Physics-Based Simulation

Conservation of Momentum

In the continuous setting, forces are categorized into body forces (also known as external forces, such as gravity) and surface forces (or internal forces, typically stress-based, like those arising from elasticity). We define stress-based forces through a traction field, whose existence is assumed. The traction, or force per unit area, is represented by the field $\mathbf{T}(\cdot ,\mathbf{N},t):{\Omega }^0\to {\mathbb{R}}^d$ and is defined by the equation: $${\text{force}}_S(B_{ϵ}^0)={\int }_{\partial B_{ϵ}^0}\mathbf{T}(\mathbf{X},\mathbf{N}(\mathbf{X}))ds(\mathbf{X}),$$ where $\mathbf{N}$ represents the outward-pointing normal direction in the material space. Here, ${\text{force}}_S(B_{ϵ}^0)$ denotes the net force exerted from the material outside $\partial B_{ϵ}^0$ on the material inside $B_{ϵ}^0$ through their interface. The function $\mathbf{T}(\mathbf{X},\mathbf{N},t)$ quantifies the force per unit area ($d=3$) or length ($d=2$) that material on the ${\mathbf{N}}^{+}$ side exerts at point $\mathbf{X}$ on material on the ${\mathbf{N}}^{-}$ side.

It can be shown that this implies the existence of a stress field (first Piola-Kirchoff stress) $\mathbf{P}(\cdot ,t):{\Omega }^0\to {\mathbb{R}}^{d\times d}$ with: $$\mathbf{T}(\mathbf{X},\mathbf{N},t)=\mathbf{P}(\mathbf{X},t)\mathbf{N}.$$

Then, by applying Newton's second law on $B_{ϵ}^0$, we can express the conservation of momentum as: $$\begin{array}{rl}& {\int }_{B_{ϵ}^0}R(\mathbf{X},0)\frac{\partial \mathbf{V}}{\partial t}(\mathbf{X},t)d\mathbf{X}\\ =& {\int }_{\partial B_{ϵ}^0}\mathbf{P}(\mathbf{X},t)\mathbf{N}(\mathbf{X})ds(\mathbf{X})+{\int }_{B_{ϵ}^0}R(\mathbf{X},0){\mathbf{A}}^{\text{ext}}(\mathbf{X},t)d\mathbf{X},\end{array}\text{\hspace{1em}(16.2.1)}$$ for all $B_{ϵ}^0\subset {\Omega }^0$ and $t\ge 0$.

Applying the divergence theorem, we can transform the boundary integral in Equation (16.2.1) into a volume integral and obtain: $$\begin{array}{rl}& {\int }_{B_{ϵ}^0}R(\mathbf{X},0)\frac{\partial \mathbf{V}}{\partial t}(\mathbf{X},t)d\mathbf{X}\\ =& {\int }_{B_{ϵ}^0}{\nabla }^{\mathbf{X}}\cdot \mathbf{P}(\mathbf{X},t)d\mathbf{X}+{\int }_{B_{ϵ}^0}R(\mathbf{X},0){\mathbf{A}}^{\text{ext}}(\mathbf{X},t)d\mathbf{X},\end{array}\text{\hspace{1em}(16.2.2)}$$ for all $B_{ϵ}^0\subset {\Omega }^0$ and $t\ge 0$.

Definition 16.2.1 (Divergence Theorem for Vectors). For a vector-valued function $\mathbf{f}(\mathbf{x}):\Omega \to {\mathbb{R}}^d$ defined on a closed domain $\Omega$, let $\mathbf{n}(\mathbf{x})$ be the outward-pointing normal on the boundary of this domain, the following equality holds: $${\int }_{\partial \Omega }\mathbf{f}\cdot \mathbf{n}ds(\mathbf{x})={\int }_{\Omega }\nabla \cdot \mathbf{f}d\mathbf{x}.$$ This theorem allows us to conveniently transform between boundary and volume integrals.

Here the divergence operator $\nabla \cdot$ acts on every row vector of $\mathbf{P}$ independently and results in a column vector: $({\nabla }^{\mathbf{X}}\cdot \mathbf{P}{)}_i={\sum }_j{P}_{ij,j}$. Since Equation (16.2.2) also holds for arbitrary $B_{ϵ}^0$, we arrive at the strong form of the force balance equation by removing the integration: $$\begin{array}{r}R(\mathbf{X},0)\frac{\partial \mathbf{V}}{\partial t}(\mathbf{X},t)={\nabla }^{\mathbf{X}}\cdot \mathbf{P}(\mathbf{X},t)+R(\mathbf{X},0){\mathbf{A}}^{\text{ext}}(\mathbf{X},t),\\ \forall \mathbf{X}\in {\Omega }^0\text{and}t\ge 0.\end{array}\text{\hspace{1em}(16.2.3)}$$

Remark 16.2.1 (Momentum Conservation in Solid Simulation). Conservation of momentum is the primary governing equation we use to simulate solids. As discussed previously, both the acceleration, denoted by $\frac{\partial \mathbf{V}}{\partial t}(\mathbf{X},t)$, and the internal force, expressed as ${\nabla }^{\mathbf{X}}\cdot \mathbf{P}(\mathbf{X},t)$, can be described using world space coordinates $\mathbf{x}$. With all other relevant quantities established, we incrementally solve for $\mathbf{x}$ to get dynamic motions step by step.