PDEModels Overview
Acoustics
Acoustic PDEs and Boundary Conditions Guide
Acoustics in the Frequency Domain
Introduction
Time Harmonic Analysis
Eigenfrequency Analysis
Helmholtz Equation
Introduction to Helmholtz Equation
Derivation of the frequency acoustics model from the time domain model
Model Parameter Setup
Source Types
Monopole Sources
Dipole Sources
Sound Propagation in Lossy Media
Sound Propagation in Multi-material Media
A Comparison of Time Domain and Frequency Domain Modeling
Wave Equation: Time Domain Modeling
Helmholtz Equation: Frequency Domain Modeling
Accuracy Comparison
Acoustic Boundary Conditions
Impedance Boundary Conditions
Formulation
Derivation
Impedance Boundary Conditions in Time Harmonic Analysis
Absorbing Boundary Conditions
Formulation
Derivation
Absorbing Boundary Conditions in Time Harmonic Analysis
Sound Hard Boundary Conditions - Walls
Formulation
Derivation
Sound Hard Boundary Conditions in Time Harmonic Analysis
Sound Hard Boundary Conditions in Eigenfrequency Analysis
Normal Velocity Boundary Conditions
Formulation
Derivation
Normal Velocity Boundary Conditions in Time Harmonic Analysis
Sound Soft Boundary Conditions
Formulation
Derivation
Sound Soft Boundary Conditions in Time Harmonic Analysis
Sound Soft Boundary Conditions in Eigenfrequency Analysis
Pressure Source Boundary Conditions
Formulation
Derivation
Pressure Source Boundary Conditions in Time Harmonic Analysis
Radiation Boundary Conditions
Formulation
Derivation
Radiation Boundary Conditions in Time Harmonic Analysis
Floquet Periodic Boundary Conditions
Formulation
Derivation
Floquet Periodic Boundary Conditions in Time Harmonic Analysis
Perfectly Matched Layer (PML)
Perfectly Matched Layer in Time Harmonic Analysis
Nomenclature
References
Acoustics in the Time Domain
Introduction
Wave Equation
Introduction to the Wave Equation
Model Parameter Setup
Wave Types
Time Harmonic Waves
Time Inharmonic Waves
Source Types
Monopole Sources
Dipole Sources
The Wave Equation as a System of First Order Equations
Sound Propagation in Lossy Media
Sound Propagation in Multi-material Media
Acoustic Boundary Conditions
Impedance Boundary Conditions
Formulation
Derivation
Absorbing Boundary Conditions
Formulation
Derivation
Sound Hard Boundary Conditions - Walls
Formulation
Derivation
Normal Velocity Boundary Conditions
Formulation
Derivation
Sound Soft Boundary Conditions
Formulation
Derivation
Pressure Source Boundary Conditions
Formulation
Derivation
Radiation Boundary Conditions
Formulation
Derivation
Periodic Boundary Conditions
Formulation
Derivation
Perfectly Matched Layer (PML)
Appendix
Perfectly Matched Layer Derivation
Attenuation using an Artificial Complex Dimension
PML Coordinate Transformations on the Coupled 1st Order Wave Equation
NeumannValues on Time Derivatives
Radiated Sound Waves with Oblique Incidence
Nomenclature
References
Acoustics Models
Acoustic Cloak
Acoustic Horn
Acoustic Mirror
Acoustic Muffler
Acoustic Wave Diffraction
Electric Motor Noise Analysis
Room Eigenfrequencies
Acoustics Examples
Acoustic Eigenmodes in a Car
Electromagnetics
Electromagnetics Models
Single Aperture Scalar Diffraction
Multiple Aperture Vector Diffraction
Electromagnetics Examples
Shielded Micro Strip
Motor
Fluid Dynamics
Fluid Dynamics Models
Buoyancy-Driven Flow
Heat Exchanger
Thermal Decomposition
Fluid Dynamics Examples
Stokes Flow
Navier-Stokes Equation
Time dependent Navier-Stokes Equation
Fluid Dynamics with Heat Transfer
Heat Transfer
Heat Transfer PDEs and Boundary Conditions Guide
Heat Transfer
Introduction
Heat Equation
Introduction to Heat Equation
Heat Equation Derivation
Heat Transfer Model Setup
Model Parameter Setup
Basic Heat Transfer Example
Source Types
Volumetric Heat Source
Layer Heat Source
Point Heat Source
Anisotropic and Orthotropic Heat Transfer
Nonlinear Heat Transfer
Heat Transfer with Events
Heat Transfer in Porous Media
Heat Transfer with Mixed Dimensions
Boundary Conditions in Heat Transfer
Temperature Surface Boundary Condition
Purpose
Formulation
Derivation
Heat Flux Boundary Condition
Purpose
Formulation
Derivation
Thermally Insulated Boundary Condition
Purpose
Formulation
Derivation
Symmetry Boundary Condition
Purpose
Formulation
Derivation
Outflow Boundary Condition
Purpose
Formulation
Derivation
Convective Boundary Condition
Purpose
Formulation
Derivation
Thermal Radiation Boundary Condition
Purpose
Formulation
Derivation
Periodic Boundary Condition
Purpose
Formulation
Derivation
Appendix
Special Cases of the Heat Equation
Stationary case
Heat Equation in Cylindrical Coordinates
Heat Equation in Spherical Coordinates
The Smoothing Characteristic of the Diffusion Equation
Modeling Heat Sources using Element Markers
Conservation Laws with Discontinuous PDE coefficients
Nomenclature
References
Heat Transfer Verification Tests
Heat Transfer Models
Buoyancy Driven Flow
Heat Exchanger
Joule Heating
Laser Welding
Shrink Fitting
Thermal Load
Heat Transfer Examples
Heat Transfer with Events - Thermostat
Heat Transfer with Fluid Dynamics - Energy Transport
Heat Transfer with Model Order Reduction - Fast Time Integration
Nonlinear Heat Transfer Verification Test 1
Nonlinear Heat Transfer Verification Test 2
Mass Transport
Mass Transport PDEs and Boundary Conditions Guide
Mass Transport
Introduction
Mass Balance Equation
Mass Balance Equation Introduction
Mass Balance Equation Derivation
Mass Transport Model Setup
Initial Mass Transport Example
Mass Transport with a Chemical Reaction
Anisotropic and Orthotropic Mass Diffusion
Variable Mass Diffusion Coefficient
Temperature Dependence of the Diffusion Coefficient
Concentration Dependence of the Diffusion Coefficient
Boundary Conditions in Mass Transport
Neumann Values for Conservative and Non-conservative Models
Model Parameter Setup
Concentration Boundary Condition
Purpose
Formulation
Derivation
Example
Visualization
Outflow Boundary Condition
Purpose
Formulation
Derivation
Example
Visualization of an Outflow Boundary Condition
Mass Flux Boundary Condition
Purpose
Formulation
Derivation
Example
Visualization of a Mass Flux Boundary Condition
Impermeable Boundary Condition
Purpose
Formulation
Derivation
Example
Visualization of an Impermeable Boundary Condition
Surrounding Flux Boundary Condition
Purpose
Formulation
Derivation
Example
Visualization of a Surrounding Flux Boundary Condition
Symmetry Boundary Condition
Purpose
Formulation
Derivation
Example
Periodic Boundary Condition
Purpose
Formulation
Derivation
Example
Visualization of a Periodic Boundary Condition
Appendix
Special Cases of the Mass Balance Equation
Stationary Case
Mass Transport by Diffusion Only
Mass Transport by Convection Only
Mass Balance Equation in Cylindrical Coordinates
Mass Balance in Spherical Coordinates
Nomenclature
References
Mass Transport Models
Catalyst Deactivation
Catalytic Converter
Gas Absorption
Mass Transport Examples
Cyclic Voltammetry
Fokker-Planck equation
Smoluchowski diffusion equation
Multiphysics
Multiphysics Models
Buoyancy-Driven Flow
Heat Exchanger
Joule Heating
Thermal Decomposition
Thermal Load
Structural Mechanics
Structural Mechanics Models
Thermal Load
Structural Mechanics Examples
Plane Stress
Plane Stress Eigenmodes
Plane Strain
Rayleigh damping
3D Stress