A convergent finite element scheme for a fourth-order liquid crystal model

In this manuscript we propose and analyse a fully discrete, unconditionally stable finite element scheme for a recently developed director model for liquid crystalline flows (Metzger, S. (2020) On a novel approach for modeling liquid crystalline flows. Commun. Math. Sci., 18, 359–378). The model consists of nonlinear fourth-order partial differential equations describing the evolution of the director field and Navier–Stokes equations governing the velocity field. We employ a stable splitting approach to reduce the computational complexity by decoupling the update of the director field from the update of the velocity field. We also perform a rigorous passage to the limit as the spatial and temporal discretization parameters simultaneously tend to zero, and show that a subsequence of finite element approximations converges towards a weak solution of the original model. Passing to the limit in the nonlinear terms requires us to derive the strong convergence of the gradient of the director field from uniform bounds for its discrete Laplacian. Furthermore, we present simulations underlining the practicability of the proposed scheme, investigate its convergence properties and discuss the differences between the underlying model and already established Ericksen–Leslie-type models.

REDUCING SHORT CIRCUIT LEVELS BY USING SYSTEM SPLITTING STRATEGIES

Preliminary studies on Iraqi power system show a significant increase in the short circuit level at some of the grid substations and some power stations. This increasing results from the growth of the power generation and transmission systems in size and complexity. Islanding or splitting is dividing the power system into several islands inorder to reduce short circuit levels and avoiding blackouts. The main islanding problem is determining the location of proper splitting points and load balance and satisfaction of transmission capacity constraints for each islands.This paper mainly introduces new proposed splitting strategies of large-scale power systems by using (PSS™E version 30.3 PACKAGE PROGRAME), such that, make re-interconnection of 400KV super high voltage substation based on three-phase load flow to be minimum flow at splitting point and infeed fault current details method to control short circuit levels in Iraq power system without islanding the power system into isolated islands. Controlled islanding or splitting scheme is frequently considered as the final solution to avoid blackouts of power system.Simulation IEEE-25 bus and Iraqi power system used as the test systems for this method. Furthermore, simulation results show significant effectiveness on reducing short circuit levels with same time give stable splitting islands with same frequency for preventing the system blackouts.

Curve and Surface Smoothing Using a Modified Cahn-Hilliard Equation

We present a new method using the modified Cahn-Hilliard (CH) equation for smoothing piecewise linear shapes of two- and three-dimensional objects. The CH equation has good smoothing dynamics and it is coupled with a fidelity term which keeps the original given data; that is, it does not produce significant shrinkage. The modified CH equation is discretized using a linearly stable splitting scheme in time and the resulting scheme is solved by using a Fourier spectral method. We present computational results for both curve and surface smoothing problems. The computational results demonstrate that the proposed algorithm is fast and efficient.