Fully Implicit, Coupled Procedures in Computational Fluid Dynamics

Navier-Stokes-Based Regularization for 4d Flow MRI Super-Resolution

“…Dirichlet conditions were applied on every border with a noslip condition on the wall and interpolated data velocity on the inlet and outlet. A common strategy to get a first order approximation of these equations is to use the finite-volume method, which relies on equations ( 3) and ( 4) integration over the voxels' volume [14,10,16]. Consequently, equations ( 3) and ( 4) can be linearized at any point X k , leading to a quadratic regularization term N S k :…”

“…where S X k is the convection-diffusion matrix computed from X k and b contains the boundary conditions either on the fieldof-view or on the pre-established segmentation borders. Contrary to the proposed approach in Rispoli et al [10], all the velocity components are updated simultaneously by using a coupled velocity-pressure formulation [16]. In that respect, an iterative scheme is proposed to solve the non-linear problem (1) by defining the weighted least-squares criterion for each X k :…”

“…Moreover, the Navier-Stokes equations are accounted for in a penalization approach rather than a hard constraint so as to reduce the computation time. The linearization of the Navier-Stokes equations is performed using the finite-volume method [10,16]. For the sake of simplicity, this study is led on 2D sections of simulated synthetic data and experimental 4D flow MRI measurements on a phantom (see section 3).…”

Segmentation-Free Super-Resolved 4D flow MRI Reconstruction Exploiting Navier-Stokes Equations and Spatial Regularization

“…The Navier-Stokes equations are applied on the whole field-of-view (FOV) and Dirichlet conditions are used on its borders, in which interpolated velocity is applied on the inlet and the outlet while the no-slip condition is enforced on the other walls. Finite-volume method is generally used to discretize fluid mechanics problems [11,18,6], and especially to obtain a first order approximation. Akin to previous contributions [12,18], a velocity-pressure coupling formulation enables the direct computation of all velocity components which differ from a segregated algorithm used in Rispoli et al [11] solution.…”

“…Finite-volume method is generally used to discretize fluid mechanics problems [11,18,6], and especially to obtain a first order approximation. Akin to previous contributions [12,18], a velocity-pressure coupling formulation enables the direct computation of all velocity components which differ from a segregated algorithm used in Rispoli et al [11] solution. Thus, the regularization term N S (X), based on equations ( 4) and ( 5), is linearized in the vicinity of any X k such as the quadratic norm of the fluid mechanic equations becomes:…”