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.
November 10, 2025
Тетяна Анатоліївна Олійник,
Людмила Василівна Скляр,
Дмитро Олександрович Румницький
На прикладі визначення швидкості вільного та стисненого падіння частинок встановлена можливість використання методів обчислювальної гідродинаміки для моделювання процесу осадження мінеральних частинок у рідині та визначення швидкості їх падіння. Показано, що ефективність розділення у спіральних сепараторах суттєво залежить від взаємозв’язку розміру та густини частинок. Збільшення діаметра частинок зменшує вплив турбулентних коливань потоку, що сприяє стабілізації їх траєкторій і забезпечує більш чіткий поділ фракцій. Проведене чисельне моделювання із застосуванням методу скінченних елементів (FEM) у середовищі FreeFEM++ дало змогу визначити гідродинамічні та кінематичні характеристики потоку, які можуть бути використані для оптимізації конструктивних параметрів та режимів роботи спіральних сепараторів задля підвищення селективності процесу гравітаційного збагачення магнетитових руд. Побудова тривимірної моделі потоку відкрила можливості для детального аналізу зон турбулентності, локального збагачення та розподілу концентрацій по ширині каналу. Такі дослідження можуть бути розширені у програмних комплексах OpenFOAM або COMSOL, що підтримують обмін даними з FreeFEM++ через формат mesh-файлів. Результати моделювання були підтверджені фізичними експериментами на спіральному сепараторі СВШ-2-1000, який має параболічну геометрію поперечного перерізу, та сепараторі лабораторного масштабу з профілем поперечного перерізу у вигляді кривої слабко похилої. Лабораторний сепаратор забезпечує отримання концентрату з вищими показниками виходу, вмісту та вилучення магнетиту (до 85,4 %). Втрати магнетиту з хвостами становили 12,5 % для лабораторного апарата СВШ-2-1000 і не перевищували 3 %. Отримані результати свідчать про високу узгодженість чисельних розрахунків і експериментальних даних. Підтверджують перспективність використання обчислювальної гідродинаміки для дослідження та оптимізації процесів гравітаційного збагачення у спіральних сепараторах.
October 30, 2025
Phi Hùng Phạm,
Thị Tuyết Nhung Mai,
Thị Thanh Mơ Tạ
Trong bài báo này chúng tôi trình bày phương pháp phần tử hữu hạn không-thời gian (space-time finite element method) giải xấp xỉ phương trình phản ứng khuếch tán. Khác với các phương pháp số truyền thống phân tách xấp xỉ miền thời gian và không gian riêng biệt, phương pháp này rời rạc đồng thời miền Q = Ω × (0, T ) trên cùng một cấu trúc lưới, giúp tối ưu chi phí tính toán và nâng cao độ chính xác của nghiệm xấp xỉ. Cách tiếp cận của phương pháp là đưa bài toán ban đầu về dạng biến phân (bài toán yếu). Tính đặt chỉnh của bài toán yếu được chứng minh thông qua việc áp dụng định lí Banach–Nečas–Babuška, đảm bảo sự tồn tại, duy nhất nghiệm. Phân tích đánh giá sai số tiên nghiệm (a priori error estimates) cho thấy phương pháp đạt tốc độ hội tụ tối ưu trong không gian tương ứng. Tính hiệu quả và độ chính xác của phương pháp được kiểm chứng qua các thí nghiệm số được xây dựng trên phần mềm mã nguồn mở FreeFEM++.
October 08, 2025
J. Villa‐Morales
We study a stochastic differential model for the dynamics of institutional corruption, extending a deterministic three-variable system—corruption perception, proportion of sanctioned acts, and policy laxity—by incorporating Gaussian perturbations into key parameters. We prove global existence and uniqueness of solutions in the physically relevant domain, and we analyze the linearization around the asymptotically stable equilibrium of the deterministic system. Explicit mean square bounds for the linearized process are derived in terms of the spectral properties of a symmetric matrix, providing insight into the temporal validity of the linear approximation. To investigate global behavior, we relate the first exit time from the domain of interest to backward Kolmogorov equations and numerically solve the associated elliptic and parabolic PDEs with FreeFEM, obtaining estimates of expectations and survival probabilities. An application to the case of Mexico highlights nontrivial effects: while the spectral structure governs local stability, institutional volatility can non-monotonically accelerate global exit, showing that highly reactive interventions without effective sanctions increase uncertainty. Policy implications and possible extensions are discussed.
September 23, 2025
Pascal Ventura,
F. Hecht,
Michel Potier-Ferry,
H. Zahrouni,
Fan Xu,
Hamza Azzayani,
Michael Brun,
Anh-Khoa Chau
The main purposes of the present paper are to present the mathematical and algorithmic aspects of the ANM/FEM numerical model and to show how it is applied to analyze elastic and thermo-elastic nonlinear solid mechanical problems. ANM is a robust continuation method based on a perturbation technique for solving nonlinear problems dependent on a loading parameter. Historically, this technique has been successfully applied to problems in various fields of solid and fluid mechanics. This paper shows how ANM is used to solve nonlinear elastic and nonlinear thermo-elastic problems involving elastic behavior and geometrical nonlinearities. The implementation of ANM for FEM in the FreeFEM++ language is then presented. The FEM software development platform, called FreeFEM++, is structured to work with variational formulations and, therefore, is well adapted to implement ANM for instability problems in solid mechanics. In order to illustrate the great efficiency of FreeFEM++, scripts will be presented for computing the different steps of ANM continuation for solid elastic structures, considering simple geometries subjected to conservative loading. For the purpose of validation, the problem of a cantilever subjected to an applied force is presented. Next, the new numerical model is applied to study wrinkles appearing in a planar film/substrate system that is subjected to compressive surface forces at the lateral faces of the film. Finally, the model is applied to a spherical film/substrate system subjected to thermo-elastic shrinkage. In both cases, the ANM/FEM prediction method, together with a Newton–Riks correction (if needed), identifies the equilibrium paths efficiently, especially after the post-buckling regime.
September 22, 2025
Christopher M. Douglas,
P. Jolivet
Nonlinear PDEs give rise to complex dynamics that are often difficult to analyze in state space due to their relatively large numbers of degrees of freedom, ill-conditioned operators, and changing spatial and parameter resolution requirements. This work introduces ff-bifbox: a new open-source toolbox for performing numerical branch tracing, stability/bifurcation analysis, resolvent analysis, and time integration of large, time-dependent nonlinear PDEs discretized on adaptively refined meshes in two and three spatial dimensions. Spatial discretization is handled using finite elements in FreeFEM, with the discretized operators manipulated in a distributed framework via PETSc. Following a summary of the underlying theory and numerics, results from three examples are presented to validate the implementation and demonstrate its capabilities. The considered examples, which are provided with runnable ff-bifbox code, include: a 3-D Brusselator system, a 3-D plate buckling system, and a 2-D compressible Navier--Stokes system. In addition to reproducing results from prior studies, novel results are presented for each system.
