## Key Features

• Structural analysis, including linear static, dynamic, and modal analysis
• Heat transfer analysis for conduction-dominant problems
• General linear and nonlinear PDEs for stationary, time-dependent, and eigenvalue problems
• 2D and 3D geometry import from STL files and mesh data
• Automatic meshing using triangular and tetrahedral elements with linear or quadratic basis functions
• User-defined functions for specifying PDE coefficients, boundary conditions, and initial conditions
• Plotting and animating results, as well as derived and interpolated values

## Structural Analysis

With structural analysis, you can predict how components behave under loading, vibration, and other physical effects. This helps you design robust mechanical components by validating designs through simulation and reducing the need for physical testing.

Using linear static analysis to compute displacement, stress, and strain under load, you can evaluate a component’s mechanical strength and behavior.

Using functions in Partial Differential Equation Toolbox to read 3D geometry of a bracket, and to automatically mesh, solve, and visualize the solution.
Apply a uniaxial tension and find stress concentrations in a thin plate with a hole by using the plane-stress approximation.

Mechanical components can resonate, which can result in deformation and potentially dangerous and damaging large-amplitude vibrations. Partial Differential Equation Toolbox™ lets you perform modal analysis to find natural frequencies and mode shapes to identify and prevent potential resonances.

Calculate the vibration modes and frequencies of a 3-D simply supported square elastic plate.

For time-varying loads, you can perform transient dynamic simulation to compute displacement, velocity, acceleration, stress, and strain. You can plot or animate the deformed shapes and compute reaction forces.

Perform modal analysis to find modes and frequencies and simulate the transient dynamics by exciting the fundamental mode.

## Thermal Analysis

Address challenges with thermal management by analyzing the temperature distributions of components based on material properties, external heat sources, and internal heat generation. Partial Differential Equation Toolbox lets you solve conduction-dominant heat transfer problems with convection and radiation occurring at the boundaries. You can perform steady-state and transient heat transfer analysis.

Model steady-state conduction-dominant heat transfer in a robotics component.
Use a custom MATLAB function to define temperature-dependent thermal conductivity.

## General PDEs

You can use Partial Differential Equation Toolbox to solve engineering problems described by second-order PDEs, including electrostatics, magnetostatics, diffusion, and other custom applications. For more information, see equations you can solve with the toolbox.

Solve the 2D wave equation represented by the time-independent elliptic Helmholtz equation.
Solve a coupled elasticity-electrostatics problem using an equation-based formulation.

## FEA Workflows in MATLAB

Partial Differential Equation Toolbox uses the finite element method to solve problems. A typical workflow consists of importing geometry; generating a mesh; defining the physics, including materials as well as boundary and initial conditions; and then solving and visualizing your results. The toolbox provides functions for each step in the workflow, allowing you to perform finite element analysis (FEA) in just a few lines of code.

Perform structural analysis on a plate with a hole to determine peak deformations and stress concentration factors.

The MATLAB® language makes it easy to customize, automate, and integrate your FEA applications. Partial Differential Equation Toolbox integrates with other MATLAB products, allowing you to build and share custom applications with MATLAB Compiler™, run design of experiments in parallel with Parallel Computing Toolbox™, and leverage high-fidelity simulation in Simulink® and Simscape™.

## Geometry and Meshing

You can import 2D or 3D geometry in STL format or create geometry directly from mesh data.

Import a geometry from a STL file which is a common file format used in CAD software.
Import mesh data from another FEA software.

Partial Differential Equation Toolbox provides functions to create simple 3D multidomain geometries as well as multiple ways to create 2D geometries.

Create simple parameterized shapes for use in analysis including spheres, cylinders, and cubes.

You can automatically generate a finite element mesh using triangular elements in 2D and tetrahedral elements in 3D. You can specify either piecewise linear or quadratic basis functions when meshing, depending on the required accuracy of your solution. You can inspect and analyze the mesh for quality and compute useful metrics like area or volume. For more information on generating finite element meshes, see Meshing.

Measure accuracy in FEA by measuring shape quality of mesh elements.

## Visualization and Postprocessing

### Plots and Animations

You can use Partial Differential Equation Toolbox and MATLAB graphics to visualize your solution by creating plots and animations. You can plot the geometry, mesh, results, and derived and interpolated quantities. You can also create multiple subplots and easily customize plot properties.

Custom plot of volume slices from a 3D diffusion problem.
Transient heat transfer visualized at multiple points in time with custom colormaps.
Animations of the transient solution to the 2D Helmholtz wave equation.

### Postprocessing

Partial Differential Equation Toolbox provides you with solutions and their gradients at the mesh nodes and lets you interpolate them within the domain. MATLAB provides extensive functionality for further statistical postprocessing and data analysis.

A fast Fourier transform (FFT) is used on the displacement time series to see that the vibrational frequency is close to the fundamental frequency of the component.