MATLAB Partial Differential Equation Toolbox™ User's Guide

Getting Started
	Partial Differential Equation Toolbox Product Description
		Key Features
	Equations You Can Solve Using PDE Toolbox
	Solve 2-D PDEs Using the PDE Modeler App
		Tips
	Poisson’s Equation with Complex 2-D Geometry: PDE Modeler App
	Finite Element Method Basics
	Deflection Analysis of Bracket
	Heat Transfer in Block with Cavity
Setting Up Your PDE
	Solve Problems Using PDEModel Objects
	Geometry and Mesh Components
	2-D Geometry Creation at Command Line
		Three Elements of Geometry
		Basic Shapes
		Rectangle with Circular End Cap and Another Circular Excision
		Decomposed Geometry Data Structure
	Parametrized Function for 2-D Geometry Creation
		Required Syntax
		Relation Between Parametrization and Region Labels
		Geometry Function for a Circle
		Arc Length Calculations for a Geometry Function
		Geometry Function Example with Subdomains and a Hole
		Nested Function for Geometry with Additional Parameters
	Geometry from polyshape
	STL File Import
	STEP File Import
	Geometry from Triangulated Mesh
		3-D Geometry from a Finite Element Mesh
		2-D Multidomain Geometry
	Geometry from alphaShape
	Cuboids, Cylinders, and Spheres
	Sphere in Cube
	3-D Multidomain Geometry from 2-D Geometry
	Multidomain Geometry Reconstructed from Mesh
	Put Equations in Divergence Form
		Coefficient Matching for Divergence Form
		Boundary Conditions Can Affect the c Coefficient
		Coefficient Conversion with Symbolic Math Toolbox
		Some Equations Cannot Be Converted
	f Coefficient for specifyCoefficients
	c Coefficient for specifyCoefficients
		Overview of the c Coefficient
		Definition of the c Tensor Elements
		Some c Vectors Can Be Short
		Functional Form
	m, d, or a Coefficient for specifyCoefficients
		Coefficients m, d, or a
		Short m, d, or a vectors
		Nonconstant m, d, or a
	View, Edit, and Delete PDE Coefficients
		View Coefficients
		Delete Existing Coefficients
		Change a Coefficient Assignment
	Set Initial Conditions
		What Are Initial Conditions?
		Constant Initial Conditions
		Nonconstant Initial Conditions
		Nodal Initial Conditions
	Nonlinear System with Cross-Coupling Between Components
	Set Initial Condition for Model with Fine Mesh Using Solution Obtained with Coarser Mesh
	View, Edit, and Delete Initial Conditions
		View Initial Conditions
		Delete Existing Initial Conditions
		Change an Initial Conditions Assignment
	No Boundary Conditions Between Subdomains
	Identify Boundary Labels
	Specify Boundary Conditions
		Dirichlet Boundary Conditions
		Neumann Boundary Conditions
		Mixed Boundary Conditions
		Nonconstant Boundary Conditions
		Additional Arguments in Functions for Nonconstant Boundary Conditions
	Solve PDEs with Constant Boundary Conditions
	Specify Nonconstant Boundary Conditions
		Geometry and Mesh
		Scalar PDE Problem with Nonconstant Boundary Conditions
		Anonymous Functions for Nonconstant Boundary Conditions
		Additional Arguments
		System of PDEs
	Specify Nonconstant PDE Coefficients
		Geometry and Mesh
		Function for Nonconstant Coefficient f
		Anonymous Function for a PDE Coefficient
		Additional Arguments
	View, Edit, and Delete Boundary Conditions
		View Boundary Conditions
		Delete Existing Boundary Conditions
		Change a Boundary Conditions Assignment
	Generate Mesh
	Find Mesh Elements and Nodes by Location
	Assess Quality of Mesh Elements
	Mesh Data as [p,e,t] Triples
	Mesh Data
Solving PDEs
	von Mises Effective Stress and Displacements: PDE Modeler App
	Clamped Square Isotropic Plate with Uniform Pressure Load
	Deflection of Piezoelectric Actuator
	Dynamics of Damped Cantilever Beam
	Dynamic Analysis of Clamped Beam
	Reduced-Order Modeling Technique for Beam with Point Load
	Modal and Frequency Response Analysis for Single Part of Kinova Gen3 Robotic Arm
	Thermal Stress Analysis of Jet Engine Turbine Blade
	Finite Element Analysis of Electrostatically Actuated MEMS Device
	Deflection Analysis of Bracket
	Vibration of Square Plate
	Structural Dynamics of Tuning Fork
	Modal Superposition Method for Structural Dynamics Problem
	Stress Concentration in Plate with Circular Hole
	Thermal Deflection of Bimetallic Beam
	Axisymmetric Thermal and Structural Analysis of Disc Brake
	Electrostatic Potential in Air-Filled Frame
	Electrostatic Potential in Air-Filled Frame: PDE Modeler App
	Electrostatic Analysis of Transformer Bushing Insulator
	Magnetic Flux Density in H-Shaped Magnet
	Magnetic Flux Density in Electromagnet
	Linear Elasticity Equations
		Summary of the Equations of Linear Elasticity
		3D Linear Elasticity Problem
		Plane Stress
		Plane Strain
	Magnetic Field in Two-Pole Electric Motor
	Magnetic Field in Two-Pole Electric Motor: PDE Modeler App
	Helmholtz Equation on Disk with Square Hole
	Electrostatics and Magnetostatics
		Electrostatics
		Magnetostatics
		Magnetostatics with Permanent Magnets
	DC Conduction
	Harmonic Electromagnetics
	Current Density Between Two Metallic Conductors
	Skin Effect in Copper Wire with Circular Cross Section: PDE Modeler App
	Current Density Between Two Metallic Conductors: PDE Modeler App
	Heat Transfer Between Two Squares Made of Different Materials: PDE Modeler App
	Temperature Distribution in Heat Sink
		Create 2-D Geometry in PDE Modeler App
		Extrude 2-D Geometry into 3-D Geometry of Heat Sink
		Perform Thermal Analysis
	Nonlinear Heat Transfer in Thin Plate
	Poisson's Equation on Unit Disk: PDE Modeler App
	Poisson's Equation on Unit Disk
	Scattering Problem
	Scattering Problem: PDE Modeler App
	Minimal Surface Problem
	Minimal Surface Problem: PDE Modeler App
	Poisson's Equation with Point Source and Adaptive Mesh Refinement
	Heat Transfer in Block with Cavity: PDE Modeler App
	Heat Transfer in Block with Cavity
	Heat Transfer Problem with Temperature-Dependent Properties
	Heat Conduction in Multidomain Geometry with Nonuniform Heat Flux
	Inhomogeneous Heat Equation on Square Domain
	Heat Distribution in Circular Cylindrical Rod
	Thermal Analysis of Disc Brake
	Heat Distribution in Circular Cylindrical Rod: PDE Modeler App
	Wave Equation on Square Domain
	Wave Equation on Square Domain: PDE Modeler App
	Eigenvalues and Eigenmodes of L-Shaped Membrane
	Eigenvalues and Eigenmodes of L-Shaped Membrane: PDE Modeler App
	L-Shaped Membrane with Rounded Corner: PDE Modeler App
	Eigenvalues and Eigenmodes of Square
	Eigenvalues and Eigenmodes of Square: PDE Modeler App
	Vibration of Circular Membrane
	Solution and Gradient Plots with pdeplot and pdeplot3D
	2-D Solution and Gradient Plots with MATLAB Functions
	3-D Solution and Gradient Plots with MATLAB Functions
		Types of 3-D Solution Plots Available in MATLAB
		2-D Slices Through 3-D Geometry
		Contour Slices Through 3-D Solution
		Plots of Gradients and Streamlines
	Dimensions of Solutions, Gradients, and Fluxes
PDE Modeler App
	Open the PDE Modeler App
	2-D Geometry Creation in PDE Modeler App
		Create Basic Shapes
		Select Several Shapes
		Rotate Shapes
		Create Complex Geometries
		Adjust Axes Limits and Grid
		Create Geometry with Rounded Corners
	Specify Boundary Conditions in the PDE Modeler App
	Specify Coefficients in PDE Modeler App
		Coefficients for Scalar PDEs
		Coefficients for Systems of PDEs
		Coefficients That Depend on Time and Space
	Specify Mesh Parameters in the PDE Modeler App
	Adjust Solve Parameters in the PDE Modeler App
		Elliptic Equations
		Parabolic Equations
		Hyperbolic Equations
		Eigenvalue Equations
		Nonlinear Equations
	Plot the Solution in the PDE Modeler App
		Additional Plot Control Options
		Tooltip Displays for Mesh and Plots
Functions
	Visualize PDE Results
	adaptmesh
	addCell
	addFace
	addVertex
	addVoid
	AnalyticGeometry
	applyBoundaryCondition
	area
	assema
	assemb
	assembleFEMatrices
	assempde
	cellEdges
	cellFaces
	BodyLoadAssignment
	BoundaryCondition
	CoefficientAssignment
	createpde
	createPDEResults
	csgdel
	decsg
	DiscreteGeometry
	dst
	idst
	EigenResults
	ElectromagneticModel
	electromagneticBC
	ElectromagneticBCAssignment
	electromagneticProperties
	ElectromagneticMaterialAssignment
	electromagneticSource
	ElectromagneticSourceAssignment
	ElectrostaticResults
	MagnetostaticResults
	HarmonicResults
	ConductionResults
	evaluate
	evaluateCGradient
	evaluateGradient
	evaluateHeatFlux
	evaluateHeatRate
	evaluatePrincipalStrain
	evaluatePrincipalStress
	evaluateReaction
	evaluateStrain
	evaluateStress
	evaluateTemperatureGradient
	evaluateVonMisesStress
	FEMesh
	extrude
	faceEdges
	facesAttachedToEdges
	findBodyLoad
	findBoundaryConditions
	findCoefficients
	findElectromagneticBC
	findElectromagneticProperties
	findElectromagneticSource
	findElements
	findHeatSource
	findInitialConditions
	findNodes
	findStructuralBC
	findStructuralIC
	findStructuralProperties
	findThermalBC
	findThermalIC
	findThermalProperties
	generateMesh
	GeometricInitialConditions
	GeometricStructuralICs
	GeometricThermalICs
	NodalInitialConditions
	NodalStructuralICs
	NodalThermalICs
	geometryFromEdges
	geometryFromMesh
	HeatSourceAssignment
	hyperbolic
	importGeometry
	initmesh
	internalHeatSource
	interpolateAcceleration
	interpolateDisplacement
	interpolateElectricField
	interpolateElectricFlux
	interpolateCurrentDensity
	interpolateHarmonicField
	interpolateElectricPotential
	interpolateMagneticField
	interpolateMagneticFlux
	interpolateMagneticPotential
	interpolateSolution
	interpolateStrain
	interpolateStress
	interpolateTemperature
	interpolateVelocity
	interpolateVonMisesStress
	jigglemesh
	linearize
	linearizeInput
	linearizeOutput
	meshQuality
	meshToPet
	multicuboid
	multicylinder
	multisphere
	nearestEdge
	nearestFace
	parabolic
	pdearcl
	pdecgrad
	pdecirc
	pdecont
	pdeeig
	pdeellip
	pdegplot
	pdegrad
	pdeInterpolant
	pdeintrp
	pdejmps
	pdemesh
	PDEModel
	pdenonlin
	pdeplot
	pdeplot3D
	pdepoly
	pdeprtni
	pderect
	pdesdp
	pdesde
	pdesdt
	pdesmech
	PDESolverOptions
	pdesurf
	PDE Modeler
	pdetrg
	pdetriq
	pdeviz
	PDEVisualization
	poimesh
	poisolv
	reconstructSolution
	reduce
	ReducedStructuralModel
	ReducedThermalModel
	refinemesh
	rotate
	scale
	setInitialConditions
	solve
	solvepde
	solvepdeeig
	specifyCoefficients
	ModalStructuralResults
	FrequencyStructuralResults
	StaticStructuralResults
	TransientStructuralResults
	structuralBC
	StructuralBC
	structuralIC
	structuralDamping
	findStructuralDamping
	StructuralDampingAssignment
	StructuralMaterialAssignment
	structuralBodyLoad
	structuralBoundaryLoad
	StructuralModel
	structuralProperties
	structuralSEInterface
	StructuralSEIAssignment
	StationaryResults
	SteadyStateThermalResults
	ThermalBC
	ThermalMaterialAssignment
	ThermalModel
	thermalProperties
	translate
	TimeDependentResults
	TransientThermalResults
	ModalThermalResults
	thermalBC
	thermalIC
	tri2grid
	volume
	wbound
	wgeom