Units of Measurements

Partial Differential Equation Toolbox™ is unitless, which means you can choose to specify physical quantities in any system of units. The toolbox does not check specified physical quantities for consistency. Therefore, you must be sure to you specify all parameters consistently, using the same system of units. You can use SI (International System of Units), CGS (Centimeter-Gram-Second), US customary, or any other system. The toolbox has no default units.

You can follow the two main workflows to solve problems:

• Unified Modeling for solving structural, thermal, and electromagnetic problems and switching between these types of problems

• General PDEs for solving general linear and nonlinear PDEs for stationary, time-dependent, and eigenvalue problems

In both workflows, you typically start by creating or importing a geometry. At this step, decide which units you want to use for specifying dimensions. You can use meters, centimeters, kilometers, miles, feet, inches, parsecs, angstroms, astronomical units, and so on. You must specify all other physical quantities for the same problem using a consistent set of units.

Unified Modeling Workflow

The unified modeling workflow uses a finite element analysis object femodel, which enables you to specify physical parameters using familiar domain-specific terms for structural, thermal, and electromagnetic analysis. You must specify all parameters in consistent units using the same system of measurements. Some types of analysis require you to specify constants, such as the Stefan-Boltzmann constant or the electric permittivity and magnetic permeability of vacuum. The toolbox has no default values for such constants because these values depend on the system of units. You must specify the constants required for your analysis type in the units consistent with all other physical quantities in your problem. For example, values of the Stefan-Boltzmann constant in common systems of units are:

• SI — 5.670367e-8 W/(m²·K⁴)

• CGS — 5.6704e-5 erg/(cm²·s·K⁴)

• US customary — 1.714e-9 BTU/(hr·ft²·R⁴)

In the SI system of units, the electric permittivity of vacuum is 8.8541878128e-12 F/m, and the magnetic permeability is 1.2566370614e-6 H/m.

For problems with radiation, all temperature values must be in an absolute scale. Because the toolbox is unitless, it has no internal conversion to an absolute scale. So, for example, if you use SI units, then you must specify all temperature values in Kelvin.

General PDE Workflow

The general PDE workflow requires you to specify PDEs for your problem in a particular form, and define all parameters of a problem by specifying the number of equations, geometry, PDE coefficients, and boundary and initial conditions. For details, see Put Equations in Divergence Form. This workflow uses a PDEModel object to store the parameters of the problem.

The workflow does not make assumptions about the type of physical problem you are trying to solve. When using this workflow, start by defining your problem in consistent units using the same system of measurements. Then convert the equations defining your problem to the divergence form required by the toolbox. Note that this workflow does not specify physical constants, such as the Stefan-Boltzmann constant or the electric permittivity and magnetic permeability of vacuum. Instead, you must include any constants in the equations.