Neumann boundary conditions¶
Editor: Weipeng Xu & Copilot
Last modified: 24/01/2026
The neumann boundary condition specifies the value of the derivative of a function normal to the boundary of the domain. From the weak formulation and the finite element discretization, we can express the Neumann boundary condition term as the following boundary integral: $$ \int_{\Gamma_N}\boldsymbol{f}(\boldsymbol{x}) \, d\Gamma $$ where $\boldsymbol{f}$ is the integrand function defined on the Neumann boundary $\Gamma_N$. $\boldsymbol{x}$ represents the coordinates in the physical domain. The computation of this integral requires two main steps: mapping the physical domain to a reference domain and then to a parametric domain for numerical integration.
Physical domain to reference domain mapping¶
Since the boundary $\Gamma_N$ may have a complex shape in the physical domain, we first map it to a simpler reference domain. We denote the mapping from the reference configuration $\boldsymbol{X}$ to the physical configuration $\boldsymbol{x}$ as: $$ \boldsymbol{x} = \boldsymbol{x}(\boldsymbol{X}) $$ The deformation gradient $\boldsymbol{F}$ is defined as: $$ \boldsymbol{F} = \frac{\partial \boldsymbol{x}}{\partial \boldsymbol{X}} $$ with determinant $J = \det(\boldsymbol{F})$. The Nanson's formula tells us how the oriented surface elements transform under the mapping. Specifically, the differential area element on the boundary transforms as: $$ d\Gamma = J\|\boldsymbol{F}^{-\top}\boldsymbol{N}_r\| d\Gamma_r $$ where $J$ is the determinant of the deformation gradient $\boldsymbol{F}$, $\boldsymbol{N}_r$ is the normal vector in the reference configuration, and $d\Gamma_r$ is the differential area element in the reference domain. Thus, the integral becomes: $$ \int_{\Gamma_N}\boldsymbol{f}(\boldsymbol{x}) \, d\Gamma = \int_{\Gamma_{N,r}}\boldsymbol{f}(\boldsymbol{X}) J(\boldsymbol{X})\|\boldsymbol{F}^{-\top}(\boldsymbol{X})\boldsymbol{N}_r\| d\Gamma_r $$
Reference domain to parametric domain mapping¶
To evaluate the integral over the reference domain, we need to map it to a parametric domain suitable for numerical integration. Let $\boldsymbol{\xi}$ denote the parametric coordinates. The mapping from the parametric domain to the reference domain is given by: $$ \boldsymbol{X} = \boldsymbol{X}(\boldsymbol{\xi}) $$ The differential area element transformation can be expressed in terms of the parametric coordinates: $$ d\Gamma_r = \boldsymbol{J}_\Gamma d\Gamma_\xi $$ where $\boldsymbol{J}_\Gamma$ is the Jacobian determinant of the transformation from the parametric domain to the reference domain. The specific form of $\boldsymbol{J}_\Gamma$ depends on whether we are dealing with a 2D or 3D problem. For a 2D problem, it is given by: $$ \boldsymbol{J}_\Gamma = \left\|\frac{\partial \boldsymbol{X}}{\partial \xi}\right\| $$ For a 3D problem, it is given by: $$ \boldsymbol{J}_\Gamma = \left\|\frac{\partial \boldsymbol{X}}{\partial \xi_1} \times \frac{\partial \boldsymbol{X}}{\partial \xi_2}\right\| $$ Thus, the integral over the Neumann boundary in the parametric domain becomes: $$ \int_{\Gamma_N}\boldsymbol{f}(\boldsymbol{x}) \, d\Gamma = \int_{\Gamma_{N,\xi}}\boldsymbol{f}(\boldsymbol{\xi}) J(\boldsymbol{\xi})\|\boldsymbol{F}^{-\top}(\boldsymbol{\xi})\boldsymbol{N}_r\| \boldsymbol{J}_\Gamma d\Gamma_\xi $$
Further discussion¶
Nanson's formula¶
Assume we have a current configuration $\boldsymbol{x}$ with outward normal vector $\boldsymbol{n}$ and differential area element $da$, as well as a reference configuration $\boldsymbol{X}$ with $\boldsymbol{N}$ and $dA$. Nanson's formula relates these quantities as follows: $$ \boldsymbol{n} \, da = J \boldsymbol{F}^{-\top} \boldsymbol{N} \, dA $$ where $\boldsymbol{F}=\frac{\partial \boldsymbol{x}}{\partial \boldsymbol{X}}$ is the deformation gradient and $J=\det(\boldsymbol{F})$ is its determinant.
See here for more details on the derivation of Nanson's formula.