Linearize a model for thermal analysis and return finite element matrices.

Create a transient thermal model.

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

Add the block geometry to the thermal model by using thegeometryFromEdgesfunction. The geometry description file for this problem is calledcrackg.m.

geometryFromEdges(thermalmodel,@crackg);

Plot the geometry with the edge labels.

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

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 as100, 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);

Specify the inputs of the linearized model by calling thelinearizeInputfunction with the previously defined labels for the boundary conditions and the internal heat source. Add one label per function call.

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

Specify the outputs of the linearized model by calling thelinearizeOutputfunction to set the regions of interest for measuring temperature. Specify one region per function call. For example, specify that the output is the temperature value at all nodes on edge 2.

linearizeOutput(thermalmodel,'Edge',2);

Measure the temperature on edge 2.

sys = linearize(thermalmodel)

稀疏的连续时间与27 o状态空间模型utputs, 3 inputs, and 1363 states. Use "spy" and "showStateInfo" to inspect model structure. Type "properties('sparss')" for a list of model properties. Type "help sparssOptions" for available solver options for this model.

In the linearized model, usesys.InputNameto check that the inputs tosysare the heat source, the temperature on edge 6, and the heat flux on edge 1.

sys.InputName

ans =3x1 cell{'HeatSource'} {'TempBC' } {'FluxBC' }

In the linearized model, usesys.OutputGroupto locate the sections associated with each coordinate.

sys.OutputGroup

ans =struct with fields:Edge2: [1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 ... ]

If you do not have Control System Toolbox™, you can access the finite element matricesA,B,C, andEas follows.

mx = linearize(thermalmodel,'OutputType','matrices')

mx =struct with fields:A: [1363x1363 double] B: [1363x3 double] C: [27x1363 double] E: [1363x1363 double]

Linearize a structural model and return finite element matrices.

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 Young's modulus, Poisson's ratio, and the 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'

为了防止施加足够的边界限制rigid body motion under applied loading. Typically, you hold a tuning fork by hand or mount it on a table. A simplified approximation to this boundary condition is fixing a region near the intersection of 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 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 tuning fork.

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

Measure they-displacement of face 12 andx-displacement of face 6.

linearizeOutput(structuralmodel,'Face',12,'Component','y'); linearizeOutput(structuralmodel,'Face',6,'Component','x');

Obtain amechssmodel of the tuning fork.

sys = linearize(structuralmodel)

Sparse continuous-time second-order model with 26 outputs, 4 inputs, and 3240 degrees of freedom. Use "spy" and "showStateInfo" to inspect model structure. Type "properties('mechss')" for a list of model properties. Type "help mechssOptions" for available solver options for this model.

In the linearized model, usesys.InputNameto check that the inputs tosysare the gravity body load and the pressure pulse on a tine. The gravity body load produces three inputs because it hasx-,y-, andz-components.

sys.InputName

ans =4x1 cell{'Gravity_x'} {'Gravity_y'} {'Gravity_z'} {'Pressure' }

In the linearized model, usesys.OutputGroupto locate the sections associated with each coordinate.

sys.OutputGroup

ans =struct with fields:Face12_y: [1 2 3 4 5 6 7 8 9 10 11 12 13] Face6_x: [14 15 16 17 18 19 20 21 22 23 24 25 26]

If you do not have Control System Toolbox™, you can access the finite element matricesM,K,B, andFas follows.

mx = linearize(structuralmodel,'OutputType','matrices')

mx =struct with fields:M: [3240x3240 double] K: [3240x3240 double] B: [3240x4 double] F: [26x3240 double]