# Partial Differential Equation Toolbox

## Structural Mechanics

Solve linear static, transient, modal analysis, and frequency response problems.

### Linear Static Analysis

Compute displacement, stress, and strain under load and boundary conditions, and evaluate a component’s mechanical strength and behavior.

### Modal and Frequency Response Analysis

Find natural frequencies and mode shapes to identify and prevent potential resonances, and simulate dynamic behavior of a structure using its frequency responses.

First six mode shapes of a robotic arm.

### Transient Analysis

Compute displacement, velocity, acceleration, stress, and strain under time-varying loads.

Deflection of a beam center as function of time.

## Heat Transfer

Analyze temperature distributions of components to address thermal management challenges.

Find temperature distributions and other thermal characteristics under constant thermal loads.

### Transient Thermal Analysis

Find temperature distributions and other thermal characteristics under time-varying thermal loads, and approximate dynamic characteristics using reduced-order models.

Contour plots of temperature distribution with respect to time.

### Coupled Thermal-Stress Analysis

Analyze mechanical behavior under coupled thermal and mechanical loads.

Stress distribution under combined mechanical and thermal loads.

## Electromagnetics

Perform electromagnetic analysis for design of electrical and electronic components.

### Electrostatics and Magnetostatics

Solve Maxwell’s equations modeling electrostatic and magnetostatic problems.

Magnetic Potential and Field in Two-Pole Electric Motor.

## General PDEs

Solve PDEs that occur in common applications in engineering and science.

### Second-Order PDEs

Solve second-order linear and nonlinear PDEs for stationary, time-dependent, and eigenvalue problems.

L-shaped membrane with zero Dirichlet Boundary condition.

## Geometry and Meshing

Define geometry and discretize it to build finite element models.

### Import/Create Geometry

Reconstruct 2D and 3D geometry from imported STL, STEP, or mesh data, or create simple parameterized shapes using geometric primitives.

Import or create geometry in MATLAB.

### Generate Mesh

Generate finite element mesh using triangular elements in 2D and tetrahedral elements in 3D. Inspect and analyze mesh quality to assess accuracy of results.

Generate mesh and ensure its quality for accuracy of results.

## Visualization and Postprocessing

Compute derived and interpolated data from results as well as create plots and animations

### Plot and Animate Solutions

Visualize models and solutions by creating plots and animations of geometry, mesh, results, and derived and interpolated quantities by leveraging powerful MATLAB graphics. Create multiple subplots and easily customize plot properties.

Contour plot slices in 3D.

### Postprocessing

Analyze solutions and its gradients at mesh nodes and other interpolated locations. Leverage MATLAB’s extensive functionalities for further statistical postprocessing and data analysis using Statistics and Machine Learning Toolbox and Optimization Toolbox.

Fast Fourier Transform of tip displacement.

### Interactive Solution Plots

Plot and inspect results with the interactive controls in Visualize PDE Results Live Editor task

Visualize PDE Results Live Editor Task.

## Automate, Integrate, and Share FEA Workflows

Automate, Integrate, and Share Finite Element Analysis (FEA) Workflows in MATLAB.

### FEA Workflows

Create a typical FEA workflow in MATLAB – import or create geometries, generate mesh, define physics with load, boundary, and initial conditions, solve, and visualize results – all from one user interface.

MATLAB helps automate and integrate FEA workflows.