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,
Aziz 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
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.
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.
June 24, 2026
A. Forero-Laiton,
Y. Sarmiento-Perdomo,
Y. Trujillo-Ladino
The aim of this article is to study and apply three numerical methods for solving elliptic partial differential equations: the finite element method (FEM), the multiscale finite element method (MsFEM), and the generalized multiscale finite element method (GMsFEM), with special emphasis on the latter. To this end, a theoretical review of their variational and matrix foundations is presented, followed by computational implementations for homogeneous, heterogeneous, and high-contrast diffusion problems. The simulations were carried out using FreeFem++ and MATLAB, which made it possible to compare the performance of the methods under different scenarios. The results show that the classical multiscale method has limitations when the problem involves highly heterogeneous media or non-separable scales, whereas the generalized multiscale method provides a better representation of local behavior through the construction of basis functions obtained from local spectral problems. It is concluded that this latter method is a more suitable alternative for the numerical approximation of complex multiscale problems, especially in high-contrast settings, where reducing degrees of freedom is required without significantly compromising approximation quality.
June 23, 2026
EUREKA: Physics and Engineering
Before corrigendum (First published version)Khaldjigitov, A., Djumayozov, U., Tilovov, O., Bobonazarov, A., Khudazarov, R., Pulatov, S. (2026). EXPRESSION OF CONCERN AND CORRIGENDUM. Development of thermoelasticity equations for strains. EUREKA: Physics and Engineering, 1, 181-194. https://doi.org/10.21303/2461-4262.2026.003924
After corrigendum (Corrected version)Khaldjigitov, A., Djumayozov, U., Tilovov, O., Bobonazarov, A., Khudazarov, R., Pulatov, S. (2026). CORRIGENDUM. Development of thermoelasticity equations for strains. EUREKA: Physics and Engineering, 1, 229–246. https://doi.org/10.21303/2461-4262.2026.004305
Corrigendum notification
CORRIGENDUM NOTIFICATION to Development of thermoelasticity equations for strains (2026). EUREKA: Physics and Engneering, 1, 247–248. https://doi.org/10.21303/2461-4262.2026.004308
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A revised version of the paper has been published Khaldjigitov, A., Djumayozov, U., Tilovov, O., Bobonazarov, A., Khudazarov, R., & Pulatov, S. (2026). CORRIGENDUM. Development of thermoelasticity equations for strains. EUREKA: Physics and Engineering, 1, 00–00. https://doi.org/10.21303/2461-4262.2026.004305, adding additional references, strengthening the discussion of compatibility conditions and stress-based formulations, as well as adding new numerical comparisons and convergence illustrations.
However, the editors would like to warn readers that in the paper the admissibility and equivalence of the additional boundary conditions imposed in the strain formulation remain insufficiently justified from a rigorous mathematical standpoint. The revised manuscript demonstrates that the proposed approach may be computationally workable and numerically promising for the considered benchmark problems, but it still falls short of providing a complete theoretical foundation establishing equivalence with classical thermoelasticity.
The formulations proposed in the paper remain the subject of further theoretical verification, in particular regarding the rigorous mathematical justification of the boundary conditions and the proof of equivalence with classical thermoelasticity.
The revised version clearly reflects the authors' considerable efforts to address the criticisms. The manuscript has been significantly expanded, additional references have been added, the discussion of compatibility conditions and stress-based formulations has been strengthened, and new numerical comparisons and illustrations of convergence have been added. However, despite these improvements, some of the major methodological problems identified in the initial peer review remain only partially resolved.
The authors correctly emphasize that strain-based formulations of elasticity and thermoelasticity are admissible within continuum mechanics and that similar conceptual difficulties also arise in stress-based formulations such as the Beltrami–Michell approach. The corrigendum paper provides a broader theoretical context and cites literature discussing the dependence and independence of compatibility equations, uniqueness issues, and alternative formulations in stresses and strains. This addition improves the scientific positioning of the work and weakens the earlier criticism that the proposed formulation is fundamentally incompatible with classical thermoelasticity.
The derivation of the generalized compatibility equation, including thermal terms, in the corrigendum paper presented more clearly and consistently. The authors explicitly state that temperature enters the formulation through the Duhamel-Neumann relations and demonstrate the sequence of substitutions leading to equation (10). From the standpoint of thermoelastic theory, this derivation is acceptable. The revised text therefore adequately addresses the earlier concern that thermal effects had been introduced into the compatibility equations in an inconsistent or nonphysical manner.
However, the central issue (namely the mathematical and physical admissibility of the additional boundary conditions) remains insufficiently resolved. The corrigendum paper continues to rely on equilibrium equations imposed on the boundary as additional boundary conditions required to close the strain-based system. Although the authors now cite prior works in which similar ideas were considered for stress-based formulations, this does not constitute a rigorous proof that the proposed boundary conditions are independent, sufficient, or fully equivalent to classical traction or displacement conditions. The response letter openly acknowledges that “a rigorous theoretical proof of equivalence to displacement-based boundary value problems requires a separate investigation and constitutes a subject of future research”. This statement is important because it effectively confirms that the principal mathematical concern has not yet been fully resolved.
From a rigorous continuum mechanics perspective, the problem is not merely whether equilibrium relations can formally be written on the boundary, but whether such conditions ensure a well-posed problem in the Hadamard sense, including existence, uniqueness, and continuous dependence on the prescribed data. The corrigendum paper still does not provide such analysis. No variational framework, functional setting, or proof of equivalence between the proposed strain formulation and the classical displacement formulation is established. Consequently, the mathematical status of Problems A and B remains partially heuristic.
The discussion concerning the overdetermined nature of the original system and the subsequent reduction procedure has also been improved but not completely resolved. The authors now explicitly discuss the issue of selecting independent compatibility equations and relate their approach to the Beltrami–Michell framework. Nevertheless, the corrigendum paper still lacks a rigorous proof that the reduced Poisson-type system obtained for Problem B is mathematically equivalent to the original compatibility formulation and does not eliminate physically relevant constraints. The derivation is formally plausible, but the equivalence remains assumed rather than demonstrated.
The numerical section has been substantially strengthened compared with the original version. The addition of comparisons with FreeFEM++ finite element results, convergence plots, mesh refinement comments, and quantitative discrepancy estimates represents a clear improvement. The agreement between the finite-difference formulation and the FEM solution is indeed reasonably good for the selected benchmark example. The convergence behavior shown in Fig. 3 also supports the internal numerical consistency of the implemented scheme.
At the same time, the presented validation still has limitations. The comparisons are restricted to a single relatively simple benchmark involving a rectangular plate with a prescribed sinusoidal temperature field. The study does not include analytical benchmark solutions, error norms, sensitivity analyses, or verification against cases with nontrivial mechanical loading or mixed boundary conditions. Moreover, the claim that strain-based formulations are “more accurate” than displacement-based approaches because displacement formulations require numerical differentiation is presented too categorically and without sufficient quantitative justification. Modern finite element methods formulated in displacements are capable of highly accurate stress recovery, and the issue is more subtle than stated in the manuscript. The current discussion tends to overgeneralize the limitations of displacement-based methods while emphasizing the advantages of the proposed approach without a sufficiently broad comparative basis.
Another important point is that the corrigendum paper occasionally mixes numerical evidence with mathematical validation. Numerical agreement between different discretization certainly supports the practical consistency of the method, but it cannot replace a proof of theoretical equivalence or well-posedness. Several statements in the conclusions continue to overstate the level of validation achieved. For example, assertions that the results “confirm the correctness” and “substantiate the validity” of the proposed thermoelastic equations should be formulated more cautiously, since the theoretical foundation of the additional boundary conditions remains unresolved.
In conclusion, the corrigendum paper should not be regarded as fundamentally incorrect, and the additional numerical results substantially strengthen the practical credibility of the approach.
However, the work still contains unresolved theoretical issues that limit the strength of its claims. The paper may therefore be considered an interesting and potentially valuable exploratory contribution to alternative thermoelastic formulations, but not yet a fully validated theoretical framework.
May 29, 2026
Х. Факих,
Н. Насреддин,
С. Мансур,
Р. Мгамес
В данной статье интерес для нас представляет комплексная версия уравнения Бертоцци-Эседоглу-Жилле-Кана-Хиллиарда для восстановления черно-белых изображений, а также многокомпонентные системы Кана-Хиллиарда для восстановления изображений, т.е. расширение подхода для восстановления цветных изображений. Мы изучили корректность стационарной задачи, связанной с комплексным уравнением Бертоцци-Эседоглу-Жилле-Кана-Хиллиарда, а также с системами Бертоцци-Эседоглу-Жилле-Кана-Хиллиарда. Затем рассматривалась неявная дискретизация Эйлера по времени в обеих упомянутых выше моделях. Нам удалось доказать устойчивость неявной схемы Эйлера. Были выполнены численные эксперименты, которые подтверждают теоретические результаты и показывают эффективность схемы. Эти эксперименты проводились с использованием FreeFem++.
In this article, we are interested in the complex version of Bertozzi-Esedoglu-Gillete-Cahn-Hilliard equation for grayscale image inpainting as well as the multi-component Cahn-Hilliard systems for image inpainting, that is an extension approach for color image inpainting. We have studied well-posedness of the steady state problem associated to the complex Bertozzi-Esedoglu-Gillete-Cahn-Hilliard equation as well as to Bertozzi-Esedoglu-Gillete-Cahn Hilliard systems. Then, Backward Euler discretization on time has been considered in both models mentioned above. We were able to prove the stability of the backward Euler scheme. Finally, we do some numerical simulations that confirm the theoretical results and show the efficiency of the scheme. The simulations were done using FreeFem++.
May 10, 2026
Zhen Song,
Yulong Liu,
Cheng Wan,
Chenjun Li,
Lingfu Liu,
Yunyi Li,
C. Yuan
Execution-based evaluation of LLM-generated code implicitly treats successful execution as a proxy for correctness. In scientific simulation, this proxy is insufficient: a generated input file can run, mesh, and converge while encoding governing equations that differ from the user's intent. We call this mismatch between intended physics and generated code the comprehension-generation gap. We instantiate this in MOOSE, where Kernel and BC objects map compositionally to weak-form residual terms, enabling deterministic reconstruction of the encoded PDE and comparison against an intended contract. We formalize this comparison as the Intent Fidelity Score (IFS), a structural metric covering governing terms, BCs, ICs, coefficients, and time scheme. Building on IFS, we develop a PDE-grounded refinement loop that uses deterministic violation reports to correct generated code iteratively. We evaluate on MooseBench, a 220-case multiphysics benchmark with PDE-level ground truth released with this work. On this benchmark, our method consistently improves mean IFS over direct generation, with gains concentrated on hard cases. On the subset where direct generation falls below IFS 0.7, refinement adds +0.22 to +0.41 absolute IFS. In the deployment audit, execution-only repair improves execution success while leaving 39-40% of all 220 cases runnable but still solving the wrong physics across the three main deployment-audit models, exposing executability and intent fidelity as separable failure modes. Static proof-of-concept experiments on four PDE-oriented DSLs (UFL/FEniCS, FreeFEM, FiPy, and Devito) suggest that the reconstruction-and-comparison pattern extends beyond MOOSE. These findings reinforce that executable simulation code should be verified against the mathematical structure it is intended to encode, not accepted on execution alone.
