December 01, 2023
K. Kokars,
A. Krauze,
K. Muiznieks,
J. Virbulis,
P. Verners,
A. Gutcaits,
J. Olins
Abstract 3D printed plastic casts can be used for healing bone fractures. The main requirements for these cases are: they should be light, require little printing time, have good mechanical properties, and ensure proper skin ventilation. We present a density-based topology optimization algorithm for obtaining optimal cast shapes that fulfil these requirements. The algorithm uses a linear stress model and simplified boundary conditions to model the contact problems. The cast shapes were optimized against the influence of several sharp corners. The parametric studies showed that the mass of optimized casts was reduced by 20 %–25 % in comparison with original industrial casts, and the printing time is reduced by 1.4–1.7 h for the largest cast. A major model drawback is the use of 3D numerical volume to model the density distribution. The density distribution should be homogenized across the cast layer. The overhang problem should also be addressed. We also suggest that the cast producers collect more experimental data on the cast breakages for a better calibration of the numerical model.
November 24, 2023
M. Bonnivard,
Amina Mecherbet
In this paper, we investigate the instability of the spherical travelling wave solutions for the Transport-Stokes system in $\mathbb{R}^3$. First, a classical scaling argument ensures instability among all probability measures for the Wasserstein metric and the $L^1$ norm. Secondly, we investigate the instability among patch solutions with a perturbed surface. To this end, we study the linearized system of a contour dynamics equation derived in [18] in the case where the support of the patch is axisymmetric and is described by spherical parametrization and show well-posedness of the linearized system. The actual version does not include the investigation of the eigenvalue and will be completed in the future. Eventually we investigate numerically the instability of the travelling wave by solving the Transport-Stokes equation using a finite element method on FreeFem.
November 23, 2023
Nacer Sellila,
M. Louaked,
Waleed Mouhali,
Houari Mechkour
This work is intended as an attempt to explore the use of optimal control techniques for designing green spaces and for dealing with the environmental problems related to urban heat islands appearing in cities. A three-dimensional model is established for numerical studies of the effects of urban anthropogenic heat and wind velocity in urban and rural regions. The transport mechanism of fluid in the cities is governed by the Navier–Stokes–Forschheimer porous media system. We introduce the penalty approximation method to overcome the difficulty induced by the incompressibility constraint. The partial differential equation optimal control problem is solved by using a Spectral Projected Gradient algorithm. To validate the method, we implement a numerical scheme, based on a finite element method, employing the free software FreeFem++ 14.3. We show the results for the optimized and non-optimized situations to compare the two cases.
November 19, 2023
A. Jüngel,
Boyi Wang
A fully discrete semi-convex-splitting finite-element scheme with stabilization for a degenerate Cahn-Hilliard cross-diffusion system is analyzed. The system consists of parabolic fourth-order equations for the volume fraction of the fiber phase and the solute concentration, modeling pre-patterning of lymphatic vessel morphology. The existence of discrete solutions is proved, and it is shown that the numerical scheme is energy stable up to stabilization, conserves the solute mass, and preserves the lower and upper bounds of the fiber phase fraction. Numerical experiments in two space dimensions using FreeFEM illustrate the phase segregation and pattern formation.
November 17, 2023
S. Chabbar,
A. Habbal,
R. Aboulaich,
Nabil Ismaili,
Sanaa El Majjaoui
Prostate cancer is a hormone-dependent cancer characterized by two types of cancer cells, androgen-dependent cancer cells and androgen-resistant ones. The objective of this paper is to present a novel mathematical model for the treatment of prostate cancer under combined hormone therapy and brachytherapy. Using a system of partial differential equations, we quantify and study the evolution of the different cell densities involved in prostate cancer and their responses to the two treatments. Numerical simulations of tumor growth under different therapeutic strategies are explored and presented. The numerical simulations are carried out on FreeFem++ using a 2D finite element method.
November 09, 2023
Sahar Borzooei,
Pierre-Henri Tournier,
V. Dolean,
Christian Pichot,
N. Joachimowicz,
Hélène Roussel,
C. Migliaccio
A portable imaging system for the on-site detection of shoulder injury is necessary to identify its extent and avoid its development to severe condition. Here, firstly a microwave tomography system is introduced using state-of-the-art numerical modeling and parallel computing for imaging different tissues in the shoulder. The results show that the proposed method is capable of accurately detecting and localizing rotator cuff tears of different size. In the next step, an efficient design in terms of computing time and complexity is proposed to detect the variations in the injured model with respect to the healthy model. The method is based on finite element discretization and uses parallel preconditioners from the domain decomposition method to accelerate computations. It is implemented using the open source FreeFEM software.
September 25, 2023
Natalia M. Polyakova,
Victoria I. Tsvetkova
The structure of a symmetric two-dimensional unsteady flow between two planes having a periodic step-like irregularity is investigated by numerical methods using the finite element package Freefem++. The kinematic viscosity of the fluid is chosen as typical for turbulent flows, depending on the coordinates as 𝜇 = 𝜇0(𝜂2(𝑥) − 𝑧2), 𝜂(𝑥) = ℎ(𝑥) + 𝛿, where 𝜇0 is the characteristic viscosity, 𝛿 is the roughness, ℎ(𝑥) is the function defining the boundary 𝑧 = 𝜂(𝑥) (profiles of planes mirror-symmetric with respect to 𝑧 = 0), 𝑥, 𝑧 are longitudinal and transverse coordinates along the main flow. In contrast to the usual parabolic profile of the Poiseuille flow between the planes, with the specified choice of viscosity, the flow profile is logarithmic and has a singularity at the boundary of the region, to prevent which the roughness of the boundary is introduced. A distinctive feature of the problem under study is the setting of boundary conditions on an irregular part of the boundary. On plane sections of the boundary, we maintain the usual non-slip conditions for a viscous liquid, but on the irregular part of the boundary we set only the impermeability conditions of the boundary for the liquid (no sticking!). Computational experiment - numerical solution of the Navier-Stokes equations for a viscous incompressible fluid using a modified penalty method, showed that for a wide set of parameters, the structure of the flow described by the initial non-stationary complete problem and the quasi-stationary flow described by the asymptotic model constructed in the presented paper, for which there is an exact solution, are in good agreement - in the non-stationary flow, a stable regular system of vortices is established concentrated near those surface irregularities for which the boundary has a negative curvature (`deepening’of boundary).
June 30, 2023
Md. Masum Murshed,
Md. Morshed Bin Shiraj,
Md. Safik Ullah,
Md. Mizanur Rahman,
Md. Manik Hossain
In this study, the Bay of Bengal domain has been approximated using triangular mesh so that the finite element method (FEM) can be employed on it. The area between 15º N and 23º N Latitudes and 85º E and 95º E Longitudes is considered as the physical domain. A MATLAB routine and the cubic spline interpolation have been used to extract the coordinates of the points on the boundary of the whole domain and the points on the boundary of the islands from a colour image of the domain. A C++ routine is used to generate an edp file for triangular mesh using the extracted coordinates. Then FreeFem++ is used to create triangular mesh for the whole domain from the generated edp file. All the major islands are also incorporated in the final mesh. The obtained triangular mesh can be used to develop a storm surge prediction model for the Bay of Bengal region implementing FEM.
May 16, 2023
E. Ramos,
I. Monsivais,
F. Méndez,
J. Lizardi
In the present work, we developed a numerical analysis for an electroosmotic flow circulating in a rectangular microchannel considering electrolyte viscosity as a function of the induced electric field; which is also reflected in the slip condition imposed on the system walls, since the slip length is a function of the fluid viscosity. It should be clarified this is an entirely hydrodynamic problem, and for this reason there are no induced pressure gradients, because we are in the presence of a purely electroosmotic flow, where the fluid motion is due only to electrokinetic forces. Based on these comments, the problem is centered on high induced potentials, enabling viscoelectric effect analysis in the electroosmotic flow, which leads to significant increases in velocity and volumetric flow profiles compared to the case where the viscosity is a constant and there is no slip condition. Due to analytical analysis limitations, we implemented a dimensionless equation scheme defined by the continuity equation, the momentum equations in the x and y direction, the Poisson-Boltzmann equation, and the charge conservation equation to obtain the velocity and volumetric flow rate profiles mentioned above. This model is described in its variational form in order to implement the finite element technique using free software, FreeFem++. The results obtained show how the viscoelectric effect is relevant when working with high induced potentials; that is, for values of ζ¯>1 , when the dimensionless viscoelectric parameter f¯ increases, there is a significant decrease in the velocity profiles u, a situation that is not observed when ζ¯≤1 , where there are low induced potentials, and for this reason, as the dimensionless parameter f¯ increases, the velocity profiles remain constant. This condition is preserved for different values of the slip length δ¯ .
April 01, 2023
M. S. Khan,
Isma Hameed,
M. A. Memon,
E. Bonyah
In this article, we aim to computationally analyze the magnetic induced micropolar flow in a rectangular channel using a multiphysics finite element solver, FreeFem++. In this respect, a physical model in the framework of the micropolar continuum is taken into consideration with appropriate boundary conditions. The flow is considered laminar and incompressible and moves under the application of an external magnetic field at the boundary of the flow channel. The flow governing equations of momentum, microrotation, and induction are derived, and their weak integral forms in the context of finite elements are presented. The developed finite element model is then implemented in FreeFem++ in order to compute numerical solutions to the corresponding boundary value problem. The effects of different physical parameters are studied and discussed in detail. The main findings of this investigation pertaining to different physical aspects are summarized in the conclusion. It is interesting to find that the present problem becomes unstable with specific choices of material parameters, thereby leading to an unstable solution by the direct solver. However, numerical experimentation suggests that an iterative solver based on the generalized minimum residual method can stabilize the numerical solutions. In this connection, results are shown for varying Hartmann numbers. Moreover, it is worth mentioning that FreeFEM++ provides an efficient platform to compute and analyze magnetic induced flow within the context of a higher order continuum.