Benchmark Problems for the Numerical Discretization of the Cahn–Hilliard Equation with a Source Term

In this paper, we present benchmark problems for the numerical discretization of the Cahn–Hilliard equation with a source term. If the source term includes an isotropic growth term, then initially circular and spherical shapes should grow with their original shapes. However, there is numerical anisotropic error and this error results in anisotropic evolutions. Therefore, it is essential to use isotropic space discretization in the simulation of growth phenomenon such as tumor growth. To test numerical discretization, we present two benchmark problems: one is the growth of a disk or a sphere and the other is the growth of a rotated ellipse or a rotated ellipsoid. The computational results show that the standard discrete Laplace operator has severe grid orientation dependence. However, the isotropic discrete Laplace operator generates good results.

Propulsion simulation on fully appended ship model

This study presents the numerical investigation for the flow around the propeller of the ONR Tumblehome combatant in open water and for the flow around the same ship in the case of self-propulsion with actuator disk method. Computational Fluid Dynamics based on RANS-VOF solver have been used in order to analyse the flow. The free surface treatment is multi-phase flow approach, incompressible and nonmiscible flow phases are modelled through the use of conservation equations for each volume fraction of phase. Accuracy involves close attention to the physical modelling, particularly the effects of turbulence, as well as the numerical discretization.

A Generalized Finite Volume Method for Density Driven Flows in Porous Media

In this article, we consider a time evolution equation for solute transport, coupled with a pressure equation in space dimension 2. For the numerical discretization, we combine the generalized finite volume method SUSHI on adaptive meshes with a time semi-implicit scheme. In the first part of this article, we present numerical simulations for two problems: a rotating interface between fresh and salt water and a well-known test case proposed by Henry. In the second part, we also introduce heat transfer and perform simulations for a system from the documentation of the software SEAWAT.

Study of a Fractional-Order Chaotic System Represented by the Caputo Operator

This paper is presented on the theory and applications of the fractional-order chaotic system described by the Caputo fractional derivative. Considering the new fractional model, it is important to establish the presence or absence of chaotic behaviors. The Lyapunov exponents in the fractional context will be our fundamental tool to arrive at our conclusions. The variations of the model’s parameters will generate chaotic behavior, in general, which will be established using the Lyapunov exponents and bifurcation diagrams. For the system’s phase portrait, we will present and apply an interesting fractional numerical discretization. For confirmation of the results provided in this paper, the circuit schematic is drawn and simulated. As it will be observed, the results obtained after the simulation of the numerical scheme and with the Multisim are in good agreement.

Relaxation Limit of the Aggregation Equation with Pointy Potential

This work was devoted to the study of a relaxation limit of the so-called aggregation equation with a pointy potential in one-dimensional space. The aggregation equation is today widely used to model the dynamics of a density of individuals attracting each other through a potential. When this potential is pointy, solutions are known to blow up in final time. For this reason, measure-valued solutions have been defined. In this paper, we investigated an approximation of such measure-valued solutions thanks to a relaxation limit in the spirit of Jin and Xin. We study the convergence of this approximation and give a rigorous estimate of the speed of convergence in one dimension with the Newtonian potential. We also investigated the numerical discretization of this relaxation limit by uniformly accurate schemes.

Exciton-Related Raman Scattering, Interband Absorption and Photoluminescence in Colloidal Cdse/Cds Core/Shell Quantum Dots Ensemble

By using the numerical discretization method within the effective-mass approximation, we have theoretically investigated the exciton-related Raman scattering, interband absorption and photoluminescence in colloidal CdSe/CdS core/shell quantum dots ensemble. The interband optical absorption and photoluminescence spectra have been revealed for CdSe/CdS quantum dots, taking into account the size dispersion of the ensemble. Numerical calculation of the differential cross section has been presented for the exciton-related Stokes–Raman scattering in CdSe/CdS quantum dots ensemble with different mean sizes.

Finite element formulations for constrained spatial nonlinear beam theories

A new director-based finite element formulation for geometrically exact beams is proposed by weak enforcement of the orthonormality constraints of the directors. In addition to an improved numerical performance, this formulation enables the development of two more beam theories by adding further constraints. Thus, the paper presents a complete intrinsic spatial nonlinear theory of three kinematically different beams which can undergo large displacements and which can have precurved reference configurations. Moreover, the hyperelastic constitutive laws allow for elastic finite strain material behavior of the beams. Furthermore, the numerical discretization using concepts of isogeometric analysis is highlighted in all clarity. Finally, all presented models are numerically validated using exclusive analytical solutions, existing finite element formulations, and a complex dynamical real-world example.