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1
GUSTAFSSON, BERTIL. « Analysis and Methods in Fluid Mechanics ». International Journal of Modern Physics C 02, no 01 (mars 1991) : 75–85. http://dx.doi.org/10.1142/s0129183191000093.
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When constructing numerical methods for partial differential equations, it is important to have a thorough understanding of the continuous model and the characteristic properties of its solutions. We shall present methods of analysis for determining well-posedness of hyperbolic and mixed hyperbolic-parabolic équations which are applicable to the time-dependent Euler and Navier-Stokes equations. We shall then discuss difference- and finite volume methods and the construction of grids. The geometry of realistic problems is usually such that it is almost impossible to construct one structured grid. One way to overcome this difficulty is to use overlapping grids, where each domain has a structured grid. We discuss stability and accuracy of difference methods applied on such grids. Many problems in physics and engineering are defined in boundary domains, and artificial boundaries are introduced for computational reasons. In some cases one can construct accurate boundary conditions at these open boundaries. We shall indicate how this can be achieved, but we will also point out certain cases where accurate solutions are impossible to be obtained on limited domains. Finally some comments will be given on the difficulties arising when almost incompressible flow is computed. This corresponds to small Mach-numbers, and extra care must be taken when designing numerical methods. The theory will be complemented by numerical experiments for various flow problems in two space dimensions.
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Ramírez-Espinoza, Germán I., et Matthias Ehrhardt. « Conservative and Finite Volume Methods for the Convection-Dominated Pricing Problem ». Advances in Applied Mathematics and Mechanics 5, no 06 (décembre 2013) : 759–90. http://dx.doi.org/10.4208/aamm.12-m1216.
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AbstractThis work presents a comparison study of different numerical methods to solve Black-Scholes-type partial differential equations (PDE) in the convection-dominated case, i.e., for European options, if the ratio of the risk-free interest rate and the squared volatility-known in fluid dynamics as Péclet number-is high. For Asian options, additional similar problems arise when the “spatial” variable, the stock price, is close to zero.Here we focus on three methods: the exponentially fitted scheme, a modification of Wang’s finite volume method specially designed for the Black-Scholes equation, and the Kurganov-Tadmor scheme for a general convection-diffusion equation, that is applied for the first time to option pricing problems. Special emphasis is put in the Kurganov-Tadmor because its flexibility allows the simulation of a great variety of types of options and it exhibits quadratic convergence. For the reduction technique proposed by Wilmott, a put-call parity is presented based on the similarity reduction and the put-call parity expression for Asian options. Finally, we present experiments and comparisons with different (non)linear Black-Scholes PDEs.
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Rakhsha, Milad, Christopher E. Kees et Dan Negrut. « Lagrangian vs. Eulerian : An Analysis of Two Solution Methods for Free-Surface Flows and Fluid Solid Interaction Problems ». Fluids 6, no 12 (16 décembre 2021) : 460. http://dx.doi.org/10.3390/fluids6120460.
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As a step towards addressing a scarcity of references on this topic, we compared the Eulerian and Lagrangian Computational Fluid Dynamics (CFD) approaches for the solution of free-surface and Fluid–Solid Interaction (FSI) problems. The Eulerian approach uses the Finite Element Method (FEM) to spatially discretize the Navier–Stokes equations. The free surface is handled via the volume-of-fluid (VOF) and the level-set (LS) equations; an Immersed Boundary Method (IBM) in conjunction with the Nitsche’s technique were applied to resolve the fluid–solid coupling. For the Lagrangian approach, the smoothed particle hydrodynamics (SPH) method is the meshless discretization technique of choice; no additional equations are needed to handle free-surface or FSI coupling. We compared the two approaches for a flow around cylinder. The dam break test was used to gauge the performance for free-surface flows. Lastly, the two approaches were compared on two FSI problems—one with a floating rigid body dropped into the fluid and one with an elastic gate interacting with the flow. We conclude with a discussion of the robustness, ease of model setup, and versatility of the two approaches. The Eulerian and Lagrangian solvers used in this study are open-source and available in the public domain.
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Issakhov, Alibek, et Medina Imanberdiyeva. « Numerical Study of the Movement of Water Surface of Dam Break Flow by VOF Methods for Various Obstacles ». International Journal of Nonlinear Sciences and Numerical Simulation 21, no 5 (28 juillet 2020) : 475–500. http://dx.doi.org/10.1515/ijnsns-2018-0278.
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AbstractIn this paper, the movement of the water surface is numerically simulated when a dam is broken by the volume of fluid (VOF) method. The mathematical model is based on the Navier–Stokes equations and uses the large eddy simulation turbulent model, describing the flow of an incompressible viscous fluid and the equation for the phase. These equations are discretized by the finite-volume method. Numerical PISO (Pressure-Implicit with Splitting of Operators) algorithm was chosen for numerical solution of this equation system. The movement of the water surface is captured by using the VOF method, which leads to a strict mass conservation law. The accuracy of the three-dimensional model and the chosen numerical algorithm were tested using several laboratory experiments on dam break problem. In each of the problems, the obtained results were compared with the experimental data and several calculations by other authors and in each of the test problems, the developed model showed results close to the experimental data. Comparison of simulation results with experimental data for various turbulent models was also performed. And also two combined problems were performed which are more close to real conditions; with the help of these problems, flooding zones and flooding time were identified that would help in evacuating people from dangerous zones.
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Baliga, Bantwal R. (Rabi), et Iurii Yuri Lokhmanets. « Generalized Richardson extrapolation procedures for estimating grid-independent numerical solutions ». International Journal of Numerical Methods for Heat & ; Fluid Flow 26, no 3/4 (3 mai 2016) : 1121–44. http://dx.doi.org/10.1108/hff-10-2015-0445.
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Purpose – The purpose of this paper is to present outcomes of efforts made over the last 20 years to extend the applicability of the Richardson extrapolation procedure to numerical predictions of multidimensional, steady and unsteady, fluid flow and heat transfer phenomena in regular and irregular calculation domains. Design/methodology/approach – Pattern-preserving grid-refinement strategies are proposed for mathematically rigorous generalizations of the Richardson extrapolation procedure for numerical predictions of steady fluid flow and heat transfer, using finite volume methods and structured multidimensional Cartesian grids; and control-volume finite element methods and unstructured two-dimensional planar grids, consisting of three-node triangular elements. Mathematically sound extrapolation procedures are also proposed for numerical solutions of unsteady and boundary-layer-type problems. The applicability of such procedures to numerical solutions of problems with curved boundaries and internal interfaces, and also those based on unstructured grids of general quadrilateral, tetrahedral, or hexahedral elements, is discussed. Findings – Applications to three demonstration problems, with discretizations in the asymptotic regime, showed the following: the apparent orders of accuracy were the same as those of the numerical methods used; and the extrapolated results, measures of error, and a grid convergence index, could be obtained in a smooth and non-oscillatory manner. Originality/value – Strict or approximate pattern-preserving grid-refinement strategies are used to propose generalized Richardson extrapolation procedures for estimating grid-independent numerical solutions. Such extrapolation procedures play an indispensable role in the verification and validation techniques that are employed to assess the accuracy of numerical predictions which are used for designing, optimizing, virtual prototyping, and certification of thermofluid systems.
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de Boer, Gregory Nicholas, Adam Johns, Nicolas Delbosc, Daniel Burdett, Morgan Tatchell-Evans, Jonathan Summers et Remi Baudot. « Three computational methods for analysing thermal airflow distributions in the cooling of data centres ». International Journal of Numerical Methods for Heat & ; Fluid Flow 28, no 2 (5 février 2018) : 271–88. http://dx.doi.org/10.1108/hff-10-2016-0431.
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Purpose This aim of this work is to investigate different modelling approaches for air-cooled data centres. The study employs three computational methods, which are based on finite element, finite volume and lattice Boltzmann methods and which are respectively implemented via commercial Multiphysics software, open-source computational fluid dynamics code and graphical processing unit-based code developed by the authors. The results focus on comparison of the three methods, all of which include models for turbulence, when applied to two rows of datacom racks with cool air supplied via an underfloor plenum. Design/methodology/approach This paper studies thermal airflows in a data centre by applying different numerical simulation techniques that are able to analyse the thermal airflow distribution for a simplified layout of datacom racks in the presence of a computer room air conditioner. Findings Good quantitative agreement between the three methods is seen in terms of the inlet temperatures to the datacom equipment. The computational methods are contrasted in terms of application to thermal management of data centres. Originality/value The work demonstrates how the different simulation techniques applied to thermal management of airflow in a data centre can provide valuable design and operational understanding. Basing the analysis on three very different computational approaches is new and would offer an informed understanding of their potential for a class of problems.
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Mustafa, Mustafa Abdulsalam, Atheer Raheem Abdullah, Wajeeh Kamal Hasan, Laith J. Habeeb et Maadh Fawzi Nassar. « Two-way fluid-structure interaction study of twisted tape insert in a circular tube having integral fins with nanofluid ». Eastern-European Journal of Enterprise Technologies 3, no 8(111) (30 juin 2021) : 25–34. http://dx.doi.org/10.15587/1729-4061.2021.234125.
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This work deals with fluid-structure interaction (FSI), one of the emerging areas of numerical simulation and calculation. This research shows a numerical study investigating heat transfer enhancement and fluid-structure interaction in a circular finned tube by using alumina nanofluid as a working fluid with a typical twisted tape that has a twisting ratio of 1.85. The studied nanofluid volumes of fraction are φ=0, 3, 5 % under conditions of laminar and turbulent flow. The solution for such problems is based on the relations of continuum mechanics and is mostly done with numerical methods. FSI occurs when the flow of fluid influences the properties of a structure or vice versa. It is a computational challenge to deal with such problems due to complexity in defining the geometries, nature of the interaction between a fluid and solid, intricate physics of fluids and requirements of computational resources. CFD investigations were made based on the numerical finite volume techniques to solve the governing three-dimensional partial differential equations to get the influence of inserted twisted tape and concentration of nanofluid on heat transfer enhancement, friction loss, average Nusselt number, velocity profile, thermal performance factor characteristics, and two-way interaction in a circular tube at laminar and turbulent flow. The governing continuity, momentum and energy transfer equations are solved using Ansys Fluent and Transient Structural. The simulation results show that the deformations of two-way coupling fluctuate from side to side, with 0.004 mm, as maximum amplitude, located at the typical twisted tape center. Heat transfer dissipation improved by adding fins and as Reynolds numbers increase the heat transfer behavior increases.
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Rajapakse, R. K. N. D., et T. Senjuntichai. « Fundamental Solutions for a Poroelastic Half-Space With Compressible Constituents ». Journal of Applied Mechanics 60, no 4 (1 décembre 1993) : 847–56. http://dx.doi.org/10.1115/1.2900993.
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This paper presents a comprehensive analytical treatment of the three-dimensional response of a poroelastic half-space with compressible constituents. General solutions for equations of equilibrium expressed in terms of displacements and variation of fluid volume are derived by applying Fourier expansion, Hankel transforms, and Laplace transforms with respect to the circumferential, radial, and time coordinates, respectively. The general solutions are used to derive a set of fundamental solutions corresponding to circular ring loads and to a fluid source applied at a finite depth below the free surface of a poroelastic half-space. The circumferential variation of the ring loads and the fluid source is described by appropriate trigonometric terms. Fundamental solutions presented in this paper can be used to model arbitrarily distributed loadings and a fluid source as well as a number of other problems encountered in geomechanics and energy resource explorations. In addition, these fundamental solutions can be used as the kernel functions in the development of boundary integral equation methods for a poroelastic half-space. Solutions for buried circular patch loads/fluid sink and concentrated loads/fluid sink are also deduced. Numerical evaluation of the fundamental solutions is also discussed. Comparisons are presented for time domain solutions obtained by using the Stehfest’s and Schapery’s schemes for Laplace inversion. A selected set of numerical solutions are presented to portray the features of consolidation process in six different poroelastic materials under buried patch loads and a sink.
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Ghassemi, H., M. Mansouri et S. Zaferanlouei. « Interceptor hydrodynamic analysis for handling trim control problems in the high-speed crafts ». Proceedings of the Institution of Mechanical Engineers, Part C : Journal of Mechanical Engineering Science 225, no 11 (14 septembre 2011) : 2597–618. http://dx.doi.org/10.1177/0954406211406650.
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In this study, the effects of hydrodynamic interceptors on fast crafts are investigated to find their optimum geometrical characteristics based on numerical methods. Interceptors are vertical blades installed symmetrically at the aft of the craft. They are designed either fixed or variable. In variable mode, interceptors’ heights are adjusted by various mechanisms. They also cause changes in pressure disruption around the craft and especially at the end of the hull. The pressure variations have an effect on draft height and lifting forces which directly results in a better control of trim. Using the computational fluid dynamics, the pressure distribution around the hull and its effects on trim are computed and then discussed. The Reynolds Average Navier–Stokes equations are also applied to model the flow around the fixed flat and wedge craft with an interceptor at different heights. The model is analysed based on finite volume method and SIMPLE algorithm using dynamic mesh. The results show that the interceptor causes an intense pressure rate in its contact point. It also decreases the wet surface of the craft and drag forces coefficient. At last, they lead to a better control of trim. The height of interceptor has an important effect on its efficiency and it should be selected according to the speed of the craft.
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Okano, T., et M. Koishi. « A New Computational Procedure to Predict Transient Hydroplaning Performance of a Tire ». Tire Science and Technology 29, no 1 (1 janvier 2001) : 2–22. http://dx.doi.org/10.2346/1.2135228.
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Abstract “Hydroplaning characteristics” is one of the key functions for safe driving on wet roads. Since hydroplaning depends on vehicle velocity as well as the tire construction and tread pattern, a predictive simulation tool, which reflects all these effects, is required for effective and precise tire development. A numerical analysis procedure predicting the onset of hydroplaning of a tire, including the effect of vehicle velocity, is proposed in this paper. A commercial explicit-type FEM (finite element method)/FVM (finite volume method) package is used to solve the coupled problems of tire deformation and flow of the surrounding fluid. Tire deformations and fluid flows are solved, using FEM and FVM, respectively. To simulate transient phenomena effectively, vehicle-body-fixed reference-frame is used in the analysis. The proposed analysis can accommodate 1) complex geometry of the tread pattern and 2) rotational effect of tires, which are both important functions of hydroplaning simulation, and also 3) velocity dependency. In the present study, water is assumed to be compressible and also a laminar flow, indeed the fluid viscosity, is not included. To verify the effectiveness of the method, predicted hydroplaning velocities for four different simplified tread patterns are compared with experimental results measured at the proving ground. It is concluded that the proposed numerical method is effective for hydroplaning simulation. Numerical examples are also presented in which the present simulation methods are applied to newly developed prototype tires.
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Ehlers, Wolfgang, et Bernd Markert. « A Linear Viscoelastic Biphasic Model for Soft Tissues Based on the Theory of Porous Media ». Journal of Biomechanical Engineering 123, no 5 (25 avril 2001) : 418–24. http://dx.doi.org/10.1115/1.1388292.
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Based on the Theory of Porous Media (mixture theories extended by the concept of volume fractions), a model describing the mechanical behavior of hydrated soft tissues such as articular cartilage is presented. As usual, the tissue will be modeled as a materially incompressible binary medium of one linear viscoelastic porous solid skeleton saturated by a single viscous pore-fluid. The contribution of this paper is to combine a descriptive representation of the linear viscoelasticity law for the organic solid matrix with an efficient numerical treatment of the strongly coupled solid-fluid problem. Furthermore, deformation-dependent permeability effects are considered. Within the finite element method (FEM), the weak forms of the governing model equations are set up in a system of differential algebraic equations (DAE) in time. Thus, appropriate embedded error-controlled time integration methods can be applied that allow for a reliable and efficient numerical treatment of complex initial boundary-value problems. The applicability and the efficiency of the presented model are demonstrated within canonical, numerical examples, which reveal the influence of the intrinsic dissipation on the general behavior of hydrated soft tissues, exemplarily on articular cartilage.
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Ouchi, Hisanao, Amit Katiyar, John T. Foster et Mukul M. Sharma. « A Peridynamics Model for the Propagation of Hydraulic Fractures in Naturally Fractured Reservoirs ». SPE Journal 22, no 04 (8 mai 2017) : 1082–102. http://dx.doi.org/10.2118/173361-pa.
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Summary A novel and fully coupled hydraulic-fracturing model derived from a nonlocal continuum theory of peridynamics is presented and applied to the hydraulic-fracture (HF) propagation problem. It is shown that this modeling approach provides an alternative to finite-element and finite-volume methods for solving poroelastic and fracture-propagation problems. In this paper, we specifically investigate the interaction between an HF and natural fractures (NFs). The peridynamics model presented here allows us to simulate the propagation of multiple, nonplanar, interacting fractures and provides a novel approach to simulate the interaction between HFs and NFs. The model predictions in two dimensions have been validated by reproducing published experimental results where the interaction between an HF and an NF is controlled by the principal-stress contrast and the approach angle. A detailed parametric study involving poroelasticity and mechanical properties of the rock is performed to understand why an HF becomes arrested or crosses an NF. This analysis reveals that poroelasticity, resulting from high fracture-fluid leakoff, has a dominant influence on the interaction between an HF and an NF. In addition, the fracture toughness of the rock, the toughness of the NF, and the shear strength of the NF also affect the interaction between an HF and an NF. We also investigate the interaction of multiple completing fractures with NFs in two dimensions and demonstrate the applicability of the approach to simulate complex fracture networks on a field scale. Finally, the 3D interaction study elucidated that the height of the NF, the position of the NF, and the opening resistance of the NF all have a significant effect on the 3D interaction between an HF and an NF.
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Thompson, Karsten E., Clinton S. Willson, Christopher D. White, Stephanie Nyman, Janok P. Bhattacharya et Allen H. Reed. « Application of a New Grain-Based Reconstruction Algorithm to Microtomography Images for Quantitative Characterization and Flow Modeling ». SPE Journal 13, no 02 (1 juin 2008) : 164–76. http://dx.doi.org/10.2118/95887-pa.
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Summary X-ray computed microtomography (XMT) is used for high-resolution, nondestructive imaging and has been applied successfully to geologic media. Despite the potential of XMT to aid in formation evaluation, currently it is used mostly as a research tool. One factor preventing more widespread application of XMT technology is limited accessibility to microtomography beamlines. Another factor is that computational tools for quantitative image analysis have not kept pace with the imaging technology itself. In this paper, we present a new grain-based algorithm used for network generation. The algorithm differs from other approaches because it uses the granular structure of the material as a template for creating the pore network rather than operating on the voxel set directly. With this algorithm, several advantages emerge: the algorithm is significantly faster computationally, less dependent on image resolution, and the network structure is tied to the fundamental granular structure of the material. In this paper, we present extensive validation of the algorithm using computer-generated packings. These analyses provide guidance on issues such as accuracy and voxel resolution. The algorithm is applied to two sandstone samples taken from different facies of the Frontier Formation in Wyoming, USA, and imaged using synchrotron XMT. Morphologic and flow-modeling results are presented. Introduction Subsurface transport processes such as oil and gas production are multiscale processes. The pore scale governs many physical and chemical interactions and is the appropriate characteristic scale for the fundamental governing equations. The continuum scale is used for most core or laboratory scale measurements (e.g., Darcy velocity, phase saturation, and bulk capillary pressure). The field scale is the relevant scale for production and reservoir simulation. Multiscale modeling strategies aim to address these complexities by integrating the various length scales. While pore-scale modeling is an essential component of multiscale modeling, quantitative methods are not as well-developed as their continuum-scale counterparts. Hence, pore-scale modeling represents a weak link in current multiscale techniques. The most fundamental approach for pore-scale modeling is direct solution of the equations of motion (along with other relevant conservation equations), which can be performed using a number of numerical techniques. The finite-element method is the most general approach in terms of the range of fluid and solid mechanics problems that can be addressed. Finite-difference and finite-volume methods are more widely used in the computational fluid dynamics community. The boundary element method is very well suited for low-Reynolds number flow of Newtonian fluids (including multiphase flows). Finally, the lattice-Boltzmann method has been favored in the porous-media community because it easily adapts to the complex geometries found in natural materials. A less rigorous approach is network modeling, which gives an approximate solution to the governing equations. It requires discretization of the pore space into pores and pore throats, and transport is modeled by imposing conservation equations at the pore scale. Network modeling involves two levels of approximation. The first is the representation of the complex, continuous void space as discrete pores and throats. The second is the approximation to the fluid mechanics when solving the governing equations within the networks. The positive tradeoff for these significant simplifications is the ability to model transport over orders-of-magnitude larger characteristic scales than is possible with direct solutions of the equations of motion. Consequently, the two approaches (rigorous modeling of the conservation equations vs. network modeling) have complementary roles in the overall context of multiscale modeling. Direct methods will remain essential for studying first-principles behavior and subpore-scale processes such as diffusion boundary layers during surface reactions, while network modeling will provide the best avenue for capturing larger characteristic scales (which is necessary for modeling the pore-to-continuum-scale transition). This research addresses one of the significant hurdles for quantitative network modeling: the use of high-resolution imaging of real materials for quantitative flow modeling. We focus in particular on XMT to obtain 3D pore-scale images, and present a new technique for direct mapping of the XMT data onto networks for quantitative modeling. This direct mapping (in contrast to the generation of statistically equivalent networks) ensures that subtle spatial correlations present in the original material are retained in the network structure.
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Hibi, Shigeyuki, Kazuki Yabushita et Takayuki Tsutsui. « Study on incompressible fluid analysis by three-dimensional particle method with finite volume techniques ». EPJ Web of Conferences 269 (2022) : 01020. http://dx.doi.org/10.1051/epjconf/202226901020.
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Simulating violent free-surface flows by particle methods is effective. In recent years, the SPH (Smoothed Particle Hydrodynamics) method and the MPS(Moving Particle Simulation method) method are considered representative especially in the fields of ship engineering and civil engineering but these methods need various artificial parameters in calculation and may have unnatural large numerical oscillation in the calculated pressure values. In order to avoid these problems, the authors have proposed another particle method based on the finite volume techniques. The usefulness of the proposed method in two-dimensional problems has been verified by comparing the pressure fluctuation when a water tank of the two-dimensional model is forcibly oscillated with the corresponding experiment [1]. Therefore, in this research, the authors extended the algorithm so that the calculation formula can be applied to three-dimensional problems. The effectiveness of the calculation is demonstrated by applying to the dam break problem.
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Río-Martín, Laura, Saray Busto et Michael Dumbser. « A Massively Parallel Hybrid Finite Volume/Finite Element Scheme for Computational Fluid Dynamics ». Mathematics 9, no 18 (18 septembre 2021) : 2316. http://dx.doi.org/10.3390/math9182316.
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In this paper, we propose a novel family of semi-implicit hybrid finite volume/finite element schemes for computational fluid dynamics (CFD), in particular for the approximate solution of the incompressible and compressible Navier-Stokes equations, as well as for the shallow water equations on staggered unstructured meshes in two and three space dimensions. The key features of the method are the use of an edge-based/face-based staggered dual mesh for the discretization of the nonlinear convective terms at the aid of explicit high resolution Godunov-type finite volume schemes, while pressure terms are discretized implicitly using classical continuous Lagrange finite elements on the primal simplex mesh. The resulting pressure system is symmetric positive definite and can thus be very efficiently solved at the aid of classical Krylov subspace methods, such as a matrix-free conjugate gradient method. For the compressible Navier-Stokes equations, the schemes are by construction asymptotic preserving in the low Mach number limit of the equations, hence a consistent hybrid FV/FE method for the incompressible equations is retrieved. All parts of the algorithm can be efficiently parallelized, i.e., the explicit finite volume step as well as the matrix-vector product in the implicit pressure solver. Concerning parallel implementation, we employ the Message-Passing Interface (MPI) standard in combination with spatial domain decomposition based on the free software package METIS. To show the versatility of the proposed schemes, we present a wide range of applications, starting from environmental and geophysical flows, such as dambreak problems and natural convection, over direct numerical simulations of turbulent incompressible flows to high Mach number compressible flows with shock waves. An excellent agreement with exact analytical, numerical or experimental reference solutions is achieved in all cases. Most of the simulations are run with millions of degrees of freedom on thousands of CPU cores. We show strong scaling results for the hybrid FV/FE scheme applied to the 3D incompressible Navier-Stokes equations, using millions of degrees of freedom and up to 4096 CPU cores. The largest simulation shown in this paper is the well-known 3D Taylor-Green vortex benchmark run on 671 million tetrahedral elements on 32,768 CPU cores, showing clearly the suitability of the presented algorithm for the solution of large CFD problems on modern massively parallel distributed memory supercomputers.
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Alakashi, Abobaker Mohammed, Hamidon Bin Salleh et Bambang Basuno. « The Implementation of Cell-Centred Finite Volume Method over Five Nozzle Models ». Applied Mechanics and Materials 393 (septembre 2013) : 305–10. http://dx.doi.org/10.4028/www.scientific.net/amm.393.305.
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The continued research and development of high-order methods in Computational Fluid Dynamics (CFD) is primarily motivated by their potential to significantly reduce the computational cost and memory usage required to obtain a solution to a desired level of accuracy. The present work presents the developed computer code based on Finite Volume Methods (FVM) Cell-centred Finite Volume Method applied for the case of Quasi One dimensional Inviscid Compressible flow, namely the flow pass through a convergent divergent nozzle. In absence of the viscosity, the governing equation of fluid motion is well known as Euler equation. This equation can behave as Elliptic or as Hyperbolic partial differential equation depended on the local value of its flow Mach number. As result, along the flow domain, governed by two types of partial differential equation, in the region in which the local mach number is less than one, the governing equation is elliptic while the other part is hyperbolic due to the local Mach number is a higher than one. Such a mixed type of equation is difficult to be solved since the boundary between those two flow domains is not clear. However by treating as time dependent flow problems, in respect to time, the Euler equation becomes a hyperbolic partial differential equation over the whole flow domain. There are various Finite Volume Methods can be used for solving hyperbolic type of equation, such as Cell-centered scheme, Cusp Scheme Roe Upwind Scheme and TVD Scheme. The present work will concentrate on the case of one dimensional flow problem through five nozzle models. The results of implementation of Cell Centred Finite Volume method to these five flow nozzle problems are very encouraging. This approach able to capture the presence of shock wave with very good results.
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Zamolo, Riccardo, Davide Miotti et Enrico Nobile. « Numerical analysis of thermo-fluid problems in 3D domains by means of the RBF-FD meshless method ». Journal of Physics : Conference Series 2177, no 1 (1 avril 2022) : 012007. http://dx.doi.org/10.1088/1742-6596/2177/1/012007.
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Abstract The use of CAE (Computer Aided Engineering) software, commonly applied to the design and verification of a great variety of manufactured products, is totally reliant on accurate numerical simulations. Classic mesh-based methods, e.g., Finite Element (FEM) and Finite Volume (FVM), are usually employed for such simulations, where the role of the mesh is crucial for both accuracy and time consumption issues. This is especially true for complex 3D domains which are typically encountered in most practical problems. Meshless, or meshfree, methods have been recently introduced in order to replace the usual mesh with much simpler node distributions, thus purifying the data structures of any additional geometric information. Radial Basis Function-Finite Difference (RBF-FD) meshless methods have been shown to be able to easily solve problems of engineering relevance over complex-shaped domains with great accuracy, with particular reference to fluid flow and heat transfer problems. In this paper the RBF-FD method is employed to solve heat transfer problems with incompressible, steady-state laminar flow over 3D complex-shaped domains. The required node distributions are automatically generated by using a meshless node generation algorithm, which has been specifically developed to produce high quality node arrangements over arbitrary 3D geometries. The presented strategy represents therefore a fully-meshless approach for the accurate and automatic simulation of thermo-fluid problems over 3D domains of practical interest.
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Ushakov, V. N., et A. A. Ershov. « On the parametric dependence of the volume of integral funnels and their approximations ». Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki 32, no 3 (septembre 2022) : 447–62. http://dx.doi.org/10.35634/vm220307.
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We consider a nonlinear control system in a finite-dimensional Euclidean space and on a finite time interval, which depends on a parameter. Reachable sets and integral funnels of a differential inclusion corresponding to a control system containing a parameter are studied. When studying numerous problems of control theory and differential games, constructing their solutions and estimating errors, various theoretical approaches and associated computational methods are used. The problems mentioned above include, for example, various types of approach problems, the resolving constructions of which can be described quite simply in terms of reachable sets and integral funnels. In this paper, we study the dependence of reachable sets and integral funnels on a parameter: the degree of this dependence on a parameter is estimated under certain conditions on the control system. The degree of dependence of the integral funnels is investigated for the change in their volume with a change in the parameter. To estimate this dependence, systems of sets in the phase space are introduced that approximate the reachable sets and integral funnels on a given time interval corresponding to a finite partition of this interval. In this case, the degree of dependence of the approximating system of sets on the parameter is first estimated, and then this estimate is used in estimating the dependence of the volume of the integral funnel of the differential inclusion on the parameter. This approach is natural and especially useful in the study of specific applied control problems, in solving which, in the end, one has to deal not with ideal reachable sets and integral funnels, but with their approximations corresponding to a discrete representation of the time interval.
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Alakashi, Abobaker Mohammed, et Bambang Basuno. « Comparison between Cell-Centered Schemes Computer Code and Fluent Software for a Transonic Flow Pass through an Array of Turbine Stator Blades ». Applied Mechanics and Materials 437 (octobre 2013) : 271–74. http://dx.doi.org/10.4028/www.scientific.net/amm.437.271.
Texte intégralRésumé :
The Finite Volume Method (FVM) is a discretization method which is well suited for the numerical simulation of various types (elliptic, parabolic or hyperbolic, for instance) of conservation laws; it has been extensively used in several engineering fields. The Finite volume method uses a volume integral formulation of the problem with a finite partitioning set of volumes to discretize the equations [. the developed computer code based Cell-centered scheme and Fluent software had been used to investigate the inviscid Transonic Flow Pass Through an array of Turbine Stator Blades. The governing equation of fluid motion of the flow problem in hand is assumed governed by the compressible Euler Equation. Basically this equation behave as a mixed type of partial differential equation elliptic and hyperbolic type of partial differential equation. If the local Mach number is less than one, the governing equation will behave as elliptic type of differential equation while if the Mach number is greater than one it will behave as hyperbolic type of differential equation. To eliminate the presence a mixed type behavior, the governing equation of fluid motion are treated as the governing equation of unsteady flow although the problem in hand is steady flow problems. Presenting the Euler equation in their unsteady form makes the equation becomes hyperbolic with respect to time. There are various Finite Volume Methods can used for solving hyperbolic type of equation, such as Cell-centered scheme [, Roe Upwind Scheme [ and TVD Scheme [. The present work use a cell centered scheme applied to the case of flow pass through an array of turbine stator blades. The comparison carried out with the result provided by Fluent Software for three different value of back pressure. The developed computer code shows the result close to the Fluent software although the Fluent software use a Time Averaged Navier stokes equation as its governing equation of fluid motion.
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Ji, Qiao-ling, Xi-zeng Zhao et Sheng Dong. « Numerical Study of Violent Impact Flow Using a CIP-Based Model ». Journal of Applied Mathematics 2013 (2013) : 1–12. http://dx.doi.org/10.1155/2013/920912.
Texte intégralRésumé :
A two-phase flow model is developed to study violent impact flow problem. The model governed by the Navier-Stokes equations with free surface boundary conditions is solved by a Constrained Interpolation Profile (CIP)-based high-order finite difference method on a fixed Cartesian grid system. The free surface is immersed in the computation domain and expressed by a one-fluid density function. An accurate Volume of Fluid (VOF)-type scheme, the Tangent of Hyperbola for Interface Capturing (THINC), is combined for the free surface treatment. Results of another two free surface capturing methods, the original VOF and CIP, are also presented for comparison. The validity and utility of the numerical model are demonstrated by applying it to two dam-break problems: a small-scale two-dimensional (2D) and three-dimensional (3D) full scale simulations and a large-scale 2D simulation. Main attention is paid to the water elevations and impact pressure, and the numerical results show relatively good agreement with available experimental measurements. It is shown that the present numerical model can give a satisfactory prediction for violent impact flow.
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Yaghobi Moghaddam, M., S. Z. Shafaei Tonkaboni, M. Noaparast et F. Doulati Ardejani. « A mathematical model to simulate Heap (bio)-leaching process : An exact conceptual model, Homotopy theory and comparative insights with conventional methods ». International Journal of Modeling, Simulation, and Scientific Computing 08, no 01 (10 janvier 2017) : 1750018. http://dx.doi.org/10.1142/s1793962317500180.
Texte intégralRésumé :
This paper attempts to address some nonlinear differential equations which describe main mechanisms governing heap (bio) leaching process as an important metallurgical facility in mining and mineral processing industries. The Homotopy Perturbation Method (HPM), Finite Volume Method and Analytical (Laplace) Method have been employed to provide proper solutions for these equations. Comparison was made between the methods and agreement was close; considering the fact that the proposed solution in comparison with the others provided a remarkable accuracy in dealing with nonlinear problems associated with mining and mineral processing industries. The maximum error of HPM in relation to the analytical solution was 0.02. The numerical finite volume method incorporating a computational fluid dynamics model termed PHOENICS provided rational and accurate results; describing that many chemical and biological processes extremely affect the transportation mechanism of the aqueous compounds in a heap structure and subsequently on the process efficiency. Besides, all solution methods presented to simulate heap leaching process provided valuable information related to the time dependence concentrations of dissolved compounds. The results obtained from this study can be effectively applied to manage the heap leaching costs to make the process feasible.
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Edelmann, P. V. F., L. Horst, J. P. Berberich, R. Andrassy, J. Higl, G. Leidi, C. Klingenberg et F. K. Röpke. « Well-balanced treatment of gravity in astrophysical fluid dynamics simulations at low Mach numbers ». Astronomy & ; Astrophysics 652 (août 2021) : A53. http://dx.doi.org/10.1051/0004-6361/202140653.
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Context. Accurate simulations of flows in stellar interiors are crucial to improving our understanding of stellar structure and evolution. Because the typically slow flows are merely tiny perturbations on top of a close balance between gravity and the pressure gradient, such simulations place heavy demands on numerical hydrodynamics schemes. Aims. We demonstrate how discretization errors on grids of reasonable size can lead to spurious flows orders of magnitude faster than the physical flow. Well-balanced numerical schemes can deal with this problem. Methods. Three such schemes were applied in the implicit, finite-volume SEVEN-LEAGUE HYDRO code in combination with a low-Mach-number numerical flux function. We compare how the schemes perform in four numerical experiments addressing some of the challenges imposed by typical problems in stellar hydrodynamics. Results. We find that the α-β and deviation well-balancing methods can accurately maintain hydrostatic solutions provided that gravitational potential energy is included in the total energy balance. They accurately conserve minuscule entropy fluctuations advected in an isentropic stratification, which enables the methods to reproduce the expected scaling of convective flow speed with the heating rate. The deviation method also substantially increases accuracy of maintaining stationary orbital motions in a Keplerian disk on long timescales. The Cargo–LeRoux method fares substantially worse in our tests, although its simplicity may still offer some merits in certain situations. Conclusions. Overall, we find the well-balanced treatment of gravity in combination with low Mach number flux functions essential to reproducing correct physical solutions to challenging stellar slow-flow problems on affordable collocated grids.
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Kirihara, Soshu, Katsuya Noritake, Satoko Tasaki et Hiroya Abe. « Smart Processing of Solid Electrolyte Dendrites with Ordered Porous Structures for Fuel Cell Miniaturizations ». Additional Conferences (Device Packaging, HiTEC, HiTEN, and CICMT) 2011, CICMT (1 septembre 2011) : 000017–22. http://dx.doi.org/10.4071/cicmt-2011-ta11.
Texte intégralRésumé :
Solid electrolyte dendrites of yttria stabilized zirconia with spatially ordered porous structures were successfully fabricated for fuel cell miniaturizations by using micro patterning stereolithography. Micrometer order ceramic lattices with the coordination numbers 4, 6, 8 and 12 were propagated spatially in computer graphic space. Aspect ratios of the lattice diameters and lengths were designed between 1.0 and 2.0 to value the porosities in higher levels from 50 to 80 %. On the fabrication process, nanometer sized yttria stabilized zirconia were dispersed in to photo sensitive liquid resins at 30 % in volume fraction to obtain thixotropic slurries. The paste material was spread on a grass substrate with 10 μm in layer thickness by using mechanic knife edge movements, and an ultra violet micro pattern was exposed on the surface to create cross sectional solid layer with 2 μm in part accuracy. After the layer stacking process, the ceramic dispersed resin lattices of 100 μm in diameter were obtained exactly. These composite precursors were dewaxed and sintered at 600 and 1500 °C in an air atmosphere, respectively, and the fine ceramic lattices of 98 % in relative density were created. Gaseous fluid profiles and pressure distributions in the formed ceramic lattices with the various coordination numbers and porosity percents were visualized and analyzed by using finite element method. The fabricated solid electrolytes with the extremely high porosities and wide surface areas are expect to be applied to novel electrodes in the compact fuel cells. The smart processing of the solid electrolytes by utilizing computer aided design, manufacturing and evaluation methods will be demonstrated.
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SARMENTO, C. V. S., A. O. C. FONTE, L. J. PEDROSO et P. M. V. RIBEIRO. « From numerical prototypes to real models : a progressive study of aerodynamic parameters of nonconventional concrete structures with Computational Fluid Dynamics ». Revista IBRACON de Estruturas e Materiais 13, no 3 (juin 2020) : 628–43. http://dx.doi.org/10.1590/s1983-41952020000300012.
Texte intégralRésumé :
Abstract The practical evaluation of aerodynamic coefficients in unconventional concrete structures requires specific studies, which are small-scale models evaluated in wind tunnels. Sophisticated facilities and special sensors are needed, and the tendency is for modern and slender constructions to arise with specific demands on their interaction with the wind. On the other hand, the advances obtained in modern multi-core processors emerge as an alternative for the construction of sophisticated computational models, where the Navier-Stokes differential equations are solved for fluid flow using numerical methods. Computations of this kind require specialized theoretical knowledge, efficient computer programs, and high-performance computers for large-scale calculations. This paper presents recent results involving two real-world applications in concrete structures, where the aerodynamic parameters were estimated with the aid of computational fluid dynamics. Conventional quad-core computers were applied in simulations with the Finite Volume Method and a progressive methodology is presented, highlighting the main aspects of the simulation and allowing its generalization to other types of problems. The results confirm that the proposed methodology is promising in terms of computational cost, drag coefficient estimation and versatility of simulation parameters. These results also indicate that mid-performance computers can be applied for preliminary studies of aerodynamic parameters in design offices.
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Kadioglu, Samet Y. « A Second-Order IMEX Method for Multi-Phase Flow Problems ». International Journal of Computational Methods 14, no 05 (22 novembre 2016) : 1750056. http://dx.doi.org/10.1142/s0219876217500566.
Texte intégralRésumé :
We present a fully second order IMplicit/EXplicit (IMEX) time integration technique for solving incompressible multi-phase flow problems. A typical incompressible multi-phase flow model consists of the Navier–Stokes equations plus an interface dynamics equation (e.g., the level set equation). Our IMEX strategy is applied to such a model in the following manner. The hyperbolic terms of the Navier–Stokes equations together with the interface dynamics equation are solved explicitly (Explicit Block) making use of the well-understood explicit numerical schemes [Leveque, R. J. [1998] Finite Volume Methods for Hyperbolic Problems, “Texts in Applied Mathematics”, (Cambridge University Press); Thomas, J. W. [1999] Numerical Partial Differential Equations II (Conservation Laws and Elliptic Equations), “Texts in Applied Mathematics” (Springer-Verlag, New York)]. On the other hand, the nonhyperbolic (stiff) parts of the flow equations are solved implicitly (Implicit Block) within the framework of the Jacobian-Free Newton Krylov (JFNK) method [Knoll, D. A. and Keyes, D. E. [2004] Jacobian-free Newton Krylov methods: A survey of approaches and applications. J. Comput. Phys. 193, 357–397; Saad, Y. [2003] Iterative Methods for Sparse Linear Systems (Siam); Kelley, C. T. [2003] Solving Nonlinear Equations with Newton’s Method (Siam)]. In our algorithm implementation, the explicit block is embedded in the implicit block in a way that it is always part of the nonlinear function evaluation. In this way, there exists a continuous interaction between the implicit and explicit algorithm blocks meaning that the improved solutions (in terms of time accuracy) at each nonlinear iteration are immediately felt by the explicit block and the improved explicit solutions are readily available to form the next set of nonlinear residuals. This continuous interaction between the two algorithm blocks results in an implicitly balanced algorithm in that all nonlinearities due to coupling of different time terms are converged with the desired numerical time accuracy. In other words, we obtain a self-consistent IMEX method that eliminates the possible order reductions in time convergence that is quite common in certain types of nonlinearly coupled systems. We remark that an incompressible multi-phase flow model can be a highly nonlinearly coupled system with the involvement of very stiff surface tension source terms. These kinds of flow problems are difficult to tackle numerically. In other words, highly nonlinear surface terms may remain unconverged leading to time inaccuracies or time order reductions to the first order even though the overall numerical scheme is designed as high order (second-order or higher) [Sussman, M. and Ohta, M. [2009] A stable and efficient method for treating surface tension in incompressible two-phase flow, SIAM J. Sci. Comput. 31(4), 2447–2471; Zheng, W., Zhu, B., Kim, B. and Fedkiw, R. [2015] A new incompressibility discretization for a hybrid particle MAC grid representation with surface tension, J. Comput. Phys. 280, 96–142]. These and few more issues are addressed in this paper. We have numerically tested our newly proposed scheme by solving several multi-phase flow settings such as an air bubble rising in water, a Rayleigh–Taylor instability problem that is initiated by placing a heavy fluid on top of a lighter one, and a droplet problem in which a water droplet hits the pool of water. Our numerical results show that we have achieved the second-order time accuracy without any order reductions. Moreover, the interfaces between the fluids are captured reasonably well.
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Acs, Gabor, Sandor Doleschall et Eva Farkas. « General Purpose Compositional Model ». Society of Petroleum Engineers Journal 25, no 04 (1 août 1985) : 543–53. http://dx.doi.org/10.2118/10515-pa.
Texte intégralRésumé :
Abstract A direct sequential method has been developed to simulate isothermal compositional systems. The solution technique is the same as that of the implicit pressure, explicit saturation (IMPES) method: one pressure is treated implicitly and (instead of the phase saturation) the component masses/moles are treated explicitly. A "volume balance" equation is used to obtain the pressure equation. A weighted sum of the conservation equations is used to eliminate the nonlinear saturation/concentration terms from the accumulation term of the pressure equation. The partial mass/mole volumes are used as "constants" to partial mass/mole volumes are used as "constants" to weight the mass/mole conservation equations. The method handles uniformly a range of cases from the simplified compositional (i.e., black-oil) models to the most complicated multiphase compositional models of incompressible and compressible fluid systems. The numerical solution is based on the integrated finite-difference method that allows one- (1D), two- (2D), and three-dimensional (3D) grids of regular or irregular volume elements to be handled with the same ease. The mathematical model makes it possible to develop modular versatile computer realizations; thus the model is highly suitable as a basis for general-purpose models. Introduction During the last three decades reservoir simulators have been well developed. The enormous progress in computer techniques has strongly contributed to the development of increasingly effective and sophisticated computer models. The key numerical techniques of modeling conventional displacement methods had been elaborated upon by the beginning of the 1970's, and it was possible to develop a single simulation model capable of addressing most reservoir problems encountered. Since the 1970's, however, because of the sharp rise in oil prices, the need for new enhanced recovery processes has forced reservoir-simulation experts to develop newer computer models that account for completely unknown effects of the new displacement mechanisms. The proliferation of recovery methods since the 1970's has resulted in a departure from the single-model concept because individual models tend to be developed to simulate each of the new recovery schemes. This proliferation of models, however, seems to be a less than ideal situation because of the expense involved in the development, maintenance, and applications training for the multiple new models. In addition, when different models are applied to simulate various enhanced recovery methods, no common basis exists to help survey, compare, and thus understand the different recovery mechanisms. The importance of a single, general simulator capable of modeling all or most recovery processes of interest was emphasized by Coats, who worked out a model as a step in this direction. Economic restrictions have also forced various companies to develop multiple-application reservoir models. The multiple-application reservoir simulator (MARS) program presented by Kendall et al. is one realization of the goal: a single program for multiple application. From a mathematical point of view, reservoir simulators consist of a set of partial differential equations and a set of algebraic equations, both with the appropriate initial and boundary conditions. In isothermal cases the partial differential equations, taking into account Darcy's law, describe the mass/mole/normal-volume conservation for each component of the reservoir fluid system. Phase and/or component transport caused by capillarity, gravity, and/or diffusion also can be taken into account. The algebraic equations describe the thermodynamic properties of the reservoir fluid/rock system. The existence of properties of the reservoir fluid/rock system. The existence of local and instant thermodynamic equilibria is a generally accepted assumption of reservoir simulation. This means that the number of mass/mole/normal-volume conservation equations is equal to the number of components used to describe the reservoir fluid/rock system. During the simulation the reservoir examined is divided into volume elements by a 1D, 2D, or 3D grid. Each of the volume elements is characterized by the appropriate reservoir properties and the displacement process is described by properties and the displacement process is described by a series of thermodynamic equilibria for each volume element. The difference between the simulators of conventional and enhanced recovery methods essentially arises from how many components are chosen as a means of appropriately describing the displacement process, and how the thermodynamic equilibria (thermodynamic properties) of the reservoir fluid/rock system are characterized. In cases of conventional technologies a simplified (black-oil) approach of the hydrocarbon system by a pseudogas and a pseudo-oil component generally is accepted, and the pseudo-oil component generally is accepted, and the thermodynamic properties of the given system depend only on the pressure. This approximation made it possible to develop the direct sequential IMPES solution technique, taking into account the advantage of black-oil models wherein the number of components is equal to the number of phases and thus the number of phases is equal to the number of conservation equations. SPEJ P. 543
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Gassner, Gregor J., et Andrew R. Winters. « A Novel Robust Strategy for Discontinuous Galerkin Methods in Computational Fluid Mechanics : Why ? When ? What ? Where ? » Frontiers in Physics 8 (29 janvier 2021). http://dx.doi.org/10.3389/fphy.2020.500690.
Texte intégralRésumé :
In this paper we will review a recent emerging paradigm shift in the construction and analysis of high order Discontinuous Galerkin methods applied to approximate solutions of hyperbolic or mixed hyperbolic-parabolic partial differential equations (PDEs) in computational physics. There is a long history using DG methods to approximate the solution of partial differential equations in computational physics with successful applications in linear wave propagation, like those governed by Maxwell’s equations, incompressible and compressible fluid and plasma dynamics governed by the Navier-Stokes and the Magnetohydrodynamics equations, or as a solver for ordinary differential equations (ODEs), e.g., in structural mechanics. The DG method amalgamates ideas from several existing methods such as the Finite Element Galerkin method (FEM) and the Finite Volume method (FVM) and is specifically applied to problems with advection dominated properties, such as fast moving fluids or wave propagation. In the numerics community, DG methods are infamous for being computationally complex and, due to their high order nature, as having issues with robustness, i.e., these methods are sometimes prone to crashing easily. In this article we will focus on efficient nodal versions of the DG scheme and present recent ideas to restore its robustness, its connections to and influence by other sectors of the numerical community, such as the finite difference community, and further discuss this young, but rapidly developing research topic by highlighting the main contributions and a closing discussion about possible next lines of research.
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Löhner, Rainald, Lingquan Li, Orlando Antonio Soto et Joseph David Baum. « An arbitrary Lagrangian–Eulerian method for fluid–structure interactions due to underwater explosions ». International Journal of Numerical Methods for Heat & ; Fluid Flow, 10 mars 2023. http://dx.doi.org/10.1108/hff-08-2022-0502.
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Purpose This study aims to evaluate blast loads on and the response of submerged structures. Design/methodology/approach An arbitrary Lagrangian–Eulerian method is developed to model fluid–structure interaction (FSI) problems of close-in underwater explosions (UNDEX). The “fluid” part provides the loads for the structure considers air, water and high explosive materials. The spatial discretization for the fluid domain is performed with a second-order vertex-based finite volume scheme with a tangent of hyperbola interface capturing technique. The temporal discretization is based on explicit Runge–Kutta methods. The structure is described by a large-deformation Lagrangian formulation and discretized via finite elements. First, one-dimensional test cases are given to show that the numerical method is free of mesh movement effects. Thereafter, three-dimensional FSI problems of close-in UNDEX are studied. Finally, the computation of UNDEX near a ship compartment is performed. Findings The difference in the flow mechanisms between rigid targets and deforming targets is quantified and evaluated. Research limitations/implications Cavitation is modeled only approximately and may require further refinement/modeling. Practical implications The results demonstrate that the proposed numerical method is accurate, robust and versatile for practical use. Social implications Better design of naval infrastructure [such as bridges, ports, etc.]. Originality/value To the best of the authors’ knowledge, this study has been conducted for the first time.
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Oldenburg, Jan, Finja Borowski, Alper Öner, Klaus-Peter Schmitz et Michael Stiehm. « Geometry aware physics informed neural network surrogate for solving Navier–Stokes equation (GAPINN) ». Advanced Modeling and Simulation in Engineering Sciences 9, no 1 (21 juin 2022). http://dx.doi.org/10.1186/s40323-022-00221-z.
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AbstractMany real world problems involve fluid flow phenomena, typically be described by the Navier–Stokes equations. The Navier–Stokes equations are partial differential equations (PDEs) with highly nonlinear properties. Currently mostly used methods solve this differential equation by discretizing geometries. In the field of fluid mechanics the finite volume method (FVM) is widely used for numerical flow simulation, so-called computational fluid dynamics (CFD). Due to high computational costs and cumbersome generation of the discretization they are not widely used in real time applications. Our presented work focuses on advancing PDE-constrained deep learning frameworks for more real-world applications with irregular geometries without parameterization. We present a Deep Neural Network framework that generate surrogates for non-geometric boundaries by data free solely physics driven training, by minimizing the residuals of the governing PDEs (i.e., conservation laws) so that no computationally expensive CFD simulation data is needed. We named this method geometry aware physics informed neural network—GAPINN. The framework involves three network types. The first network reduces the dimensions of the irregular geometries to a latent representation. In this work we used a Variational-Auto-Encoder (VAE) for this task. We proposed the concept of using this latent representation in combination with spatial coordinates as input for PINNs. Using PINNs we showed that it is possible to train a surrogate model purely driven on the reduction of the residuals of the underlying PDE for irregular non-parametric geometries. Furthermore, we showed the way of designing a boundary constraining network (BCN) to hardly enforce boundary conditions during training of the PINN. We evaluated this concept on test cases in the fields of biofluidmechanics. The experiments comprise laminar flow (Re = 500) in irregular shaped vessels. The main highlight of the presented GAPINN is the use of PINNs on irregular non-parameterized geometries. Despite that we showed the usage of this framework for Navier Stokes equations, it should be feasible to adapt this framework for other problems described by PDEs.
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Chaudry, Mohsin Ali, Christian Woitzik, Alexander Düster et Peter Wriggers. « A multiscale DEM–FEM coupled approach for the investigation of granules as crash-absorber in ship building ». Computational Particle Mechanics, 5 avril 2021. http://dx.doi.org/10.1007/s40571-021-00401-5.
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AbstractThis paper covers a numerical analysis of a novel approach to increasing the crashworthiness of double hull ships. As proposed in Schöttelndreyer (Füllstoffe in der Konstruktion: ein Konzept zur Verstärkung vonSchiffsseitenhüllen, Technische Uni-versitt Hamburg, Hamburg, 2015), it involves the usage of granular materials in the cavity of the double hull ship. For the modeling of this problem, the discrete element method (DEM) is used for the granules while the finite element method is used for the ship’s structure. In order to account for the structural damage caused by collision, a gradient-enhanced ductile damage model is implemented. In addition to avoid locking, an enhanced strain-based formulation is used. For large-scale problems such as the one in the current study, modeling of all granules with realistic size can be computationally expensive. A two-scale model based on the work of Wellmann and Wriggers (Comput Methods Appl Mech Eng 205:46–58, 2012) is applied—and the region of significant localization is modeled with the DEM, while a continuum model is used for the other regions. The coupling of both discretization schemes is based on the Arlequin method. Numerical homogenization is used to estimate the material parameters of the continuum region with the granules. This involves the usage of meshless interpolation functions for the projection of particle displacement and stress onto a background mesh. Later, the volume-averaged stress and strain within the representative volume element is used to estimate the material parameters. At the end, the results from the combined numerical model are compared with the results from the experiments given in Woitzik and Düster (Ships Offshore Struct 1–12, 2020). This validates both the accuracy of the numerical model and the proposed idea of increasing the crashworthiness of double hull vessels with the granular materials.
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Meindlhumer, Martin, Astrid Pechstein et Bernhard Jakoby. « Mixed finite elements applied to acoustic wave problems in compressible viscous fluids under piezoelectric actuation ». Acta Mechanica, 29 avril 2022. http://dx.doi.org/10.1007/s00707-022-03195-6.
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AbstractIn the present contribution, we develop a mixed finite element method capable of the coupled multi-field simulation of a viscous fluid actuated by a piezoelectric resonator. Several challenges are met with in this setting, among which are the necessity of correct interface coupling, near incompressibility of the fluid, adverse geometric dimensions of flat piezoelectric transducers and different length scales of shear and pressure wave. Assuming small deformations and velocities, we present a mixed variational formulation with consistent interface coupling conditions in (mechanic) frequency domain. Both fluid and piezoelectric solid domain are discretized using Tangential-Displacement Normal-Normal-Stress elements. These elements model not only the deformation, but add an independent tensor-valued stress approximation. The method has been rigorously proven to be free from shear locking for flat prismatic or hexahedral elements. Thus, modeling of the flat geometry of piezoelectric resonators as well as resolution of the fastly decaying shear wave are facilitated. To circumvent the problem of volume locking due to the near incompressibility of the fluid, an additional independent pressure field is introduced. We present computational results indicating the capability of the method.