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Partial Differential Equation Toolbox

Solve partial differential equations using finite element analysis

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 perform electrostatic and magnetostatic analyses, and also solve other standard problems using 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.

Get Started:

Structural Mechanics
Heat Transfer
Electromagnetics
General PDEs
Geometry and Meshing
Visualization and Postprocessing
Automate, Integrate, and Share FEA Workflows

What Is 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.

Structural Mechanics Workflow

Deflection Analysis of Bracket

Stress Concentration in Plate With Circular Hole

Finite Element Analysis in MATLAB, Part 1: Structural Analysis Using Finite Element Method in MATLAB

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.

Vibration of Square Plate

Modal and Frequency Response Analysis for Single Part of Kinova^® Gen3 Robotic Arm

First six mode shapes of a robotic arm.

Transient Analysis

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

Structural Dynamics of Tuning Fork

Dynamics of Damped Cantilever Beam

Deflection of a beam center as function of time.

Heat Transfer

Analyze temperature distributions of components to address thermal management challenges.

Steady-State Thermal Analysis

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

Heat Transfer Workflow

Heat Transfer Problem With Temperature-Dependent Properties

Finite Element Analysis in MATLAB, Part 2: Heat Transfer Using Finite Element Method in MATLAB

Transient Thermal Analysis

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

Heat Distribution in Circular Cylindrical Rod

Heat Conduction in Multidomain Geometry with Nonuniform Heat Flux

Reduce Thermal Model

Contour plots of temperature distribution with respect to time.

Coupled Thermal-Stress Analysis

Analyze mechanical behavior under coupled thermal and mechanical loads.

Thermal Stress Analysis of Jet Engine Turbine Blade

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.

Electrostatic Potential in Air-Filled Frame

Magnetic Field in Two-Pole Electric Motor

Electromagnetic Equations

DC conduction analysis

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.

Equations You Can Solve Using PDE Toolbox

Eigenvalues and Eigenmodes of L-Shaped Membrane

Nonlinear Heat Transfer in Thin Plate

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 Geometry From STL Data

Import Geometry From STEP Data

Create Geometry From Mesh

Create Geometric Primitives

Create Multiple Domains With Different Properties

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.

Create Triangular or Tetrahedral Mesh

Evaluate Shape Quality of Mesh Elements

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.

Plot 3D Solutions and Their Gradients

Interpolate Stress Near Circular Boundary

Scattering Problem

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.

Structural Dynamics of Tuning Fork

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

Deflection analysis of bracket

Modal and frequency response of robotic arm

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.

Automate FEA simulations by using MATLAB^® language, execute simulations faster leveraging Parallel Computing Toolbox™
Integrate with other MATLAB products such as Simscape™ Multibody™ and Simulink^® to build an end-to-end workflow
Share custom applications using MATLAB Compiler™ and App Designer as a standalone application or as a web app

Deflection Analysis of Bracket

Structural Dynamics of Tuning Fork

Reduced-Order Modeling Technique for Beam with Point Load

Modeling Systems Governed by PDEs in Simulink