9. Symbols

Table 9.1 List Of Symbols
Symbol	Description	First Appearance
$\extSpeed$	Extension speed	Level Set
$\heaviSmooth$	Smooth Heaviside	Level Set
$\symRes$	Solution Residual	Governing Equations
$\symDomain$	Volumetric Domain	Domain Definition
$\levelSet$	Level set variable	Level Set
$\symCoordVect$	Spatial Coordinate Vector	Domain Definition
$\dHeaviSmooth$	Smooth Heaviside	Level Set
$\levelSetCurv$	Level set curvature	Level Set
$\Surf$	Generic Surface	Domain Definition
$\Vol$	Generic Volume	Domain Definition
$\symDomainBoundary$	Boundary of domain $\symDomain$	Domain Definition
$\symDomainInterface$	Interface of domains $\symDomain_i$ and $\symDomain_j$	Domain Definition
$\symSlnVar$	Solution Variables	Governing Equations
$\symWeightFunc$	Finite Element Weight Function	Governing Equations
$\surfaceTensionCoeff$	Surface tension coefficient	Level Set/CDFEM
$\symJac$	Jacobian Matrix	Solution Strategy
$\heaviWidth$	Smooth Heaviside interface thickness	Level Set
$\surfaceTensionBodyForce$	Surface tension force vector	Level Set/CDFEM
$\identityTensor$	Identity tensor	Level Set/CDFEM
$\surfaceTensionTensorStab$	Surface tension stress tensor stabilization	Level Set/CDFEM
$\surfaceTensionTensor$	Surface tension stress tensor	Level Set/CDFEM
$\symSlnIncr$	Solution Increment	Solution Strategy
$\symResVec$	Residual Vector	Solution Strategy
$\symSlnVec$	Solution Vector	Solution Strategy
$\surfaceTensionSurfaceForce$	Surface tension force vector	Level Set/CDFEM
$\levelSetNormal$	Level set normal	Level Set

Table 9.2 General Notation
Expression	Meaning
$\interior{\mathcal{R}}$	interior of $\mathcal{R}$
$\boundary{\mathcal{R}}$	boundary of $\mathcal{R}$
$\mathcal{R} \cup \mathcal{F}$	union of $\mathcal{R}$ and $\mathcal{F}$
$\mathcal{R} \cap \mathcal{F}$	intersection of $\mathcal{R}$ and $\mathcal{F}$
$\mathcal{R} \subset \mathcal{F}$	$\mathcal{R}$ is a subset of $\mathcal{F}$
$\mathcal{R}\setminus \mathcal{F}$	set complement
$x \in \mathcal{R}$	$x$ is an element of the set $\mathcal{R}$
$f : \mathcal{R} \rightarrow \mathcal{F}$	$f$ maps the set $\mathcal{R}$ into the set $\mathcal{F}$ ; $\mathcal{R}$ is the domain, $\mathcal{F}$ the codomain
$x \mapsto f(x)$	mapping that carries $x$ into $f(x)$ ; e.g. $x \mapsto x^2$ is the mapping that carries every real number $x$ into its square
$f \circ g$	composition of the mappings $f$ and $g$ ; i.e. $(f \circ g)(x) = f(g(x))$
$\{x\,\|\,R(x) \; \text{holds}\}$	the set of all $x$ such that $R(x)$ holds; e.g. $\{x\,\|\,0 \leq x \leq 1\}$ is the interval $[0,1]$
$\delta(\bfx - \bfx_0)$	Dirac delta function centered at $\bfx_0$
$\ljump \cdot \rjump$	measure of jump in a quantity across an interface
$\mathbb{R}$	real numbers
$\mathbb{N}$	natural numbers