NeumannBoundaryUnitNormal
NeumannBoundaryUnitNormal[{x,y,…}]
represents an outward-pointing unit normal vector at the point on the boundary of a filled region.
Details
- NeumannBoundaryUnitNormal can be used to construct partial differential equation boundary conditions that depend on the unit normal vector of the boundary.
- NeumannBoundaryUnitNormal can be used with NeumannValue, DirichletCondition and NIntegrate with the finite element method.
- NeumannBoundaryUnitNormal is used when non-normal flux values can be specified, like in AcousticRadiationValue or in ElectricCurrentDensityValue.
- NeumannBoundaryUnitNormal is generated by nonconservative boundary conditions for mass transport, like MassOutflowValue.
- For finite element approximations, the PDE is multiplied with a test function and integrated over . Integration by parts gives . The integrand in the boundary integral is replaced with the NeumannValue .
- NeumannBoundaryUnitNormal can be used to model a boundary integration term of the form by specifying the NeumannValue as .
- Conversely, when a PDE specifies a Neumann value as , NeumannBoundaryUnitNormal can be used to model a boundary integration term of the form instead by specifying the NeumannValue as .
- NeumannBoundaryUnitNormal will evaluate to a vector of length of the embedding dimension of the region when the boundary condition is discretized.
- NeumannBoundaryUnitNormal can be used to derive the tangent line (2D) and tangent plane (3D).
- Components of the boundary unit normal can be accessed with Indexed.
- At internal boundaries of a region, the boundary unit normal is not uniquely defined.
- The value of the boundary unit normal will be computed by solving with a Dirichlet condition of on all boundaries including internal boundaries over the entire region . The boundary unit normal is then the gradient of normalized with .
Examples
open allclose allBasic Examples (2)
Set up a symbolic electric current density boundary condition with a non-surface normal current density:
Specify a differential equation operator :
On the left, a NeumannValue is set up. The default Neumann boundary integrand for this equation is . To model a boundary integrand of the form , a NeumannValue is set up:
Scope (5)
For nonconservative mass transport, boundary conditions like MassImpermeableBoundaryValue can produce NeumannBoundaryUnitNormal. Set up an impermeable boundary condition for a nonconservative model:
NeumannBoundaryUnitNormal can be used in NIntegrate to compute the flux through a boundary. Solve a Poisson equation on a unit Disk:
Compute the total flux through the boundary of the region through the boundary region:
Compute the total flux through the boundary of a subregion:
Specify a time-dependent differential equation operator :
On the left, a NeumannValue is set up. The default Neumann boundary integrand for this equation is . To model a boundary integrand of the form , a NeumannValue is set up:
Use a Neumann 0 boundary condition and solve the equation again:
Inspect how the solutions start to differ over time:
Create a tangential for a NeumannValue:
Make use of an Indexed component, the component, of a NeumannBoundaryUnitNormal to compute a NeumannValue:
Applications (1)
The AcousticRadiationValue makes use of a NeumannBoundaryUnitNormal to automatically compute the sound direction vector. Define model variables vars for a frequency domain acoustic pressure field with model parameters pars:
Set up the equation with a radiation boundary at the left end and an acoustic absorbing boundary at the right end:
Convert the solution to the time domain and visualize the solution in the frequency domain at various frequencies :
Properties & Relations (1)
The boundary unit normal is computed by solving a Poisson equation over the region and specifying 0 Dirichlet conditions. Compute a Poisson equation over a unit Disk:
Text
Wolfram Research (2025), NeumannBoundaryUnitNormal, Wolfram Language function, https://reference.wolfram.com/language/ref/NeumannBoundaryUnitNormal.html.
CMS
Wolfram Language. 2025. "NeumannBoundaryUnitNormal." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/NeumannBoundaryUnitNormal.html.
APA
Wolfram Language. (2025). NeumannBoundaryUnitNormal. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/NeumannBoundaryUnitNormal.html