July 02, 2026
А.А. Халджигитов,
А.А. Бобоназаров,
Р. 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.
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.
April 15, 2026
Khizar Hayat Khan,
Muhammad Sabeel Khan,
Nazim Hussain Hanjano,
Farxod Akmalovich Ixtiyarov,
Saeed Islam,
Amna,
Ghulam Khadija,
M. Bilal
The interplay between microrotation, microinertia, and viscoelastic relaxation significantly influences the hydrodynamic performance of non-Newtonian cavity flows. This study investigates the combined effects of the Eringen number, micropolar coupling constant, and fluid relaxation factor on drag force, microrotation, and flow behavior in a two-dimensional lid-driven cavity. The governing nonlinear partial differential equations are formulated using Eringen’s micropolar theory with relaxation-time-dependent stress, and the characteristic Galerkin finite element method is employed for numerical solution using a custom FreeFEM++ implementation. The results reveal a strong coupling between microrotation and viscoelastic relaxation effects. Increasing the relaxation factor enhances flow elasticity, leading to variations in drag force depending on the flow regime. The Eringen number stabilizes the flow and suppresses drag oscillations, while the micropolar coupling constant modulates shear-layer thickness and vortex intensity. Higher Eringen number increases kinetic energy and produces more ordered streamline patterns, whereas larger values of micropolar constant reduces drag while amplifying microrotation. Overall, tuning Eringen number, micropolar constant and time relaxation provides an effective mechanism for controlling drag and optimizing flow stability. These findings offer valuable insights for the design of advanced microfluidic and rheological systems where microstructural and viscoelastic effects coexist.
February 13, 2026
Caroline Pascal,
Pierre Marchand,
Alexandre Chapoutot,
Olivier Doaré
The identification and reconstruction of acoustic fields radiated by unknown structures is usually performed using either Sound Field Estimation or Near-field Acoustic Holography techniques. The latter turns out to be especially useful when data is only available close to the source, but information throughout the whole space is needed.
Yet, the lack of amendable and efficient implementations of state-of-the-art solutions, as well as the laborious and often lengthy deployment of acoustic measurements continue to be significant obstacles to the practical application of such methods.
The purpose of this work is to address both problems. First, a completely automated metrology setup is proposed, in which a robotic arm is used to gather extensive and accurately positioned acoustic data without any human intervention. The impact of the robot on acoustic pressure measurements is cautiously evaluated, and proved to remain limited below 1 kHz.
The Sound Field Estimation is then tackled using the Boundary Element Method, and implemented using the FreeFEM software. Numerically simulated measurements have allowed us to assess the method accuracy, which matches theoretically expected results and proves to remain robust against positioning inaccuracies, provided that the robot is carefully calibrated.
The overall solution has been successfully tested using actual robotized measurements of an unknown loudspeaker, with a reconstruction error of less than 30%.
December 31, 2025
Роман Николаевич Голых,
Александр Романович Барсуков,
С.Г. Ильясов,
Г.Т. Суханов,
Г.В. Пышнограй,
Л. Ф. Комарова,
А.Н. Блазнов,
Александр Григорьевич Овчаренко
Предложен подход к реализации метода конечных элементов (МКЭ) для решения многопараметрических и многоитерационных краевых задач. Для создания подхода проанализированы издержки алгоритма МКЭ, имеющиеся в стандартном программном пакете на примере FreeFEM++. Исследованы возможности снижения затрат на производительность за счет устранения дублирования промежуточных данных и, тем самым, уменьшения количества операций копирования, выделения и освобождения памяти. Разработан способ индексации и модифицированный алгоритм решения конечно-элементных задач. Результаты расчетов с использованием разработанного алгоритма были сопоставлены с аналитическим решением (относительное отклонение от аналитического решения составило менее 10−4). С помощью разработанной библиотеки достигнуто ускорение до 3-х раз и более без использования новых вычислительных ядер. Полученные результаты по ускорению расчётов могут быть применены применены при численном моделировании процессов массообмена в химико-технологических системах, где требуется многократное решение сопряжённых нестационарных задач переноса вещества.
An approach to implementing the finite element method (FEM) for solving multiparameter and multi-iteration boundary value problems is proposed. To develop this approach, the costs of the FEM algorithm available in the standard software package were analyzed using FreeFEM++ as an example. Possibilities of reducing performance costs by eliminating duplication of intermediate data, thereby reducing the number of copying, allocation, and deallocation operations of memory, were investigated. An indexing method and a modified algorithm for solving finite element problems were developed. The calculation results using the developed algorithm were compared with the analytical solution (the relative deviation from the analytical solution was less than 10−4). Using the developed library, an acceleration of up to 3 times or more was achieved without the use of new computing cores. The obtained results in accelerating the calculations can be applied in the numerical modeling of mass transfer processes in chemical engineering systems, where multiple solutions of conjugate non-stationary mass transfer problems are required.
December 26, 2025
Á. Arós,
Ariel L. Lombardi,
L. Venturato
We consider a model for Koiter linear elastic elliptic shells in contact with a deformable obstacle and we study the convergence of the solution of this model towards the solution of the corresponding model for elastic elliptic membrane shells when the small parameter of the model (thickness) tends to zero. Furthermore, we propose a numerical scheme for this kind of contact problems for Koiter shells and show numerical simulations after implementation by using the free software package FreeFem++.
November 30, 2025
Д. Н. Романов,
М.В. Урев
В данной работе на примере уравнения Пуассона рассматриваются вопросы численного решения методом конечных элементов однородной краевой задачи Дирихле для эллиптического уравнения в двумерной многоугольной выпуклой области Ω с сингулярной правой частью в виде дельта-функции Дирака. Доказана теорема существования и единственности обобщенного решения в дробном гильбертовом пространстве Соболева Hs(Ω), где 1/2 < s < 1. Предложен и изучен подход к дискретному анализу задачи методом конечных элементов. Приведены результаты численных экспериментов по решению методической задачи с помощью пакета FreeFem++, подтверждающие полученную оценку уклонения дискретного решения от точного.
A numerical solution by the finite element method of a homogeneous Dirichlet boundary value problem for an elliptic equation is examined (using a Poisson equation as an example) in a two-dimensional convex polygonal domain Ω with a singular right-hand side given by the Dirac delta function. A theorem on the existence and uniqueness of a generalized solution in the fractional Sobolev space Hs(Ω), 1/2 < s < 1, is proved. An approach to discrete analysis of the problem using the finite element method is proposed and investigated. The results of numerical experiments for a model problem, obtained using the FreeFem++ software, are presented. They confirm the error estimate of the difference between the discrete and exact solutions derived in the paper.
