Introduction

Simulations in Allsolve are solved based on chosen physics, or more specifically, the underlying mathematical formulations that are used to model the physics.

On this page, a general approach to formulating physics is introduced, along with the necessary mathematical identitities and theorems.

Vector calculus identities

Divergence of a curl

\begin{align} \nabla \cdot (\nabla \times \boldsymbol{A}) = 0 \end{align}

Leibniz rule for a nabla operator

\begin{align} \nabla \cdot (a \boldsymbol{b}) = \nabla a \cdot \boldsymbol{b} + a(\nabla \cdot \boldsymbol{b}) \end{align}

Divergence of a cross product

\begin{align} \nabla \cdot (\boldsymbol{a} \times \boldsymbol{b}) = (\nabla \times \boldsymbol{a}) \cdot \boldsymbol{b} - \boldsymbol{a} \cdot (\nabla \times \boldsymbol{b}) \end{align}

Scalar triple product rule

\begin{align} \boldsymbol{a} \cdot (\boldsymbol{b} \times \boldsymbol{c}) = \boldsymbol{c} \cdot (\boldsymbol{a} \times \boldsymbol{b}) = \boldsymbol{b} \cdot (\boldsymbol{c} \times \boldsymbol{a}) \end{align}

Time derivative of a curl

\begin{align} \frac{\partial}{\partial t} (\nabla \times \boldsymbol{A}) = \nabla \times \frac{\partial \boldsymbol{A}}{\partial t} \end{align}

Per partes theorem for scalars

Let $U$ and $V$ be scalar functions and $\Omega$ a set with boundary $\Gamma$ . Then let $\boldsymbol{n}$ be a unit normal vector pointing outwards on $\Gamma$ . Now

\begin{align} \int_{\Omega}\; U\nabla V\; d \Omega = \int_{\Gamma}\; UV\boldsymbol{n}\; d \Gamma - \int_{\Omega}\; V\nabla U\; d \Omega. \end{align}

The Per partes theorem is also known as integration by parts.

Per partes theorem for vectors

Let $\boldsymbol{A}$ be a vector field and $f$ a scalar function defined on a domain $\Omega$ with boundary $\Gamma$ . Then let $\boldsymbol{n}$ be a unit normal vector pointing outwards on $\Gamma$ . Now

\begin{align} \int_{\Omega}\; (\boldsymbol{A} \cdot \nabla f)\; d \Omega = \int_{\Gamma}\; f\; (\boldsymbol{A} \cdot \boldsymbol{n})\; d \Gamma - \int_{\Omega}\; f\; (\nabla \cdot \boldsymbol{A})\; d \Omega. \end{align}

The Per partes theorem is also known as integration by parts.

Divergence theorem

Let $\Omega$ be a set with boundary $\Gamma$ , let $\boldsymbol{n}$ be a unit normal vector pointing outwards on $\Gamma$ and let $\boldsymbol{T}$ be a vector field or a second-order tensor. Now

\begin{align} \int_{\Omega} \nabla \cdot \boldsymbol{T}\; d \Omega = \int_{\Gamma}\; \boldsymbol{T} \cdot \boldsymbol{n}\; d \Gamma. \end{align}

Strong and weak formulation

A common example of a PDE (partial differential equation) is the Poisson’s equation,

\begin{align} \Delta u &= f\; \text{on}\ \Omega. \end{align}

This is the strong formulation of the equation, which refers to a precise mathematical representation of the problem with very strict requirements for a solution.

The weak formulation is an alternative mathematical representation that relaxes the requirements for a solution. The weak formulation is computationally preferable as it provides the foundation for numerical methods such as the finite element method (FEM).

To derive the weak formulation of Poisson’s equation, we start by multiplying both sides of the strong formulation with a test function $u^{\prime}$ and integrating over $\Omega$ .

\begin{align} \int_{\Omega}\; (\Delta uu^{\prime} - fu^{\prime})\; d \Omega = 0 \end{align}

Let’s apply Per partes theorem on the first term on the left hand side.

\begin{align} \int_{\Omega}\; \Delta u\; u^{\prime}\; d \Omega = \int_{\Omega}\; (\nabla \cdot \nabla u)\; u^{\prime}\; d \Omega = \int_{\Gamma}\; (\nabla u \cdot \boldsymbol{n})\; u^{\prime}\; d \Gamma - \int_{\Omega}\; \nabla u \cdot \nabla u^{\prime}\; d \Omega \end{align}

Here, $U = u^{\prime}$ , $V = \nabla u$ and $\boldsymbol{n}$ is the unit normal vector. Overall we get a weak formulation of $(3)$ .

\begin{align} \int_{\Gamma}\; (\nabla u\ \cdot \boldsymbol{n})\; u^{\prime}\; d \Gamma - \int_{\Omega}\; (\nabla u \cdot \nabla u^{\prime} + f\; u^{\prime})\; d \Omega = 0 \end{align}

Now, we have less strict regularity requirements for $u$ and $f$ . Certainly, we know that a solution to $(3)$ is also a solution to $(7)$ and every weak solution smooth enough is a solution to $(3)$ .