reconstructSolution
Recover full-model transient solution from reduced-order model (ROM)
Syntax
Description
recovers the full transient thermal solution from the reduced-order model
thermalresults
= reconstructSolution(Rtherm
,u_therm
,tlist
)Rtherm
, temperature in modal coordinates
u_therm
, and the time-steps tlist
that you used to
solve the reduced model.
Examples
Reconstruct Structural Solution from ROM Results
Knowing the solution in terms of the interface degrees of freedom (DoFs) and modal DoFs, reconstruct the solution for the full structural transient analysis.
Define Parameters for Structural Analysis
Create a square cross-section beam geometry.
gm = multicuboid(0.05,0.003,0.003);
Plot the geometry, displaying face and edge labels.
figure
pdegplot(gm,FaceLabels="on",FaceAlpha=0.5)
view([71 4])
figure
pdegplot(gm,EdgeLabels="on",FaceAlpha=0.5)
view([71 4])
Add a vertex at the center of face 3.
centerVertex = addVertex(gm,Coordinates=[0.025 0.0 0.0015]);
Create an femodel
object for transient structural analysis and include the geometry in the model.
model = femodel(AnalysisType="structuralTransient", ... Geometry=gm);
Specify Young's modulus, Poisson's ratio, and the mass density of the material.
model.MaterialProperties = ... materialProperties(YoungsModulus=210E9, ... PoissonsRatio=0.3, ... MassDensity=7800);
Fix one end of the beam.
model.EdgeBC([2 8 11 12]) = edgeBC(Constraint="fixed");
Generate a mesh.
model = generateMesh(model);
Apply a sinusoidal concentrated force in the z-direction on the new vertex. First, define a sinusoidal load function, sinusoidalLoad
, to model a harmonic load. This function accepts the load magnitude (amplitude), location
and state
structure arrays, frequency, and phase. Because the function depends on time, it must return a matrix of NaN
of the correct size when state.time
is NaN
. Solvers check whether a problem is nonlinear or time-dependent by passing NaN
state values and looking for returned NaN
values.
function Tn = sinusoidalLoad(load,location,state,Frequency,Phase) if isnan(state.time) normal = [location.nx location.ny]; if isfield(location,"nz") normal = [normal location.nz]; end Tn = NaN*normal; return end if isa(load,"function_handle") load = load(location,state); else load = load(:); end % Transient model excited with harmonic load Tn = load.*sin(Frequency.*state.time + Phase); end
Now apply the force on the new vertex.
Force = [0 0 10];
Frequency = 6000;
Phase = 0;
forcePulse = @(location,state) ...
sinusoidalLoad(Force,location,state,Frequency,Phase);
model.VertexLoad(centerVertex) = vertexLoad(Force=forcePulse);
Specify zero initial conditions.
model.CellIC = cellIC(Velocity=[0 0 0],Displacement=[0 0 0]);
Reduce Model
Specify the fixed and loaded boundaries as structural superelement interfaces by creating a romInterface
object for each superelement interface. The reduced-order model technique retains the DoFs on the superelement interfaces while condensing all other DoFs to a set of modal DoFs. For better performance, use the set of edges bounding face 5 instead of using the entire face.
romObj1 = romInterface(Edge=[2 8 11 12]); romObj2 = romInterface(Vertex=centerVertex);
Assign a vector of interface objects to the ROMInterfaces
property of the model.
model.ROMInterfaces = [romObj1,romObj2];
Reduce the structure, retaining all fixed interface modes up to 5e5
.
rom = reduce(model,FrequencyRange=[-0.1,5e5]);
Simulate Transient Dynamics Using ROM
Next, use the reduced-order model to simulate the transient dynamics. Use the ode15s
function directly to integrate the reduced system of ordinary differential equations. Take the loaded and modal DoFs for time-integration, and leave the fixed DoFs aside because the solution remains zero for those DoFs.
Working with the reduced model requires indexing into the reduced system matrices rom.K
and rom.M
. The arrangement of DoFs in reduced system is such that the physical DoFs corresponding to retained interfaces appear first followed by the generalized model DoFs. DoFs in a structural problem correspond to translational displacements. If the number of mesh points in a model is Nn
, then the software assigns the IDs to the DoFs as follows: the first 1
to Nn
are x-displacements, Nn+1
to 2*Nn
are y-displacements, and 2Nn+1
to 3*Nn
are z-displacements. Only the subset of these 3*Nn
DoFs corresponding to ROMInterfaces
is retained in the reduced model. The reduced model object rom
contains these IDs for the retained DoFs in rom.RetainedDoF
.
Create a function that returns DoF IDs given node IDs and the number of nodes.
getDoF = @(x,numNodes) [x(:); x(:) + numNodes; x(:) + 2*numNodes];
Find the node at the loaded vertex.
loadedNode = findNodes(rom.Mesh,"region",Vertex=centerVertex);
Find the DoF of the loaded nodes using the helper function getDoF
.
numNodes = size(rom.Mesh.Nodes,2); loadDoFs = getDoF(loadedNode,numNodes);
Knowing the DoF IDs for the given node IDs, use rom.RetainedDoF
and the intersect
function to find the required indices corresponding to those DoF in the reduced matrices.
[~,loadNodeROMIds] = intersect(rom.RetainedDoF,loadDoFs);
In the reduced matrices rom.K
and rom.M
, generalized modal DoFs appear after the retained DoFs. Find the indices of modal DoFs in rom
matrices.
modelDoFIDs = ((numel(rom.RetainedDoF) + 1):size(rom.K,1))';
Find the indices for the ODE DoFs in reduced matrices. Because fixed-end DoFs are not a part of the ODE system, these indices are as follows.
odeDoFs = [loadNodeROMIds;modelDoFIDs];
Find the relevant components of rom.K
and rom.M
for time integration.
Kconstrained = rom.K(odeDoFs,odeDoFs); Mconstrained = rom.M(odeDoFs,odeDoFs); numODE = numel(odeDoFs);
Now you have a second-order system of ODEs. To use ode15s
, you must convert this system into a system of first-order ODEs by applying linearization. This type of a first-order system is twice the size of the second-order system.
Mode = [eye(numODE,numODE), zeros(numODE,numODE); ... zeros(numODE,numODE), Mconstrained]; Kode = [zeros(numODE,numODE), -eye(numODE,numODE); ... Kconstrained, zeros(numODE,numODE)]; Fode = zeros(2*numODE,1);
The specified concentrated force load in the full system is along the z-direction, which is the third DoF in the ODE system. Accounting for the linearization, obtain the first-order system to get the loaded ODE DoF.
loadODEDoF = numODE + 3;
Specify the mass matrix and the Jacobian for the ODE solver.
odeoptions = odeset; odeoptions = odeset(odeoptions,"Jacobian",-Kode); odeoptions = odeset(odeoptions,"Mass",Mode);
Specify zero initial conditions.
u0 = zeros(2*numODE,1);
Solve the reduced system by using ode15s and the helper function.
function f = CMSODEf(t,u,Kode,Fode,centerVertex) Fode(centerVertex) = 10*sin(6000*t); f = -Kode*u +Fode; end tlist = 0:0.00005:3E-3; sol = ode15s(@(t,y) CMSODEf(t,y,Kode,Fode,loadODEDoF), ... tlist,u0,odeoptions);
Compute the values of the ODE variable and the time derivatives.
[displ,vel] = deval(sol,tlist);
Reconstruct Solution for Full Model
Knowing the solution in terms of the interface DoFs and modal DoFs, you can reconstruct the solution for the full model. The reconstructSolution
function requires the displacement, velocity, and acceleration at all DoFs in rom
. Create the complete solution vector, including the zero values at the fixed DoFs.
u = zeros(size(rom.K,1),numel(tlist)); ut = zeros(size(rom.K,1),numel(tlist)); utt = zeros(size(rom.K,1),numel(tlist)); u(odeDoFs,:) = displ(1:numODE,:); ut(odeDoFs,:) = vel(1:numODE,:); utt(odeDoFs,:) = vel(numODE+1:2*numODE,:);
Create a transient results object using this solution.
RTrom = reconstructSolution(rom,u,ut,utt,tlist);
Compute the displacement in the interior at the center of the beam using the reconstructed solution.
coordCenter = [0;0;0];
iDispRTrom = interpolateDisplacement(RTrom,coordCenter);
figure
plot(tlist,iDispRTrom.uz)
title("Z-Displacement at Geometric Center")
Reconstruct Thermal Solution from ROM Results
Reconstruct the solution for a full thermal transient analysis from the reduced-order model.
Create an femodel
object for transient thermal analysis, and include a unit square geometry in the model.
model = femodel(AnalysisType="thermalTransient", ... Geometry=@squareg);
Plot the geometry, displaying edge labels.
pdegplot(model,EdgeLabels="on")
xlim([-1.1 1.1])
ylim([-1.1 1.1])
Specify the thermal conductivity, mass density, and specific heat of the material.
model.MaterialProperties = ... materialProperties(ThermalConductivity=400, ... MassDensity=1300, ... SpecificHeat=600);
Set the temperature on the right edge to 100
.
model.EdgeBC(2) = edgeBC(Temperature=100);
Set an initial value of 50
for the temperature.
model.FaceIC = faceIC(Temperature=50);
Generate a mesh.
model = generateMesh(model);
Solve the model for three different values of heat source, and collect snapshots.
tlist = 0:10:600; snapShotIDs = [1:10 59 60 61]; Tmatrix = []; heatVariation = [10000 15000 20000]; for q = heatVariation model.FaceLoad = faceLoad(Heat=q); results = solve(model,tlist); Tmatrix = [Tmatrix,results.Temperature(:,snapShotIDs)]; end
Switch the thermal model analysis type to modal.
model.AnalysisType = "thermalModal";
Compute the POD modes.
RModal = solve(model,Snapshots=Tmatrix);
Reduce the thermal model.
Rtherm = reduce(model,ModalResults=RModal)
Rtherm = ReducedThermalModel with properties: K: [6x6 double] M: [6x6 double] F: [6x1 double] InitialConditions: [6x1 double] Mesh: [1x1 FEMesh] ModeShapes: [1529x5 double] SnapshotsAverage: [1529x1 double]
Next, use the reduced-order model to simulate the transient dynamics. Use the ode15s
function directly to integrate the reduced system ODE. Specify the mass matrix and the Jacobian for the ODE solver.
odeoptions = odeset;
odeoptions = odeset(odeoptions,Mass=Rtherm.M);
odeoptions = odeset(odeoptions,JConstant="on");
f = @(t,u) -Rtherm.K*u + Rtherm.F;
df = -Rtherm.K;
odeoptions = odeset(odeoptions,Jacobian=df);
Solve the reduced system by using ode15s
.
sol = ode15s(f,tlist,Rtherm.InitialConditions,odeoptions);
Compute the values of the ODE variable.
u = deval(sol,tlist);
Reconstruct the solution for the full model.
R = reconstructSolution(Rtherm,u,tlist);
Plot the temperature distribution at the last time step.
pdeplot(R.Mesh,XYData=R.Temperature(:,end))
Input Arguments
Rcb
— Structural results obtained using Craig-Bampton order reduction method
ReducedStructuralModel
object
Structural results obtained using the Craig-Bampton order reduction method,
specified as a ReducedStructuralModel
object.
u
— Displacement
matrix
Displacement, specified as a matrix. The number of rows in the matrix must equal the
sum of the numbers of interface degrees of freedom and the number of modes. The
x-displacements at the retained degrees of freedom must appear
first, then the y-displacements, and, for a 3-D geometry,
z-displacements, followed by the generalized modal degrees of
freedom. The number of columns must equal the number of elements in
tlist
.
Data Types: double
ut
— Velocity
matrix
Velocity, specified as a matrix. The number of rows in the matrix must equal the sum
of the numbers of interface degrees of freedom and the number of modes. The
x-velocities at the retained degrees of freedom must appear first,
then the y-velocities, and, for a 3-D geometry,
z-velocities, followed by the generalized modal degrees of freedom.
The number of columns must equal the number of elements in
tlist
.
Data Types: double
utt
— Acceleration
matrix
Acceleration, specified as a matrix. The number of rows in the matrix must equal the
sum of the numbers of interface degrees of freedom and the number of modes. The
x-accelerations at the retained degrees of freedom must appear
first, then the y-accelerations, and, for a 3-D geometry,
z-accelerations, followed by the generalized modal degrees of
freedom. The number of columns must equal the number of elements in
tlist
.
Data Types: double
tlist
— Solution times for solving reduced-order model
real vector
Solution times for solving the reduced-order model, specified as a real vector.
Data Types: double
Rtherm
— Reduced-order thermal model
ReducedThermalModel
object
Reduced-order thermal model, specified as a ReducedThermalModel
object.
u_therm
— Temperature in modal coordinates
matrix
Temperature in modal coordinates, specified as a matrix. The number of rows in the
matrix must equal the number of modes. The number of columns must equal the number of
elements in tlist
.
Data Types: double
Output Arguments
structuralresults
— Transient structural analysis results
TransientStructuralResults
object
Transient structural analysis results, returned as a TransientStructuralResults
object. The object contains the displacement,
velocity, and acceleration values at the nodes of the triangular or tetrahedral mesh
generated by generateMesh
.
thermalresults
— Transient thermal analysis results
TransientThermalResults
object
Transient thermal analysis results, returned as a TransientThermalResults
object. The object contains the temperature and
gradient values at the nodes of the triangular or tetrahedral mesh generated by
generateMesh
.
Version History
Introduced in R2019bR2022a: ROM support for thermal analysis
reconstructSolution
now also reconstructs transient thermal
solutions.
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