Bibliografías temáticas / Explicit numerical simulation

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Literatura académica sobre el tema "Explicit numerical simulation"

Autor: Grafiati
Publicado: 7 de diciembre de 2024

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Artículos de revistas sobre el tema "Explicit numerical simulation"

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Tutueva, Aleksandra y Denis Butusov. "Stability Analysis and Optimization of Semi-Explicit Predictor–Corrector Methods". Mathematics 9, n.º 19 (3 de octubre de 2021): 2463. http://dx.doi.org/10.3390/math9192463.

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The increasing complexity of advanced devices and systems increases the scale of mathematical models used in computer simulations. Multiparametric analysis and study on long-term time intervals of large-scale systems are computationally expensive. Therefore, efficient numerical methods are required to reduce time costs. Recently, semi-explicit and semi-implicit Adams–Bashforth–Moulton methods have been proposed, showing great computational efficiency in low-dimensional systems simulation. In this study, we examine the numerical stability of these methods by plotting stability regions. We explicitly show that semi-explicit methods possess higher numerical stability than the conventional predictor–corrector algorithms. The second contribution of the reported research is a novel algorithm to generate an optimized finite-difference scheme of semi-explicit and semi-implicit Adams–Bashforth–Moulton methods without redundant computation of predicted values that are not used for correction. The experimental part of the study includes the numerical simulation of the three-body problem and a network of coupled oscillators with a fixed and variable integration step and finely confirms the theoretical findings.
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Azadani, L. N. y A. E. Staples. "Large-Eddy Simulation of Turbulent Barotropic Flows in Spectral Space on a Sphere". Journal of the Atmospheric Sciences 72, n.º 5 (1 de mayo de 2015): 1727–42. http://dx.doi.org/10.1175/jas-d-14-0183.1.

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Abstract Numerical simulations of atmospheric circulation models are limited by their finite spatial resolution, so large-eddy simulation (LES) is a preferred approach to study these models. In LES, a low-pass filter is applied to the flow field to separate the large- and small-scale motions. In implicitly filtered LES, the computational mesh and discretization schemes are considered to be the low-pass filter, while in the explicitly filtered LES approach, the filtering procedure is separated from the grid and discretization operators and allows for better control of the numerical errors. The aim of this paper is to study and compare implicitly filtered and explicitly filtered LES of atmospheric circulation models in spectral space. To achieve this goal, the results of implicitly filtered and explicitly filtered LES of a barotropic atmosphere circulation model on a sphere in spectral space are presented and compared with the results obtained from direct numerical simulation (DNS). Different numerical experiments are performed to investigate the efficiency of explicit filtering over implicit filtering under different dissipation terms and rotation rates. The study shows that explicit filtering increases the accuracy of the computations and improves the results, particularly where the location of coherent structures is concerned, a topic of particular importance in LES of atmospheric flows for climate and weather applications.
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Bannikova, E. Yu y A. T. Kotvytskiy. "Three Einstein rings: explicit solution and numerical simulation". Monthly Notices of the Royal Astronomical Society 445, n.º 4 (7 de noviembre de 2014): 4435–42. http://dx.doi.org/10.1093/mnras/stu2068.

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Goel, M. D., KrishnaPrasad Kallada y I. L. Muthreja. "Numerical Simulation of Bunker Buster Slab under Projectile Impact". Proceedings of the 12th Structural Engineering Convention, SEC 2022: Themes 1-2 1, n.º 1 (19 de diciembre de 2022): 1073–79. http://dx.doi.org/10.38208/acp.v1.623.

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In the present investigation, 3-D finite element modelling is carried out to predict the penetration depth and residual velocity of a projectile impacting a target. For the purpose of numerical simulation, target is a concrete slab of planar dimension 5 m × 4 m with an unconfined compressive strength of 50 MPa. This target is considered to be part of protective structure like bunker buster slab. The explicit algorithm of ABAQUS/Explicit® is employed in the simulation of projectile impact. The projectile is modelled as a rigid body and the concrete slab as a deformable body using Lagrangian formulation. The energy dissipation, penetration depth and residual velocities of the projectile impacting at different velocities are compared with the empirical equation as available by simulating a real impact scenario numerically. Investigation is carried out over a wide range of impacting velocities under sub-hydrodynamic range.
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Oleksik, Valentin, Radu Breaz, Gabriel Racz, Paul Dan Brindasu y Octavian Bologa. "Advanced Techniques used in Numerical Simulation for Deep-drawing Process". MATEC Web of Conferences 290 (2019): 03012. http://dx.doi.org/10.1051/matecconf/201929003012.

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The present paper analyse the main characteristics of the numerical simulation by finite element method of the deep-drawing processes. Also the authors’ highlights the mathematical apparatus and the calculus method used for numerical simulations of metal forming processes in many of the current simulation software. The authors present the capabilities of the inverse analysis, direct analysis, implicit analysis (for springback simulation) and the optimisation analysis applied to explicit formulations.
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Chen, Lei. "Comparisons of Explicit and Implicit Finite Element Methods for Sheet Metal Forming". Advanced Materials Research 936 (junio de 2014): 1836–39. http://dx.doi.org/10.4028/www.scientific.net/amr.936.1836.

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Sheet metal forming is one of the most commonly practiced fabrication processes in industry. Numerical simulations of the complex parts are possible by finite element method in the past thirty years. The most important problem of the simulation is the reliability of the model. Static implicit method (SI) and dynamic explicit method (DE) were used to simulation sheet metal forming process. It was found that simulation speed in dynamic explicit software has large effect on the simulation results. The best simulation speed is 5~10 m/s. Compared with the simulation and experimental results of thickness, draw-in and CPU time, the DE method is preferred.
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Humby, S. J., M. J. Biggs y U. Tüzün. "Explicit numerical simulation of fluids in reconstructed porous media". Chemical Engineering Science 57, n.º 11 (junio de 2002): 1955–68. http://dx.doi.org/10.1016/s0009-2509(02)00103-3.

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Liu, Heng y Zhenpeng Liao. "An explicit method for numerical simulation of wave equations". Earthquake Engineering and Engineering Vibration 8, n.º 1 (marzo de 2009): 17–28. http://dx.doi.org/10.1007/s11803-009-8132-6.

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Phantu, Suganya, Yupaporn Areepong y Saowanit Sukparungsee. "Double Moving Average Control Chart for Time Series Data with Poisson INARCH(1)". WSEAS TRANSACTIONS ON BUSINESS AND ECONOMICS 21 (23 de febrero de 2024): 694–707. http://dx.doi.org/10.37394/23207.2024.21.58.

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The objectives of this research are to find the explicit formulas of the average run length (ARL) of a double moving average (DMA) control chart for first-order integer-valued autoregressive conditional heteroscedasticity (INARCH1))) of Poisson count data. In addition, the numerical results obtained from the proposed explicit formulas are compared with those obtained from Monte Carlo simulations (MC) for the Poisson INARCH(1) counting process. An out-of-control ARL (ARL1) is the criteria for measuring the performance of control charts. The numerical results found that the values of both ARL0 and ARL1 obtained from explicit formulas agree with the numerical results obtained from the Monte Carlo simulation (MC), but the latter is very timeconsuming.
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ANDREUCCI, DANIELE, ANTONIO FASANO, MARIO PRIMICERIO, MAURIZIO PAOLINI y CLAUDIO VERDI. "NUMERICAL SIMULATION OF POLYMER CRYSTALLIZATION". Mathematical Models and Methods in Applied Sciences 04, n.º 01 (febrero de 1994): 135–45. http://dx.doi.org/10.1142/s0218202594000091.

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We consider a mathematical model for solidification of semicrystalline polymers, describing the evolution of temperature, crystalline volume fraction, number and average size of crystals. In turn, the model couples a suitable kinetics of nonisothermal crystallization, taking into account both formation and growth of nuclei, with the thermal energy balance equation. We also present a model of secondary crystallization. The numerical approximation is performed by semiexplicit finite differences in time and finite elements in space. The fully discrete scheme amounts to solve, at any time step, a symmetric positive definite linear system preceded by an elementwise explicit computation. The computed numerical crystal structures match qualitatively the experimental ones.
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Tesis sobre el tema "Explicit numerical simulation"

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Ertem-Müller, Senem [Verfasser]. "Numerical Efficiency of Implicit and Explicit Methods with Multigrid for Large Eddy Simulation in Complex Geometries / Senem Ertem-Müller". Aachen : Shaker, 2003. http://d-nb.info/1181602696/34.

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2

Gouillou, Franck. "Comportement dynamique des composites à fibres naturelles et résines thermoplastiques : étude de la sensibilité à la vitesse de déformation". Electronic Thesis or Diss., Ecole nationale des Mines d'Albi-Carmaux, 2024. http://www.theses.fr/2024EMAC0004.

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Ces travaux visent à explorer le comportement des composites à matrice thermoplastique renforcée de fibres naturelles en examinant leur réponse à la vitesse de déformation lors de sollicitations en traction, en se concentrant particulièrement sur la variation du module d'élasticité longitudinal pour les composites à 0° et du module d'élasticité en cisaillement pour les stratifiés à ±45°. En utilisant des fibres de bambou, de lin et de basalte comme renforts unidirectionnels et le polyamide-11 semi-cristallin ainsi que la résine amorphe ELIUM comme matrices, diverses campagnes expérimentales ont été menées pour caractériser les matériaux. Des tests de traction statiques et dynamiques sont effectués à l'aide d'équipements électromécaniques, DMA et tours de chute. De nombreuses simulations numériques explicites sont utilisées afin d'analyser les données expérimentales. Les résultats révèlent que les composites avec des plis à 0° montrent une sensibilité limitée aux vitesses de déformation inférieurs à 100 s-1, les composites en bambou et en lin présentant peu de variations du module de Young, tandis que les composites renforcés de basalte montrent une légère augmentation. Cependant, les composites avec un empilement à ±45° affichent une sensibilité accrue, principalement influencée par la résine. Cette investigation exhaustive fournit des informations sur le comportement de ces composites à fibres naturelles et polymères thermoplastiques sous différentes vitesses de chargement, guidant leur optimisation et application
These works aim to explore the behavior of thermoplastic matrix composites reinforced with natural fibers by examining their strain rate response under tensile loading, focusing particularly on the variation of the longitudinal elastic modulus for 0° composites and the shear modulus for ±45° laminates. Using bamboo, flax, and basalt fibers as unidirectional reinforcements and semi-crystalline polyamide-11 as well as amorphous ELIUM resin as matrices, various experimental campaigns were conducted to characterize the materials. Static and dynamic tensile tests were performed using electromechanical equipment, DMA, and drop towers. Numerous explicit numerical simulations were employed to analyze the experimental data. The results reveal that 0° plies exhibit limited sensitivity to strain rates below 100 s⁻¹, with bamboo and flax composites showing little variation in Young's modulus, while basalt-reinforced composites display a slight increase. However, composites with ±45° stacking exhibit increased sensitivity, primarily influenced by the resin. This comprehensive investigation provides insights into the behavior of these natural fiber-reinforced thermoplastic polymer composites under different loading rates, guiding their optimization and application
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Profota, Martin. "Pevnostní návrh ostruhy letadla". Master's thesis, Vysoké učení technické v Brně. Fakulta strojního inženýrství, 2017. http://www.nusl.cz/ntk/nusl-318638.

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This master thesis deals with computational stress-strain analysis of the tailskid of airplane L410 NG with main focus firstly the check current design of the tailskid and then the design another design solution with the able to absorb as much as possible the deformation energy. Solution of this problem is performed using computational modeling utilizing numerical simulation of quasi-static and crash deformation load of the tailskid with using explicit Finite Element Method (FEM) in program ABAQUS v6.14. After the introduction with problem situation and tailskid assembly introductory part is devoted to the research study of various designs of the tailskid for different types of airplanes. There follows these theoretical general principles of thin-walled structures and buckling of them. Before the creating of the computational model itself, the explicit form of the Finite Element Method is better described. The conclusion of this thesis deals with the mutual comparison of the most advantageous design variants of the tailskid and the selection of the most suitable one of them for the airplane L410 NG.
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Tatalák, Adam. "Deformačně-napěťová analýza tenkostěnné skříně vystavené rázovému zatížení od výbuchu". Master's thesis, Vysoké učení technické v Brně. Fakulta strojního inženýrství, 2016. http://www.nusl.cz/ntk/nusl-241688.

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This master thesis deals with stress-strain analysis of simplified model of the thin wall transformer case subjected to impact load of electrical blast. Electrical blast is replaced by chemical blast (detonation of high explosive). The problem is solved using computational modeling utilizing the Finite Element Method (FEM) and LS-DYNA solver. After the introduction where detonation and shock wave propagation is explained the analytical approach is presented. This approach serves to results verification. In the next chapter is conducted research of applicable methods from which ALE method is chosen. In preliminary study is performed the mesh size analysis that is focused on finding the size of element which is both computational effective and gives accurate results. Next the infulence of input conditions (shape, location and parametres of high explosive, location of detonation point, boundary conditions) on distribution and time progress of pressure is investigated. Then influence of the opening on upper side of the case on overall pressure redistribution and strain and stress of the case is analysed. The stress-strain analysis of the case´s door which are connected to case by various types of contact models is performed as well as stiffness analysis of these types of contact.
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Bauer, Frédéric. "Transport et production dans les écoulements turbulents de paroi à des nombres de Reynolds modérés". Thesis, Université Grenoble Alpes (ComUE), 2015. http://www.theses.fr/2015GREAI033/document.

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L'approche de simulation numérique directe est utilisée pour la simulation d'un écoulement en canal pleinement turbulent afin d'étudier l'influence des grandes échelles de l'écoulement ainsi que la dynamique du transport des contraintes de Reynolds et de la vorticité. Les simulations sont réalisées sur un domaine de calcul de grande taille afin de pouvoir capturer l'intégralité des grandes structures de l'écoulement, et portent sur une gamme relativement étendue de nombres de Reynolds (Reτ =180, 395, 590 et 1100) allant des écoulements faiblement turbulents à des écoulements modérément turbulents. L'invariance remarquable des fluctuations de vorticité normale est expliquée à travers une analyse spectrale de la vorticité. L'étude des différents termes du transport de l'intensité turbulente de la vorticité révèle par ailleurs que le pic de production de la vorticité transverse est situé à proximité immédiate de la paroi et pourrait ouvrir la voie à des stratégies de réduction de la traînée basées sur la réduction de la vorticité transverse. Le transport des contraintes de Reynolds dans la couche interne et dans la couche de recouvrement est également étudié. A proximité des parois, la dépendance des termes de transport avec le nombre de Reynolds dans les échelles internes montre que ces dernières ne suffisent pas à caractériser la dynamique des contraintes de Reynolds dans cette zone. Cette insuffisance des échelles internes nous a amenés à nous intéresser plus particulièrement au processus de production à travers les statistiques de la production conditionnées par le passage par niveau des fluctuations de la vitesse normale ou longitudinale. Cette étude nous a permis d'identifier les fluctuations qui contribuent le plus à la production et celles qui sont à l'origine de la dépendance avec le nombre de Reynolds
The direct numerical simulations of a fully turbulent channel flow are investigated to study the large scales effects on the flow quantities such as the Reynolds stresses and vorticity transport processes. Large computational domains are used so as to cover the largest scales of the flow. The simulations are performed in a wide range of Reynolds numbers (Reτ=180, 395, 590 and 1100) going from weakly to moderately high Reynolds number turbulent flows. The invariance of the wall-normal vorticity fluctuations scaled in wall variables in the inner layer versus the Reynolds number is analyzed using a spectral analysis. The vorticity transport equations are investigated in detail, presumably for the first time. The transport mechanism of the Reynolds shear stresses are subsequently analyzed in the inner layer and the overlapping zone. In the wall layer, different terms of the Reynolds stresses transport expressed in inner scales depend on the Reynolds number. This scaling failure lead us to focus on the statistics of the production when the streamwise or normal velocity fluctuations cross a given level, through the conditional Palm statistics. The main aim is to identify those amplitudes of the fluctuations that contribute more to the production and those which are responsible for the production Reynolds dependence
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6

Di, Stasio Jean. "The CD-Lagrange scheme, a robust explicit time-integrator for impact dynamics : A new singular mass formulation, and an extension to deformable-deformable contact". Thesis, Lyon, 2021. http://www.theses.fr/2021LYSEI029.

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Les pneumatiques sont complexes à simuler car les matériaux y sont hétérogènes, incompressibles et non-linéaires. De plus la géométrie descend jusqu’à l’échelle millimétrique pour les sculptures de la bande de roulement, ce qui requiert un maillage fin. Le modèle éléments finis présente donc un grand nombre de degrés de liberté, reliés par des équations non-linéaires. En dynamique, la simulation est d’autant plus compliquée avec des chocs. Néanmoins elle est cruciale dans le processus de conception pneumatique, où elle apporte une meilleur compréhension de la physique ceci sans tests réels. Les schémas explicites rendent possible les simulations de chocs, car ils résolvent facilement les non-linéarités avec un coup calcul bas. Associés à une formulation de contact précise , ils forment des schémas robustes, précis et efficaces pour la dynamique non-linéaire avec impacts. Ce travail vise à choisir et un tel schéma, et l’améliorer pour la simulation de chocs sur pneumatiques.La première partie est un benchmark identifiant le schéma CD-Lagrange. L’intégration temporelle est réalisée par le schéma de la différence centrée, et le contact imposé par multiplicateurs de Lagrange sur la vitesse. Deux possibilités d’amélioration sont identifiées. La première est d’atteindre un impact conservatif, seul instant où le schéma n’est pas symplectique. La seconde amélioration est d’étendre la formulation au contact déformable-déformable.La deuxième partie vise à atteindre la conservation de l’énergie à l’impact en adaptant la méthode de la masse singulière au CD-Lagrange. Une première formulation 1D est construite. Elle démontre une amélioration majeure du bilan d’énergie. Deux formulations 3D sont ensuite explorées.La troisième partie introduit les méthodes mortier dans le CD-Lagrange. Elles permettent de traiter un contact déformable-déformable de manière robuste, même en présence de friction et de grands glissements. Une technique d’accélération est proposée pour résoudre le problème de contact, ceci sans perte de précision
Tyres are complex structures to simulate. The materials are heterogeneous and incompressible with non-linear responses. The geometry goes to the millimetre scales for tread patterns. For a finite elements simulation a precise mesh is then required. The model has then a large number of degrees of freedom and non-linear material laws. In dynamics, the simulation becomes even more challenging especially with impacts. Nevertheless it is crucial in the tire design process because it brings a deeper comprehension of the tire and avoids test on real structures. The explicit time-integration make feasible the impact simulations. They handle easily the non-linearities with a very low computational cost for a time-step. Merged with a precise contact formulation, they form robust, accurate and efficient schemes for addressing impact simulations. This work aims to choose and improve an explicit scheme for non-linear dynamics with impacts. The first part is a benchmark for selecting a scheme and enhance its possibilities of improvement. The selected one is the CD-Lagrange: an explicit scheme based on central difference method, a contact enforcement by Lagrange multipliers, and a contact condition on velocity. Two mains improvements are identified and explored. Firstly, the energy conservation at impact would make the scheme symplectic for deformable bodies. Secondly the formulation must be enlarged to deformable–deformable contact. The second part aims then to achieve the conservation of energy by adapting the singular mass matrix to the CD-Lagrange. The formulation is firstly built in 1D, and shows a major improvement for the energy balance. Then two possible extensions are explored for the 3D cases. The third part presents the CD-Lagrange scheme with a mortar formulation for deformable-deformable contact. It handles with stability and accuracy large sliding and friction. An acceleration technique is proposed for solving the contact problem, without any loss of accuracy
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Stauffert, Maxime. "Simulation numérique d'écoulements compressibles complexes par des méthodes de type Lagrange-projection : applications aux équations de Saint-Venant". Thesis, Université Paris-Saclay (ComUE), 2018. http://www.theses.fr/2018SACLV045/document.

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On étudie dans le cadre de la thèse une famille de schémas numériques permettant de résoudre les équations de Saint-Venant. Ces schémas utilisent une décomposition d'opérateur de type Lagrange-projection afin de séparer les ondes de gravité et les ondes de transport. Un traitement implicite du système acoustique (relié aux ondes de gravité) permet aux schémas de rester stable avec de grands pas de temps. La correction des flux de pression rend possible l'obtention d'une solution approchée précise quel que soit le régime d'écoulement vis-à-vis du nombre de Froude. Une attention toute particulière est portée sur le traitement du terme source qui permet la prise en compte de l'influence de la topographie. On obtient notamment la propriété dite équilibre permettant de conserver exactement certains états stationnaires, appelés état du "lac au repos". Des versions 1D et 2D sur maillages non-structurés de ces méthodes ont été étudiées et implémentées dans un cadre volumes finis. Enfin, une extension vers des méthodes ordres élevés Galerkin discontinue a été proposée en 1D avec des limiteurs classiques ainsi que combinée avec une boucle MOOD de limitation a posteriori
In this thesis we study a family of numerical schemes solving the shallow water equations system. These schemes use a Lagrange-projection like splitting operator technique in order to separate the gravity waves and the transport waves. An implicit-explicit treatment of the acoustic system (linked to the gravity waves) allows the schemes to stay stable with large time step. The correction of the pressure fluxes enables the obtain of a precise approximation solution whatever the regime flow is with respect to the Froude number. A particular attention has been paid over the source term treatment which permits to take the topography into account. We especially obtain the so-called well-balanced property giving the exact conservation of some steady states, namely the "lake at rest" state. 1D and 2D versions of this methods have been studied and implemented in the finite volumes framework. Finally, a high order discontinuous Galerkin extension has been proposed in 1D with classical limiters along with a combined MOOD loop a posteriori limiting strategy
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8

Boilevin-Kayl, Ludovic. "Modeling and numerical simulation of implantable cardiovascular devices". Thesis, Sorbonne université, 2019. http://www.theses.fr/2019SORUS039.

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Cette thèse, réalisée dans le cadre du projet Mivana, est consacrée à la modélisation et à la simulation numérique de dispositifs cardiaques implantables. Ce projet est mené par les start-up Kephalios et Epygon, concepteurs de solutions chirurgicales non invasives pour le traitement de la régurgitation mitrale. La conception et la simulation de tels dispositifs nécessitent des méthodes numériques efficaces et précises capables de calculer correctement l’hémodynamique cardiaque. C’est le but principal de cette thèse. Dans la première partie, nous décrivons le système cardiovasculaire et les valves cardiaques avant de présenter quelques éléments de théorie concernant la modélisation mathématique de l’hémodynamique cardiaque. En fonction du degré de complexité adopté pour la modélisation des feuillets de la valve, deux approches sont identifiées : le modèle de surfaces résistives immergées et le modèle complet d’interaction fluide-structure. Dans la deuxième partie, nous étudions la première approche qui consiste à combiner une modélisation réduite de la dynamique des valves avec un découplage cinématique de l’hémodynamique cardiaque et de l’électromécanique. Nous l’enrichissons de données physiologiques externes pour la simulation correcte des phases isovolumétriques, pierres angulaires du battement cardiaque, permettant d’obtenir un modèle relativement précis qui évite la complexité des problèmes entièrement couplés. Ensuite, une série d’essais numériques sur des géométries 3D physiologiques, impliquant la régurgitation mitrale et plusieurs configurations de valves immergées, illustre la performance du modèle proposé. Dans la troisième et dernière partie, des modèles complets d’interaction fluide-structure sont considérés. Ce type de modélisation est nécessaire pour étudier des problèmes plus complexes où la précédente approche n’est plus satisfaisante, comme par exemple le prolapsus de la valve mitrale ou la fermeture d’une valve mécanique. D’un point de vue numérique, le développement de méthodes précises et efficaces est indispensable pour pouvoir simuler de tels cas physiologiques. Nous considérons alors une étude numérique complète dans laquelle plusieurs méthodes de maillages non compatibles sont comparées. Puis, nous présentons un nouveau schéma de couplage explicite dans le cadre d’une méthode de type domaine fictif pour lequel la stabilité inconditionnelle au sens de la norme en énergie est démontrée. Plusieurs exemples numériques en 2D sont proposés afin d’illustrer les propriétés et les performances de ce schéma. Enfin, cette méthode est finalement utilisée pour la simulation numérique 2D et 3D de dispositifs cardiovasculaires implantables dans un modèle complet d’interaction fluide-structure
This thesis, taking place in the context of the Mivana project, is devoted to the modeling and to the numerical simulation of implantable cardiovascular devices. This project is led by the start-up companies Kephalios and Epygon, conceptors of minimally invasive surgical solutions for the treatment of mitral regurgitation. The design and the simulation of such devices call for efficient and accurate numerical methods able to correctly compute cardiac hemodynamics. This is the main purpose of this thesis. In the first part, we describe the cardiovascular system and the cardiac valves before presenting some standard material for the mathematical modeling of cardiac hemodynamics. Based on the degree of complexity adopted for the modeling of the valve leaflets, two approaches are identified: the resistive immersed surfaces model and the complete fluidstructure interaction model. In the second part, we investigate the first approach which consists in combining a reduced modeling of the valves dynamics with a kinematic uncoupling of cardiac hemodynamics and electromechanics. We enhance it with external physiological data for the correct simulation of isovolumetric phases, cornerstones of the heartbeat, resulting in a relatively accurate model which avoids the complexity of fully coupled problems. Then, a series of numerical tests on 3D physiological geometries, involving mitral regurgitation and several configurations of immersed valves, illustrates the performance of the proposed model. In the third and final part, complete fluid-structure interaction models are considered. This type of modeling is necessary when investigating more complex problems where the previous approach is no longer satisfactory, such as mitral valve prolapse or the closing of a mechanical valve. From the numerical point of view, the development of accurate and efficient methods is mandatory to be able to compute such physiological cases. We then consider a complete numerical study in which several unfitted meshes methods are compared. Next, we present a new explicit coupling scheme in the context of the fictitious domain method for which the unconditional stability in the energy norm is proved. Several 2D numerical examples are provided to illustrate the properties and the performance of this scheme. Last, this method is finally used for 2D and 3D numerical simulation of implantable cardiovascular devices in a complete fluid-structure interaction framework
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Gonzalez-Ramirez, Noemi. "Simulating Flood Propagation in Urban Areas using a Two-Dimensional Numerical Model". Scholarly Repository, 2010. http://scholarlyrepository.miami.edu/oa_dissertations/648.

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A two-dimensional numerical model (RiverFLO-2D) has been enhanced to simulate flooding of urban areas by developing an innovative wet and dry surface algorithm, accounting for variable rainfall, and recoding the model computer program for parallel computing. The model formulation is based on the shallow water equations solved with an explicit time-stepping element-by-element finite element method. The dry-wet surface algorithm is based on a local approximation of the continuity and momentum equations for elements that are completely dry. This algorithm achieves global volume conservation in the finite element, even for flows over complex topographic surfaces. A new module was implemented to account for variable rainfall in space and time using NEXRAD precipitation estimates. The resulting computer code was parallelized using OpenMP Application Program Interface, which allows the model to run up to 5 times faster on multiple core computers. The model was verified with analytical solutions and validated with laboratory and field data. Model application to the Malpasset dam break and Sumacarcel flooding event show that the model accurately predicts flood wave travel times and water depths for these numerically demanding real cases. To illustrate the predictive capability of the enhanced model, an application was made of the city of Sweetwater flooding in Miami-Dade County, FL caused by the Hurricane Irene. The simulation starts with dry bed and rainfall is provided by NEXRAD estimates. Integrating NEXRAD rainfall estimates, developing a novel dry-wet area algorithm and parallelizing RiverFLO-2D code, this dissertation presents a proof of concept to accurately and efficiently predict floods in urban areas, identifying future improvements along this line of research.
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Zemzemi, Nejib. "Étude théorique et numérique de l'activité électrique du cœur: Applications aux électrocardiogrammes". Phd thesis, Université Paris Sud - Paris XI, 2009. http://tel.archives-ouvertes.fr/tel-00470375.

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La modélisation du vivant, en particulier la modélisation de l'activité cardiaque, est devenue un défi scientifique majeur. Le but de cette thématique est de mieux comprendre les phénomènes physiologiques et donc d'apporter des solutions à des problèmes cliniques. Nous nous intéressons dans cette thèse à la modélisation et à l'étude numérique de l'activité électrique du cœur, en particulier l'étude des électrocardiogrammes (ECGs). L'onde électrique dans le cœur est gouvernée par un système d'équations de réaction-diffusion appelé modèle bidomaine ce système est couplé à une EDO représentant l'activité cellulaire. Afin simuler des ECGs, nous tenons en compte la propagation de l'onde électrique dans le thorax qui est décrite par une équation de diffusion. Nous commençons par une démonstrer l'existence d'une solution faible du système couplé cœur-thorax pour une classe de modèles ioniques phénoménologiques. Nous prouvons ensuite l'unicité de cette solution sous certaines conditions. Le plus grand apport de cette thèse est l'étude et la simulation numérique du couplage électrique cœur-thorax. Les résultats de simulations sont représentés à l'aide des ECGs. Dans une première partie, nous produisons des simulations pour un cas normal et pour des cas pathologiques (blocs de branche gauche et droit et des arhythmies). Nous étudions également l'impact de certaines hypothèses de modélisation sur les ECGs (couplage faible, utilisation du modèle monodomaine, isotropie, homogénéité cellulaire, comportement résistance-condensateur du péricarde,. . . ). Nous étudions à la fin de cette partie la sensibilité des ECGs par apport aux paramètres du modèle. En deuxième partie, nous effectuons l'analyse numérique de schémas du premier ordre en temps découplant les calculs du potentiel d'action et du potentiel extérieur. Puis, nous combinons ces schémas en temps avec un traîtement explicite du type Robin-Robin des conditions de couplage entre le cœur et le thorax. Nous proposons une analyse de stabilité de ces schémas et nous illustrons les résultats avec des simulations numériques d'ECGs. La dernière partie est consacrée à trois applications. Nous commençons par l'estimation de certains paramètres du modèle (conductivité du thorax et paramètres ioniques). Dans la deuxième application, qui est d'originie industrielle, nous utilisons des méthodes d'apprentissage statistique pour reconstruire des ECGs à partir de mesures ('électrogrammes). Enfin, nous présentons des simulations électro-mécaniques du coeur sur une géométrie réelle dans diverses situations physiologiques et pathologiques. Les indicateurs cliniques, électriques et mécaniques, calculés à partir de ces simulations sont très similaires à ceux observés en réalité.
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Libros sobre el tema "Explicit numerical simulation"

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H, Carpenter Mark, Lewis R. Michael y Langley Research Center, eds. Low-storage, explicit Runge-Kutta schemes for the compressible Navier-Stokes equations. Hampton, Va: National Aeronautics and Space Administration, Langley Research Center, 1999.

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Low-storage, explicit Runge-Kutta schemes for the compressible Navier-Stokes equations. Hampton, Va: National Aeronautics and Space Administration, Langley Research Center, 1999.

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Low-storage, explicit Runge-Kutta schemes for the compressible Navier-Stokes equations. Hampton, Va: National Aeronautics and Space Administration, Langley Research Center, 1999.

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Coolen, A. C. C., A. Annibale y E. S. Roberts. Graphs with hard constraints: further applications and extensions. Oxford University Press, 2017. http://dx.doi.org/10.1093/oso/9780198709893.003.0007.

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This chapter looks at further topics pertaining to the effective use of Markov Chain Monte Carlo to sample from hard- and soft-constrained exponential random graph models. The chapter considers the question of how moves can be sampled efficiently without introducing unintended bias. It is shown mathematically and numerically that apparently very similar methods of picking out moves can give rise to significant differences in the average topology of the networks generated by the MCMC process. The general discussion in complemented with pseudocode in the relevant section of the Algorithms chapter, which explicitly sets out some accurate and practical move sampling approaches. The chapter also describes how the MCMC equilibrium probabilities can be purposely deformed to, for example, target desired correlations between degrees of connected nodes. The mathematical exposition is complemented with graphs showing the results of numerical simulations.
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Coolen, Ton, Alessia Annibale y Ekaterina Roberts. Generating Random Networks and Graphs. Oxford University Press, 2017. http://dx.doi.org/10.1093/oso/9780198709893.001.0001.

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This book supports researchers who need to generate random networks, or who are interested in the theoretical study of random graphs. The coverage includes exponential random graphs (where the targeted probability of each network appearing in the ensemble is specified), growth algorithms (i.e. preferential attachment and the stub-joining configuration model), special constructions (e.g. geometric graphs and Watts Strogatz models) and graphs on structured spaces (e.g. multiplex networks). The presentation aims to be a complete starting point, including details of both theory and implementation, as well as discussions of the main strengths and weaknesses of each approach. It includes extensive references for readers wishing to go further. The material is carefully structured to be accessible to researchers from all disciplines while also containing rigorous mathematical analysis (largely based on the techniques of statistical mechanics) to support those wishing to further develop or implement the theory of random graph generation. This book is aimed at the graduate student or advanced undergraduate. It includes many worked examples, numerical simulations and exercises making it suitable for use in teaching. Explicit pseudocode algorithms are included to make the ideas easy to apply. Datasets are becoming increasingly large and network applications wider and more sophisticated. Testing hypotheses against properly specified control cases (null models) is at the heart of the ‘scientific method’. Knowledge on how to generate controlled and unbiased random graph ensembles is vital for anybody wishing to apply network science in their research.
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Capítulos de libros sobre el tema "Explicit numerical simulation"

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Li, Pu, Daixing Lu, Robert Schmoll y Bernhard Schweizer. "Explicit Co-simulation Approach with Improved Numerical Stability". En IUTAM Symposium on Solver-Coupling and Co-Simulation, 153–201. Cham: Springer International Publishing, 2019. http://dx.doi.org/10.1007/978-3-030-14883-6_9.

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Ren, Hui y Ping Zhou. "A Fast Explicit Integrator for Numerical Simulation of Multibody System Dynamics". En Multibody Dynamics 2019, 348–55. Cham: Springer International Publishing, 2019. http://dx.doi.org/10.1007/978-3-030-23132-3_42.

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Mota, Magno T., Eduardo de M. R. Fairbairn, Fernando L. B. Ribeiro, Pierre Rossi, Jean-Louis Tailhan, Henrique C. C. Andrade y Mariane R. Rita. "Numerical Simulation of Concrete Fracture by Means of a 3D Probabilistic Explicit Cracking Model". En RILEM Bookseries, 248–57. Cham: Springer International Publishing, 2022. http://dx.doi.org/10.1007/978-3-031-07746-3_25.

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Macedo, Antonini Puppin y Antonio C. P. Brasil. "A Coupled Monte Carlo/Explicit Euler Method for the Numerical Simulation of a Forest Fire Spreading Model". En Monte Carlo and Quasi-Monte Carlo Methods in Scientific Computing, 333–45. New York, NY: Springer New York, 1995. http://dx.doi.org/10.1007/978-1-4612-2552-2_21.

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Zwick, Benjamin, Grand Roman Joldes, Adam Wittek y Karol Miller. "Numerical Algorithm for Simulation of Soft Tissue Swelling and Shrinking in a Total Lagrangian Explicit Dynamics Framework". En Computational Biomechanics for Medicine, 37–46. Cham: Springer International Publishing, 2015. http://dx.doi.org/10.1007/978-3-319-15503-6_4.

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Wheeler, A. B., R. S. Jones y T. N. Phillips. "Numerical Simulation of Fibre Reorientation in a Squeezing Flow and other Flow Geometries using an Explicit Projection Method". En Computer Aided Design in Composite Material Technology III, 177–88. Dordrecht: Springer Netherlands, 1992. http://dx.doi.org/10.1007/978-94-011-2874-2_12.

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Aumann, Quirin, Peter Benner, Jens Saak y Julia Vettermann. "Model Order Reduction Strategies for the Computation of Compact Machine Tool Models". En Lecture Notes in Production Engineering, 132–45. Cham: Springer International Publishing, 2023. http://dx.doi.org/10.1007/978-3-031-34486-2_10.

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AbstractThe deviation of the tool center point (TCP) of a machine tool from its desired location needs to be assessed correctly to ensure an accurate and safe operation of the machine. A major source of TCP deviation are thermal loads, which are constantly changing during operation. Numerical simulation models help predicting these loads, but are typically large and expensive to solve. Especially in (real-time feedback) control settings, but also to ensure an efficient design phase of machine tools, it is inevitable to use compact reduced-order surrogate models which approximate the behavior of the original system but are much less computationally expensive to evaluate. Model order reduction (MOR) methods generate computationally efficient surrogates. Classic intrusive methods require explicit access to the assembled system matrices. However, commercial software packages, which are typically used for the design of machine tools, do not always allow an unrestricted access to the required matrices. Non-intrusive data-driven methods compute surrogates requiring only input and output data of a dynamical system and are therefore independent of the discretization method. We evaluate the performance of such data-driven approaches to compute cheap-to-evaluate surrogate models of machine tools and compare their efficacy to intrusive MOR strategies. A focus is put on modeling the machine tool via individual substructures, which can be reduced independently of each other.
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Taube, Arne, Gregor Gassner y Claus-Dieter Munz. "Explicit One-Step Discontinuous Galerkin Schemes for Unsteady Flow Simulations". En Notes on Numerical Fluid Mechanics and Multidisciplinary Design, 53–65. Berlin, Heidelberg: Springer Berlin Heidelberg, 2010. http://dx.doi.org/10.1007/978-3-642-03707-8_5.

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Taube, Arne, Gregor Gassner y Claus-Dieter Munz. "Cavity Simulations Using an Explicit Discontinuous Galerkin Scheme with Local Time-Stepping". En Notes on Numerical Fluid Mechanics and Multidisciplinary Design, 689–97. Berlin, Heidelberg: Springer Berlin Heidelberg, 2013. http://dx.doi.org/10.1007/978-3-642-35680-3_82.

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Holm, Darryl D., Ruiao Hu y Oliver D. Street. "Geometric Theory of Perturbation Dynamics Around Non-equilibrium Fluid Flows". En Mathematics of Planet Earth, 87–113. Cham: Springer Nature Switzerland, 2024. http://dx.doi.org/10.1007/978-3-031-70660-8_5.

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AbstractThe present work investigates the evolution of linear perturbations of time-dependent ideal fluid flows with advected quantities, expressed in terms of the second order variations of the action corresponding to a Lagrangian defined on a semidirect product space. This approach is related to Jacobi fields along geodesics and several examples are given explicitly to elucidate our approach. Numerical simulations of the perturbation dynamics are also presented.
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Actas de conferencias sobre el tema "Explicit numerical simulation"

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David, Matthew, Rodney Thomson, Thomas Billac, Christof Kindervater, Mark Battley, Tom Allen y Raj Das. "Validation of Numerical Methods for Multi-terrain Impact Simulations of a Crashworthy Composite Helicopter Subfloor". En Vertical Flight Society 70th Annual Forum & Technology Display, 1–13. The Vertical Flight Society, 2014. http://dx.doi.org/10.4050/f-0070-2014-9475.

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Helicopters typically operate over a large variety of terrains. Designing a crashworthy subfloor that can perform efficiently in most impact environments requires the use of accurate simulation tools with a good confidence in the modeling approach and the representation of the soft surfaces. The present work investigates the accuracy of the most commonly used equations of state to numerically model water with explicit Finite Element and Smoothed Particle Hydrodynamics methods. Both analytical and numerical investigations indicated that the Smoothed Particle Hydrodynamics method together with the Murnaghan equation of state provide the best compromise between accuracy and efficiency in predicting the dynamics of a structure impacting onto water. The modeling approach also showed superior accuracy to the more common Finite Element method in representing a soft ground like soil. Based on the use of trapezoidal energy absorbing elements, a composite subfloor was then designed and simulations using the previously validated technique indicated its good crashworthy potential in impact situations over different surfaces.
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Lotfollahi, Mehrdad, Mohammad Mehdi Alinia, Ertugrul Taciroglu, Theodore E. Simos, George Psihoyios, Ch Tsitouras y Zacharias Anastassi. "Inelastic Buckling Simulation of Steel Braces through Explicit Dynamic Analyses". En NUMERICAL ANALYSIS AND APPLIED MATHEMATICS ICNAAM 2011: International Conference on Numerical Analysis and Applied Mathematics. AIP, 2011. http://dx.doi.org/10.1063/1.3637010.

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Dong, Wei y Peng Li. "Parallelizable stable explicit numerical integration for efficient circuit simulation". En the 46th Annual Design Automation Conference. New York, New York, USA: ACM Press, 2009. http://dx.doi.org/10.1145/1629911.1630012.

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Schafer, Nick, Radu Serban y Dan Negrut. "An Investigation on New Numerical Methods for Molecular Dynamics Simulation". En ASME 2007 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference. ASMEDC, 2007. http://dx.doi.org/10.1115/detc2007-35519.

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Explicit integrators have found common use in Molecular Dynamics (MD) simulations because they are easy to implement and work well under many conditions. However, in other classical mechanics applications that require the numerical solution of the equations of motion for complex systems, explicit methods have encountered major difficulties. In these cases, the state of the art relies on implicit methods, which are stable under large time steps and therefore can be used to decrease the number of integration steps necessay for a simulation. This in turn results in an overall reduction of CPU time that opens the door to an increase in the dimension of the problem that can be considered. The premise of this work is that numerical methods that are suitable for efficient simulation of mechanical systems will lead to significant gains when used in MD. The goal of the proposed work is to investigate this assumption by comparing in terms of accuracy and efficiency the Hilber-Hughes-Taylor (HHT) integrator against current explicit MD integrators for a set of two benchmark problems.
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Graf, A. y U. Riedel. "Numerical simulation of supersonic reactive flows using explicit Runge-Kutta methods". En 38th Aerospace Sciences Meeting and Exhibit. Reston, Virigina: American Institute of Aeronautics and Astronautics, 2000. http://dx.doi.org/10.2514/6.2000-438.

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Ozcer, Isik, Guido S. Baruzzi, Miraj Desai y Maged Yassin. "Numerical Simulation of Aircraft and Variable-Pitch Propeller Icing with Explicit Coupling". En International Conference on Icing of Aircraft, Engines, and Structures. 400 Commonwealth Drive, Warrendale, PA, United States: SAE International, 2019. http://dx.doi.org/10.4271/2019-01-1954.

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Chen, M., E. Cormier-Michel, C. G. R. Geddes, David L. Bruhwiler, L. L. Yu, E. Esarey, C. B. Schroeder y W. P. Leemans. "Numerical simulation of laser tunneling ionization in explicit particle-in-cell codes". En 42ND ANNUAL REVIEW OF PROGRESS IN QUANTITATIVE NONDESTRUCTIVE EVALUATION: Incorporating the 6th European-American Workshop on Reliability of NDE. AIP Publishing LLC, 2012. http://dx.doi.org/10.1063/1.4788987.

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Carvalho, Laise Lima De, Creto Augusto Vidal, Joaquim Bento Cavalcante-Neto y Suzana Matos Franca de Oliveira. "Dynamic Cloth Simulation: A Comparative Study of Explicit and Implicit Numerical Integration". En 2012 14th Symposium on Virtual and Augmented Reality (SVR). IEEE, 2012. http://dx.doi.org/10.1109/svr.2012.11.

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Ji-long, Yin. "Simulation of Roll Forming With Dynamic Explicit Finite Element Method". En NUMISHEET 2005: Proceedings of the 6th International Conference and Workshop on Numerical Simulation of 3D Sheet Metal Forming Process. AIP, 2005. http://dx.doi.org/10.1063/1.2011217.

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Hsu, Kwen. "Numerical Performances of Explicit Cartesian Methods for Compressible Moving-Boundary Flow Problems". En ASME 2006 2nd Joint U.S.-European Fluids Engineering Summer Meeting Collocated With the 14th International Conference on Nuclear Engineering. ASMEDC, 2006. http://dx.doi.org/10.1115/fedsm2006-98086.

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Cartesian method, which in some places is mentioned as Volume-Tracking Method, is one of the popular methods used in simulating transient flow problems involving complex moving boundaries. It has the advantage of being time-saving, efficient and robust for certain types of fluid-structure interaction problems. This method is featured by a Cartesian background Euler mesh discretizing the flow domain and a moving surface cutting through it. The most critical operation of this method is treating the cells cut by the moving boundaries accurately and stably. When the Cartesian methods are applied, the temporal discretization of the governing equations of the flow can be either implicit or explicit. For simulations cases in which time-accurately capturing wave propagation or flow evolution is essential, explicit approach still plays an important role among the researchers and currently available simulation codes. The current study is focused on the numerical performances of the explicit type of Cartesian methods when applied on the compressible flow cases. The accuracy of the simulation results, stability and grid-convergence problems resulted from a moving, impermeable boundary cutting through the background mesh are addressed. Example problems include the one-dimensional piston problem and the expanding sphere flow problems. In one case the sphere expands supersonically thus a spherical shock is generated. In another case it expands at a subsonic speed and works as a monople impulse noise source. To the best knowledge of the author, the problem of expanding-sphere generated acoustic impulse has not been reported anywhere else. Simple theoretical analyses are included and results of numerical experiments are reported.
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Informes sobre el tema "Explicit numerical simulation"

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Chung, T. J. Flowfield-Dependent Mixed Explicit-Implicit(FDMEI) Algorithm Toward Direct Numerical Simulation in High Speed Flows. Fort Belvoir, VA: Defense Technical Information Center, julio de 1997. http://dx.doi.org/10.21236/ada329549.

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Nadal-Caraballo, Norberto C., Madison C. Yawn, Luke A. Aucoin, Meredith L. Carr, Jeffrey A. Melby, Efrain Ramos-Santiago, Victor M. Gonzalez et al. Coastal Hazards System–Louisiana (CHS-LA). US Army Engineer Research and Development Center, agosto de 2022. http://dx.doi.org/10.21079/11681/45286.

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The US Army Engineer Research and Development Center (ERDC), Coastal and Hydraulics Laboratory (CHL) expanded the Coastal Hazards System (CHS) to quantify storm surge and wave hazards for coastal Louisiana. The CHS Louisiana (CHS-LA) coastal study was sponsored by the Louisiana Coastal Protection and Restoration Authority (CPRA) and the New Orleans District (MVN), US Army Corps of Engineers (USACE) to support Louisiana’s critical coastal infrastructure and to ensure the effectiveness of coastal storm risk management projects. The CHS-LA applied the CHS Probabilistic Coastal Hazard Analysis (PCHA) framework to quantify tropical cyclone (TC) responses, leveraging new atmospheric and hydrodynamic numerical model simulations of synthetic TCs developed explicitly for the Louisiana region. This report focuses on documenting the PCHA conducted for the CHS-LA, including details related to the characterization of storm climate, storm sampling, storm recurrence rate estimation, marginal distributions, correlation and dependence structure of TC atmospheric-forcing parameters, development of augmented storm suites, and assignment of discrete storm weights to the synthetic TCs. As part of CHS-LA, coastal hazards were estimated within the study area for annual exceedance frequencies (AEFs) over the range of 10 yr-1 to 1×10-4 yr-1.
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Nadal-Caraballo, Norberto, Madison Yawn, Luke Aucoin, Meredith Carr, Jeffrey Melby, Efrain Ramos-Santiago, Fabian Garcia-Moreno et al. Coastal Hazards System–Puerto Rico and US Virgin Islands (CHS-PR). Engineer Research and Development Center (U.S.), diciembre de 2022. http://dx.doi.org/10.21079/11681/46200.

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The South Atlantic Coastal Study (SACS) was completed by the US Army Corps of Engineers to quantify storm surge and wave hazards allowing for the expansion of the Coastal Hazards System (CHS) to the South Atlantic Division (SAD) domain. The goal of the CHS-SACS was to quantify coastal storm hazards for present conditions and future sea level rise (SLR) scenarios to aid in reducing flooding risk and increasing resiliency in coastal environments. CHS-SACS was completed for three regions within the SAD domain, and this report focuses on the Coastal Hazards System–Puerto Rico and US Virgin Islands (CHS-PR). This study applied the CHS Probabilistic Coastal Hazard Analysis (PCHA) framework for quantifying tropical cyclone (TC) responses, leveraging new atmospheric and hydrodynamic numerical model simulations of synthetic TCs developed explicitly for the CHS-PR region. This report focuses on documenting the PCHA conducted for CHS-PR, including the characterization of storm climate, storm sampling, storm recurrence rate estimation, marginal distributions, correlation and dependence structure of TC atmospheric-forcing parameters, development of augmented storm suites, and assignment of discrete storm weights to the synthetic TCs. As part of CHS-PR, coastal hazards were estimated for annual exceedance frequencies over the range of 10 yr⁻¹ to 10⁻⁴ yr⁻¹.
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NUMERICAL ANALYSIS AND EVALUATION OF EFFECTIVE SLAB WIDTH OF COMPOSITE CONTINUOUS BEAMS WITH SEMI-RIGID JOINT. The Hong Kong Institute of Steel Construction, diciembre de 2021. http://dx.doi.org/10.18057/ijasc.2021.17.4.1.

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The characterization of the structural behavior of composite beams is directly affected by the determination of the effective slab width. Various codes propose their own definitions of the effective width based on the beam span and the slab width parameters. However, the evaluation of the effective width may be influenced by other parameters. The aim of this work is to determine the most important factors affecting effective width for continuous composite beams with semi-rigid joints using numerical simulations. A three-dimensional finite element model of a composite continuous beam using explicit-solver available in ABAQUS is developed. The proposed model is validated through comparisons to available experimental results. A modified model is proposed based on the so-validated model to study the influence of the composite beam-column joint stiffness on the effective width. Then, both numerical models are used to perform an extensive parametric study to investigate the influence of various parameters on the estimation of the effective slab width. The influence of slab width, the shear connection degree, and composite joint stiffness are particularly analyzed to find out the most important parameters influencing the effective width so that simplified equations for the calculation of the effective slab width are proposed.
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