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

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