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.

Temperature distribution on cylindrical battery cells showing a hotter trigger cell with surface-to-surface radiation.

Heat Transfer

Perform steady-state, transient, modal, or coupled stress-thermal analysis to compute temperature distributions and other thermal characteristics. Analyze spatial or temporal thermal behavior for applications like battery thermal management.

Documentation | Examples

Six mode shapes of a robotic arm shoulder link showing deformation patterns at different natural frequencies.

Structural Mechanics

Perform linear static, transient, modal, and frequency response analyses. Evaluate mechanical strength by computing displacements, stresses, and strains or simulate dynamic behavior of mechanical systems.

Documentation | Examples

Magnetic potential and field lines in a two pole electric motor showing flux patterns across the stator, rotor, and air gap.

Electromagnetics

Analyze electrostatic, magnetostatic, DC conduction, or harmonic problems and design electrical and electronic components.

Documentation | Examples

Plot of discharge voltage vs time for 0.2C, 0.5C, 1C, and 2C; higher C rates cause faster voltage drop.

Battery P2D Modeling

Simulate the electrochemical behavior of Li-ion batteries using the pseudo-2D (P2D) model, also known as the Doyle-Fuller-Newman (DFN) model. Capture electrolyte diffusion through thickness and solid-phase diffusion in electrode particles to simulate voltage, concentration, and current density under cycling.

Documentation | Examples

Surface plot of the twelfth eigenmode of the L-Shaped membrane showing nodal patterns and displacement variation.

General PDEs

Solve second-order linear and nonlinear PDEs for stationary, time-dependent, and eigenvalue problems commonly arising in engineering and science.

Documentation | Examples

Diagram depicting a closed loop CPU cooling system with a thermal ROM plant model with fan control and temperature feedback.

Reduced Order Modeling and Scientific Machine Learning

Create fast surrogate models using Reduced Order Modeling (ROM) and Scientific Machine Learning (SciML) techniques to enable system-level simulation, control, physical modeling, and rapid design exploration and optimization of systems governed by PDEs. Create and train AI-based PDE solvers such as Physics-Informed Neural Networks, Graph Neural Networks, and Fourier Neural Operators (with Deep Learning Toolbox).

ROM for Thermal Behavior of Battery

Documentation | Examples

Transparent 3 D view of an imported forearm link geometry from an STL file with coordinate axes.

Geometry and Meshing

Define a 2D or 3D geometry by importing STL, STEP, or mesh data or by creating parameterized shapes using geometric primitives. Modify geometries using operations like extrusion and Booleans, then generate finite element meshes using triangular elements in 2D and tetrahedral elements in 3D.

Documentation | Examples

Visualization and Postprocessing

Visualize models and solutions by leveraging powerful MATLAB graphics and graphics functions created specifically for PDE problems. Plot and animate results and their derived and interpolated quantities, as well as meshes and geometries.

Documentation | Examples

Workflow diagram showing geometry setup, model development, PDE analysis, integration with MATLAB and Simulink products, and application sharing.

Automation, Integration, and Sharing FEA Workflows

Automate FEA simulations using MATLAB and integrate with other MATLAB and Simulink products to build complete workflows. Share custom applications using App Designer and MATLAB Compiler.

Modeling PDE-based systems in Simulink

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