Nomenclature
Symbol	Meaning
$\mathrm{A}$	${\mathrm{A} = 6\mathsf{\xi }\mathrm{k}}_B\mathrm{T}$ in BD method
${\mathbb{A}}_{ij}$	maximum repulsion between bead $i$ and bead $j$ in DPD method
${\mathrm{a}}_i$	acceleration of $i$th particle
$B^A$	atomistic domain in concurrent simulations
$B^C$	continuum domain in concurrent simulations
$B^H$	handshake region in concurrent simulations
$B^I$	interfacial region in concurrent simulations
$B^P$	padding region in concurrent simulations
${\mathrm{b}}_i$	fitting parameter
${\mathrm{c}}_i$	fitting parameter
${\mathrm{D}}^{\mathsf{\vartheta }}$	the diffusion term of $\mathsf{\vartheta }$
${\mathrm{D}}_{\mathit{cm}}$	center-of-mass self-diffusion coefficient
$e$	element
$\mathsf{𝕖}$	absolute unit charge of an electron
$E_f$	Young’s modulus
${\mathrm{E}}_i$	energy of atom, particle, or node $i$
$\overline{{\mathrm{E}}_i}$	energy of the $i$th representative atom in QC method
${\mathrm{E}}_k$	eigenstate of energy
${\mathrm{E}}_{k_{el}}$	eigenstate energy of an electron
${\mathrm{E}}_{k_n}$	eigenstate energy of a nucleon
${\mathrm{E}}^{tot}$	total energy
${\Delta \mathrm{F}(\mathsf{\psi }}_{\alpha })$	free energy difference in H-AdResS method
${\mathrm{F}}_{ij}^{\mathrm{C}}$	conservative force between bead $i$ and its neighboring bead $j$ within the force cutoff radius $r_{cut}$
${\mathrm{F}}_{ij}^{\mathrm{D}}$	dissipative force between bead $i$ and its neighboring bead $j$ within the force cutoff radius $r_{cut}$
${\mathrm{F}}_{ij}^{\mathrm{R}}$	random forces between bead $i$ and its neighboring bead $j$ within the force cutoff radius $r_{cut}$
${\mathrm{F}}_{\alpha }^{\mathit{drift}}$	drift force of molecule $\alpha$
$\overline{\mathrm{f}}$	vector of applied forces in the FE region of a concurrent simulation
${\mathrm{f}}_i$	force acting on the $i$th atom, particle, or node
${\mathrm{f}}_{\alpha \beta }$	force acting between molecules $\alpha$ and $\beta$
${\mathrm{f}}^{th}$	thermodynamic force
${\mathrm{f}}_i^{\mathrm{B}}$	Brownian random force acting on the $i$th particle
${\mathrm{f}}_{\alpha \beta }^{\mathrm{AA}}$	atomistic forces acting on molecule $\alpha$ due to the interaction with molecule $\beta$
${ \mathrm{f}}_{\alpha \beta }^{\mathrm{CG}}$	CG forces acting on molecule $\alpha$ due to the interaction with molecule $\beta$
$\mathrm{G}\prime$	storage modulus
$\mathrm{G}″$	loss modulus
${\mathrm{H}(\mathsf{\Gamma }}_i)$	Hamiltonian of the system at system state ${\mathsf{\Gamma }}_i$
$\hat{\mathrm{H}}$	modified Hamiltonian of the H-AdResS method
${\Delta \mathrm{H}(\mathsf{\Gamma }}_{i\to j})$	change in the system Hamiltonian for going from system state ${\mathsf{\Gamma }}_i$ to ${\mathsf{\Gamma }}_j$
${\Delta \mathrm{H}(\mathsf{\psi }}_{\alpha })$	compensation term in the Hamiltonian of the H-AdResS method
${\mathrm{H}}_{\mathit{FE}}({\mathrm{u}}_{\alpha },{\dot{\mathrm{u}}}_{\alpha })$	Hamiltonian of the FE region as a function of the nodal displacements ${\mathrm{u}}_{\alpha }$, and time rate of nodal displacements ${\dot{\mathrm{u}}}_{\alpha }$
${\mathrm{H}}_{\mathit{FE}/\mathit{MD}}({\mathrm{r}}_j,{\mathrm{v}}_j,{\mathrm{u}}_{\alpha },{\dot{\mathrm{u}}}_{\alpha })$	Hamiltonian of the FE/MD handshake region as a function of the atomic positions ${\mathrm{r}}_j$, atomic velocities ${\mathrm{v}}_j$, nodal displacements ${\mathrm{u}}_{\alpha }$, and time rate of nodal displacements ${\dot{\mathrm{u}}}_{\alpha }$
${\mathrm{H}}_{MD}({\mathrm{r}}_j{,\mathrm{v}}_j)$	Hamiltonian of the MD region as a function of the atomic positions ${\mathrm{r}}_j$, and atomic velocities ${\mathrm{v}}_j$
${\mathrm{H}}_{MD/TB}({\mathrm{r}}_j{,\mathrm{v}}_j)$	Hamiltonian of the MD/TB handshake region as a function of the atomic positions ${\mathrm{r}}_j$, and atomic velocities ${\mathrm{v}}_j$
${\mathrm{H}}_{TB}({\mathrm{r}}_j{,\mathrm{v}}_j)$	Hamiltonian of the TB region as a function of the atomic positions ${\mathrm{r}}_j$, and atomic velocities ${\mathrm{v}}_j$
${\mathrm{H}}_{tot}$	total Hamiltonian
$\mathrm{h}$	Planck’s constant
${\mathrm{J}}^{\mathsf{\vartheta },\text{C}}$	convection flux term in FVM formulation
${\mathrm{J}}^{\mathsf{\vartheta },\mathrm{D}}$	diffusion flux term in FVM formulation
$\mathrm{K}$	the all-atom kinetic energy of the molecules
${\mathrm{k}}_B$	Boltzmann’s constant
${\mathrm{k}}_T$	isothermal compressibility
$l$	bond length
$\mathrm{M}$, ${\mathrm{M}}_w$	molecular weight
$\mathrm{m}$	mass of an atom or particle
${\mathrm{m}}_{el}$	mass of an electron
${\mathrm{m}}_n$	mass of a nucleon
$\mathrm{N}$	number of atoms, particles, or nodes
${\mathrm{N}}_c$	number of monomers per chain
$N_e$	number of elements
$N_q$	number of quadrature points in the numerical integration
${\mathrm{N}}_r$	number of representative atoms in QC method
$\mathrm{P}$	the projection matrix
${\Delta \mathrm{p}(\mathsf{\psi }}_{\alpha })$	pressure difference along the interface in H-AdResS method
${\mathrm{p}}_{i\to j}$	probability of accepting a new configuration for going from system state ${\mathsf{\Gamma }}_i$ to ${\mathsf{\Gamma }}_j$
${\mathrm{p}}^{\mathrm{R}}$	probability distribution function
${\mathrm{p}}_{\mathit{target}}^{\mathrm{R}}$	the target probability distribution function of AA simulations
${\mathrm{Q}}^{\mathsf{\vartheta }}$	the generation/destruction of $\mathsf{\vartheta }$ within the control volume per unit volume
$\mathrm{R}(\mathrm{u})$	residual form of a partial differential equation in terms of the unknown function $\mathrm{u}$ in FEM scheme
${\mathrm{R}}_g$	radius of gyration
${\mathrm{R}}_i$	center of mass coordinates of the $i$th molecule
$\mathbf{r}$	coordinates vector of an atom, or particle, or node
$\mathrm{r}$	distance
$r_{cut}$	force cutoff radius
${\mathrm{r}}_{{el}_i}$	spatial coordinates of an electron
${\hat{\mathrm{r}}}_{ij}$	unit vector pointing from the center of bead $j$ to that of bead $i$
${\mathrm{r}}_{n_j}$	spatial coordinates of a nucleon
${\mathrm{r}}_e^{\mathit{cent}}$	coordinates of the Gauss point in element $e$ taken at the centroid of the triangular elements
${\mathrm{r}}_e^q$	position of quadrature point $q$ of element $e$ in the reference configuration
${\mathsf{\delta }\mathrm{r}}_i^{\mathrm{B}}(\mathrm{t}+\Delta \mathrm{t})$	random displacement of the $i$th particle due to the random forces during time step $\Delta \mathrm{t}$
$\mathrm{S}$	surface vector
${\mathrm{S}}_i$	$i$th subregion
$\left\{{\mathrm{S}}_{\mathsf{\varrho }}\right\}$	set of weighting functions in FEM
${\mathrm{s}}_{\mathit{entropy}}$	rescaling factor for the entropy change
${\mathrm{s}}_{\mathit{friction}}$	rescaling factor for the friction change
$\mathrm{T}$	temperature
$\mathrm{t}$	time
$∆\mathrm{t}$	time step
$\mathrm{U}(\mathrm{r})$	potential energy
${\mathrm{U}}^A$	potential energies of the atomistic region
${\mathrm{U}}^{\mathit{atom}}$	energy functional of a systems assuming it is entirely modelled using atoms
${\mathrm{U}}^C$	potential energies of the continuum region
${\mathrm{U}}^{\mathrm{CG}}(\mathrm{r},l,\mathsf{\theta },\mathsf{\mho })$	general form of the CG potential function in IBI method
${\mathrm{U}}^{\mathit{FE}}$	energy functional of a systems assuming it is entirely modelled using FEM
${\mathrm{U}}^H$	potential energies of the handshake region
${\mathrm{U}}^{\mathit{int}}$	energy of internal interactions
${\mathrm{U}}^{tot}$	total potential energy of the entire system
${\mathrm{U}}_{\mathit{angle} }^{\mathrm{CG}}(\mathsf{\theta })$	bond angle potential in the blob model
${\mathrm{U}}_{\mathit{bond} }^{\mathrm{CG}}(l)$	bond potential in the blob model
${\mathrm{U}}_{\mathit{nonbonded} }^{\mathrm{CG}}(\mathrm{r})$	potential of nonbonded interactions in the blob model
${\mathrm{U}}_{\alpha }^{\mathrm{AA}}$	potential energy of molecule $\alpha$ in the AA representation
${\mathrm{U}}_{\alpha }^{\mathrm{CG}}$	potential energy of molecule $\alpha$ in the CG representation
$\mathrm{u}$	vector of nodal displacements in the FE region of a concurrent simulation
$\mathrm{u}(\mathrm{r})$	the unknown function in FEM which one needs to find
${\mathrm{u}}_h(\mathrm{r})$	approximation of the function $\mathrm{u}(\mathrm{r})$ under consideration in FEM
${\mathrm{u}}_{\alpha }$	displacements of atom, particle, or node $\alpha$
${\dot{\mathrm{u}}}_{\alpha }$	rate of displacements of atom, particle, or node $\alpha$
${\mathrm{u}}^n$	values of the function ${\mathrm{u}}_h$ at node $n$ of the mesh
${\mathrm{V}}_e$	volume of element $e$
$\mathrm{dV}$	volume element of the simulation domain in FEM
$\partial {\mathrm{V}}_e$	surfaces surrounding the volume ${\mathrm{v}}_e$ of element $e$
$\mathrm{v}$	macroscopic velocity magnitude
$\mathsf{𝕧}$	$\mathsf{𝕧} = \sqrt{3 }{\mathsf{𝕧}}_{\mathrm{s}}$ in LB method
$\mathrm{v}(\mathrm{r},\mathrm{t})$	macroscopic local velocity at node $\mathrm{r}$ at time $\mathrm{t}$ in LB
$\tilde{\mathrm{v}}(\mathrm{t}+∆\mathrm{t})$	estimated velocity in the next time step using a predictor method in DPD velocity-Verlet algorithm
${\mathsf{\delta }\mathrm{v}}_i^{\mathrm{B}}(\mathrm{t}+\Delta \mathrm{t})$	Random velocity change of the $i$th particle due to the random forces during time step $\Delta \mathrm{t}$
${\mathrm{v}}_i$	velocity of $i$th atom, particle, or node
$\left|{\mathsf{𝕧}}_i\right|$	velocity magnitude in $i$-direction in LB method
$\left\{{\mathsf{𝕧}}_k\right\}$	set of prescribed velocity vectors connecting the neighboring nodes in LB method
${\mathsf{𝕧}}_s$	speed of sound
$\mathrm{W}$	a function of deformation gradient $\Delta$
${\mathrm{w}}_i$	weighting constants used in LB method
${\mathrm{z}}_n\mathsf{𝕖}$	positive unit charge of a nucleon
${\mathsf{\Gamma }}_i$	system state in a phase space at position $i$
$\mathsf{\gamma }$	exact solution in the projection method
$\dot{\mathsf{\gamma }}$	shear-rate
$\overline{\mathsf{\gamma }}({\mathrm{r}}_{\alpha })$	coarse scale solution of a problem in the projection method
$\mathsf{\gamma }\prime$	fine scale solution of a problem in the projection method
$\Delta$	deformation gradient
$\mathsf{\delta }$	delta function
$∆\mathsf{\mu }({\mathsf{\psi }}_{\alpha })$	chemical potential gradient in H-AdResS method
$\epsilon$	neighboring cells of a specific element in FVM
$\mathsf{\zeta }$	random number between 0 and 1 which is to determine the acceptance or rejection of a new configuration
${\mathsf{\zeta }}_{ij}$	a Gaussian random number with zero mean and unit variance used in the definition of the random forces between beads $i$ and $j$ in DPD method
$\mathsf{\eta }$	viscosity
$\mathsf{\Theta }$	a weighting function to link FE and atomistic models in concurrent simulations
$\mathsf{\theta }$	bond angle
${\mathsf{\theta }}_{\mathrm{ave}}$	averaged initial orientation angle
${\mathsf{\Lambda }}_{ik}$	collision matrix used in LB method
$\mathsf{\lambda }$	multiplication parameter in in DPD velocity-Verlet algorithm
$\mathsf{\mu }$	fitting parameter
$\mathsf{\nu }$	fitting parameter
$\mathsf{\vartheta }$	a general conserved scalar variable in FVM scheme
$\mathsf{\xi }$	friction coefficient between atoms or particles
${\mathsf{\xi }}_{ij}$	friction coefficient between bead $i$ and bead $j$ in DPD method
${\mathsf{\xi }}_m$	friction coefficient between particles of freely-rotating chains
$\mathsf{\varpi }$	wave function of electrons
$\mathsf{\rho }$	fluid density in CFD
$\mathsf{\rho }(\mathrm{r},\mathrm{t})$	macroscopic local density at node $\mathrm{r}$ at time $\mathrm{t}$ in LB method
${\mathsf{\rho }}_i(\mathrm{r})$	molecular density profile in the $i$th iteration step as a function of the position in the direction perpendicular to the interface, in AdResS method
${\mathsf{\rho }}^{*}$	reference molecular density
${\mathsf{\varrho }}_i$	$i$th weighting function in FEM
${\mathsf{\sigma }}_{ij}$	noise amplitude between bead $i$ and bead $j$ in DPD method
${\mathsf{\varsigma }}_i^{\alpha }$	shape function of node $i$ evaluated at the point with coordinates ${\mathrm{r}}_{\alpha }$
$\mathsf{\tau }$	characteristic collision time in LB method
$\mathsf{\Phi }(\mathrm{u})$	integral form of the weighted residuals in FEM
${\mathsf{\phi }(\mathrm{r})}_k$	wave function in Schrödinger’s equation
$\mathsf{\varphi }$	wave function of the nuclei
${\mathsf{\chi }}_{ij}$	a parameter in DPD formulation which equals 1 for beads with a distance less than $r_{cut}$ and equals 0 otherwise
${\mathsf{\Psi }}_i(\mathrm{r},\mathrm{t})$	particle distribution function used in LB at node $\mathrm{r}$ at time $\mathrm{t}$ moving with velocity ${\mathsf{𝕧}}_i$ In the $i$-direction
${\mathsf{\Psi }}_i^{\mathrm{eq}}(\mathrm{r},\mathrm{t})$	equilibrium particle distribution function used in LB at node $\mathrm{r}$ at time $\mathrm{t}$ moving with velocity ${\mathsf{𝕧}}_i$ In the $i$-direction
$\mathsf{\psi }$	spatial interpolation function in AdResS method
${\mathsf{\psi }}_n(\mathrm{r})$	interpolation functions in FEM for node $n$
${\mathsf{\psi }}_n^e(\mathrm{r})$	interpolation functions in FEM for node $n$ in element $e$
$\mathsf{\Omega }$	simulation domain in FEM
$\partial \mathsf{\Omega }$	boundaries of the simulation domain in FEM
$\mathsf{\mho }$	dihedral angle
$\mathsf{\omega }$	Frequency
${\mathsf{\omega }}_i$	quadrature weight signifying how many atoms a given representative atom stands for in the description of the total energy, in QC method
${\mathsf{\omega }}^{\mathrm{D}}({\mathrm{r}}_{ij})$	dissipative weight function in DPD method
${\mathsf{\omega }}^q$	associated Gauss quadrature weights of quadrature point $q$ of element $e$
${\mathsf{\omega }}^{\mathrm{R}}({\mathrm{r}}_{ij})$	random weight function in DPD method