Mathematical and Computational Applications

2023

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3 pages, 177 KiB

Open AccessEditorial

Feature Paper Collection of Mathematical and Computational Applications—2022

by Gianluigi Rozza, Oliver Schütze and Nicholas Fantuzzi

Math. Comput. Appl. 2023, 28(1), 16; https://doi.org/10.3390/mca28010016 - 28 Jan 2023

Viewed by 1492

Abstract

This Special Issue comprises the first collection of papers submitted by the Editorial Board Members (EBMs) of the journal Mathematical and Computational Applications (MCA), as well as outstanding scholars working in the core research fields of MCA [...] Full article

2022

Jump to: 2023

16 pages, 968 KiB

Open AccessFeature PaperArticle

A New Material Model for Agglomerated Cork

by Gabriel Thomaz de Aquino Pereira, Ricardo J. Alves de Sousa, I-Shih Liu, Marcello Goulart Teixeira and Fábio A. O. Fernandes

Math. Comput. Appl. 2022, 27(6), 92; https://doi.org/10.3390/mca27060092 - 9 Nov 2022

Cited by 1 | Viewed by 1491

Abstract

It is increasingly necessary to promote means of production that are less polluting and less harmful to the environment following the UN 2030 agenda for sustainable development. Using natural cellular materials in structural applications can be essential for enabling a future in this [...] Read more.

It is increasingly necessary to promote means of production that are less polluting and less harmful to the environment following the UN 2030 agenda for sustainable development. Using natural cellular materials in structural applications can be essential for enabling a future in this direction. Cork is a natural cellular material with an excellent energy absorption capacity. Its use in engineering applications and products has grown over time, so predicting its mechanical response through numerical tools is crucial. Classical cork modeling uses a model developed for foam material, including an adjustment function that does not have a clear physical interpretation. This work presents a new material model for an agglomerated cork based solely on well-known hypotheses of continuum mechanics using fewer parameters than the classical model and further a finite element framework to validate the new model against experimental data. Full article

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13 pages, 693 KiB

Open AccessArticle

Entropy Analysis for Hydromagnetic Darcy–Forchheimer Flow Subject to Soret and Dufour Effects

by Sohail A. Khan and Tasawar Hayat

Math. Comput. Appl. 2022, 27(5), 80; https://doi.org/10.3390/mca27050080 - 19 Sep 2022

Cited by 1 | Viewed by 1775

Abstract

Here, our main aim is to examine the impacts of Dufour and Soret in a radiative Darcy–Forchheimer flow. Ohmic heating and the dissipative features are outlined. The characteristics of the thermo-diffusion and diffusion-thermo effects are addressed. A binary chemical reaction is deliberated. To [...] Read more.

Here, our main aim is to examine the impacts of Dufour and Soret in a radiative Darcy–Forchheimer flow. Ohmic heating and the dissipative features are outlined. The characteristics of the thermo-diffusion and diffusion-thermo effects are addressed. A binary chemical reaction is deliberated. To examine the thermodynamical system performance, we discuss entropy generation. A non-linear differential system is computed by the finite difference technique. Variations in the velocity, concentration, thermal field and entropy rate for the emerging parameters are scrutinized. A decay in velocity is observed for the Forchheimer number. Higher estimation of the magnetic number has the opposite influence for the velocity and temperature. The velocity, concentration and thermal field have a similar effect on the suction variable. The temperature against the Dufour number is augmented. A decay in the concentration is found against the Soret number. A similar trend holds for the entropy rate through the radiation and diffusion variables. An augmentation in the entropy rate is observed for the diffusion variable. Full article

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22 pages, 416 KiB

Open AccessArticle

Spectral Analysis of the Finite Element Matrices Approximating 3D Linearly Elastic Structures and Multigrid Proposals

by Quoc Khanh Nguyen, Stefano Serra-Capizzano, Cristina Tablino-Possio and Eddie Wadbro

Math. Comput. Appl. 2022, 27(5), 78; https://doi.org/10.3390/mca27050078 - 14 Sep 2022

Cited by 1 | Viewed by 2226

Abstract

The so-called material distribution methods for topology optimization cast the governing equation as an extended or fictitious domain problem, in which a coefficient field represents the design. In practice, the finite element method is typically used to approximate that kind of governing equations [...] Read more.

The so-called material distribution methods for topology optimization cast the governing equation as an extended or fictitious domain problem, in which a coefficient field represents the design. In practice, the finite element method is typically used to approximate that kind of governing equations by using a large number of elements to discretize the design domain, and an element-wise constant function approximates the coefficient field in that domain. This paper presents a spectral analysis of the coefficient matrices associated with the linear systems stemming from the finite element discretization of a linearly elastic problem for an arbitrary coefficient field in three spatial dimensions. The given theoretical analysis is used for designing and studying an optimal multigrid method in the sense that the (arithmetic) cost for solving the problem, up to a fixed desired accuracy, is linear in the corresponding matrix size. Few selected numerical examples are presented and discussed in connection with the theoretical findings. Full article

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12 pages, 354 KiB

Open AccessFeature PaperArticle

Analytical Solutions of Microplastic Particles Dispersion Using a Lotka–Volterra Predator–Prey Model with Time-Varying Intraspecies Coefficients

by Lindomar Soares Dos Santos, José Renato Alcarás, Lucas Murilo Da Costa, Mateus Mendonça Ramos Simões and Alexandre Souto Martinez

Math. Comput. Appl. 2022, 27(4), 66; https://doi.org/10.3390/mca27040066 - 3 Aug 2022

Cited by 2 | Viewed by 2356

Abstract

Discarded plastic is subjected to weather effects from different ecosystems and becomes microplastic particles. Due to their small size, they have spread across the planet. Their presence in living organisms can have several harmful consequences, such as altering the interaction between prey and [...] Read more.

Discarded plastic is subjected to weather effects from different ecosystems and becomes microplastic particles. Due to their small size, they have spread across the planet. Their presence in living organisms can have several harmful consequences, such as altering the interaction between prey and predator. Huang et al. successfully modeled this system presenting numerical results of ecological relevance. Here, we have rewritten their equations and solved a set of them analytically, confirming that microplastic particles accumulate faster in predators than in prey and calculating the time values from which it happens. Using these analytical solutions, we have retrieved the Lotka–Volterra predator–prey model with time-varying intraspecific coefficients, allowing us to interpret ecological quantities referring to microplastics dispersion. After validating our equations, we solved analytically particular situations of ecological interest, characterized by extreme effects on predatory performance, and proposed a second-order differential equation as a possible next step to address this model. Our results open space for further refinement in the study of predator–prey models under the effects of microplastic particles, either exploring the second-order equation that we propose or modify the Huang et al. model to reduce the number of parameters, embedding in the time-varying intraspecies coefficients all the adverse effects caused by microplastic particles. Full article

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21 pages, 3038 KiB

Open AccessFeature PaperArticle

Using the Theory of Functional Connections to Solve Boundary Value Geodesic Problems

by Daniele Mortari

Math. Comput. Appl. 2022, 27(4), 64; https://doi.org/10.3390/mca27040064 - 27 Jul 2022

Cited by 5 | Viewed by 2241

Abstract

This study provides a least-squares-based numerical approach to estimate the boundary value geodesic trajectory and associated parametric velocity on curved surfaces. The approach is based on the Theory of Functional Connections, an analytical framework to perform functional interpolation. Numerical examples are provided for [...] Read more.

This study provides a least-squares-based numerical approach to estimate the boundary value geodesic trajectory and associated parametric velocity on curved surfaces. The approach is based on the Theory of Functional Connections, an analytical framework to perform functional interpolation. Numerical examples are provided for a set of two-dimensional quadrics, including ellipsoid, elliptic hyperboloid, elliptic paraboloid, hyperbolic paraboloid, torus, one-sheeted hyperboloid, Moëbius strips, as well as on a generic surface. The estimated geodesic solutions for the tested surfaces are obtained with residuals at the machine-error level. In principle, the proposed approach can be applied to solve boundary value problems in more complex scenarios, such as on Riemannian manifolds. Full article

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16 pages, 3197 KiB

Open AccessFeature PaperArticle

Resolving Boundary Layers with Harmonic Extension Finite Elements

by Harri Hakula

Math. Comput. Appl. 2022, 27(4), 57; https://doi.org/10.3390/mca27040057 - 8 Jul 2022

Cited by 2 | Viewed by 1924

Abstract

In recent years, the standard numerical methods for partial differential equations have been extended with variants that address the issue of domain discretisation in complicated domains. Sometimes similar requirements are induced by local parameter-dependent features of the solutions, for instance, boundary or internal [...] Read more.

In recent years, the standard numerical methods for partial differential equations have been extended with variants that address the issue of domain discretisation in complicated domains. Sometimes similar requirements are induced by local parameter-dependent features of the solutions, for instance, boundary or internal layers. The adaptive reference elements are one way with which harmonic extension elements, an extension of the p-version of the finite element method, can be implemented. In combination with simple replacement rule-based mesh generation, the performance of the method is shown to be equivalent to that of the standard p-version in problems where the boundary layers dominate the solution. The performance over a parameter range is demonstrated in an application of computational asymptotic analysis, where known estimates are recovered via computational means only. Full article

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14 pages, 2635 KiB

Open AccessArticle

In Vivo Validation of a Cardiovascular Simulation Model in Pigs

by Moriz A. Habigt, Jonas Gesenhues, Maike Stemmler, Marc Hein, Rolf Rossaint and Mare Mechelinck

Math. Comput. Appl. 2022, 27(2), 28; https://doi.org/10.3390/mca27020028 - 18 Mar 2022

Cited by 2 | Viewed by 2596

Abstract

Many computer simulation models of the cardiovascular system, of varying complexity and objectives, have been proposed in physiological science. Every model needs to be parameterized and evaluated individually. We conducted a porcine animal model to parameterize and evaluate a computer simulation model, recently [...] Read more.

Many computer simulation models of the cardiovascular system, of varying complexity and objectives, have been proposed in physiological science. Every model needs to be parameterized and evaluated individually. We conducted a porcine animal model to parameterize and evaluate a computer simulation model, recently proposed by our group. The results of an animal model, on thirteen healthy pigs, were used to generate consistent parameterization data for the full heart computer simulation model. To evaluate the simulation model, differences between the resulting simulation output and original animal data were analysed. The input parameters of the animal model, used to individualize the computer simulation, showed high interindividual variability (range of coefficient of variation: 10.1–84.5%), which was well-reflected by the resulting haemodynamic output parameters of the simulation (range of coefficient of variation: 12.6–45.7%). The overall bias between the animal and simulation model was low (mean: −3.24%, range: from −26.5 to 20.1%). The simulation model used in this study was able to adapt to the high physiological variability in the animal model. Possible reasons for the remaining differences between the animal and simulation model might be a static measurement error, unconsidered inaccuracies within the model, or unconsidered physiological interactions. Full article

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16 pages, 3150 KiB

Open AccessArticle

Image Segmentation with a Priori Conditions: Applications to Medical and Geophysical Imaging

by Guzel Khayretdinova, Christian Gout, Théophile Chaumont-Frelet and Sergei Kuksenko

Math. Comput. Appl. 2022, 27(2), 26; https://doi.org/10.3390/mca27020026 - 11 Mar 2022

Cited by 3 | Viewed by 2638

Abstract

In this paper, we propose a method for semi-supervised image segmentation based on geometric active contours. The main novelty of the proposed method is the initialization of the segmentation process, which is performed with a polynomial approximation of a user defined initialization (for [...] Read more.

In this paper, we propose a method for semi-supervised image segmentation based on geometric active contours. The main novelty of the proposed method is the initialization of the segmentation process, which is performed with a polynomial approximation of a user defined initialization (for instance, a set of points or a curve to be interpolated). This work is related to many potential applications: the geometric conditions can be useful to improve the quality the segmentation process in medicine and geophysics when it is required (weak contrast of the image, missing parts in the image, non-continuous contour…). We compare our method to other segmentation algorithms, and we give experimental results related to several medical and geophysical applications. Full article

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35 pages, 2658 KiB

Open AccessFeature PaperArticle

Stochastic Neural Networks for Automatic Cell Tracking in Microscopy Image Sequences of Bacterial Colonies

by Sorena Sarmadi, James J. Winkle, Razan N. Alnahhas, Matthew R. Bennett, Krešimir Josić, Andreas Mang and Robert Azencott

Math. Comput. Appl. 2022, 27(2), 22; https://doi.org/10.3390/mca27020022 - 2 Mar 2022

Cited by 2 | Viewed by 3468

Abstract

Our work targets automated analysis to quantify the growth dynamics of a population of bacilliform bacteria. We propose an innovative approach to frame-sequence tracking of deformable-cell motion by the automated minimization of a new, specific cost functional. This minimization is implemented by dedicated [...] Read more.

Our work targets automated analysis to quantify the growth dynamics of a population of bacilliform bacteria. We propose an innovative approach to frame-sequence tracking of deformable-cell motion by the automated minimization of a new, specific cost functional. This minimization is implemented by dedicated Boltzmann machines (stochastic recurrent neural networks). Automated detection of cell divisions is handled similarly by successive minimizations of two cost functions, alternating the identification of children pairs and parent identification. We validate the proposed automatic cell tracking algorithm using (i) recordings of simulated cell colonies that closely mimic the growth dynamics of E. coli in microfluidic traps and (ii) real data. On a batch of 1100 simulated image frames, cell registration accuracies per frame ranged from 94.5% to 100%, with a high average. Our initial tests using experimental image sequences (i.e., real data) of E. coli colonies also yield convincing results, with a registration accuracy ranging from 90% to 100%. Full article

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44 pages, 8758 KiB

Open AccessArticle

Arbitrarily Accurate Analytical Approximations for the Error Function

by Roy M. Howard

Math. Comput. Appl. 2022, 27(1), 14; https://doi.org/10.3390/mca27010014 - 9 Feb 2022

Cited by 5 | Viewed by 3960

Abstract

A spline-based integral approximation is utilized to define a sequence of approximations to the error function that converge at a significantly faster manner than the default Taylor series. The real case is considered and the approximations can be improved by utilizing the approximation [...] Read more.

A spline-based integral approximation is utilized to define a sequence of approximations to the error function that converge at a significantly faster manner than the default Taylor series. The real case is considered and the approximations can be improved by utilizing the approximation

\erf (x) \approx 1

for

| x | > x_{o}

and with

x_{o}

optimally chosen. Two generalizations are possible; the first is based on demarcating the integration interval into m equally spaced subintervals. The second, is based on utilizing a larger fixed subinterval, with a known integral, and a smaller subinterval whose integral is to be approximated. Both generalizations lead to significantly improved accuracy. Furthermore, the initial approximations, and those arising from the first generalization, can be utilized as inputs to a custom dynamic system to establish approximations with better convergence properties. Indicative results include those of a fourth-order approximation, based on four subintervals, which leads to a relative error bound of 1.43 × 10⁻⁷ over the interval

[0, \infty]

. The corresponding sixteenth-order approximation achieves a relative error bound of 2.01 × 10⁻¹⁹. Various approximations that achieve the set relative error bounds of 10⁻⁴, 10⁻⁶, 10⁻¹⁰, and 10⁻¹⁶, over

[0, \infty]

, are specified. Applications include, first, the definition of functions that are upper and lower bounds, of arbitrary accuracy, for the error function. Second, new series for the error function. Third, new sequences of approximations for

\exp (- x^{2})

that have significantly higher convergence properties than a Taylor series approximation. Fourth, the definition of a complementary demarcation function

e_{C} (x)

that satisfies the constraint

e_{C}^{2} (x) + \erf^{2} (x) = 1

. Fifth, arbitrarily accurate approximations for the power and harmonic distortion for a sinusoidal signal subject to an error function nonlinearity. Sixth, approximate expressions for the linear filtering of a step signal that is modeled by the error function. Full article

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15 pages, 5406 KiB

Open AccessFeature PaperArticle

Approximating the Steady-State Temperature of 3D Electronic Systems with Convolutional Neural Networks

by Monika Stipsitz and Hèlios Sanchis-Alepuz

Math. Comput. Appl. 2022, 27(1), 7; https://doi.org/10.3390/mca27010007 - 14 Jan 2022

Cited by 6 | Viewed by 3141

Abstract

Thermal simulations are an important part of the design process in many engineering disciplines. In simulation-based design approaches, a considerable amount of time is spent by repeated simulations. An alternative, fast simulation tool would be a welcome addition to any automatized and simulation-based [...] Read more.

Thermal simulations are an important part of the design process in many engineering disciplines. In simulation-based design approaches, a considerable amount of time is spent by repeated simulations. An alternative, fast simulation tool would be a welcome addition to any automatized and simulation-based optimisation workflow. In this work, we present a proof-of-concept study of the application of convolutional neural networks to accelerate thermal simulations. We focus on the thermal aspect of electronic systems. The goal of such a tool is to provide accurate approximations of a full solution, in order to quickly select promising designs for more detailed investigations. Based on a training set of randomly generated circuits with corresponding finite element solutions, the full 3D steady-state temperature field is estimated using a fully convolutional neural network. A custom network architecture is proposed which captures the long-range correlations present in heat conduction problems. We test the network on a separate dataset and find that the mean relative error is around 2% and the typical evaluation time is 35 ms per sample (2 ms for evaluation, 33 ms for data transfer). The benefit of this neural-network-based approach is that, once training is completed, the network can be applied to any system within the design space spanned by the randomized training dataset (which includes different components, material properties, different positioning of components on a PCB, etc.). Full article

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