solve

Solve heat transfer, structural analysis, or electromagnetic analysis problem

Syntax

structuralStaticResults = solve(structuralStatic)

structuralTransientResults = solve(structuralTransient,tlist)

structuralFrequencyResponseResults = solve(structuralFrequencyResponse,flist)

structuralModalResults = solve(structuralModal,"FrequencyRange",[omega1,omega2])

structuralTransientResults = solve(structuralTransient,tlist,"ModalResults",structuralModalR)

structuralFrequencyResponseResults = solve(structuralFrequencyResponse,flist,"ModalResults",structuralModalR)

thermalSteadyStateResults = solve(thermalSteadyState)

thermalTransientResults = solve(thermalTransient,tlist)

thermalModalResults = solve(thermalModal,"DecayRange",[lambda1,lambda2])

thermalModalResults = solve(thermalModal,"Snapshots",Tmatrix)

thermalTransientResults = solve(thermalTransient,"ModalResults",thermalModalR)

emagStaticResults = solve(emagmodel)

emagHarmonicResults = solve(emagmodel,"Frequency",omega)

Description

structuralStaticResults = solve(structuralStatic) returns the solution to the static structural analysis model represented in structuralStatic.

structuralTransientResults = solve(structuralTransient,tlist) returns the solution to the transient structural dynamics model represented in structuralTransient at the times specified in tlist.

structuralFrequencyResponseResults = solve(structuralFrequencyResponse,flist) returns the solution to the frequency response model represented in structuralFrequencyResponse at the frequencies specified in flist.

structuralModalResults = solve(structuralModal,"FrequencyRange",[omega1,omega2]) returns the solution to the modal analysis model for all modes in the frequency range [omega1,omega2]. Define omega1 as slightly lower than the lowest expected frequency and omega2 as slightly higher than the highest expected frequency. For example, if the lowest expected frequency is zero, then use a small negative value for omega1.

structuralTransientResults = solve(structuralTransient,tlist,"ModalResults",structuralModalR) and structuralFrequencyResponseResults = solve(structuralFrequencyResponse,flist,"ModalResults",structuralModalR) solves a transient and a frequency response structural model, respectively, by using the modal superposition method to speed up computations. First, perform modal analysis to compute natural frequencies and mode shapes in a particular frequency range. Then, use this syntax to invoke the modal superposition method. The accuracy of the results depends on the modes in the modal analysis results.

thermalSteadyStateResults = solve(thermalSteadyState) returns the solution to the steady-state thermal model represented in thermalSteadyState.

thermalTransientResults = solve(thermalTransient,tlist) returns the solution to the transient thermal model represented in thermalTransient at the times specified in tlist.

thermalModalResults = solve(thermalModal,"DecayRange",[lambda1,lambda2]) performs an eigen decomposition of a linear thermal model thermalModal for all modes in the decay range [lambda1,lambda2]. The resulting modes enable you to:

• Use the modal superposition method to speed up a transient thermal analysis.

• Extract the reduced modal system to use, for example, in Simulink^®.

thermalModalResults = solve(thermalModal,"Snapshots",Tmatrix) obtains the modal basis of a linear or nonlinear thermal model using proper orthogonal decomposition (POD). You can use the resulting modes to speed up a transient thermal analysis or, if your thermal model is linear, to extract the reduced modal system.

thermalTransientResults = solve(thermalTransient,"ModalResults",thermalModalR) solves a transient thermal model by using the modal superposition method to speed up computations. First, perform modal decomposition to compute mode shapes for a particular decay range. Then, use this syntax to invoke the modal superposition method. The accuracy of the results depends on the modes in the modal analysis results.

emagStaticResults = solve(emagmodel) returns the solution to the electrostatic, magnetostatic, or DC conduction model represented in emagmodel.

emagHarmonicResults = solve(emagmodel,"Frequency",omega) returns the solution to the harmonic electromagnetic analysis model represented in emagmodel at the frequencies specified in omega.

Examples

Solution to Static Structural Model

Solve a static structural model representing a bimetallic cable under tension.

Create a static structural model for a solid (3-D) problem.

structuralmodel = createpde("structural","static-solid");

Create the geometry and include it in the model. Plot the geometry.

gm = multicylinder([0.01 0.015],0.05);
structuralmodel.Geometry = gm;
pdegplot(structuralmodel,"FaceLabels","on", ...
                         "CellLabels","on", ...
                         "FaceAlpha",0.5)

Specify Young's modulus and Poisson's ratio for each metal.

structuralProperties(structuralmodel,"Cell",1,"YoungsModulus",110E9, ...
                                              "PoissonsRatio",0.28);
structuralProperties(structuralmodel,"Cell",2,"YoungsModulus",210E9, ...
                                              "PoissonsRatio",0.3);

Specify that faces 1 and 4 are fixed boundaries.

structuralBC(structuralmodel,"Face",[1,4],"Constraint","fixed");

Specify the surface traction for faces 2 and 5.

structuralBoundaryLoad(structuralmodel,"Face",[2,5], ...
                                       "SurfaceTraction",[0;0;100]);

Generate a mesh and solve the problem.

generateMesh(structuralmodel);
structuralresults = solve(structuralmodel)

structuralresults = 
  StaticStructuralResults with properties:

      Displacement: [1x1 FEStruct]
            Strain: [1x1 FEStruct]
            Stress: [1x1 FEStruct]
    VonMisesStress: [22402x1 double]
              Mesh: [1x1 FEMesh]

The solver finds the values of the displacement, stress, strain, and von Mises stress at the nodal locations. To access these values, use structuralresults.Displacement, structuralresults.Stress, and so on. The displacement, stress, and strain values at the nodal locations are returned as FEStruct objects with the properties representing their components. Note that properties of an FEStruct object are read-only.

structuralresults.Displacement

ans = 
  FEStruct with properties:

           ux: [22402x1 double]
           uy: [22402x1 double]
           uz: [22402x1 double]
    Magnitude: [22402x1 double]

structuralresults.Stress

ans = 
  FEStruct with properties:

    sxx: [22402x1 double]
    syy: [22402x1 double]
    szz: [22402x1 double]
    syz: [22402x1 double]
    sxz: [22402x1 double]
    sxy: [22402x1 double]

structuralresults.Strain

ans = 
  FEStruct with properties:

    exx: [22402x1 double]
    eyy: [22402x1 double]
    ezz: [22402x1 double]
    eyz: [22402x1 double]
    exz: [22402x1 double]
    exy: [22402x1 double]

Plot the deformed shape with the z-component of normal stress.

pdeplot3D(structuralmodel, ...
          "ColorMapData",structuralresults.Stress.szz, ...
          "Deformation",structuralresults.Displacement)

Solution to Transient Structural Model

Solve for the transient response of a thin 3-D plate under a harmonic load at the center.

Create a transient dynamic model for a 3-D problem.

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

Create the geometry and include it in the model. Plot the geometry.

gm = multicuboid([5,0.05],[5,0.05],0.01);
structuralmodel.Geometry = gm;
pdegplot(structuralmodel,"FaceLabels","on","FaceAlpha",0.5)

Zoom in to see the face labels on the small plate at the center.

figure
pdegplot(structuralmodel,"FaceLabels","on","FaceAlpha",0.25)
axis([-0.2 0.2 -0.2 0.2 -0.1 0.1])

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

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

Specify that all faces on the periphery of the thin 3-D plate are fixed boundaries.

structuralBC(structuralmodel,"Constraint","fixed","Face",5:8);

Apply a sinusoidal pressure load on the small face at the center of the plate.

structuralBoundaryLoad(structuralmodel,"Face",12, ...
                                       "Pressure",5E7, ...
                                       "Frequency",25);

Generate a mesh with linear elements.

generateMesh(structuralmodel,"GeometricOrder","linear","Hmax",0.2);

Specify zero initial displacement and velocity.

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

Solve the model.

tlist = linspace(0,1,300);
structuralresults = solve(structuralmodel,tlist);

The solver finds the values of the displacement, velocity, and acceleration at the nodal locations. To access these values, use structuralresults.Displacement, structuralresults.Velocity, and so on. The displacement, velocity, and acceleration values are returned as FEStruct objects with the properties representing their components. Note that properties of an FEStruct object are read-only.

structuralresults.Displacement

ans = 
  FEStruct with properties:

           ux: [1873x300 double]
           uy: [1873x300 double]
           uz: [1873x300 double]
    Magnitude: [1873x300 double]

structuralresults.Velocity

ans = 
  FEStruct with properties:

           vx: [1873x300 double]
           vy: [1873x300 double]
           vz: [1873x300 double]
    Magnitude: [1873x300 double]

structuralresults.Acceleration

ans = 
  FEStruct with properties:

           ax: [1873x300 double]
           ay: [1873x300 double]
           az: [1873x300 double]
    Magnitude: [1873x300 double]

Frequency Response Analysis

Perform frequency response analysis of a tuning fork.

First, create a structural model for modal analysis of a solid tuning fork.

model = createpde("structural","frequency-solid");

Import the tuning fork geometry.

importGeometry(model,"TuningFork.stl");

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(model,"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(model,"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. To create a simple approximation of this boundary condition, fix a region near the intersection of tines and the handle (faces 21 and 22).

structuralBC(model,"Face",[21,22],"Constraint","fixed");

Specify the pressure loading on a tine (face 11) as a short rectangular pressure pulse. In the frequency domain, this pressure pulse is a unit load uniformly distributed across all frequencies.

structuralBoundaryLoad(model,"Face",11,"Pressure",1);
flist = linspace(0,4000,150);
mesh = generateMesh(model,"Hmax",0.005); 
R = solve(model,2*pi*flist);

Plot the vibration frequency of the tine tip, which is face 12. Find nodes on the tip face and plot the y-component of the displacement over the frequency, using one of these nodes.

excitedTineTipNodes = findNodes(mesh,"region","Face",12);
tipDisp = R.Displacement.uy(excitedTineTipNodes(1),:);

figure 
plot(flist,abs(tipDisp)) 
xlabel("Frequency");
ylabel("|Y-Displacement|");

Solution to Modal Analysis Structural Model

Find the fundamental (lowest) mode of a 2-D cantilevered beam, assuming prevalence of the plane-stress condition.

Specify geometric and structural properties of the beam, along with a unit plane-stress thickness.

length = 5;
height = 0.1;
E = 3E7;
nu = 0.3;
rho = 0.3/386;

Create a modal plane-stress model, assign a geometry, and generate a mesh.

structuralmodel = createpde("structural","modal-planestress");
gdm = [3;4;0;length;length;0;0;0;height;height];
g = decsg(gdm,'S1',('S1')');
geometryFromEdges(structuralmodel,g);

Define a maximum element size (five elements through the beam thickness).

hmax = height/5;
msh=generateMesh(structuralmodel,"Hmax",hmax);

Specify the structural properties and boundary constraints.

structuralProperties(structuralmodel,"YoungsModulus",E, ...
                                     "MassDensity",rho, ... 
                                     "PoissonsRatio",nu);
structuralBC(structuralmodel,"Edge",4,"Constraint","fixed");

Compute the analytical fundamental frequency (Hz) using the beam theory.

I = height^3/12;
analyticalOmega1 = 3.516*sqrt(E*I/(length^4*(rho*height)))/(2*pi)

analyticalOmega1 = 126.9498

Specify a frequency range that includes an analytically computed frequency and solve the model.

modalresults = solve(structuralmodel,"FrequencyRange",[0,1e6])

modalresults = 
  ModalStructuralResults with properties:

    NaturalFrequencies: [32x1 double]
            ModeShapes: [1x1 FEStruct]
                  Mesh: [1x1 FEMesh]

The solver finds natural frequencies and modal displacement values at nodal locations. To access these values, use modalresults.NaturalFrequencies and modalresults.ModeShapes.

modalresults.NaturalFrequencies/(2*pi)

ans = 32×1
105 ×

    0.0013
    0.0079
    0.0222
    0.0433
    0.0711
    0.0983
    0.1055
    0.1462
    0.1930
    0.2455
      ⋮

modalresults.ModeShapes

ans = 
  FEStruct with properties:

           ux: [6511x32 double]
           uy: [6511x32 double]
    Magnitude: [6511x32 double]

Plot the y-component of the solution for the fundamental frequency.

pdeplot(structuralmodel,"XYData",modalresults.ModeShapes.uy(:,1))
title(['First Mode with Frequency ', ...
        num2str(modalresults.NaturalFrequencies(1)/(2*pi)),' Hz'])
axis equal

Solution to Transient Structural Model Using Modal Superposition Method

Solve for the transient response at the center of a 3-D beam under a harmonic load on one of its corners.

Modal Analysis

Create a modal analysis model for a 3-D problem.

structuralmodel = createpde("structural","modal-solid");

Create the geometry and include it in the model. Plot the geometry and display the edge and vertex labels.

gm = multicuboid(0.05,0.003,0.003);
structuralmodel.Geometry = gm;
pdegplot(structuralmodel,"EdgeLabels","on", ...
                         "VertexLabels","on");
view([95 5])

Generate a mesh.

msh = generateMesh(structuralmodel);

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

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

Specify minimal constraints on one end of the beam to prevent rigid body modes. For example, specify that edge 4 and vertex 7 are fixed boundaries.

structuralBC(structuralmodel,"Edge",4,"Constraint","fixed");
structuralBC(structuralmodel,"Vertex",7,"Constraint","fixed");

Solve the problem for the frequency range from 0 to 500,000. The recommended approach is to use a value that is slightly lower than the expected lowest frequency. Thus, use -0.1 instead of 0.

Rm = solve(structuralmodel,"FrequencyRange",[-0.1,500000]);

Transient Analysis

Switch the analysis type of the model to transient.

structuralmodel.AnalysisType = "transient-solid";

Apply a sinusoidal force on the corner opposite of the constrained edge and vertex.

structuralBoundaryLoad(structuralmodel,"Vertex",5, ...
                                       "Force",[0,0,10], ...
                                       "Frequency",7600);

Specify zero initial displacement and velocity.

structuralIC(structuralmodel,"Velocity",[0;0;0], ...
                             "Displacement",[0;0;0]);

Specify the relative and absolute tolerances for the solver.

structuralmodel.SolverOptions.RelativeTolerance = 1E-5;
structuralmodel.SolverOptions.AbsoluteTolerance = 1E-9;

Solve the model using the modal results.

tlist = linspace(0,0.004,120);
Rdm = solve(structuralmodel,tlist,"ModalResults",Rm);

Interpolate and plot the displacement at the center of the beam.

intrpUdm = interpolateDisplacement(Rdm,0,0,0.0015);

plot(Rdm.SolutionTimes,intrpUdm.uz)
grid on
xlabel("Time");
ylabel("Center of beam displacement")

Expansion of Cantilever Beam Under Thermal Load

Find the deflection of a 3-D cantilever beam under a nonuniform thermal load. Specify the thermal load on the structural model using the solution from a transient thermal analysis on the same geometry and mesh.

Transient Thermal Model Analysis

Create a transient thermal model.

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

Create and plot the geometry.

gm = multicuboid(0.5,0.1,0.05);
thermalmodel.Geometry = gm;
pdegplot(thermalmodel,"FaceLabels","on","FaceAlpha",0.5)

Generate a mesh.

mesh = generateMesh(thermalmodel);

Specify the thermal properties of the material.

thermalProperties(thermalmodel,"ThermalConductivity",5e-3, ...
                               "MassDensity",2.7*10^(-6), ...
                               "SpecificHeat",10);

Specify the constant temperatures applied to the left and right ends of the beam.

thermalBC(thermalmodel,"Face",3,"Temperature",100);
thermalBC(thermalmodel,"Face",5,"Temperature",0);

Specify the heat source over the entire geometry.

internalHeatSource(thermalmodel,10);

Set the initial temperature.

thermalIC(thermalmodel,0);

Solve the model.

tlist = [0:1e-4:2e-4];
thermalresults = solve(thermalmodel,tlist)

thermalresults = 
  TransientThermalResults with properties:

      Temperature: [3870x3 double]
    SolutionTimes: [0 1.0000e-04 2.0000e-04]
       XGradients: [3870x3 double]
       YGradients: [3870x3 double]
       ZGradients: [3870x3 double]
             Mesh: [1x1 FEMesh]

Plot the temperature distribution for each time step.

for n = 1:numel(thermalresults.SolutionTimes)
    figure
    pdeplot3D(thermalmodel, ...
                "ColorMapData", ...
                thermalresults.Temperature(:,n))
    title(["Temperature at Time = " ...
                num2str(tlist(n))])
    caxis([0 100])
end

Structural Analysis with Thermal Load

Create a static structural model.

structuralmodel = createpde("structural","static-solid");

Include the same geometry as for the thermal model.

structuralmodel.Geometry = gm;

Use the same mesh that you used to obtain the thermal solution.

structuralmodel.Mesh = mesh;

Specify Young's modulus, Poisson's ratio, and the coefficient of thermal expansion.

structuralProperties(structuralmodel,"YoungsModulus",1e10, ...
                                     "PoissonsRatio",0.3, ...
                                     "CTE",11.7e-6);

Apply a fixed boundary condition on face 5.

structuralBC(structuralmodel,"Face",5,"Constraint","fixed");

Apply a body load using the transient thermal model solution. By default, structuralBodyLoad uses the solution for the last time step.

structuralBodyLoad(structuralmodel,"Temperature",thermalresults);

Specify the reference temperature.

structuralmodel.ReferenceTemperature = 10;

Solve the structural model.

thermalstressresults = solve(structuralmodel);

Plot the deformed shape of the beam corresponding to the last step of the transient thermal model solution.

pdeplot3D(structuralmodel, ...
            "ColorMapData", ...
             thermalstressresults.Displacement.Magnitude, ...
            "Deformation", ...
             thermalstressresults.Displacement)
title(["Thermal Expansion at Solution Time = " ...
                         num2str(tlist(end))])
caxis([0 3e-3])

Now specify the body loads as the thermal model solutions for all time steps. For each body load, solve the structural model and plot the corresponding deformed shape of the beam.

for n = 1:numel(thermalresults.SolutionTimes)
    structuralBodyLoad(structuralmodel, ...
                         "Temperature", ...
                          thermalresults, ...
                         "TimeStep",n);
    thermalstressresults = solve(structuralmodel);
    figure
    pdeplot3D(structuralmodel, ...
               "ColorMapData", ...
                thermalstressresults.Displacement.Magnitude, ...
               "Deformation", ...
                thermalstressresults.Displacement)
    title(["Thermal Results at Solution Time = " ...
                             num2str(tlist(n))])
    caxis([0 3e-3])
end

Solution to Steady-State Thermal Model

Solve a 3-D steady-state thermal problem.

Create a thermal model for this problem.

thermalmodel = createpde("thermal");

Import and plot the block geometry.

importGeometry(thermalmodel,"Block.stl"); 
pdegplot(thermalmodel,"FaceLabel","on","FaceAlpha",0.5)
axis equal

Assign material properties.

thermalProperties(thermalmodel,"ThermalConductivity",80);

Apply a constant temperature of 100 °C to the left side of the block (face 1) and a constant temperature of 300 °C to the right side of the block (face 3). All other faces are insulated by default.

thermalBC(thermalmodel,"Face",1,"Temperature",100);
thermalBC(thermalmodel,"Face",3,"Temperature",300);

Mesh the geometry and solve the problem.

generateMesh(thermalmodel);
thermalresults = solve(thermalmodel)

thermalresults = 
  SteadyStateThermalResults with properties:

    Temperature: [12691x1 double]
     XGradients: [12691x1 double]
     YGradients: [12691x1 double]
     ZGradients: [12691x1 double]
           Mesh: [1x1 FEMesh]

The solver finds the temperatures and temperature gradients at the nodal locations. To access these values, use thermalresults.Temperature, thermalresults.XGradients, and so on. For example, plot temperatures at the nodal locations.

pdeplot3D(thermalmodel,"ColorMapData",thermalresults.Temperature)

Solution to Transient Thermal Model

Solve a 2-D transient thermal problem.

Create a transient thermal model for this problem.

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

Create the geometry and include it in the model.

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';
dl = decsg(gd,sf,ns);
geometryFromEdges(thermalmodel,dl);
pdegplot(thermalmodel,"EdgeLabels","on","FaceLabels","on")
xlim([-1.5 4.5])
ylim([-0.5 3.5])
axis equal

For the square region, assign these thermal properties:

• Thermal conductivity is $10\mathrm{W}/\left(\mathrm{m}{\cdot }^{\circ }\mathrm{C}\right)$

• Mass density is $2\mathrm{kg}/{\mathrm{m}}^3$

• Specific heat is $0.1\mathrm{J}/\left({\mathrm{kg}\cdot }^{\circ }\mathrm{C}\right)$

thermalProperties(thermalmodel,"ThermalConductivity",10, ...
                               "MassDensity",2, ...
                               "SpecificHeat",0.1, ...
                               "Face",1);

For the diamond region, assign these thermal properties:

• Thermal conductivity is $2\mathrm{W}/\left(\mathrm{m}{\cdot }^{\circ }\mathrm{C}\right)$

• Mass density is $1\mathrm{kg}/{\mathrm{m}}^3$

• Specific heat is $0.1\mathrm{J}/\left({\mathrm{kg}\cdot }^{\circ }\mathrm{C}\right)$

thermalProperties(thermalmodel,"ThermalConductivity",2, ...
                               "MassDensity",1, ...
                               "SpecificHeat",0.1, ...
                               "Face",2);

Assume that the diamond-shaped region is a heat source with a density of $4\mathrm{W}/{\mathrm{m}}^2$.

internalHeatSource(thermalmodel,4,"Face",2);

Apply a constant temperature of 0 °C to the sides of the square plate.

thermalBC(thermalmodel,"Temperature",0,"Edge",[1 2 7 8]);

Set the initial temperature to 0 °C.

thermalIC(thermalmodel,0);

Generate the mesh.

generateMesh(thermalmodel);

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,-1,10). This command returns 10 logarithmically spaced solution times between 0.01 and 0.1.

tlist = logspace(-2,-1,10);

Solve the equation.

thermalresults = solve(thermalmodel,tlist);

Plot the solution with isothermal lines by using a contour plot.

T = thermalresults.Temperature;
pdeplot(thermalmodel,"XYData",T(:,10),"Contour","on","ColorMap","hot")

Solution to Transient Thermal Model 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

First, create a modal thermal model.

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

Create the geometry and include it in the model.

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';
dl = decsg(gd,sf,ns);
geometryFromEdges(thermalmodel,dl);
pdegplot(thermalmodel,"EdgeLabels","on","FaceLabels","on")
xlim([-1.5 4.5])
ylim([-0.5 3.5])
axis equal

For the square region, assign these thermal properties:

• Thermal conductivity is $10\mathrm{W}/\left(\mathrm{m}{\cdot }^{\circ }\mathrm{C}\right)$.

• Mass density is $2\mathrm{kg}/{\mathrm{m}}^3$.

• Specific heat is $0.1\mathrm{J}/\left({\mathrm{kg}\cdot }^{\circ }\mathrm{C}\right)$.

thermalProperties(thermalmodel,"ThermalConductivity",10, ...
                               "MassDensity",2, ...
                               "SpecificHeat",0.1, ...
                               "Face",1);

For the diamond region, assign these thermal properties:

• Thermal conductivity is $2\mathrm{W}/\left(\mathrm{m}{\cdot }^{\circ }\mathrm{C}\right)$.

• Mass density is $1\mathrm{kg}/{\mathrm{m}}^3$.

• Specific heat is $0.1\mathrm{J}/\left({\mathrm{kg}\cdot }^{\circ }\mathrm{C}\right)$.

thermalProperties(thermalmodel,"ThermalConductivity",2, ...
                               "MassDensity",1, ...
                               "SpecificHeat",0.1, ...
                               "Face",2);

Assume that the diamond-shaped region is a heat source with a density of $4\mathrm{W}/{\mathrm{m}}^2$.

internalHeatSource(thermalmodel,4,"Face",2);

Apply a constant temperature of 0 °C to the sides of the square plate.

thermalBC(thermalmodel,"Temperature",0,"Edge",[1 2 7 8]);

Set the initial temperature to 0 °C.

thermalIC(thermalmodel,0);

Generate the mesh.

generateMesh(thermalmodel);

Compute eigenmodes of the thermal model in the decay range [100,10000] $$s^{-1}$$.

RModal = solve(thermalmodel,"DecayRange",[100,10000])

RModal = 
  ModalThermalResults with properties:

    DecayRates: [164x1 double]
    ModeShapes: [1481x164 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 thermal model analysis type to transient.

thermalmodel.AnalysisType = "transient";

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(thermalmodel,tlist,"ModalResults",RModal);

Plot the solution with isothermal lines by using a contour plot.

T = Rtransient.Temperature;
pdeplot(thermalmodel,"XYData",T(:,end), ...
                     "Contour","on", ...
                     "ColorMap","hot")

Snapshots for Proper Orthogonal Decomposition

Obtain POD modes of a linear thermal model using several instances of the transient solution (snapshots).

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 to 100.

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

Set an initial value of 0 for the temperature.

thermalIC(thermalmodel,0);

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];
for q = 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)

RModal = 
  ModalThermalResults with properties:

          DecayRates: [6x1 double]
          ModeShapes: [1541x6 double]
    SnapshotsAverage: [1541x1 double]
            ModeType: "PODModes"
                Mesh: [1x1 FEMesh]

Solution to 2-D Electrostatic Analysis Model

Solve an electromagnetic problem and find the electric potential and field distribution for a 2-D geometry representing a plate with a hole.

Create an electromagnetic model for electrostatic analysis.

emagmodel = createpde("electromagnetic","electrostatic");

Import and plot the geometry representing a plate with a hole.

importGeometry(emagmodel,"PlateHolePlanar.stl");
pdegplot(emagmodel,"EdgeLabels","on")

Specify the vacuum permittivity value in the SI system of units.

emagmodel.VacuumPermittivity = 8.8541878128E-12;

Specify the relative permittivity of the material.

electromagneticProperties(emagmodel,"RelativePermittivity",1);

Apply the voltage boundary conditions on the edges framing the rectangle and the circle.

electromagneticBC(emagmodel,"Voltage",0,"Edge",1:4);
electromagneticBC(emagmodel,"Voltage",1000,"Edge",5);

Specify the charge density for the entire geometry.

electromagneticSource(emagmodel,"ChargeDensity",5E-9);

Generate the mesh.

generateMesh(emagmodel);

Solve the model.

R = solve(emagmodel)

R = 
  ElectrostaticResults with properties:

      ElectricPotential: [1218x1 double]
          ElectricField: [1x1 FEStruct]
    ElectricFluxDensity: [1x1 FEStruct]
                   Mesh: [1x1 FEMesh]

Plot the electric potential and field.

pdeplot(emagmodel,"XYData",R.ElectricPotential, ...
                  "FlowData",[R.ElectricField.Ex ...
                              R.ElectricField.Ey])
axis equal

Solution to 3-D Magnetostatic Analysis Model

Solve a 3-D electromagnetic problem on a geometry representing a plate with a hole in its center. Plot the resulting magnetic potential and field distribution.

Create an electromagnetic model for magnetostatic analysis.

emagmodel = createpde("electromagnetic","magnetostatic");

Import and plot the geometry representing a plate with a hole.

importGeometry(emagmodel,"PlateHoleSolid.stl");
pdegplot(emagmodel,"FaceLabels","on","FaceAlpha",0.3)

Specify the vacuum permeability value in the SI system of units.

emagmodel.VacuumPermeability = 1.2566370614e-6;

Specify the relative permeability of the material.

electromagneticProperties(emagmodel,"RelativePermeability",5000);

Apply the magnetic potential boundary conditions on the side faces and the face bordering the hole.

electromagneticBC(emagmodel,"MagneticPotential",[0;0;0],"Face",3:6); 
electromagneticBC(emagmodel,"MagneticPotential",[0;0;0.01],"Face",7);

Specify the current density for the entire geometry.

electromagneticSource(emagmodel,"CurrentDensity",[0;0;0.5]);

Generate the mesh.

generateMesh(emagmodel);

Solve the model.

R = solve(emagmodel)

R = 
  MagnetostaticResults with properties:

      MagneticPotential: [1x1 FEStruct]
          MagneticField: [1x1 FEStruct]
    MagneticFluxDensity: [1x1 FEStruct]
                   Mesh: [1x1 FEMesh]

Plot the z-component of the magnetic potential.

pdeplot3D(emagmodel,"ColormapData",R.MagneticPotential.Az)

Plot the magnetic field.

pdeplot3D(emagmodel,"FlowData",[R.MagneticField.Hx ...
                                R.MagneticField.Hy ...
                                R.MagneticField.Hz])

Solution to 3-D DC Conduction Model

Solve a DC conduction problem on a geometry representing a 3-D plate with a hole in its center. Plot the electric potential and the components of the current density.

Create an electromagnetic model for DC conduction analysis.

emagmodel = createpde("electromagnetic","conduction");

Import and plot a geometry representing a plate with a hole.

gm = importGeometry(emagmodel,"PlateHoleSolid.stl");
pdegplot(gm,"FaceLabels","on","FaceAlpha",0.3)

Specify the conductivity of the material.

electromagneticProperties(emagmodel,"Conductivity",6e4);

Apply the voltage boundary conditions on the left, right, top, and bottom faces of the plate.

electromagneticBC(emagmodel,"Voltage",0,"Face",3:6);

Specify the surface current density on the face bordering the hole.

electromagneticBC(emagmodel,"SurfaceCurrentDensity",100,"Face",7);

Generate the mesh.

generateMesh(emagmodel);

Solve the model.

R = solve(emagmodel)

R = 
  ConductionResults with properties:

    ElectricPotential: [4359x1 double]
        ElectricField: [1x1 FEStruct]
       CurrentDensity: [1x1 FEStruct]
                 Mesh: [1x1 FEMesh]

Plot the electric potential.

figure
pdeplot3D(emagmodel,"ColorMapData",R.ElectricPotential)

Plot the x-component of the current density.

figure
pdeplot3D(emagmodel,"ColorMapData",R.CurrentDensity.Jx)
title("x-Component of Current Density")

Plot the y-component of the current density.

figure
pdeplot3D(emagmodel,"ColorMapData",R.CurrentDensity.Jy)
title("y-Component of Current Density")

Plot the z-component of the current density.

figure
pdeplot3D(emagmodel,"ColorMapData",R.CurrentDensity.Jz)
title("z-Component of Current Density")

DC Conduction Solution as Current Density for Magnetostatic Model

Use a solution obtained by performing a DC conduction analysis to specify current density for a magnetostatic model.

Create an electromagnetic model for DC conduction analysis.

emagmodel = createpde("electromagnetic","conduction");

Import and plot a geometry representing a plate with a hole.

gm = importGeometry(emagmodel,"PlateHoleSolid.stl");
pdegplot(gm,"FaceLabels","on","FaceAlpha",0.3)

Specify the conductivity of the material.

electromagneticProperties(emagmodel,"Conductivity",6e4);

Apply the voltage boundary conditions on the left, right, top, and bottom faces of the plate.

electromagneticBC(emagmodel,"Voltage",0,"Face",3:6);

Specify the surface current density on the face bordering the hole.

electromagneticBC(emagmodel,"SurfaceCurrentDensity",100,"Face",7);

Generate the mesh.

generateMesh(emagmodel);

Solve the model.

R = solve(emagmodel);

Change the analysis type of the model to magnetostatic.

emagmodel.AnalysisType = "magnetostatic";

This model already has a quadratic mesh that you generated for the DC conduction analysis. For a 3-D magnetostatic model, the mesh must be linear. Generate a new linear mesh. The generateMesh function creates a linear mesh by default if the model is 3-D and magnetostatic.

generateMesh(emagmodel);

Specify the vacuum permeability value in the SI system of units.

emagmodel.VacuumPermeability = 1.2566370614e-6;

Specify the relative permeability of the material.

electromagneticProperties(emagmodel,"RelativePermeability",5000);

Apply the magnetic potential boundary conditions on the side faces and the face bordering the hole.

electromagneticBC(emagmodel,"MagneticPotential",[0;0;0],"Face",3:6);
electromagneticBC(emagmodel,"MagneticPotential",[0;0;0.01],"Face",7);

Specify the current density for the entire geometry using the DC conduction solution.

electromagneticSource(emagmodel,"CurrentDensity",R);

Solve the model.

Rmagnetostatic = solve(emagmodel);

Plot the x- and z-components of the magnetic potential.

pdeplot3D(emagmodel,"ColormapData",Rmagnetostatic.MagneticPotential.Ax)

pdeplot3D(emagmodel,"ColormapData",Rmagnetostatic.MagneticPotential.Az)

Solution to 2-D Magnetostatic Model with Permanent Magnet

Solve a magnetostatic model of a copper square with a permanent neodymium magnet in its center.

Create the unit square geometry with a circle in its center.

L = 0.8;
r = 0.25;
sq = [3 4 -L L L -L -L -L L L]';
circ = [1 0 0 r 0 0 0 0 0 0]';
gd = [sq,circ];
sf = "sq + circ";
ns = char('sq','circ');
ns = ns';
g = decsg(gd,sf,ns);

Plot the geometry with the face and edge labels.

pdegplot(g,"FaceLabels","on","EdgeLabels","on")

Create a magnetostatic model and include the geometry in the model.

emagmodel = createpde("electromagnetic","magnetostatic");
geometryFromEdges(emagmodel,g);

Specify the vacuum permeability value in the SI system of units.

emagmodel.VacuumPermeability = 1.2566370614e-6;

Specify the relative permeability of the copper for the square.

electromagneticProperties(emagmodel,"Face",1, ...
                                    "RelativePermeability",1);

Specify the relative permeability of the neodymium for the circle.

electromagneticProperties(emagmodel,"Face",2, ...
                                    "RelativePermeability",1.05);

Specify the magnetization magnitude for the neodymium magnet.

M = 1e6;

Specify magnetization on the circular face in the positive x-direction. Magnetization for a 2-D model is a column vector of two elements.

dir = [1;0];
electromagneticSource(emagmodel,"Face",2,"Magnetization",M*dir);

Apply the magnetic potential boundary conditions on the edges framing the square.

electromagneticBC(emagmodel,"Edge",1:4,"MagneticPotential",0);

Generate the mesh with finer meshing near the edges of the circle.

generateMesh(emagmodel,"Hedge",{5:8,0.007});
figure
pdemesh(emagmodel)

Solve the model, and find the resulting magnetic fields B and H. Here, $\mathit{B}=\mu \mathit{H}+{\mu }_0\mathit{M}$, where $\mu$ is the absolute magnetic permeability of the material, ${\mu }_0$ is the vacuum permeability, and $\mathit{M}$ is the magnetization.

R = solve(emagmodel);
Bmag = sqrt(R.MagneticFluxDensity.Bx.^2 + R.MagneticFluxDensity.By.^2);
Hmag = sqrt(R.MagneticField.Hx.^2 + R.MagneticField.Hy.^2);

Plot the magnetic field B.

figure
pdeplot(emagmodel,"XYData",Bmag, ...
                  "FlowData",[R.MagneticFluxDensity.Bx ...
                              R.MagneticFluxDensity.By])

Plot the magnetic field H.

figure
pdeplot(emagmodel,"XYData",Hmag, ...
                  "FlowData",[R.MagneticField.Hx R.MagneticField.Hy])

Solution to 2-D Harmonic Electromagnetic Model

For an electromagnetic harmonic analysis problem, find the x- and y-components of the electric field. Solve the problem on a domain consisting of a square with a circular hole.

Create an electromagnetic model for harmonic analysis.

emagmodel = createpde("electromagnetic","harmonic");

Define a circle in a square, place them in one matrix, and create a set formula that subtracts the circle from the square.

SQ = [3,4,-5,-5,5,5,-5,5,5,-5]';
C = [1,0,0,1]';
C = [C;zeros(length(SQ) - length(C),1)];
gm = [SQ,C];
sf = 'SQ-C';

Create the geometry.

ns = char('SQ','C');
ns = ns';
g = decsg(gm,sf,ns);

Include the geometry in the model and plot the geometry with the edge labels.

geometryFromEdges(emagmodel,g);
pdegplot(emagmodel,"EdgeLabels","on")
xlim([-5.5 5.5])
ylim([-5.5 5.5])

Specify the vacuum permittivity and permeability values as 1.

emagmodel.VacuumPermittivity = 1;
emagmodel.VacuumPermeability = 1;

Specify the relative permittivity, relative permeability, and conductivity of the material.

electromagneticProperties(emagmodel,"RelativePermittivity",1, ...
                                    "RelativePermeability",1, ...
                                    "Conductivity",0);

Apply the absorbing boundary condition with a thickness of 2 on the edges of the square. Use the default attenuation rate for the absorbing region.

electromagneticBC(emagmodel,"Edge",1:4, ...
                            "FarField","absorbing", ...
                            "Thickness",2);

Specify an electric field on the edges of the hole.

E = @(location,state) [1;0]*exp(-1i*2*pi*location.y); 
electromagneticBC(emagmodel,"Edge",5:8,"ElectricField",E);

Generate a mesh.

generateMesh(emagmodel,"Hmax",1/2^3);

Solve the model for a frequency of $2\pi$.

result = solve(emagmodel,"Frequency",2*pi);

Plot the real part of the x-component of the resulting electric field.

figure
pdeplot(emagmodel,"XYData",real(result.ElectricField.Ex));
title("Real Part of x-Component of Electric Field")

Plot the real part of the y-component of the resulting electric field.

figure
pdeplot(emagmodel,"XYData",real(result.ElectricField.Ey));
title("Real Part of y-Component of Electric Field")

Input Arguments

structuralStatic — Static structural analysis model

StructuralModel object

Static structural analysis model, specified as a StructuralModel object. The model contains the geometry, mesh, structural properties of the material, body loads, boundary loads, and boundary conditions.

Example: structuralmodel = createpde("structural","static-solid")

structuralTransient — Transient structural analysis model

StructuralModel object

Transient structural analysis model, specified as a StructuralModel object. The model contains the geometry, mesh, structural properties of the material, body loads, boundary loads, and boundary conditions.

Example: structuralmodel = createpde("structural","transient-solid")

structuralFrequencyResponse — Frequency response analysis structural model

StructuralModel object

Frequency response analysis structural model, specified as a StructuralModel object. The model contains the geometry, mesh, structural properties of the material, body loads, boundary loads, and boundary conditions.

Example: structuralmodel = createpde("structural","frequency-solid")

structuralModal — Modal analysis structural model

StructuralModel object

Modal analysis structural model, specified as a StructuralModel object. The model contains the geometry, mesh, structural properties of the material, body loads, boundary loads, and boundary conditions.

Example: structuralmodel = createpde("structural","modal-solid")

tlist — Solution times for structural or thermal transient analysis

real vector

Solution times for structural or thermal transient analysis, specified as a real vector of monotonically increasing or decreasing values.

Example: 0:20

Data Types: double

flist — Solution frequencies for frequency response structural analysis

real vector

Solution frequencies for a frequency response structural analysis, specified as a real vector of monotonically increasing or decreasing values.

Example: linspace(0,4000,150)

Data Types: double

[omega1,omega2] — Frequency range for structural modal analysis

vector of two elements

Frequency range for a structural modal analysis, specified as a vector of two elements. Define omega1 as slightly lower than the lowest expected frequency and omega2 as slightly higher than the highest expected frequency. For example, if the lowest expected frequency is zero, then use a small negative value for omega1.

Example: [-0.1,1000]

Data Types: double

structuralModalR — Modal analysis results for structural model

ModalStructuralResults object

Modal analysis results for a structural model, specified as a ModalStructuralResults object.

Example: structuralModalR = solve(structuralmodel,"FrequencyRange",[0,1e6])

thermalSteadyState — Steady-state thermal analysis model

ThermalModel object

Steady-state thermal analysis model, specified as a ThermalModel object. ThermalModel contains the geometry, mesh, thermal properties of the material, internal heat source, Stefan-Boltzmann constant, boundary conditions, and initial conditions.

Example: thermalmodel = createpde("thermal","steadystate")

thermalTransient — Transient thermal analysis model

ThermalModel object

Transient thermal analysis model, specified as a ThermalModel object. ThermalModel contains the geometry, mesh, thermal properties of the material, internal heat source, Stefan-Boltzmann constant, boundary conditions, and initial conditions.

thermalModal — Modal thermal analysis model

ThermalModel object

Modal thermal analysis model, specified as a ThermalModel object. ThermalModel contains the geometry, mesh, thermal properties of the material, internal heat source, Stefan-Boltzmann constant, boundary conditions, and initial conditions.

[lambda1,lambda2] — Decay range for modal thermal analysis

vector of two elements

Decay range for modal thermal analysis, specified as a vector of two elements. The solve function solves a modal thermal analysis model for all modes in the decay range.

Data Types: double

Tmatrix — Thermal model solution snapshots

matrix

Thermal model solution snapshots, specified as a matrix.

Data Types: double

thermalModalR — Modal analysis results for thermal model

ModalThermalResults object

Modal analysis results for a thermal model, specified as a ModalThermalResults object.

Example: thermalModalR = solve(thermalmodel,"DecayRange",[0,1000])

emagmodel — Electromagnetic model for electrostatic, magnetostatic, or DC conduction analysis

ElectromagneticModel object

Electromagnetic model for electrostatic, magnetostatic, or DC conduction analysis, specified as an ElectromagneticModel object. The model contains the geometry, mesh, material properties, electromagnetic sources, and boundary conditions.

Example: emagmodel = createpde("electromagnetic","magnetostatic")

omega — Solution frequencies for harmonic electromagnetic analysis

nonnegative number | vector of nonnegative numbers

Solution frequencies for a harmonic electromagnetic analysis, specified as a nonnegative number or a vector of nonnegative numbers.

Data Types: double

Output Arguments

structuralStaticResults — Static structural analysis results

StaticStructuralResults object

Static structural analysis results, returned as a StaticStructuralResults object.

structuralTransientResults — Transient structural analysis results

TransientStructuralResults object

Transient structural analysis results, returned as a TransientStructuralResults object.

structuralFrequencyResponseResults — Frequency response structural analysis results

FrequencyStructuralResults object

Frequency response structural analysis results, returned as a FrequencyStructuralResults object.

structuralModalResults — Modal structural analysis results

ModalStructuralResults object

Modal structural analysis results, returned as a ModalStructuralResults object.

thermalSteadyStateResults — Steady-state thermal analysis results

SteadyStateThermalResults object

Steady-state thermal analysis results, returned as a SteadyStateThermalResults object.

thermalTransientResults — Transient thermal analysis results

TransientThermalResults object

Transient thermal analysis results, returned as a TransientThermalResults object.

thermalModalResults — Modal thermal analysis results

ModalThermalResults object

Modal thermal analysis results, returned as a ModalThermalResults object.

emagStaticResults — Electrostatic, magnetostatic or DC conduction analysis results

ElectrostaticResults object | MagnetostaticResults object | ConductionResults object

Electrostatic, magnetostatic, or DC conduction analysis results, returned as an ElectrostaticResults, MagnetostaticResults, or ConductionResults object.

emagHarmonicResults — Harmonic electromagnetic analysis results

HarmonicResults object

Harmonic electromagnetic analysis results, returned as a HarmonicResults object.

Tips

• When you use modal analysis results to solve a transient structural dynamics model, the modalresults argument must be created in Partial Differential Equation Toolbox™ from R2019a or newer.

• For a frequency response model with damping, the results are complex. Use functions such as abs and angle to obtain real-valued results, such as the magnitude and phase.

Version History

Introduced in R2017a

R2022b: DC conduction and permanent magnets

You can now solve stationary current distribution in conductors due to applied voltage. You also can solve electromagnetic problems accounting for magnetization of materials.

R2022a: Harmonic electromagnetic analysis

You can now solve 2-D and 3-D time-harmonic Maxwell’s equations (the Helmholtz equation).

R2022a: Reduced-order modeling for thermal analysis

You can now compute modes of a thermal model using eigenvalue or proper orthogonal decomposition. You also can speed up computations for a transient thermal model by using the computed modes.

R2021b: 3-D electrostatic and magnetostatic problems

You can now solve 3-D electrostatic and magnetostatic problems.

R2021a: 2-D electrostatic and magnetostatic problems

You can now solve 2-D electrostatic and magnetostatic problems.

R2020a: Axisymmetric analysis

You can now solve axisymmetric structural and thermal problems. Axisymmetric analysis simplifies 3-D thermal problems to 2-D using their symmetry around the axis of rotation.

R2019b: Frequency response structural analysis

You can now solve frequency response structural problems and find displacement, velocity, acceleration, and solution frequencies at nodal locations of the mesh. To speed up computations, you can use modal analysis results for frequency response analysis. The ModalResults argument triggers the solve function to use the modal superposition method.

R2019b: Lanczos algorithm for structural modal analysis problems

You can now specify the maximum number of Lanczos shifts and the block size for block Lanczos recurrence by using the SolverOptions property of StructuralModel. For details, see PDESolverOptions Properties.

R2019a: Modal superposition method for transient structural analysis

The new ModalResults argument triggers the solve function to switch to the modal transient solver instead of using the direct integration approach.

R2018b: Thermal stress

The solver now solves accounts for mechanical and thermal effects when solving a static structural analysis model. The function returns a displacement, stress, strain, and von Mises stress induced by both mechanical and thermal loads.

R2018a: Transient and modal structural analyses

You can now solve dynamic linear elasticity problems and find displacement, velocity, and acceleration at nodal locations of the mesh.

You also can solve modal analysis problems and find natural frequencies and mode shapes of a structure. When solving a modal analysis model, the solver requires a frequency range parameter and returns the modal solution in that frequency range.

R2017b: Static structural analysis

You can now solve static linear elasticity problems and find displacement, stress, strain, and von Mises stress at nodal locations of the mesh.