In this study, the perturbation method was implemented to analytically solve the governing equations relevant to both hydrodynamically and thermally fully developed power-law fluid and plug flows through parallel-plates and circular microchannels under constant isoflux thermal and slip boundary condition.
The implementation of the adjoint method requires access to the source code - either to implement the analytically derived adjoint equations into the code (so-called continuous adjoint) or to perform a manual or automatic differentiation of the primal code (discrete adjoint, see [5] for a comparison).
The images were reconstructed by a 2-dimensional MLEM algorithm implemented on an analytically derived system matrix [36].
A general-purpose method is presented and implemented to express analytically one stationary configuration of an ideal 3D elastic rod when the end-to-end relative position and orientation are imposed.
Consequently, the necessary Tykhonov matrix, which can be used in the regularized estimation of the Euler vector, is not available and has to be developed analytically as implemented in this study.
Except in rare cases, one cannot analytically implement this update.
The dynamic model that results from replacing the canopy structure component of a standard forest gap model (SORTIE: [2], [24]) with the ITD, is also rapid to implement, and becomes analytically tractable for such metrics as equilibrium density, basal area and biomass, patterns of self thinning, and the invasion of monocultures by other species (Strigul et al. in review).
The cracked element was analytically formulated and implemented in the finite element code ABAQUS Standard as a User-defined Element (UEL) subroutine.
Exploiting the existence of a system of action and angular variables for trapped and circulating particles, a functional, extremum in the turbulent electromagnetic field, is analytically established and implemented in the code TORRID.
An analytically based and numerically implemented method for the prediction of the behavior of cable networks has been developed for the analysis of the in-plane free-vibration problem of a set of interconnected taut cable elements.
