Physics & Boundary Conditions

Elementa uses a plugin architecture for physics modules. Each module declares the governing equation it solves, the material properties it needs, and the boundary condition types it supports.

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Electrostatics

Governing equation (Poisson):

\[-\nabla \cdot (\varepsilon_0 \varepsilon_r \nabla \varphi) = \rho\]

where \(\varepsilon_0 = 8.854 \times 10^{-12}\) F/m.

Material property required: Relative Permittivity \(\varepsilon_r\) (dimensionless)

Supported study types: Stationary

Boundary Conditions

Type	Description	Properties
Electric Potential	Dirichlet — fixes the voltage	value (V)
Ground	Shorthand for Electric Potential = 0	—
Surface Charge Density	Neumann — prescribes normal flux	value (C/m²)
Zero Charge	Homogeneous Neumann (default on unset boundaries)	—

Domain Features

Feature	Description	Properties
Charge Density	Volumetric free charge \(\rho\)	value (C/m³)

Result Fields

Key	Label	Type	Unit
phi	V (Potential)	Scalar	V
E	E (Electric Field)	Vector	V/m

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Heat Transfer

Governing equation:

\[\rho C_p \frac{\partial T}{\partial t} - \nabla \cdot (k \nabla T) = Q\]

For stationary analysis the time derivative is dropped.

Material properties required:

Property	Symbol	Unit
Thermal Conductivity	\(k\)	W/(m·K)
Density	\(\rho\)	kg/m³
Heat Capacity	\(C_p\)	J/(kg·K)

Supported study types: Stationary, Time Dependent

Boundary Conditions

Type	Description	Properties
Temperature	Dirichlet — fixed temperature	value (K)
Heat Flux	Neumann — prescribed normal heat flux	value (W/m²)
Convection	Robin — convective heat transfer	h (W/m²K), T_ext (K)
Thermal Insulation	Homogeneous Neumann (zero flux)	—

The convective boundary condition applies:

\[-k \nabla T \cdot \hat{n} = h (T - T_\text{ext})\]

Domain Features

Feature	Description	Properties
Heat Source	Volumetric heat generation \(Q\)	value (W/m³)

Result Fields

Key	Label	Type	Unit
T	T (Temperature)	Scalar	K
q	q (Heat Flux)	Vector	W/m²

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Time Dependent Studies

When Time Dependent is selected in the Study panel, configure:

Setting	Description	Default
t_start	Start time (s)	0.0
t_end	End time (s)	1.0
dt	Time step (s)	0.1

Elementa uses Backward Euler time integration. After solving, a time slider in the Results panel lets you step through the computed solutions.

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Material Properties and Physics

Enable Use Material Properties in the Physics settings panel to apply domain-specific material values from the Materials panel. When disabled, the uniform values entered in the Physics config apply everywhere.

Built-in Material Library

Material	\(\varepsilon_r\)	\(\sigma\) (S/m)	\(k\) (W/mK)	\(\rho\) (kg/m³)	\(C_p\) (J/kgK)
Air	1.0	0	0.025	1.225	1006
Vacuum	1.0	0	0	0	0
Water	80.1	5.5×10⁻⁶	0.598	997	4184
Silicon	11.68	1.56×10⁻³	148	2330	710
Copper	1.0	5.96×10⁷	400	8960	385