MATLAB Partial Differential Equation Toolbox – User ‘ s Guide

MATLAB Partial Differential Equation Toolbox – User ‘ s Guide
Getting Started
1
Partial Differential Equation Toolbox Product Description 1-2
Key Features 1-2
Equations You Can Solve Using PDE Toolbox . 1-3
Solve 2-D PDEs Using the PDE Modeler App 1-5
Tips . 1-5
Poisson’s Equation with Complex 2-D Geometry: PDE Modeler App . 1-7
Finite Element Method Basics 1-11
Deflection Analysis of Bracket 1-14
Heat Transfer in Block with Cavity 1-25
Setting Up Your PDE
2
Solve Problems Using PDEModel Objects 2-2
Geometry and Mesh Components . 2-4
2-D Geometry Creation at Command Line . 2-16
Three Elements of Geometry . 2-16
Basic Shapes . 2-16
Rectangle with Circular End Cap and Another Circular Excision . 2-17
Decomposed Geometry Data Structure 2-20
Parametrized Function for 2-D Geometry Creation . 2-22
Required Syntax . 2-22
Relation Between Parametrization and Region Labels . 2-22
Geometry Function for a Circle 2-23
Arc Length Calculations for a Geometry Function 2-24
Geometry Function Example with Subdomains and a Hole . 2-33
Nested Function for Geometry with Additional Parameters . 2-35
Geometry from polyshape 2-39
STL File Import 2-43
v
ContentsSTEP File Import . 2-59
Geometry from Triangulated Mesh 2-63
3-D Geometry from a Finite Element Mesh . 2-63
2-D Multidomain Geometry . 2-64
Geometry from alphaShape 2-66
Cuboids, Cylinders, and Spheres 2-68
Sphere in Cube 2-75
3-D Multidomain Geometry from 2-D Geometry 2-79
Multidomain Geometry Reconstructed from Mesh 2-83
Put Equations in Divergence Form 2-87
Coefficient Matching for Divergence Form . 2-87
Boundary Conditions Can Affect the c Coefficient 2-88
Coefficient Conversion with Symbolic Math Toolbox 2-88
Some Equations Cannot Be Converted 2-89
f Coefficient for specifyCoefficients 2-90
c Coefficient for specifyCoefficients . 2-92
Overview of the c Coefficient . 2-92
Definition of the c Tensor Elements . 2-92
Some c Vectors Can Be Short . 2-94
Functional Form 2-104
m, d, or a Coefficient for specifyCoefficients . 2-107
Coefficients m, d, or a 2-107
Short m, d, or a vectors 2-107
Nonconstant m, d, or a . 2-108
View, Edit, and Delete PDE Coefficients . 2-111
View Coefficients . 2-111
Delete Existing Coefficients . 2-112
Change a Coefficient Assignment . 2-113
Set Initial Conditions 2-114
What Are Initial Conditions? . 2-114
Constant Initial Conditions 2-114
Nonconstant Initial Conditions . 2-114
Nodal Initial Conditions 2-115
Nonlinear System with Cross-Coupling Between Components . 2-117
Set Initial Condition for Model with Fine Mesh Using Solution Obtained
with Coarser Mesh . 2-121
View, Edit, and Delete Initial Conditions . 2-123
View Initial Conditions . 2-123
Delete Existing Initial Conditions . 2-124
Change an Initial Conditions Assignment . 2-125
vi ContentsNo Boundary Conditions Between Subdomains . 2-126
Identify Boundary Labels . 2-128
Specify Boundary Conditions 2-129
Dirichlet Boundary Conditions . 2-129
Neumann Boundary Conditions 2-130
Mixed Boundary Conditions . 2-132
Nonconstant Boundary Conditions 2-132
Additional Arguments in Functions for Nonconstant Boundary Conditions
2-134
Solve PDEs with Constant Boundary Conditions 2-135
Specify Nonconstant Boundary Conditions . 2-139
Geometry and Mesh . 2-139
Scalar PDE Problem with Nonconstant Boundary Conditions 2-140
Anonymous Functions for Nonconstant Boundary Conditions 2-142
Additional Arguments 2-142
System of PDEs 2-143
Specify Nonconstant PDE Coefficients 2-146
Geometry and Mesh . 2-146
Function for Nonconstant Coefficient f . 2-146
Anonymous Function for a PDE Coefficient 2-148
Additional Arguments 2-148
View, Edit, and Delete Boundary Conditions 2-150
View Boundary Conditions 2-150
Delete Existing Boundary Conditions 2-152
Change a Boundary Conditions Assignment . 2-152
Generate Mesh 2-154
Find Mesh Elements and Nodes by Location 2-162
Assess Quality of Mesh Elements . 2-168
Mesh Data as [p,e,t] Triples . 2-172
Mesh Data . 2-175
Solving PDEs
3
von Mises Effective Stress and Displacements: PDE Modeler App 3-3
Clamped Square Isotropic Plate with Uniform Pressure Load . 3-7
Deflection of Piezoelectric Actuator . 3-11
viiDynamics of Damped Cantilever Beam . 3-21
Dynamic Analysis of Clamped Beam . 3-28
Reduced-Order Modeling Technique for Beam with Point Load . 3-33
Modal and Frequency Response Analysis for Single Part of Kinova Gen3
Robotic Arm . 3-40
Thermal Stress Analysis of Jet Engine Turbine Blade . 3-52
Finite Element Analysis of Electrostatically Actuated MEMS Device . 3-59
Deflection Analysis of Bracket 3-74
Vibration of Square Plate 3-85
Structural Dynamics of Tuning Fork . 3-90
Modal Superposition Method for Structural Dynamics Problem 3-99
Stress Concentration in Plate with Circular Hole 3-102
Thermal Deflection of Bimetallic Beam 3-110
Axisymmetric Thermal and Structural Analysis of Disc Brake . 3-119
Electrostatic Potential in Air-Filled Frame . 3-130
Electrostatic Potential in Air-Filled Frame: PDE Modeler App . 3-133
Electrostatic Analysis of Transformer Bushing Insulator . 3-135
Magnetic Flux Density in H-Shaped Magnet 3-141
Magnetic Flux Density in Electromagnet 3-146
Linear Elasticity Equations 3-156
Summary of the Equations of Linear Elasticity . 3-156
3D Linear Elasticity Problem 3-157
Plane Stress . 3-159
Plane Strain . 3-160
Magnetic Field in Two-Pole Electric Motor . 3-161
Magnetic Field in Two-Pole Electric Motor: PDE Modeler App . 3-166
Helmholtz Equation on Disk with Square Hole 3-170
Electrostatics and Magnetostatics 3-176
Electrostatics 3-176
Magnetostatics . 3-176
Magnetostatics with Permanent Magnets . 3-177
viii ContentsDC Conduction 3-178
Harmonic Electromagnetics . 3-179
Current Density Between Two Metallic Conductors 3-181
Skin Effect in Copper Wire with Circular Cross Section: PDE Modeler App
3-184
Current Density Between Two Metallic Conductors: PDE Modeler App
3-192
Heat Transfer Between Two Squares Made of Different Materials: PDE
Modeler App 3-195
Temperature Distribution in Heat Sink 3-199
Create 2-D Geometry in PDE Modeler App 3-199
Extrude 2-D Geometry into 3-D Geometry of Heat Sink . 3-200
Perform Thermal Analysis . 3-203
Nonlinear Heat Transfer in Thin Plate 3-210
Poisson’s Equation on Unit Disk: PDE Modeler App . 3-218
Poisson’s Equation on Unit Disk . 3-224
Scattering Problem 3-232
Scattering Problem: PDE Modeler App 3-237
Minimal Surface Problem . 3-241
Minimal Surface Problem: PDE Modeler App . 3-247
Poisson’s Equation with Point Source and Adaptive Mesh Refinement 3-249
Heat Transfer in Block with Cavity: PDE Modeler App . 3-254
Heat Transfer in Block with Cavity . 3-258
Heat Transfer Problem with Temperature-Dependent Properties . 3-262
Heat Conduction in Multidomain Geometry with Nonuniform Heat Flux
3-270
Inhomogeneous Heat Equation on Square Domain 3-277
Heat Distribution in Circular Cylindrical Rod 3-281
Thermal Analysis of Disc Brake 3-287
Heat Distribution in Circular Cylindrical Rod: PDE Modeler App . 3-295
Wave Equation on Square Domain 3-298
ixWave Equation on Square Domain: PDE Modeler App 3-302
Eigenvalues and Eigenmodes of L-Shaped Membrane 3-305
Eigenvalues and Eigenmodes of L-Shaped Membrane: PDE Modeler App
3-311
L-Shaped Membrane with Rounded Corner: PDE Modeler App 3-314
Eigenvalues and Eigenmodes of Square . 3-317
Eigenvalues and Eigenmodes of Square: PDE Modeler App . 3-322
Vibration of Circular Membrane . 3-325
Solution and Gradient Plots with pdeplot and pdeplot3D . 3-329
2-D Solution and Gradient Plots with MATLAB Functions . 3-338
3-D Solution and Gradient Plots with MATLAB Functions . 3-344
Types of 3-D Solution Plots Available in MATLAB . 3-344
2-D Slices Through 3-D Geometry . 3-344
Contour Slices Through 3-D Solution 3-347
Plots of Gradients and Streamlines 3-351
Dimensions of Solutions, Gradients, and Fluxes 3-356
PDE Modeler App
4
Open the PDE Modeler App . 4-2
2-D Geometry Creation in PDE Modeler App . 4-3
Create Basic Shapes . 4-3
Select Several Shapes 4-4
Rotate Shapes . 4-4
Create Complex Geometries . 4-4
Adjust Axes Limits and Grid . 4-5
Create Geometry with Rounded Corners 4-8
Specify Boundary Conditions in the PDE Modeler App 4-12
Specify Coefficients in PDE Modeler App . 4-14
Coefficients for Scalar PDEs 4-14
Coefficients for Systems of PDEs 4-16
Coefficients That Depend on Time and Space . 4-18
Specify Mesh Parameters in the PDE Modeler App . 4-24
Adjust Solve Parameters in the PDE Modeler App 4-26
Elliptic Equations 4-26
x ContentsParabolic Equations 4-28
Hyperbolic Equations . 4-29
Eigenvalue Equations . 4-29
Nonlinear Equations 4-30
Plot the Solution in the PDE Modeler App 4-31
Additional Plot Control Options 4-33
Tooltip Displays for Mesh and Plots 4-35
Functions

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