October 16, 2026
G. Krivovichev,
Dana A. Naumova
The paper is devoted to the construction and analysis of a projection-difference scheme for numerical simulation of blood flow in vascular systems. An implicit scheme is proposed for a hyperbolic system obtained by averaging the hydrodynamic equations over the vessel crosssection. The unconditional stability of the proposed scheme is demonstrated. For practical implementation, a method of splitting with respect to physical processes is proposed. The first stage of the method involves computation of the deformed configuration of the vessel for a given flow velocity. The second stage involves computation of the fluid velocity in the deformed vessel. Riemann invariants are used to obtain the values at the boundaries. The FreeFEM++ package is used for the software development. As a result of computations for the scheme with piecewise-linear nodal functions, it is demonstrated that the convergence with second order takes place. As a special case, a scheme with weights is considered. It is demonstrated, that the variations of the weight parameter lead to monotonization of the numerical solutions.
October 16, 2026
Bing-Zhou Chen,
Amelia Deakin,
Andy Adler,
P. Ledger,
William R. B. Lionheart
Objective. Medical electrical impedance tomography systems prescribe electrode currents. Accurate voltages, even on passive electrodes, require the complete electrode model (CEM), the most accurate model known for skin electrodes. Its current constraints specify total electrode currents, whereas standard finite element boundary interfaces impose local conditions. We describe how to implement the CEM through Robin boundary conditions, which are available in standard FEM packages. Approach. Prescribed electrode potentials give Robin problems whose unit-potential responses define an electrode transadmittance matrix. We prove equivalence to the conventional augmented CEM and formalise the discrete energy, equivalence and reference arguments in Lean 4. Numerical comparisons use two-dimensional P1 finite elements in DOLFINx, FreeFEM, GetFEM, MFEM and NGSolve, with the same nodal reference potential as EIDORS. Main results. For identical assembled blocks, the two procedures have the same solution set whenever the Robin matrix is invertible. Across the five packages in dimensionless test cases, maximum absolute electrode-potential differences from EIDORS were below $5\times10^{-13}$ (2692 nodes), and errors against an exact rational-arithmetic reference were below $1.1\times10^{-13}$ (2535 nodes). Significance. Standard Robin solves and consistent electrode-current integration provide a portable route to the current-driven CEM, with current constraints enforced through a small electrode system.
October 13, 2026
A. Bobonazarov,
A. Khaldjigitov,
U. Djumayozov,
С. П. Пулатов,
Maftuna N. Abdirakhmonova,
Yulduz Sh. Narmatova
This study considers a two-dimensional dynamic problem of linear elasticity formulated directly in terms of the stress tensor components. A coupled system of differential equations for two normal stress components and one shear stress component is derived from the equations of motion, kinematic relations, and Hooke’s law. Initial and traction boundary conditions are specified for the resulting system. Spatial discretization is performed using the Galerkin method with linear triangular finite elements. Element mass and stiffness matrices accounting for the coupling between the stress components are derived. Time integration is implemented using the Newmark method in a C++ program. An independent computational implementation is developed in FreeFEM++ using an implicit finite-difference approximation of the time derivatives. The sensitivity of the solution to the finite element size and time step is investigated. Successive refinement of the spatial and temporal discretizations leads to stabilization of the computed stresses. The method is validated using two benchmark problems. For a rectangular domain subjected to a time-dependent surface load, the stress histories computed in C++ and FreeFEM++ are in agreement. For transient wave propagation from a circular cavity, the results are compared with an analytical solution, the QL boundary element method, and the classical displacement-based finite element formulation. The minimum circumferential stress is −1.2250, compared with the analytical value of −1.2253, corresponding to a difference of approximately 0.025%. The direct stress formulation eliminates the need to recover stresses by differentiating the displacement field and yields a smoother time history for the discretization considered. The results confirm the applicability of the developed finite element scheme to dynamic elasticity problems formulated in terms of stresses.
September 24, 2026
Vigneswaran Ravi,
Sophias Boko,
C. Coillot,
S. Nanot
The Hall effect, which occurs in metals and semiconductors, is used in numerous applications, particularly in the fabrication of magnetic field sensors. This phenomenon is influenced by the physical properties of the material (electron mobility, doping), the geometry of the device, and the size and position of the metallic contacts. The analytical model has limitations in accurately capturing all these phenomena. Therefore, in this work, we propose a finite element model using FreeFEM++ to accurately model the Hall effect. The validity of this model is assessed by comparing it to analytical ones. The effect of physical defects on the electrical measurements, which cannot be analytically calculated, is explored to illustrate the usefulness of this model under realistic conditions. These simulations can be used both as teaching tools, to illustrate this physical effect and as research tools, as demonstrated in the case of a graphene-based transistor. The reported simulation script is openly shared.
September 21, 2026
Romain Gervalle,
Jorge F. M. Delgado,
E. dos Santos Costa,
C. Herdeiro,
Víctor Jaramillo,
Raimon Luna,
Eugen Radu
Boson stars are self-gravitating solitons of the Einstein-Klein-Gordon equations for a massive complex scalar field, and arguably the simplest horizonless compact objects arising in General Relativity. Beyond spherical symmetry, their construction requires solving a system of nonlinear elliptic partial differential equations, a difficult task per se and that each group has traditionally addressed with its own numerical implementation. A systematic comparison of different methods applied to the same physical model has so far been lacking. In this work, we carry out such a comparison for the two simplest non-spherical configurations: rotating boson stars and static dipolar boson stars. We employ three independent codes, based respectively on finite differences (FIDISOL/CADSOL), finite elements (FreeFem), and spectral methods (Kadath), and assess their accuracy through different diagnostics. The three solvers agree on the global observables to typically eight significant digits. We further compare the runtime of each code on identical hardware and provide precise benchmark values of the relevant physical quantities, intended to serve as reference data for future numerical studies of boson stars. As a further proof of concept, we also present the use of physics-informed neural networks to construct a rotating Q-ball solution.
September 19, 2026
A. Khaldjigitov,
A. Bobonazarov,
U. Djumayozov,
O. Tilovov,
С. П. Пулатов,
Maftuna N. Abdirakhmonova,
Fazilat S. Ochilova
This paper proposes a finite element approach to the numerical solution of boundary value problems of linear elasticity theory formulated directly in terms of the stress tensor components. In contrast to the classical finite element formulation, in which displacements are the primary unknowns, the approach considered here treats the stress components as the sought quantities. Two forms of the boundary value problem are investigated. The first is based on the joint use of the equilibrium equations and the Beltrami–Michell equations, while the second is a transformed system of Poisson-type equations for the stress tensor components. Variational relations based on the Galerkin method are obtained for both formulations. Using linear basis functions on triangular finite elements, local matrices are constructed, and global systems of algebraic equations are formed, whose unknowns are the nodal values of the stresses. The numerical implementation of the developed schemes is carried out in an in-house C++ program and in the FreeFEM++ software environment. To verify the reliability of the proposed mathematical and numerical models, the classical Kirsch problem of a stretched elastic plate with a circular hole—characterized by a pronounced stress concentration near the edge of the hole—is solved. The numerical values of the stress components are compared with the analytical solution obtained using the Airy stress function method, as well as with the results of calculations performed with the FreeFEM++ software package. The comparison shows good agreement between the obtained solutions and confirms the possibility of determining stresses directly, without first computing the displacement field. A mesh-refinement study further showed a systematic reduction in the numerical error: on the finest mesh considered, the relative error decreased to 3.189% for formulation A and to 7.415% for formulation B. The proposed approach extends the applicability of the finite element method to boundary value problems of elasticity theory formulated in terms of stresses and can be used to study problems with complex geometry and local stress concentration.
August 17, 2026
A. Khaldjigitov,
U. Djumayozov,
A. Kalandarov,
A. Bobonazarov,
O. Tilovov,
Robiya Rakhmonova,
Zebo Khasanova
Boundary value problems in the theory of elasticity are traditionally formulated in terms of displacements, while stresses and strains are calculated from the displacements. Usually, the calculation of stresses and strains is accompanied by approximation errors. In this paper, alternative formulations of elasticity problems in stresses and strains are developed. It is shown that, to formulate boundary value problems (BVP) in stresses, it suffices to consider the three off-diagonal Beltrami–Michell equations together with the three equilibrium equations. Similarly, the off-diagonal strain compatibility equations are considered in conjunction with the three equilibrium equations expressed in terms of strains. In addition, the Beltrami–Michell equations and the strain compatibility equations are reduced to the Poisson equations for the stress and strain tensors, respectively. For the numerical solution of the formulated BVP in stresses and strains, the finite difference method and the finite element method within the FreeFEM++ framework were applied. A comparison of numerical results for the problems of stretching a rectangular plate under a parabolic load and of a dam subjected to hydrostatic pressure and self-weight with known solutions demonstrates good agreement, thereby confirming the validity of the proposed boundary value problems regarding stresses and strains.
July 02, 2026
А. А. Халджигитов,
A. Bobonazarov,
Р. А. Рахмонова
The paper addresses the formulation of plane elasticity problems in terms of stresses and their numerical solution by the Galerkin finite element method. Based on the equilibrium equations, geometric relations, and physical laws of elasticity, a system of differential equations describing the plane stress state of an elastic medium is derived. The Galerkin method yields the weak form of the boundary value problem and a discrete model suitable for computer implementation. The bilinear forms and local finite element matrices required to assemble the global algebraic system are constructed. The algorithms are implemented in C++ and FreeFEM++. The approach is verified on the classical Kirsch problem of stress distribution around a circular hole in an elastic plate: both implementations agree well with each other and with known analytical solutions, confirming the correctness of the formulation and the applicability of the method.
July 01, 2026
I. Monsivais,
J. Lizardi,
F. Méndez,
Edgar A. Ramos
In the present work, a numerical study of the incompressible laminar flow of a Newtonian fluid circulating through a rectangular microchannel of parallel plates with permeable walls is carried out considering slip conditions. For this purpose, the corresponding governing equations, i.e., the mass and momentum conservation equations, are solved using the finite element technique with the free software FreeFEM++ to analyze the hydrodynamics of the flow under consideration, obtaining the velocity and pressure profiles. The main results show that by increasing the dimensionless filtration parameter \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta$$\end{document}, the transverse velocity increases, which causes the longitudinal velocity to decrease due to mass conservation. This behavior is maintained even considering the influence of the dimensionless slip parameter \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\delta$$\end{document}, which is also reflected in the volumetric flow rate, since as this parameter increases, the volumetric flow rate in the longitudinal direction is slightly enhanced.
July 01, 2026
Yong-Xing Wang
We present a comprehensive review of finite element methods (FEM) for solving fluid–structure interaction (FSI) problems. Our aim is to systematically categorise and compare these methods based on three key aspects: mesh type, coupling strategy, and the choice of solved variables. For each method, we highlight its respective advantages and limitations. In addition, we formulate and compare the finite element weak forms of the various approaches, employing Newton’s method to linearise nonlinear terms at the continuous functional level. We implement six FSI methods in the open-source software package FreeFEM++, and compare these methods based on three benchmarks.
