linearizeInput

Specify inputs to linearized model

Syntax

linearizeInput(model,labeltext)

input = linearizeInput(model,labeltext)

Description

linearizeInput(model,labeltext) adds inputs for the boundary condition, constraint, load, or source with the label labeltext. In the linearized model, the input value u = 1 corresponds to a unit boundary condition acting on the entire region specified by labeltext. In other words, simulating the linearized model with the input value u(t) = 25 is equivalent to setting the boundary condition value to 25 in the thermal or structural model in Partial Differential Equation Toolbox™. For more information, see Algorithms.

For a structural analysis model, the following boundary conditions, constraints, and loads can become inputs of the linearized model:

• A structural boundary constraint. Use the structuralBC function with the Constraint argument.

• A displacement or a displacement component on the boundary. Use the structuralBC function with the Displacement, XDisplacement, YDisplacement, or ZDisplacement argument.

• A structural boundary load. Use the structuralBoundaryLoad function with the Pressure, Force, or SurfaceTraction argument.

• A structural body load. Use the structuralBodyLoad function with the GravitationalAcceleration argument.

The boundary conditions, loads, or constraints with x-, y-, and z- components produce one input channel per component.

For a thermal analysis model, the following boundary conditions and sources can become inputs of the linearized model:

• A temperature or heat flux on the boundary. Use the thermalBC function with the Temperature or HeatFlux argument.

• An internal heat source. Use the internalHeatSource function.

Each selected condition or source produces a single scalar input in the linearized model.

To make a condition, constraint, load, or source available as a linearization input, always label it upon creation. For example, specify an internal heat source for a thermal model as follows:

internalHeatSource(thermalmodel,25,'Label','HeatSource');

The remaining boundary conditions are set to zero for linearization purposes, regardless of their value in the structural or thermal model. Ensure that you label all nonzero boundary conditions and pass them as inputs using linearizeInput.

Use linearizeInput and linearizeOutput together with the linearize function to extract sparse linear models from structural and thermal models.

input = linearizeInput(model,labeltext) returns a structure array input with the linearization input description.

Examples

Thermal Boundary Conditions and Internal Heat Source as Inputs for linearize Function

Use labels to pass the parameters of a 2-D thermal analysis model to the linearize function. This function extracts sparse linear models for use with Control System Toolbox™.

Create a transient thermal model.

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

Add the block geometry to the thermal model by using the geometryFromEdges function. The geometry description file for this problem is called crackg.m.

geometryFromEdges(thermalmodel,@crackg);

Plot the geometry, displaying edge labels.

pdegplot(thermalmodel,'EdgeLabels','on')
ylim([-1,1])
axis equal

Generate a mesh.

generateMesh(thermalmodel);

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

thermalProperties(thermalmodel,'ThermalConductivity',1, ...
                               'MassDensity',1, ...
                               'SpecificHeat',1); 

Specify the temperature on the left edge as 100, and constant heat flow to the exterior through the right edge as -10. Add a unique label to each boundary condition.

thermalBC(thermalmodel,'Edge',6,'Temperature',100,'Label','TempBC');
thermalBC(thermalmodel,'Edge',1,'HeatFlux',-10,'Label','FluxBC');

Specify that the entire geometry generates heat and add a unique label to this assignment.

internalHeatSource(thermalmodel,25,'Label','HeatSource');

Set an initial value of 0 for the temperature.

thermalIC(thermalmodel,0);

Call the linearizeInput function with the previously defined labels for the boundary conditions and the internal heat source to set the inputs for the linearize function. Add one label per function call.

linearizeInput(thermalmodel,'HeatSource');
linearizeInput(thermalmodel,'TempBC');
linearizeInput(thermalmodel,'FluxBC');

The LinearizeInputs property of thermalmodel stores the inputs.

thermalmodel.LinearizeInputs

ans=1×3 struct array with fields:
    RegionType
    RegionID
    Label

Pressure and Gravity as Inputs for linearize Function

Create inputs for gravity and a short pressure pulse on tuning fork.

Create a structural transient analysis model.

structuralmodel = createpde('structural','transient-solid');

Import and plot the tuning fork geometry.

importGeometry(structuralmodel,'TuningFork.stl');
pdegplot(structuralmodel)

Generate a mesh.

generateMesh(structuralmodel,'Hmax',0.005);

Specify the Young's modulus, Poisson's ratio, and mass density to model linear elastic material behavior. Specify all physical properties in consistent units.

structuralProperties(structuralmodel,'YoungsModulus',210E9, ...
                                     'PoissonsRatio',0.3, ...
                                     'MassDensity',8000);

Identify faces for applying boundary constraints and loads by plotting the geometry with the face labels.

figure('units','normalized','outerposition',[0 0 1 1])
pdegplot(structuralmodel,'FaceLabels','on')
view(-50,15)
title 'Geometry with Face Labels'

Impose sufficient boundary constraints to prevent rigid body motion under applied loading. Typically, you hold a tuning fork by hand or mount it on a table. As a simplified approximation of this boundary condition, fix a region near the intersection of the tines and the handle (faces 21 and 22).

structuralBC(structuralmodel,'Face',[21,22],'Constraint','fixed');

Specify the pressure loading on a tine as a short rectangular pressure pulse.

structuralBoundaryLoad(structuralmodel,'Face',11, ...
                                       'Pressure',5E6, ...
                                       'EndTime',1e-3, ...
                                       'Label','Pressure');

Specify the acceleration due to gravity as a body load.

structuralBodyLoad(structuralmodel, ...
                   'GravitationalAcceleration',[0 0 -1], ...
                                     'Label','Gravity');

Create inputs for gravity and the pressure pulse on the tuning fork.

linearizeInput(structuralmodel,'Gravity');
linearizeInput(structuralmodel,'Pressure');

The LinearizeInputs property of structuralmodel stores the inputs.

structuralmodel.LinearizeInputs

ans=1×2 struct array with fields:
    RegionType
    RegionID
    Label

Input Arguments

model — Structural or linear thermal model

StructuralModel object | ThermalModel object

Structural or linear thermal model, specified as a StructuralModel object or a ThermalModel object. The linearize function does not support nonlinear thermal analysis.

labeltext — Label for boundary condition

character vector | string

Label for boundary condition, specified as a character vector or a string.

Data Types: char | string

Output Arguments

input — Linearization input description

structure array

Linearization input description, returned as a structure array.

Algorithms

The linearize function constructs a linear model whose inputs are a subset of the boundary conditions, loads, or sources applied to the thermal or structural model in Partial Differential Equation Toolbox and whose outputs are the resulting values at the selected DoFs. For example, if you designate the heat source

internalHeatSource(model,25,'Face',2,'Label','heatSource')

as a linearization input

linearizeInput(model,'heatSource')

and designate the temperatures on face X as linearization outputs

linearizeOutput(model,’Face’,X)

then the response of the linearized model to the constant input u(t) = 25 (the heat source value in the thermal model) matches the Partial Differential Equation Toolbox simulation results for face X.

tlist = 1:10;
u = repmat(25,size(tlist));
ysp = lsim(linsys,uLoad,tlist);

Note that loads and boundary conditions not included as linearization inputs are assumed to be zero in the linearized model regardless of their values in the structural or thermal model in Partial Differential Equation Toolbox. Simulation results can differ in this case.