ModalThermalResults
Description
A ModalThermalResults
object contains the eigenvalues and
eigenvector matrix of a thermal analysis problem, and average of snapshots used for proper
orthogonal decomposition (POD).
Creation
Solve a modal thermal problem using the solve
function. This function returns a modal thermal solution as a ModalThermalResults
object.
Properties
DecayRates
— Eigenvalues of thermal problem
column vector
This property is read-only.
Eigenvalues of a thermal problem, returned as a column vector.
Data Types: double
ModeShapes
— Eigenvector matrix
matrix
This property is read-only.
Eigenvector matrix, returned as a matrix.
Data Types: double
SnapshotsAverage
— Average of snapshots used for POD
column vector
This property is read-only.
Average of snapshots used for POD, returned as a column vector.
Data Types: double
ModeType
— Type of modes
"EigenModes"
| "PODModes"
This property is read-only.
Type of modes, returned as "EigenModes"
or
"PODModes"
.
Data Types: string
Mesh
— Finite element mesh
FEMesh
object
This property is read-only.
Finite element mesh, returned as an FEMesh
object.
Examples
Solve Transient Thermal Problem Using Modal Superposition Method
Solve a transient thermal problem by first obtaining mode shapes for a particular decay range and then using the modal superposition method.
Modal Decomposition
Create a geometry representing a square plate with a diamond-shaped region in its center.
SQ1 = [3; 4; 0; 3; 3; 0; 0; 0; 3; 3]; D1 = [2; 4; 0.5; 1.5; 2.5; 1.5; 1.5; 0.5; 1.5; 2.5]; gd = [SQ1 D1]; sf = 'SQ1+D1'; ns = char('SQ1','D1'); ns = ns'; g = decsg(gd,sf,ns); pdegplot(g,EdgeLabels="on",FaceLabels="on") xlim([-1.5 4.5]) ylim([-0.5 3.5]) axis equal
Create an femodel
object for modal thermal analysis and include the geometry.
model = femodel(AnalysisType="thermalModal", ... Geometry=g);
For the square region, assign these thermal properties:
Thermal conductivity is .
Mass density is .
Specific heat is .
model.MaterialProperties(1) = ... materialProperties(ThermalConductivity=10, ... MassDensity=2, ... SpecificHeat=0.1);
For the diamond region, assign these thermal properties:
Thermal conductivity is .
Mass density is .
Specific heat is .
model.MaterialProperties(2) = ... materialProperties(ThermalConductivity=2, ... MassDensity=1, ... SpecificHeat=0.1);
Assume that the diamond-shaped region is a heat source with a density of .
model.FaceLoad(2) = faceLoad(Heat=4);
Apply a constant temperature of 0 °C to the sides of the square plate.
model.EdgeBC([1 2 7 8]) = edgeBC(Temperature=0);
Set the initial temperature to 0 °C.
model.FaceIC = faceIC(Temperature=0);
Generate the mesh.
model = generateMesh(model);
Compute eigenmodes of the model in the decay range [100,10000] .
RModal = solve(model,DecayRange=[100,10000])
RModal = ModalThermalResults with properties: DecayRates: [171x1 double] ModeShapes: [1461x171 double] ModeType: "EigenModes" Mesh: [1x1 FEMesh]
Transient Analysis
Knowing the mode shapes, you can now use the modal superposition method to solve the transient thermal problem. First, switch the model analysis type to thermal transient.
model.AnalysisType = "thermalTransient";
The dynamics for this problem are very fast. The temperature reaches a steady state in about 0.1 second. To capture the most active part of the dynamics, set the solution time to logspace(-2,-1,100)
. This command returns 100 logarithmically spaced solution times between 0.01 and 0.1.
tlist = logspace(-2,-1,10);
Solve the equation.
Rtransient = solve(model,tlist,ModalResults=RModal);
Plot the solution with isothermal lines by using a contour plot.
msh = Rtransient.Mesh
msh = FEMesh with properties: Nodes: [2x1461 double] Elements: [6x694 double] MaxElementSize: 0.1697 MinElementSize: 0.0849 MeshGradation: 1.5000 GeometricOrder: 'quadratic'
T = Rtransient.Temperature; pdeplot(msh,XYData=T(:,end),Contour="on", ... ColorMap="hot")
Snapshots for Proper Orthogonal Decomposition
Obtain POD modes of a linear thermal problem using several instances of the transient solution (snapshots).
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.Geometry,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 0
for the temperature.
model.FaceIC = faceIC(Temperature=0);
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 model analysis type to thermal modal.
model.AnalysisType = "thermalModal";
Compute the POD modes.
RModal = solve(model,Snapshots=Tmatrix)
RModal = ModalThermalResults with properties: DecayRates: [6x1 double] ModeShapes: [1529x6 double] SnapshotsAverage: [1529x1 double] ModeType: "PODModes" Mesh: [1x1 FEMesh]
Version History
Introduced in R2022a
See Also
Functions
Objects
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