same amount.
mesh.Animation.AutoScale = false;
mesh.Animation.Scale = 10;
Figure 3-19: Animation of the assembled system
The resulting CMS model after assembly consists of just under 3500 DoFs. A modal solution can be computed from the CMS model using vibes.CMSModel.toModalModel.
We would like to constrain the building to the ground. In order to achieve this, we will select all boundary DoFs in the coupled system with a Z-coordinate below 0.1m. These DoFs will be constrained in the computation of the modal solution. Constraints can be provided using the optional parameter ‘ConstrainedDoFs’.
% First, select the boundary DoFs to constrain
u_constr = cms.BoundaryDoFs.select(‘Z’,'<‘,0.1);
% Compute the constrained modal solution for the first 50 modes
mm = cms.toModalModel(50,’Symmetrize’,’avg’,’ConstrainedDoFs’,u_constr);
The modal solution we just calculated consists of the same DoF set as the CMS model it was calculated from. In order to obtain a solution that describes the displacements of the complete node set, we like to expand the solution to the original DoFs. For this, we can use the reduction matrix, which is stored in the T field of the vibes.CMSModel class.
% Select the appropriate transformation matrix from the ‘T’ field
T = cms.T.select(‘Name’,’Reduction’);
% Multiply the modal model with the transformation matrix
mm_full = T * mm;
Plotting a mesh of this full modal model allows you to visualize mode shapes of the complete, coupled system.
mesh = mm_full.plotMesh();
% Plot the constrained nodes
Figure 3-20: Solving the assembled vibration modes
Now that we have a complete model of the coupled substructures, it is very easy to calculate the frequency response at the boundary DoFs of the system. This can be done by matrix-inversion of the reduced dynamic stiffness.
% First, add some damping
We will select three of the boundary DoFs as inputs and outputs for our FRF matrix: a Z-direction on floor 1 to an X-direction on a wall on the same side of the building and a Z-direction on the other side of the building.
% Get boundary nodes
bnodes = cms.BoundaryNodes();
n1 = bnodes.selectNearest([0 2 3]);
n2 = bnodes.selectNearest([5 2 6]);
n3 = bnodes.selectNearest([35 2 9]);
dofs(1) = cms.BoundaryDoFs.select(‘Node’,n1,’Direction’,’==’,[1 0 0],’first’);
dofs(2) = cms.BoundaryDoFs.select(‘Node’,n2,’Direction’,’==’,[0 0 1],’first’);
dofs(3) = cms.BoundaryDoFs.select(‘Node’,n3,’Direction’,’==’,[0 0 1],’first’);
% Calculate FRFs for displacement over force between 1 and 15 Hz.
frftype = 0;
freq = 1:15;
Y = cms.toFRFMatrix(dofs,dofs,freq,frftype,’ConstrainedDoFs’,u_constr);
Plot the generated FRF.
% Open a new figure
% Create a plot
Figure 3-21: Plotting the generated FRFs
Indicate the channels in the 3D viewer.
% Make building transparent
Figure 3-22: Showing the channels in the 3D viewer
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