# Partial Differential Equation Toolbox

## Solve Partial Differential Equations using Finite Element Method (FEM)

Partial Differential Equation Toolbox provides functions for solving structural mechanics, heat transfer, and general partial differential equations (PDEs) using finite element analysis.

You can perform linear static analysis to compute deformation, stress, and strain. For modeling structural dynamics and vibration, the toolbox provides a direct time integration solver. You can analyze a component’s structural characteristics by performing modal analysis to find natural frequencies and mode shapes. You can model conduction-dominant heat transfer problems to calculate temperature distributions, heat fluxes, and heat flow rates through surfaces. You can also solve standard problems such as diffusion, electrostatics, and magnetostatics, as well as custom PDEs.

Partial Differential Equation Toolbox lets you import 2D and 3D geometries from STL or mesh data. You can automatically generate meshes with triangular and tetrahedral elements. You can solve PDEs by using the finite element method, and postprocess results to explore and analyze them.

## 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.

### Transient Analysis

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

## 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.

### Coupled Thermal-Stress Analysis

Analyze mechanical behavior under coupled thermal and mechanical loads.

## 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.

### Electromagnetics

Solve electrostatics, magnetostatics, DC conduction, and wave propagation problems.

## Geometry and Meshing

Define geometry and discretize it to build finite element models.

### Import/Create Geometry

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

### 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.

## 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.

### 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.

## 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.

## Latest Features

### Reduced Order Modeling

Approximate dynamic characteristics of a structural model for faster execution

### Simscape Multibody Models

Generate dynamic substructure for flexible multibody simulations

### Frequency Response Analysis

Determine dynamic response of a structure in the frequency domain

### Structural Analysis

Set concentrated boundary conditions at arbitrary locations on geometry surfaces

### Eigenvalue Solver Options

Specify parameters for Lanczos algorithm

See release notes for details on any of these features and corresponding functions.