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Mathematics
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Mathematical Modeling and Numerical Analysis with Applications in Various Fields
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Mathematical Modeling and Numerical Analysis with Applications in Various Fields

A special issue of Mathematics (ISSN 2227-7390). This special issue belongs to the section "E: Applied Mathematics".

Deadline for manuscript submissions: 31 January 2026 | Viewed by 6246

Special Issue Editor

Dr. Irina G. Nizovtseva

E-Mail Website
Guest Editor
Otto Schott Institute of Materials Research, Friedrich Schiller University, 07743 Jena, Germany
Interests: mathematical modeling; structural-phase transformations; phase-field modeling; intelligent control systems; analytical and numerical solutions; bioreactors

Special Issue Information

Dear Colleagues,

Lately, an increasing demand for the scientific community to obtain effective mathematical tools for describing and solving problems in biotechnology, medicine, engineering, and physical chemistry has been observed. It is worth mentioning that finding methodology approaches for solving the aforementioned problems demands both the proper use of a developed mathematical apparatus, including analytical solutions of partial differential equations and numerical analysis approaches, but also the use of modern advances in computational, high-performance, and artificial intelligence tools. We cordially invite authors to share their recent contributions and new results within the scope of this Special Issue, focusing on a mathematical formulation of a problem and corresponding solutions. Inspiration can be drawn from various areas, including but not limited to the broad range of physical, chemical, and biology sciences, as well as engineering disciplines and medicine.

Dr. Irina G. Nizovtseva
Guest Editor

Manuscript Submission Information

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Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 2600 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

Keywords

  • mathematical modeling
  • analytical solutions
  • numerical analysis
  • aproximation
  • multiscale problems
  • stability
  • convergence
  • complexity

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Published Papers (7 papers)

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Research

12 pages, 3571 KiB  
Article
Frequency-Based Finite Element Updating Method for Physics-Based Digital Twin
by Youngjae Jeon, Geomji Choi, Kwanghyun Ahn, Kang-Heon Lee and Seongmin Chang
Mathematics 2025, 13(5), 738; https://doi.org/10.3390/math13050738 - 24 Feb 2025
Viewed by 464
Abstract
This study proposes a frequency-based finite element updating method for an effective physics-based digital twin (DT). One approach to constructing a physics-based DT is to develop a mechanics-based mathematical model that accurately simulates the behavior of an actual structure. The proposed method utilizes [...] Read more.
This study proposes a frequency-based finite element updating method for an effective physics-based digital twin (DT). One approach to constructing a physics-based DT is to develop a mechanics-based mathematical model that accurately simulates the behavior of an actual structure. The proposed method utilizes finite element updating, adjusting model parameters to improve model accuracy. Unlike simple modal analysis, which focuses on vibration characteristics, this method recognizes that accurate dynamic transient-based vibration analysis requires considering the damping effect, as well as mass and stiffness, during the updating process. Moreover, a frequency-based analysis is employed instead of the computationally expensive time-based analysis for more efficient dynamic modeling. By transforming data into the frequency domain, the method efficiently represents dynamic behavior within relevant frequency ranges. We further enhance the computational efficiency using the model reduction technique. To validate the method’s accuracy and efficiency, we compare the analysis results and computation time using a numerical example of the control rod drive mechanism. The proposed method shows significantly reduced computation time, by a factor of 8.9 compared to conventional time-based methods, while preserving high accuracy. Therefore, the proposed method can effectively support the development of physics-based DTs. Full article
(This article belongs to the Special Issue Mathematical Modeling and Numerical Analysis with Applications in Various Fields)
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18 pages, 776 KiB  
Article
Mathematical Modeling of High-Energy Shaker Mill Process with Lumped Parameter Approach for One-Dimensional Oscillatory Ball Motion with Collisional Heat Generation
by Kwon Joong Son
Mathematics 2025, 13(3), 446; https://doi.org/10.3390/math13030446 - 28 Jan 2025
Viewed by 626
Abstract
This study presents an advanced mathematical model for the high-energy shaker mill process, incorporating thermal interactions among the milling ball, shaker mill vial, and the air contained within. Unlike previous models focusing solely on the ball’s temperature, this research emphasizes the heat produced [...] Read more.
This study presents an advanced mathematical model for the high-energy shaker mill process, incorporating thermal interactions among the milling ball, shaker mill vial, and the air contained within. Unlike previous models focusing solely on the ball’s temperature, this research emphasizes the heat produced by impacts and the thermal exchange among all three components. Incorporating these thermal interactions allows the model to provide a more comprehensive depiction of the energy dynamics within the system, leading to more precise predictions of temperature changes. Utilizing a lumped parameter method, the study simplifies complex airflow dynamics and non-uniform temperature distributions in the milling system, enabling efficient numerical analysis. Hamilton’s equations are extended to include supplementary state variables that account for the internal energies of both the vial and the air, in addition to the thermomechanical state variables of the ball. High-energy milling techniques are essential in mechanochemical synthesis and various industrial applications, where the optimization of heat transfer and energy efficiency is crucial. Numerical simulations computed using the Bogacki–Shampine integration algorithm significantly align with experimental data, confirming the model’s accuracy. This comprehensive framework enhances understanding of heat transfer in one-dimensional ball motion, optimizing milling parameters for better performance. The mathematical model facilitates the computation of heat production due to collisions, based on operational parameters like shaking frequency and amplitude, thereby allowing for the anticipation of chemical reaction activation potential in mechanochemistry. Full article
(This article belongs to the Special Issue Mathematical Modeling and Numerical Analysis with Applications in Various Fields)
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23 pages, 6226 KiB  
Article
Three-Dimensional Modeling of the Behavior of a Blood Clot Using Different Mechanical Properties of a Blood Vessel
by Mantas Brusokas, Sergejus Borodinas and Raimondas Jasevičius
Mathematics 2025, 13(2), 285; https://doi.org/10.3390/math13020285 - 17 Jan 2025
Viewed by 740
Abstract
In this work, the behavior of a 3D blood clot located inside a vein under the influence of the mechanical effect of blood flow was analyzed. It has been observed that the mechanical properties of the blood vessel play an important role in [...] Read more.
In this work, the behavior of a 3D blood clot located inside a vein under the influence of the mechanical effect of blood flow was analyzed. It has been observed that the mechanical properties of the blood vessel play an important role in the behavior of a blood clot. When the blood vessel changes its shape and/or diameter over time, the position and orientation of the clot in space and time is not constant, and consequently, it influences the blood flow. Moreover, the changed lumen of the blood vessel has a direct impact on the blood velocity, and thus the pressure is exerted not only on the blood vessel wall but also on the thrombus itself. Under these different conditions, it is important to understand the behavior of the blood clot, where each factor with a mechanical influence could potentially lead to clot detachment. Therefore, several variants of numerical simulations were analyzed, including models with different blood vessel properties, considering when the blood vessel wall has (flexible) or does not have (fixed) elastic properties. The results show the blood flow velocity, vessel wall, and blood clot deformations and/or stresses using different vessel wall rigidity levels as well as different blood clot viscoelasticity parameters. Full article
(This article belongs to the Special Issue Mathematical Modeling and Numerical Analysis with Applications in Various Fields)
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14 pages, 1881 KiB  
Article
Novel Framework for Artificial Bubble Image Generation and Boundary Detection Using Superformula Regression and Computer Vision Techniques
by Pavel Mikushin, Nickolay Martynenko, Irina Nizovtseva, Ksenia Makhaeva, Margarita Nikishina, Dmitrii Chernushkin, Sergey Lezhnin and Ilya Starodumov
Mathematics 2025, 13(1), 127; https://doi.org/10.3390/math13010127 - 31 Dec 2024
Viewed by 826
Abstract
Bubble multiphase systems are crucial in industries such as biotechnology, medicine, oil and gas, and water treatment. Optical data analysis provides critical insights into bubble characteristics, such as the shape and size, complementing physical sensor data. Existing detection techniques rely on classical computer [...] Read more.
Bubble multiphase systems are crucial in industries such as biotechnology, medicine, oil and gas, and water treatment. Optical data analysis provides critical insights into bubble characteristics, such as the shape and size, complementing physical sensor data. Existing detection techniques rely on classical computer vision algorithms and neural network models. While neural networks achieve a higher accuracy, they require extensive annotated datasets, and classical methods often struggle with complex systems due to their lower accuracy. This study proposes a novel framework to address these limitations. Using Superformula parameter regression, we introduce an advanced border detection method for accurately identifying gas inclusions and complex-shaped objects in multiphase environments. The framework also includes a new approach for generating realistic artificial bubble images based on physical flow conditions, leveraging the Superformula to create extensive, labeled datasets without manual annotation. Tested on real bubble flows in mass transfer equipment, the algorithms enable bubble classification by shape and size, enhance detection accuracy, and reduce development time for neural network solutions. This work provides a robust method for object detection and dataset generation in multiphase systems, paving the way for more precise modeling and analysis. Full article
(This article belongs to the Special Issue Mathematical Modeling and Numerical Analysis with Applications in Various Fields)
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23 pages, 10308 KiB  
Article
Dynamic Condensation-Based Reduction Method for Precise Broadband Frequency Analysis
by Geomji Choi, Juhwan Lee and Seongmin Chang
Mathematics 2025, 13(1), 81; https://doi.org/10.3390/math13010081 - 28 Dec 2024
Viewed by 706
Abstract
In this paper, we propose a degree-of-freedom-based adaptive reduction method that ensures accuracy over a wide band. In the conventional dynamic condensation method, a single reduced model consisting of low-order modes is used throughout the analysis. This results in low accuracy in the [...] Read more.
In this paper, we propose a degree-of-freedom-based adaptive reduction method that ensures accuracy over a wide band. In the conventional dynamic condensation method, a single reduced model consisting of low-order modes is used throughout the analysis. This results in low accuracy in the high-frequency band because it does not reflect the characteristics of the frequencies. To address this issue, we implemented a reduced model for each frequency using a Taylor series. This method converts the transformation matrix into a frequency-independent form, which allows for a simple interpolation of the reduction model by updating the differences between frequencies. Numerical examples were adopted to examine the accuracy and efficiency of the proposed method. Full article
(This article belongs to the Special Issue Mathematical Modeling and Numerical Analysis with Applications in Various Fields)
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15 pages, 777 KiB  
Article
The Role of a Two-Phase Region in Directional Crystallization of Binary Liquids
by Dmitri V. Alexandrov, Irina V. Alexandrova, Alexander A. Ivanov and Liubov V. Toropova
Mathematics 2024, 12(14), 2178; https://doi.org/10.3390/math12142178 - 11 Jul 2024
Cited by 2 | Viewed by 1027
Abstract
Motivated by the widespread occurrence of directional crystallization in nature, laboratory experiments and industrial facilities, we consider how a two-phase (mushy) region filled simultaneously with liquid and solid material influences the process and changes the solute concentration in both the phases. A mushy [...] Read more.
Motivated by the widespread occurrence of directional crystallization in nature, laboratory experiments and industrial facilities, we consider how a two-phase (mushy) region filled simultaneously with liquid and solid material influences the process and changes the solute concentration in both the phases. A mushy layer arising as a result of constitutional supercooling in binary liquids drastically changes all process parameters in comparison with the frequently used approximation of a macroscopically planar phase interface. The heat and mass transfer problem with a moving mushy region is replaced by the equivalent model with a discontinuity interface that divides the liquid and solid phases and inherits the properties of a mushy layer. Analytical solutions that describe both crystallization modes with a planar phase interface and discontinuity interface (representing a mushy layer) are constructed for the steady-state and self-similar conditions. The switching time of the crystallization model with a planar phase interface to the model with a two-phase layer is determined. Our calculations, based on analytical solutions, show that the presence of a mushy layer can change the solute concentration in liquid and solid phases to a few tens of percent as compared to the planar interface model. This explains the importance of accounting for the two-phase region when describing the crystallization of supercooled binary liquids. Full article
(This article belongs to the Special Issue Mathematical Modeling and Numerical Analysis with Applications in Various Fields)
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11 pages, 2544 KiB  
Article
A Shape-Preserving Variational Spline Approximation Problem for Hole Filling in Generalized Offset Surfaces
by Abdelouahed Kouibia, Miguel Pasadas and Loubna Omri
Mathematics 2024, 12(11), 1736; https://doi.org/10.3390/math12111736 - 3 Jun 2024
Viewed by 938
Abstract
In the study of some real cases, it is possible to encounter well-defined geometric conditions, of an industrial or design type—for example, the case of a specific volume within each of several holes. In most of these cases, it is recommended to fulfil [...] Read more.
In the study of some real cases, it is possible to encounter well-defined geometric conditions, of an industrial or design type—for example, the case of a specific volume within each of several holes. In most of these cases, it is recommended to fulfil a function defined in a domain in which information is missing in one or more sub-domains (holes) of the global set, where the function data are not known. The problem of filling holes or completing a surface in three dimensions appears in many fields of computing, such as computer-aided geometric design (CAGD). A method to solve the shape-preserving variational spline approximation problem for hole filling in generalized offset surfaces is presented. The existence and uniqueness of the solution of the studied method are established, as well as the computation, and certain convergence results are analyzed. A graphic and numerical example complete this study to demonstrate the effectiveness of the presented method. This manuscript presents the resolution of a complicated problem due to the study of some criteria that can be traduced via an approximation problem related to generalized offset surfaces with holes and also the preservation of the shape of such surfaces. Full article
(This article belongs to the Special Issue Mathematical Modeling and Numerical Analysis with Applications in Various Fields)
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