# reconstructSolution

Recover full-model transient solution from reduced-order model (ROM)

## Syntax

``structuralresults = reconstructSolution(Rcb,u,ut,utt,tlist)``
``thermalresults = reconstructSolution(Rtherm,u_therm,tlist)``

## Description

````structuralresults = reconstructSolution(Rcb,u,ut,utt,tlist)` recovers the full transient structural solution from the reduced-order model `Rcb`, displacement `u`, velocity `ut`, and acceleration `utt`. Typically, the displacement, velocity, and acceleration are the values returned by Simscape™.```

example

````thermalresults = reconstructSolution(Rtherm,u_therm,tlist)` recovers the full transient thermal solution from the reduced-order model `Rtherm`, temperature in modal coordinates `u_therm`, and the time-steps `tlist` that you used to solve the reduced model.```

example

## Examples

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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 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 5`0` 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

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Structural results obtained using the Craig-Bampton order reduction method, specified as a `ReducedStructuralModel` object.

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`

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`

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`

Solution times for solving the reduced-order model, specified as a real vector.

Data Types: `double`

Reduced-order thermal model, specified as a `ReducedThermalModel` object.

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

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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`.

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 R2019b

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