reconstructSolution

Knowing the solution in terms of the interface degrees of freedom (DoFs) and modal DoFs, reconstruct the solution for the full structural transient model.

Create a structural model for transient analysis.

modelT = createpde("structural","transient-solid");

Create a square cross-section beam geometry and include it in the model.

gm = multicuboid(0.05,0.003,0.003); modelT.Geometry = gm;

Plot the geometry, displaying face and edge labels.

figure pdegplot(modelT,"FaceLabels","on","FaceAlpha",0.5) view([71 4])

figure pdegplot(modelT,"EdgeLabels","on","FaceAlpha",0.5) view([71 4])

Specify Young's modulus, Poisson's ratio, and the mass density of the material.

structuralProperties(modelT,"YoungsModulus",210E9,..."PoissonsRatio",0.3,..."MassDensity",7800);

Fix one end of the beam.

structuralBC(modelT,"Edge",[2 8 11 12],"Constraint","fixed");

Add a vertex at the center of face 3.

loadedVertex = addVertex(gm,"Coordinates",[0.025 0.0 0.0015]);

Generate a mesh.

generateMesh(modelT);

Apply a sinusoidal concentrated force in thez-direction on the new vertex.

structuralBoundaryLoad(modelT,"Vertex",loadedVertex,..."Force",[0;0;10],"Frequency",6000);

Specify zero initial conditions.

structuralIC(modelT,"Velocity",[0 0 0],"Displacement",[0 0 0]);

Define superelement interfaces using the fixed and loaded boundaries. In this case, the reduced-order model retains the DoFs on the fixed face and the loaded vertex while condensing all other DoFs in favor of modal DoFs. For better performance, use the set of edges bounding face 5 instead of using the entire face.

structuralSEInterface(modelT,"Edge",[2 8 11 12]); structuralSEInterface(modelT,"Vertex",loadedVertex);

Reduce the structure, retaining all fixed interface modes up to5e5.

rom = reduce(modelT,"FrequencyRange",[-0.1,5e5]);

Next, use the reduced-order model to simulate the transient dynamics. Use theode15sfunction directly to integrate the reduced system ODE. Working with the reduced model requires indexing into the reduced system matricesrom.Kandrom.M. First, construct mappings of indices ofKandMto loaded and fixed DoFs by using the data available inrom.

DoFs correspond to translational displacements. If the number of mesh points in a model isNn, then the toolbox assigns the IDs to the DoFs as follows: the first1toNnarex-displacements,Nn+1to2*Nnarey-displacements, and2Nn+1to3*Nnarez-displacements. The reduced model objectromcontains these IDs for the retained DoFs inrom.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];

Knowing the DoF IDs for the given node IDs, use theintersectfunction to find the required indices.

numNodes = size(rom.Mesh.Nodes,2); loadedNode = findNodes(rom.Mesh,"region","Vertex",loadedVertex); loadDoFs = getDoF(loadedNode,numNodes); [~,loadNodeROMIds,~] = intersect(rom.RetainedDoF,loadDoFs);

In the reduced matricesrom.Kandrom.M, generalized modal DoFs appear after the retained DoFs.

fixedIntModeIds = (numel(rom.RetainedDoF) + 1:size(rom.K,1))';

Because fixed-end DoFs are not a part of the ODE system, the indices for the ODE DoFs in reduced matrices are as follows.

odeDoFs = [loadNodeROMIds;fixedIntModeIds];

The relevant components ofrom.Kandrom.Mfor time integration are:

Kconstrained = rom.K (odeDoFs odeDoFs);Mconstrained = rom.M(odeDoFs,odeDoFs); numODE = numel(odeDoFs);

Now you have a second-order system of ODEs. To useode15s, convert this into a system of first-order ODEs by applying linearization. Such 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);

指定的集中力ad in the full system is along thez-direction, which is the third DoF in the ODE system. Accounting for the linearization to obtain the first-order system gives the loaded ODE DoF.

loadODEDoF = numODE + 3;

指定质量矩阵和O的雅可比矩阵DE 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 functionCMSODEf, which is defined at the end of this example.

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);

Knowing the solution in terms of the interface DoFs and modal DoFs, you can reconstruct the solution for the full model. ThereconstructSolutionfunction requires the displacement, velocity, and acceleration at all DoFs inrom. Construct 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,:);

Construct 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")

functionf = CMSODEf(t,u,Kode,Fode,loadedVertex) Fode(loadedVertex) = 10*sin(6000*t); f = -Kode*u +Fode;end

Reconstruct the solution for a full thermal transient model from the reduced-order model.

Create a transient thermal model.

thermalmodel = createpde("thermal","transient");

Create a unit square geometry and include it in the model.

geometryFromEdges(thermalmodel,@squareg);

Plot the geometry, displaying edge labels.

pdegplot(thermalmodel,"EdgeLabels","on") xlim([-1.1 1.1]) ylim([-1.1 1.1])

Specify the thermal conductivity, mass density, and specific heat of the material.

thermalProperties(thermalmodel,"ThermalConductivity",400,..."MassDensity",1300,..."SpecificHeat",600);

Set the temperature on the right edge to100.

thermalBC(thermalmodel,"Edge",2,"Temperature",100);

Set an initial value of 50for the temperature.

thermalIC(thermalmodel,50);

Generate a mesh.

generateMesh(thermalmodel);

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];forq = heatVariation internalHeatSource(thermalmodel,q); results = solve(thermalmodel,tlist); Tmatrix = [Tmatrix,results.Temperature(:,snapShotIDs)];end

Switch the thermal model analysis type to modal.

thermalmodel.AnalysisType ="modal";

Compute the POD modes.

RModal = solve(thermalmodel,"Snapshots",Tmatrix);

Reduce the thermal model.

Rtherm = reduce(thermalmodel,"ModalResults",RModal)

Rtherm = ReducedThermalModel with properties: K: [6x6 double] M: [6x6 double] F: [6x1 double] InitialConditions: [6x1 double] Mesh: [1x1 FEMesh] ModeShapes: [1541x5 double] SnapshotsAverage: [1541x1 double]

Next, use the reduced-order model to simulate the transient dynamics. Use theode15sfunction 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 usingode15s.

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(thermalmodel,"XYData",R.Temperature(:,end))