Notation and Conventions

This section records notation used throughout the book.

Typography and indices

Object	Convention	Typical examples
Scalar	italic lowercase or Greek letter	$u$, $T$, $k$, $\rho$
Vector	bold lowercase letter	$\bx$, $\bu$, $\bq$
Second-order tensor	bold uppercase or bold Greek letter	$\bF$, $\bC$, $\bsig$
Fourth-order tensor	blackboard-bold uppercase letter	$\mathbb{C}$
Set or function space	calligraphic or standard space symbol	$\Th$, $\V$, $H^1(\Om)$
Matrix and coefficient vector	bold uppercase letter	$\mathbf{K}$, $\mathbf{U}$

When components are useful, Latin indices range over the spatial dimensions and repeated indices are summed unless stated otherwise. Thus

\[ \boldsymbol{a}\cdot\boldsymbol{b}=a_i b_i, \qquad (\boldsymbol{A}\boldsymbol{a})_i=A_{ij}a_j, \qquad \boldsymbol{A}:\boldsymbol{B}=A_{ij}B_{ij}. \]

Geometry and integration

Symbol	Meaning	Symbol	Meaning
$d$	Spatial dimension	$\Om\subset\R^d$	Open physical domain
$\partial\Om$	Boundary of $\Om$	$\Gam_D$, $\Gam_N$	Dirichlet and Neumann portions of the boundary
$\bx$	Point in the spatial or current configuration	$\bX$	Point in a reference configuration
$\bn$	Outward unit normal	$\dx$, $\ds$	Volume and boundary integration measures

Unless a different decomposition is stated,

\[ \partial\Om=\overline{\Gam_D\cup\Gam_N}, \qquad \Gam_D\cap\Gam_N=\varnothing. \]

Differential operators

Symbol	Meaning
$\grad u = \nabla_{\bx} u$	Spatial gradient of a scalar field
$\grad\bu = \nabla_{\bx} \bu$	Spatial gradient of a vector field
$\divg\bq = \nabla_{\bx}\cdot \bq$	Spatial divergence of a vector field
$\divg\bsig = \nabla_{\bx} \bsig$	Spatial divergence of a second-order tensor field
$\Grad\bu = \nabla_{\bX} \bu$	Gradient with respect to reference coordinates
$\Div\bP = \nabla_{\bX} \cdot \bP$	Divergence with respect to reference coordinates

The gradient of a vector field follows the convention

\[ (\grad\bu)_{ij}=\frac{\partial u_i}{\partial x_j}, \]

and the divergence of a second-order tensor follows

\[ (\divg\boldsymbol{A})_i = \frac{\partial A_{ij}}{\partial x_j}. \]

Uppercase operators distinguish differentiation with respect to reference coordinates from differentiation with respect to spatial coordinates.

Function spaces and variational objects

Symbol	Meaning
$L^2(\Om)$	Square-integrable scalar functions on $\Om$
$H^m(\Om)$	Functions with square-integrable weak derivatives through order $m$
$H_0^1(\Om)$	$H^1$ functions with zero trace on the prescribed boundary
$\U$	Generic trial space
$\V$	Generic test or variational space
$\Ug$	Trial space satisfying prescribed Dirichlet data $g$
$\Vzero$	Test space satisfying the corresponding homogeneous data
$a(u,v)$	Bilinear form
$a(u;v)$	Semilinear form (nonlinear in $u$ and linear in $v$)
$\ell(v)$	Linear functional
$R(u;v)$	Residual evaluated in direction $v$
$\Pi(u)$	Energy or potential functional
$\delta\Pi(u;v)$	First variation of $\Pi$ at $u$ in direction $v$

The norm, seminorm, inner product, and duality pairing are written as

\[ \norm{v}_V, \qquad \seminorm{v}_{H^m(\Om)}, \qquad \inner{u}{v}_V, \qquad \dual{f}{v}. \]

Finite element notation

Symbol	Meaning
$\Th$	Mesh of the domain
$K\in\Th$	A mesh cell
$h_K$	Diameter or characteristic size of cell $K$
$h=\max_{K\in\Th}h_K$	Global mesh-size parameter
$(K,\mathcal P,\mathcal N)$	Cell, local function space, and degrees of freedom defining a finite element
$\Uh$, $\Vh$	Discrete trial and test spaces
$\uh$, $\vh$	Discrete trial and test functions
$\phi_i$	Finite element basis function
$\mathbf K\mathbf U=\mathbf F$	Representative assembled algebraic system

The book uses cell for a geometric member of a mesh.

Recurring physical fields

Symbol	Meaning	Symbol	Meaning
$T$	Temperature	$\bq$	Heat-flux vector
$\bu$	Displacement vector	$\beps$	Infinitesimal strain tensor
$\bsig$	Cauchy stress tensor	$\bF$	Deformation gradient
$\bC=\bF^T\bF$	Right Cauchy–Green tensor	$\bP$	First Piola–Kirchhoff stress tensor
$J = \mathrm{det}(\bF)$	Jacobian (determinant of $\bF$)	$I_1 = \mathrm{tr}(\bC)$	First invariant of $\bC$ (trace)

# Notation and Conventions {#sec-notation .unnumbered} This section records notation used throughout the book. ## Typography and indices | Object | Convention | Typical examples | |---|---|---| | Scalar | italic lowercase or Greek letter | $u$, $T$, $k$, $\rho$ | | Vector | bold lowercase letter | $\bx$, $\bu$, $\bq$ | | Second-order tensor | bold uppercase or bold Greek letter | $\bF$, $\bC$, $\bsig$ | | Fourth-order tensor | blackboard-bold uppercase letter | $\mathbb{C}$ | | Set or function space | calligraphic or standard space symbol | $\Th$, $\V$, $H^1(\Om)$ | | Matrix and coefficient vector | bold uppercase letter | $\mathbf{K}$, $\mathbf{U}$ | When components are useful, Latin indices range over the spatial dimensions and repeated indices are summed unless stated otherwise. Thus $$ \boldsymbol{a}\cdot\boldsymbol{b}=a_i b_i, \qquad (\boldsymbol{A}\boldsymbol{a})_i=A_{ij}a_j, \qquad \boldsymbol{A}:\boldsymbol{B}=A_{ij}B_{ij}. $$ ## Geometry and integration | Symbol | Meaning | Symbol | Meaning | |---|---|---|---| | $d$ | Spatial dimension | $\Om\subset\R^d$ | Open physical domain | | $\partial\Om$ | Boundary of $\Om$ | $\Gam_D$, $\Gam_N$ | Dirichlet and Neumann portions of the boundary | | $\bx$ | Point in the spatial or current configuration | $\bX$ | Point in a reference configuration | | $\bn$ | Outward unit normal | $\dx$, $\ds$ | Volume and boundary integration measures | Unless a different decomposition is stated, $$ \partial\Om=\overline{\Gam_D\cup\Gam_N}, \qquad \Gam_D\cap\Gam_N=\varnothing. $$ ## Differential operators | Symbol | Meaning | |---|---| | $\grad u = \nabla_{\bx} u$ | Spatial gradient of a scalar field | | $\grad\bu = \nabla_{\bx} \bu$ | Spatial gradient of a vector field | | $\divg\bq = \nabla_{\bx}\cdot \bq$ | Spatial divergence of a vector field | | $\divg\bsig = \nabla_{\bx} \bsig$ | Spatial divergence of a second-order tensor field | | $\Grad\bu = \nabla_{\bX} \bu$ | Gradient with respect to reference coordinates | | $\Div\bP = \nabla_{\bX} \cdot \bP$ | Divergence with respect to reference coordinates | The gradient of a vector field follows the convention $$ (\grad\bu)_{ij}=\frac{\partial u_i}{\partial x_j}, $$ and the divergence of a second-order tensor follows $$ (\divg\boldsymbol{A})_i = \frac{\partial A_{ij}}{\partial x_j}. $$ Uppercase operators distinguish differentiation with respect to reference coordinates from differentiation with respect to spatial coordinates. ## Function spaces and variational objects | Symbol | Meaning | |---|---| | $L^2(\Om)$ | Square-integrable scalar functions on $\Om$ | | $H^m(\Om)$ | Functions with square-integrable weak derivatives through order $m$ | | $H_0^1(\Om)$ | $H^1$ functions with zero trace on the prescribed boundary | | $\U$ | Generic trial space | | $\V$ | Generic test or variational space | | $\Ug$ | Trial space satisfying prescribed Dirichlet data $g$ | | $\Vzero$ | Test space satisfying the corresponding homogeneous data | | $a(u,v)$ | Bilinear form | | $a(u;v)$ | Semilinear form (nonlinear in $u$ and linear in $v$) | | $\ell(v)$ | Linear functional | | $R(u;v)$ | Residual evaluated in direction $v$ | | $\Pi(u)$ | Energy or potential functional | | $\delta\Pi(u;v)$ | First variation of $\Pi$ at $u$ in direction $v$ | The norm, seminorm, inner product, and duality pairing are written as $$ \norm{v}_V, \qquad \seminorm{v}_{H^m(\Om)}, \qquad \inner{u}{v}_V, \qquad \dual{f}{v}. $$ ## Finite element notation | Symbol | Meaning | |---|---| | $\Th$ | Mesh of the domain | | $K\in\Th$ | A mesh cell | | $h_K$ | Diameter or characteristic size of cell $K$ | | $h=\max_{K\in\Th}h_K$ | Global mesh-size parameter | | $(K,\mathcal P,\mathcal N)$ | Cell, local function space, and degrees of freedom defining a finite element | | $\Uh$, $\Vh$ | Discrete trial and test spaces | | $\uh$, $\vh$ | Discrete trial and test functions | | $\phi_i$ | Finite element basis function | | $\mathbf K\mathbf U=\mathbf F$ | Representative assembled algebraic system | The book uses *cell* for a geometric member of a mesh. ## Recurring physical fields | Symbol | Meaning | Symbol | Meaning | |---|---|---|---| | $T$ | Temperature | $\bq$ | Heat-flux vector | | $\bu$ | Displacement vector | $\beps$ | Infinitesimal strain tensor | | $\bsig$ | Cauchy stress tensor | $\bF$ | Deformation gradient | | $\bC=\bF^T\bF$ | Right Cauchy--Green tensor | $\bP$ | First Piola--Kirchhoff stress tensor | | $J = \mathrm{det}(\bF)$ | Jacobian (determinant of $\bF$) | $I_1 = \mathrm{tr}(\bC)$ | First invariant of $\bC$ (trace) |

Symbol	Meaning
\(L^2(\Om)\)	Square-integrable scalar functions on \(\Om\)
\(H^m(\Om)\)	Functions with square-integrable weak derivatives through order \(m\)
\(H_0^1(\Om)\)	\(H^1\) functions with zero trace on the prescribed boundary
\(\U\)	Generic trial space
\(\V\)	Generic test or variational space
\(\Ug\)	Trial space satisfying prescribed Dirichlet data \(g\)
\(\Vzero\)	Test space satisfying the corresponding homogeneous data
\(a(u,v)\)	Bilinear form
\(a(u;v)\)	Semilinear form (nonlinear in \(u\) and linear in \(v\))
\(\ell(v)\)	Linear functional
\(R(u;v)\)	Residual evaluated in direction \(v\)
\(\Pi(u)\)	Energy or potential functional
\(\delta\Pi(u;v)\)	First variation of \(\Pi\) at \(u\) in direction \(v\)