Mathematical and Computational Applications

37 pages, 7671 KiB

Open AccessFeature PaperEditor’s ChoiceArticle

FE² Computations with Deep Neural Networks: Algorithmic Structure, Data Generation, and Implementation

by Hamidreza Eivazi, Jendrik-Alexander Tröger, Stefan Wittek, Stefan Hartmann and Andreas Rausch

Math. Comput. Appl. 2023, 28(4), 91; https://doi.org/10.3390/mca28040091 - 16 Aug 2023

Cited by 5 | Viewed by 2529

Multiscale FE

^{2}

computations enable the consideration of the micro-mechanical material structure in macroscopical simulations. However, these computations are very time-consuming because of numerous evaluations of a representative volume element, which represents the microstructure. In contrast, neural networks as machine learning methods are [...] Read more.

Multiscale FE

^{2}

computations enable the consideration of the micro-mechanical material structure in macroscopical simulations. However, these computations are very time-consuming because of numerous evaluations of a representative volume element, which represents the microstructure. In contrast, neural networks as machine learning methods are very fast to evaluate once they are trained. Even the DNN-FE

^{2}

approach is currently a known procedure, where deep neural networks (DNNs) are applied as a surrogate model of the representative volume element. In this contribution, however, a clear description of the algorithmic FE

^{2}

structure and the particular integration of deep neural networks are explained in detail. This comprises a suitable training strategy, where particular knowledge of the material behavior is considered to reduce the required amount of training data, a study of the amount of training data required for reliable FE

^{2}

simulations with special focus on the errors compared to conventional FE

^{2}

simulations, and the implementation aspect to gain considerable speed-up. As it is known, the Sobolev training and automatic differentiation increase data efficiency, prediction accuracy and speed-up in comparison to using two different neural networks for stress and tangent matrix prediction. To gain a significant speed-up of the FE

^{2}

computations, an efficient implementation of the trained neural network in a finite element code is provided. This is achieved by drawing on state-of-the-art high-performance computing libraries and just-in-time compilation yielding a maximum speed-up of a factor of more than 5000 compared to a reference FE

^{2}

computation. Moreover, the deep neural network surrogate model is able to overcome load-step size limitations of the RVE computations in step-size controlled computations. Full article

(This article belongs to the Collection Feature Papers in Mathematical and Computational Applications 2023)

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16 pages, 3814 KiB

Open AccessArticle

A Computational Fluid Dynamics-Based Model for Assessing Rupture Risk in Cerebral Arteries with Varying Aneurysm Sizes

by Rohan Singla, Shubham Gupta and Arnab Chanda

Math. Comput. Appl. 2023, 28(4), 90; https://doi.org/10.3390/mca28040090 - 2 Aug 2023

Cited by 1 | Viewed by 2041

Abstract

A cerebral aneurysm is a medical condition where a cerebral artery can burst under adverse pressure conditions. A 20% mortality rate and additional 30 to 40% morbidity rate have been reported for patients suffering from the rupture of aneurysms. In addition to wall [...] Read more.

A cerebral aneurysm is a medical condition where a cerebral artery can burst under adverse pressure conditions. A 20% mortality rate and additional 30 to 40% morbidity rate have been reported for patients suffering from the rupture of aneurysms. In addition to wall shear stress, input jets, induced pressure, and complicated and unstable flow patterns are other important parameters associated with a clinical history of aneurysm ruptures. In this study, the anterior cerebral artery (ACA) was modeled using image segmentation and then rebuilt with aneurysms at locations vulnerable to aneurysm growth. To simulate various aneurysm growth stages, five aneurysm sizes and two wall thicknesses were taken into consideration. In order to simulate realistic pressure loading conditions for the anterior cerebral arteries, inlet velocity and outlet pressure were used. The pressure, wall shear stress, and flow velocity distributions were then evaluated in order to predict the risk of rupture. A low-wall shear stress-based rupture scenario was created using a smaller aneurysm and thinner walls, which enhanced pressure, shear stress, and flow velocity. Additionally, aneurysms with a 4 mm diameter and a thin wall had increased rupture risks, particularly at specific boundary conditions. It is believed that the findings of this study will help physicians predict rupture risk according to aneurysm diameters and make early treatment decisions. Full article

(This article belongs to the Special Issue Computational Methods for Coupled Problems in Science and Engineering)

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17 pages, 10059 KiB

Open AccessArticle

Finite Element Analysis of Hierarchical Metamaterial-Based Patterns for Generating High Expansion in Skin Grafting

by Vivek Gupta and Arnab Chanda

Math. Comput. Appl. 2023, 28(4), 89; https://doi.org/10.3390/mca28040089 - 1 Aug 2023

Viewed by 1454

Abstract

Burn injuries are very common due to heat, accidents, and fire. Split-thickness skin grafting technique is majorly used to recover the burn sites. In this technique, the complete epidermis and partial dermis layer of the skin are used to make grafts. A small [...] Read more.

Burn injuries are very common due to heat, accidents, and fire. Split-thickness skin grafting technique is majorly used to recover the burn sites. In this technique, the complete epidermis and partial dermis layer of the skin are used to make grafts. A small amount of skin is passed into the mesher to create an incision pattern for higher expansion. These grafts are transplanted into the burn sites with the help of sutures for recovering large burn areas. Presently, the maximum expansion possible with skin grafting is very less (<3), which is insufficient for covering larger burn area with a small amount of healthy skin. This study aimed to determine the possibility of employing innovative auxetic skin graft patterns and traditional skin graft patterns with three levels of hierarchy. Six different hierarchical skin graft designs were tested to describe the biomechanical properties. The meshing ratio, Poisson’s ratio, expansion, and induced stresses were quantified for each graft model. The computational results indicated that the expansion potential of the 3rd order auxetic skin graft was highest across all the models. These results are expected to improve burn surgeries and promote skin transplantation research. Full article

(This article belongs to the Special Issue Mathematical and Computational Modelling in Mechanics of Materials and Structures)

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19 pages, 1615 KiB

Open AccessArticle

Impact of Cooperation and Intra-Specific Competition of Prey on the Stability of Prey–Predator Models with Refuge

by Soumyadip Pal, Fahad Al Basir and Santanu Ray

Math. Comput. Appl. 2023, 28(4), 88; https://doi.org/10.3390/mca28040088 - 28 Jul 2023

Cited by 1 | Viewed by 2149

Abstract

The main objective of this study is to find out the influences of cooperation and intra-specific competition in the prey population on escaping predation through refuge and the effect of the two intra-specific interactions on the dynamics of prey–predator systems. For this purpose, [...] Read more.

The main objective of this study is to find out the influences of cooperation and intra-specific competition in the prey population on escaping predation through refuge and the effect of the two intra-specific interactions on the dynamics of prey–predator systems. For this purpose, two mathematical models with Holling type II functional response functions were proposed and analyzed. The first model includes cooperation among prey populations, whereas the second one incorporates intra-specific competition. The existence conditions and stability of different equilibrium points for both models were analyzed to determine the qualitative behaviors of the systems. Refuge through intra-specific competition has a stabilizing role, whereas cooperation has a destabilizing role on the system dynamics. Periodic oscillations were observed in both systems through Hopf bifurcation. From the analytical and numerical findings, we conclude that intra-specific competition affects the prey population and continuously controls the refuge class under a critical value, and thus, it never becomes too large to cause predator extinction due to food scarcity. Conversely, cooperation leads the maximal number of individuals to escape predation through the refuge so that predators suffer from low predation success. Full article

(This article belongs to the Section Natural Sciences)

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13 pages, 2393 KiB

Open AccessArticle

Buckling of Cracked Euler–Bernoulli Columns Embedded in a Winkler Elastic Medium

by José Antonio Loya, Carlos Santiuste, Josué Aranda-Ruiz and Ramón Zaera

Math. Comput. Appl. 2023, 28(4), 87; https://doi.org/10.3390/mca28040087 - 28 Jul 2023

Viewed by 1511

Abstract

This work analyses the buckling behaviour of cracked Euler–Bernoulli columns immersed in a Winkler elastic medium, obtaining their buckling loads. For this purpose, the beam is modelled as two segments connected in the cracked section by a mass-less rotational spring. Its rotation is [...] Read more.

This work analyses the buckling behaviour of cracked Euler–Bernoulli columns immersed in a Winkler elastic medium, obtaining their buckling loads. For this purpose, the beam is modelled as two segments connected in the cracked section by a mass-less rotational spring. Its rotation is proportional to the bending moment transmitted through the cracked section, considering the discontinuity of the rotation due to bending. The differential equations for the buckling behaviour are solved by applying the corresponding boundary conditions, as well as the compatibility and jump conditions of the cracked section. The proposed methodology allows calculating the buckling load as a function of the type of support, the parameter defining the elastic soil, the crack position and the initial length of the crack. The results obtained are compared with those published by other authors in works that deal with the problem in a partial way, showing the interaction and importance of the parameters considered in the buckling loads of the system. Full article

(This article belongs to the Special Issue Mathematical and Computational Approaches in Applied Mechanics: A Themed Issue Dedicated to Professor J.N. Reddy)

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22 pages, 3067 KiB

Open AccessArticle

A Quantile Functions-Based Investigation on the Characteristics of Southern African Solar Irradiation Data

by Daniel Maposa, Amon Masache and Precious Mdlongwa

Math. Comput. Appl. 2023, 28(4), 86; https://doi.org/10.3390/mca28040086 - 24 Jul 2023

Cited by 1 | Viewed by 1194

Abstract

Exploration of solar irradiance can greatly assist in understanding how renewable energy can be better harnessed. It helps in establishing the solar irradiance climate in a particular region for effective and efficient harvesting of solar energy. Understanding the climate provides planners, designers and [...] Read more.

Exploration of solar irradiance can greatly assist in understanding how renewable energy can be better harnessed. It helps in establishing the solar irradiance climate in a particular region for effective and efficient harvesting of solar energy. Understanding the climate provides planners, designers and investors in the solar power generation sector with critical information. However, a detailed exploration of these climatic characteristics has not yet been studied for the Southern African data. Very little exploration is being done through the use of measures of centrality only. These descriptive statistics may be misleading. As a result, we overcome limitations in the currently used deterministic models through the application of distributional modelling through quantile functions. Deterministic and stochastic elements in the data were combined and analysed simultaneously when fitting quantile distributional function models. The fitted models were then used to find population means as explorative parameters that consist of both deterministic and stochastic properties of the data. The application of QFs has been shown to be a practical tool and gives more information than approaches that focus separately on either measures of central tendency or empirical distributions. Seasonal effects were detected in the data from the whole region and can be attributed to the cyclical behaviour exhibited. Daily maximum solar irradiation is taking place within two hours of midday and monthly accumulates in summer months. Windhoek is receiving the best daily total mean, while the maximum monthly accumulated total mean is taking place in Durban. Developing separate solar irradiation models for summer and winter is highly recommended. Though robust and rigorous, quantile distributional function modelling enables exploration and understanding of all components of the behaviour of the data being studied. Therefore, a starting base for understanding Southern Africa’s solar climate was developed in this study. Full article

(This article belongs to the Special Issue Statistical Inference in Linear Models)

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13 pages, 321 KiB

Open AccessArticle

Moments of the Negative Multinomial Distribution

by Frédéric Ouimet

Math. Comput. Appl. 2023, 28(4), 85; https://doi.org/10.3390/mca28040085 - 24 Jul 2023

Cited by 1 | Viewed by 1289

Abstract

The negative multinomial distribution appears in many areas of applications such as polarimetric image processing and the analysis of longitudinal count data. In previous studies, general formulas for the falling factorial moments and cumulants of the negative multinomial distribution were obtained. However, despite [...] Read more.

The negative multinomial distribution appears in many areas of applications such as polarimetric image processing and the analysis of longitudinal count data. In previous studies, general formulas for the falling factorial moments and cumulants of the negative multinomial distribution were obtained. However, despite the availability of the moment generating function, no comprehensive formulas for the moments have been calculated thus far. This paper addresses this gap by presenting general formulas for both central and non-central moments of the negative multinomial distribution. These formulas are expressed in terms of binomial coefficients and Stirling numbers of the second kind. Utilizing these formulas, we provide explicit expressions for all central moments up to the fourth order and all non-central moments up to the eighth order. Full article

(This article belongs to the Topic Mathematical Modeling)

18 pages, 450 KiB

Open AccessArticle

Thermal–Structural Linear Static Analysis of Functionally Graded Beams Using Reddy Beam Theory

by Carlos Enrique Valencia Murillo, Miguel Ernesto Gutierrez Rivera and Luis David Celaya Garcia

Math. Comput. Appl. 2023, 28(4), 84; https://doi.org/10.3390/mca28040084 - 23 Jul 2023

Cited by 2 | Viewed by 1636

Abstract

In this work, a finite element model to perform the thermal–structural analysis of beams made of functionally graded material (FGM) is presented. The formulation is based on the third-order shear deformation theory. The constituents of the FGM are considered to vary only in [...] Read more.

In this work, a finite element model to perform the thermal–structural analysis of beams made of functionally graded material (FGM) is presented. The formulation is based on the third-order shear deformation theory. The constituents of the FGM are considered to vary only in the thickness direction, and the effective material properties are evaluated by means of the rule of mixtures. The volume distribution of the top constituent is modeled using the power law form. A comparison of the present finite element model with the numerical results available in the literature reveals that they are in good agreement. In addition, a routine to study functionally graded plane models in a commercial finite element code is used to verify the performance of the proposed model. In the present work, displacements for different values of the power law exponent and surface temperatures are presented. Furthermore, the normal stress variation along the thickness is shown for several power law exponents of functionally graded beams subjected to thermal and mechanical loads. Full article

(This article belongs to the Special Issue Mathematical and Computational Approaches in Applied Mechanics: A Themed Issue Dedicated to Professor J.N. Reddy)

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19 pages, 510 KiB

Open AccessArticle

A New Sine Family of Generalized Distributions: Statistical Inference with Applications

by SidAhmed Benchiha, Laxmi Prasad Sapkota, Aned Al Mutairi, Vijay Kumar, Rana H. Khashab, Ahmed M. Gemeay, Mohammed Elgarhy and Said G. Nassr

Math. Comput. Appl. 2023, 28(4), 83; https://doi.org/10.3390/mca28040083 - 16 Jul 2023

Cited by 7 | Viewed by 2491

Abstract

In this article, we extensively study a family of distributions using the trigonometric function. We add an extra parameter to the sine transformation family and name it the alpha-sine-G family of distributions. Some important functional forms and properties of the family are provided [...] Read more.

In this article, we extensively study a family of distributions using the trigonometric function. We add an extra parameter to the sine transformation family and name it the alpha-sine-G family of distributions. Some important functional forms and properties of the family are provided in a general form. A specific sub-model alpha-sine Weibull of this family is also introduced using the Weibull distribution as a parent distribution and studied deeply. The statistical properties of this new distribution are investigated and intended parameters are estimated using the maximum likelihood, maximum product of spacings, least square, weighted least square, and minimum distance methods. For further justification of these estimates, a simulation experiment is carried out. Two real data sets are analyzed to show the suggested model’s application. The suggested model performed well compares to some existing models considered in the study. Full article

(This article belongs to the Special Issue Statistical Inference in Linear Models)

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27 pages, 8536 KiB

Open AccessFeature PaperArticle

Thermal Stress Formation in a Functionally Graded Al₂O₃-Adhesive Single Lap Joint Subjected to a Uniform Temperature Field

by Mustafa Kemal Apalak and Junuthula N. Reddy

Math. Comput. Appl. 2023, 28(4), 82; https://doi.org/10.3390/mca28040082 - 11 Jul 2023

Viewed by 1847

Abstract

This study investigates the strain and stress states in an aluminum single lap joint bonded with a functionally graded Al

_{2}

O

_{3}

micro particle reinforced adhesive layer subjected to a uniform temperature field. Navier equations of elasticity theory were designated by considering [...] Read more.

This study investigates the strain and stress states in an aluminum single lap joint bonded with a functionally graded Al

_{2}

O

_{3}

micro particle reinforced adhesive layer subjected to a uniform temperature field. Navier equations of elasticity theory were designated by considering the spatial derivatives of Lamé constants and the coefficient of thermal expansion for local material composition. The set of partial differential equations and mechanical boundary conditions for a two-dimensional model was reduced to a set of linear equations by means of the central finite difference approximation at each grid point of a discretized joint. The through-thickness Al

_{2}

O

_{3}

-adhesive composition was tailored by the functional grading concept, and the mechanical and thermal properties of local adhesive composition were predicted by Mori–Tanaka’s homogenization approach. The adherend–adhesive interfaces exhibited sharp discontinuous thermal stresses, whereas the discontinuous nature of thermal strains along bi-material interfaces can be moderated by the gradient power index, which controls the through-thickness variation of particle amount in the local adhesive composition. The free edges of the adhesive layer were also critical due to the occurrence of high normal and shear strains and stresses. The gradient power index can influence the distribution and levels of strain and stress components only for a sufficiently high volume fraction of particles. The grading direction of particles in the adhesive layer was not influential because the temperature field is uniform; namely, it can only upturn the low and high strain and stress regions so that the neat adhesive–adherend interface and the particle-rich adhesive–adherend interface can be relocated. Full article

(This article belongs to the Special Issue Mathematical and Computational Approaches in Applied Mechanics: A Themed Issue Dedicated to Professor J.N. Reddy)

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12 pages, 1344 KiB

Open AccessArticle

Effect of Chrome and Vanadium on the Behavior of Hydrogen and Helium in Tungsten

by Meicong Li, Zheng Zhang, Yangyang Li, Qiang Zhao, Mei Huang and Xiaoping Ouyang

Math. Comput. Appl. 2023, 28(4), 81; https://doi.org/10.3390/mca28040081 - 3 Jul 2023

Viewed by 1396

Abstract

Tungsten is a promising material for nuclear fusion reactors, but its performance can be degraded by the accumulation of hydrogen (H) and helium (He) isotopes produced by nuclear reactions. This study investigates the effect of chrome (Cr) and vanadium (V) on the behavior [...] Read more.

Tungsten is a promising material for nuclear fusion reactors, but its performance can be degraded by the accumulation of hydrogen (H) and helium (He) isotopes produced by nuclear reactions. This study investigates the effect of chrome (Cr) and vanadium (V) on the behavior of hydrogen and helium in tungsten (W) using first-principles calculations. The results show W becomes easier to process after adding Cr and V. Stability improves after adding V. Adding Cr negatively impacts H and He diffusion in W, while V promotes it. There is attraction between H and Cr or H and V for distances over 1.769 Å but repulsion below 1.583 Å. There is always attraction between He and Cr or V. The attraction between vacancies and He is stronger than that between He and Cr or V. There is no clear effect on H when vacancies and Cr or V coexist in W. Vacancies can dilute the effects of Cr and V on H and He in W. Full article

(This article belongs to the Special Issue Computational Applications in Engineering, Experimental Optimization, and Sustainable Manufacturing)

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20 pages, 5359 KiB

Open AccessArticle

Vehicle Make and Model Recognition as an Open-Set Recognition Problem and New Class Discovery

by Diana-Itzel Vázquez-Santiago, Héctor-Gabriel Acosta-Mesa and Efrén Mezura-Montes

Math. Comput. Appl. 2023, 28(4), 80; https://doi.org/10.3390/mca28040080 - 3 Jul 2023

Cited by 2 | Viewed by 1442

Abstract

One of the main limitations of traditional neural-network-based classifiers is the assumption that all query data are well represented within their training set. Unfortunately, in real-life scenarios, this is often not the case, and unknown class data may appear during testing, which drastically [...] Read more.

One of the main limitations of traditional neural-network-based classifiers is the assumption that all query data are well represented within their training set. Unfortunately, in real-life scenarios, this is often not the case, and unknown class data may appear during testing, which drastically weakens the robustness of the algorithms. For this type of problem, open-set recognition (OSR) proposes a new approach where it is assumed that the world knowledge of algorithms is incomplete, so they must be prepared to detect and reject objects of unknown classes. However, the goal of this approach does not include the detection of new classes hidden within the rejected instances, which would be beneficial to increase the model’s knowledge and classification capability, even after training. This paper proposes an OSR strategy with an extension for new class discovery aimed at vehicle make and model recognition. We use a neuroevolution technique and the contrastive loss function to design a domain-specific CNN that generates a consistent distribution of feature vectors belonging to the same class within the embedded space in terms of cosine similarity, maintaining this behavior in unknown classes, which serves as the main guide for a probabilistic model and a clustering algorithm to simultaneously detect objects of new classes and discover their classes. The results show that the presented strategy works effectively to address the VMMR problem as an OSR problem and furthermore is able to simultaneously recognize the new classes hidden within the rejected objects. OSR is focused on demonstrating its effectiveness with benchmark databases that are not domain-specific. VMMR is focused on improving its classification accuracy; however, since it is a real-world recognition problem, it should have strategies to deal with unknown data, which has not been extensively addressed and, to the best of our knowledge, has never been considered from an OSR perspective, so this work also contributes as a benchmark for future domain-specific OSR. Full article

(This article belongs to the Collection Feature Papers in Mathematical and Computational Applications 2023)

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12 pages, 2771 KiB

Open AccessArticle

Solve High-Dimensional Reflected Partial Differential Equations by Neural Network Method

by Xiaowen Shi, Xiangyu Zhang, Renwu Tang and Juan Yang

Math. Comput. Appl. 2023, 28(4), 79; https://doi.org/10.3390/mca28040079 - 24 Jun 2023

Viewed by 1799

Abstract

Reflected partial differential equations (PDEs) have important applications in financial mathematics, stochastic control, physics, and engineering. This paper aims to present a numerical method for solving high-dimensional reflected PDEs. In fact, overcoming the “dimensional curse” and approximating the reflection term are challenges. Some [...] Read more.

Reflected partial differential equations (PDEs) have important applications in financial mathematics, stochastic control, physics, and engineering. This paper aims to present a numerical method for solving high-dimensional reflected PDEs. In fact, overcoming the “dimensional curse” and approximating the reflection term are challenges. Some numerical algorithms based on neural networks developed recently fail in solving high-dimensional reflected PDEs. To solve these problems, firstly, the reflected PDEs are transformed into reflected backward stochastic differential equations (BSDEs) using the reflected Feyman–Kac formula. Secondly, the reflection term of the reflected BSDEs is approximated using the penalization method. Next, the BSDEs are discretized using a strategy that combines Euler and Crank–Nicolson schemes. Finally, a deep neural network model is employed to simulate the solution of the BSDEs. The effectiveness of the proposed method is tested by two numerical experiments, and the model shows high stability and accuracy in solving reflected PDEs of up to 100 dimensions. Full article

(This article belongs to the Topic Numerical Methods for Partial Differential Equations)

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16 pages, 5085 KiB

Open AccessArticle

A New Method for Improving Inverse Finite Element Method Material Characterization for the Mooney–Rivlin Material Model through Constrained Optimization

by John Dean Van Tonder, Martin Philip Venter and Gerhard Venter

Math. Comput. Appl. 2023, 28(4), 78; https://doi.org/10.3390/mca28040078 - 24 Jun 2023

Viewed by 2547

Abstract

The inverse finite element method is a technique that can be used for material model parameter characterization. The literature shows that this approach may get caught in the local minima of the design space. These local minimum solutions often fit the material test [...] Read more.

The inverse finite element method is a technique that can be used for material model parameter characterization. The literature shows that this approach may get caught in the local minima of the design space. These local minimum solutions often fit the material test data with small errors and are often mistaken for the optimal solution. The problem with these sub-optimal solutions becomes apparent when applied to different loading conditions where significant errors can be witnessed. The research of this paper presents a new method that resolves this issue for Mooney–Rivlin and builds on a previous paper that used flat planes, referred to as hyperplanes, to map the error functions, isolating the unique optimal solution. The new method alternatively uses a constrained optimization approach, utilizing equality constraints to evaluate the error functions. As a result, the design space’s curvature is taken into account, which significantly reduces the amount of variation between predicted parameters from a maximum of

1.934

% in the previous paper down to

0.1882

% in the results presented here. The results of this study demonstrate that the new method not only isolates the unique optimal solution but also drastically reduces the variation in the predicted parameters. The paper concludes that the presented new characterization method significantly contributes to the existing literature. Full article

(This article belongs to the Special Issue Current Problems and Advances in Computational and Applied Mechanics)

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Math. Comput. Appl., Volume 28, Issue 4 (August 2023) – 14 articles

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