Haut PDF Numerical simulation of 3-D flows with a finite element method

Numerical simulation of 3-D flows with a finite element method

Numerical simulation of 3-D flows with a finite element method

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A new three-dimensional mixed finite element for direct numerical simulation of compressible viscoelastic flows with moving free surfaces

A new three-dimensional mixed finite element for direct numerical simulation of compressible viscoelastic flows with moving free surfaces

temperature. To account for compressibility, regarding what was done for vis- cous flows, various numerical methods have been proposed to solve compressible or incompressible problems, for what concerns independent variables, linear sys- tem solvers, and numerical stability. Several authors developed in the past unified computational methods for compressible and incompressible viscous flows [24–30], showing results for a wide range of flow speeds, but in two-dimensional simple geometries. Extensions of low to vanishing Mach number compressible flows to viscoelastic constitutive models have been studied by Webster and co-workers [32, 33, 31] in a very comprehensive work, and were compared to experimental results in [34]. However, these studies were devoted to two-dimensional flows. Since our objective remains the possibility of application of this kind of models for three- dimensional industrial applications, we have extended a previous approach involv- ing weakly compressible flows of generalized newtonian fluids [35] to viscoelastic constitutive models. We split compressible and viscoelastic effects by considering that each brings its contribution to the stress determination: compressibility was taken into account classically through a state equation (the Tait law), whereas viscoelasticity was determined through the evolution of the extra-stress tensor. The proposed formulation for the 3D non-steady flow of a viscoelastic compressible fluid will be presented as follows: in the first part, we outline the material behavior modelling, by drawing the basic features of the multi-mode Pom-Pom model, that has been implemented to treat the non-linear viscoelastic behavior, and how does the state law relates density evolution with pressure and temperature; in the next section, the computational methods used to solve the viscoelastic compressible flow problem are detailed. Two basic problems need to be treated: the presence of convective terms in constitutive equations, and strong non-linearity when we con- sider the coupled problem. A splitting method allowed separate resolution of the evolution equations (through a Space-Time Discontinuous Galerkin finite element method) from the flow equations, using mixed finite elements. Furthermore, exten- sion to free surface, that is briefly described here, increased the complexity of our problem. Validation tests performed on ’benchmark’ geometries (like the contrac- tion or contraction/ expansion flow) are analyzed, and finally, three-dimensional free-surface flows are considered in simple and complex geometries.
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eXtended Stochastic Finite Element Method for the numerical simulation of heterogenous materials with random material interfaces

eXtended Stochastic Finite Element Method for the numerical simulation of heterogenous materials with random material interfaces

1 rue de la Noë, 44321 Nantes, France SUMMARY An eXtended Stochastic Finite Element Method has been recently proposed for the numerical solution of partial dierential equations dened on random domains. This method is based on a mariage between the eXtended Finite Element Method and spectral stochastic methods. In this paper, we propose an extension of this method for the numerical simulation of random multi-phased materials. The random geometry of material interfaces is described implicitly by using random level-set functions. A xed deterministic nite element mesh, which is not conforming the random interfaces, is then introduced in order to approximate the geometry and the solution. Classical spectral stochastic nite element approximation spaces are not able to capture the irregularities of the solution eld with respect to spatial and stochastic variables, which leads to a deterioration of the accuracy and convergence properties of the approximate solution. In order to recover optimal convergence properties of the approximation, we propose an extension of the partition of unity method to the spectral stochastic framework. This technique allows the enrichment of approximation spaces with suitable functions based on an a priori knowledge of the irregularities in the solution. Numerical examples illustrate the eciency of the proposed method and demonstrate the relevance of the enrichment procedure. key words: Stochastic partial dierential equations; Random geometry; Random Level-sets; X- FEM; Spectral Stochastic Methods; Partition of Unity Method
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Numerical simulation of unsteady dense granular flows with rotating geometries

Numerical simulation of unsteady dense granular flows with rotating geometries

Fig. 4 – Sketch of the rotating drum case. As stated in step 2 a non-conformal mesh matching algo- rithm is used after rotation. Indeed, even if the matching between meshes is conformal at the first iteration, it will not be the case at the subsequent ones. Indeed, the rotor mesh will have rotated by an angle which does not necessarily coin- cide with the angular discretization of the meshes. Hence, in order to solve the governing equations, information has to be passed at the non-conforming interface between the stator and the rotor. This is done thanks to a mesh joining algorithm and has to be performed at each time step. The main idea of this method is to intersect the faces of both interfaces and split them so as to build conforming meshes. An illustration of this method is shown in Fig. 3. More information on this procedure can be found in EDF (2015).
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Numerical analysis of the mixed finite element method for the neutron diffusion eigenproblem with heterogeneous coefficients

Numerical analysis of the mixed finite element method for the neutron diffusion eigenproblem with heterogeneous coefficients

Similar results hold on subsets of R, provided the discretizations are conforming. Remark 3. If one chooses another discretization, all results presented hereafter hold provided the estimates (4.20) remain true. For instance, for the RTN [k] finite element defined on tetrahedral triangulations of R, cf. [14, §2.3.1]. To prove (4.20) in this case, one has simply to apply the results of [18, §3.2]. On the other hand, provided that the field q and its divergence are “smooth” in the sense that they belong to PH m+1 (R) for some integer m ≥ 0, using the RTN
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Preliminary assessment of the possibilities of the Particle Finite Element Method in the numerical simulation of bird impact on aeronautical structures

Preliminary assessment of the possibilities of the Particle Finite Element Method in the numerical simulation of bird impact on aeronautical structures

u 0 = 100 m/s. The pressure evolution at the center of the plate is shown in Fig. 5(a). The results obtained with the two geometries are quite similar except for the amplitude of the initial peak, which is much higher in the case of a flat impact surface. This fact has already been discussed in the literature (see e.g. [7]) and could be expected somehow since the impact is less abrupt in the case of an hemispheric impactor. In Fig. 5(b) the results obtained with an hemispheric impact surface are compared to those derived experimentally by Wilbeck [15] for a similar impact speed (117 m /s). The agreement is very good, except for the amplitude of the initial peak. Nonetheless, some considerations are in order. First of all, the correct amplitude of this peak is extremely delicate to evaluate and a value of 12 times the stagnation pressure is not unphysical, as it can be found elsewhere in the literature both in numerical and experimental results (see again [7]). Moreover, in this case the analysis is limited to two dimensions and to an incompressible fluid, while experimental data result, of course, from three-dimensional tests, and from the use of slightly compressible projectiles.
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Calculation of crankshaft twist angle using multibody simulation and finite element method

Calculation of crankshaft twist angle using multibody simulation and finite element method

Yannick Louvigny and Pierre Duysinx Figure 1: Nodal displacements (amplified). Figure 2: Twist angle evolution. isolated from the displacements related to the crankshaft motion. It had been possible because super element methods have been used in addition to the finite element method to model the crankshaft. The crankshaft twist angle, measured between the two crankpin, is then determined. Its value is obtained by calculating the difference of rotation angle between each crankpin central nodes. Figure 2 illustrates the variations of crankshaft twist angle during one complete four-stroke cycle. One notices that the twist angle is proportional to the instant torque produced by the engine as also noted in [3] and [4]. To study the dynamics of the engine in transient situation, simplified torsional models of the crankshaft (including super element models and beam element models), allowing faster simulations, are developed. To validate the accuracy of these crankshaft models, they are used in steady state simulations and the results obtained are compared to the ones obtained with the pure finite element model.
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Numerical analysis of the mixed finite element method for the neutron diffusion eigenproblem with heterogeneous coefficients

Numerical analysis of the mixed finite element method for the neutron diffusion eigenproblem with heterogeneous coefficients

P. Ciarlet Jr. 1 , L. Giret 1 , 2 , E. Jamelot 3 , * and F.D. Kpadonou 1 , 4 Abstract. We study first the convergence of the finite element approximation of the mixed diffusion equations with a source term, in the case where the solution is of low regularity. Such a situation commonly arises in the presence of three or more intersecting material components with different characteristics. Then we focus on the approximation of the associated eigenvalue problem. We prove spectral correctness for this problem in the mixed setting. These studies are carried out without, and then with a domain decomposition method. The domain decomposition method can be non-matching in the sense that the traces of the finite element spaces may not fit at the interface between subdomains. Finally, numerical experiments illustrate the accuracy of the method.
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Numerical simulation of impacts on elastic-viscoplastic solids with the flux-difference splitting finite volume method

Numerical simulation of impacts on elastic-viscoplastic solids with the flux-difference splitting finite volume method

The numerical simulation of impacts on dissipative solids has been and is again mainly per- formed with the classical finite element method coupled with centered differences or Newmark finite difference schemes in time [4]. Though this approach has well-known advantages, clas- sical time integrators introduce high frequency noise in the vicinity of discontinuities which is hard to remove with artificial viscosity without destroying the accuracy of the numerical so- lution. The finite volume method, initially developed for the simulation of gas dynamics [1], has gained recently more and more interest for problems involving impacts on solid media [5, 6, 7, 8]. This method shows some advantages to achieve an accurate tracking of wavefronts; among others (i) continuity of fields is not enforced on the mesh in its cell-centered version, that allows for capturing discontinuous solutions, (ii) the characteristic structure of hyperbolic equations can be introduced within the numerical solution, either through the explicit solution of a Riemann problem at cell interfaces, or in a implicit way through the construction of the numerical scheme, (iii) the amount of numerical viscosity introduced can be controlled locally as a function of the local regularity of the solution, so that to permit the elimination of spurious numerical oscillations while preserving a high order of accuracy in more regular zones.
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A Simple Finite-Volume Method for Compressible Isothermal Two-Phase Flows Simulation

A Simple Finite-Volume Method for Compressible Isothermal Two-Phase Flows Simulation

This work departs from the classical van der Waals type phase change mod- elling which usually requires additional terms related to very small scale effects (see [15]). These are intended to correct the core system intrinsic lack of hyperbolicity. A drawback of this approach is that it requires numerical strategies to use very fine discretizing grids. On the contrary, the present system is fully compliant with standard numerical relaxation techniques for hyperbolic systems although the model equilibrium states are compatible with the equilibium Maxwell points for a van der Waals law.
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Analysis of a finite volume-finite element method for Darcy-Brinkman two-phase flows in porous media

Analysis of a finite volume-finite element method for Darcy-Brinkman two-phase flows in porous media

5. Brenner, K., Canc`es, C., and Hilhorst, D. : Finite volume approximation for an immiscible two-phase flow in porous media with discontinuous capillary pressure. Computational Geosciences 17.3 (2013): 573-597. 6. Brezis, H. : Analyse fonctionnelle: Th´eorie et applications. Vol. 91. Paris: Dunod, 1999. 7. Brinkman, H. C. : A calculation of the viscous force exerted by a flowing fluid on a dense swarm of particles. Applied Scientific Research 1.1 (1949): 27-34.

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On the use of the eXtended Finite Element Method with Quatree/Octree meshes

On the use of the eXtended Finite Element Method with Quatree/Octree meshes

a recursive decomposition of the elements that exceed the prescribed error tolerance. However, this approach leads to so called ”hanging nodes” if an element and its neigh- bors have not the same size. Various approaches have been proposed in classical finite elements in order to enforce the compatibility of the approximation [3, 4, 5, 6]. Here, we consider the case of the eXtended Finite Element Method [7]. This method is part of Partition of Unity finite element methods [1, 7, 8], that generalize classical finite el- ements by enabling to incorporate a priori informations on the nature of the solution. The X-FEM was developed to overcome the necessity of geometrical conformity when using the finite element method. Its first application was in the context of problems ex- hibiting strong discontinuities such as fracture mechanics [7, 9]. From 2D linear elastic fracture mechanics, the method was further extended to 3D [10, 11, 12] and non-linear fracture mechanics [13, 14, 15, 16, 17]. The method was also developed to handle holes [18], material interfaces [18, 19, 20] and flows [21, 22] independently of the finite element mesh. Lately, problems involving stability conditions [15, 23, 24, 25] and error estimation [26, 27, 28, 29, 30, 31] have been considered. Among partition of unity finite element methods, one can cite also the Generalized Finite Element Method (GFEM) [32, 8] that has been used for a wide class of applications. In particular, the use handbook functions makes possible to enrich the approximation in the case where the enrichment functions are not explicitly known [33, 34]. Even if the use of the X-FEM improves the accuracy of the approximation in critical area, this does not eliminate the need for controlling the error which implies remeshing.
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Stabilized finite element method for heat transfer and turbulent flows inside industrial furnaces

Stabilized finite element method for heat transfer and turbulent flows inside industrial furnaces

In the present study, the proposed method aims to overcome this drawback. The main idea is to retain the use of the monolithic formulation and coupling it to some additional features that could allow a better and accurate resolution, in particularly at the interface between the fluid and solid. Recall that the monolithic resolution, based on the levelset approach consists in considering a single grid for both air and solid for which only one set of equations need to be solved. Consequently, different subdomains are treated as a single fluid with variable material properties. One important feature till now is that by solving the whole domain in a fully monolithic way there is no need of empirical data so as to determine the heat transfer coefficient. The heat exchange at the interface is replaced naturally by solving the convective fluid in the whole domain. Note also that different numerical methods introduced in the previous chapters could be used to solve the conjugate and coupled problem without additional efforts. Numerically, the communication between the solid and the fluid is obtained naturally without any further assumption and force modelling. In other words, there is no need for some coupling engines specifically designed to handle data exchange and algorithmic control signals between solid region and fluid region.
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Simulation of a two-slope pyramid made by SPIF using an adaptive remeshing method with solid-shell finite element

Simulation of a two-slope pyramid made by SPIF using an adaptive remeshing method with solid-shell finite element

Name and hardening type Swift parameters Back-stress Swift Isotropic hardening K=472.19MPa; n=0.171; ε 0 =0.001 C x =51.65; X sat =5.3 Figure 6 illustrates two distinct meshes refinements. Due to the square geometry in XY plane (see Figure 5.a) and to benefit the computation time reduction, only half of the sheet is modelled. This simplification also can provide a similar result as a full mesh (Henrard, 2008). The initial refined mesh (reference mesh) is composed by 2048 elements disposed in one layer of RESS finite element in thickness direction. The coarse mesh used with adaptive remeshing method is modelled by 128 elements on the sheet plane with one layer of RESS finite element in thickness direction. However, the nodes at the top layer of both meshes define the contact element layer at the surface. The contact modelling is based on a penalty approach and on a Coulomb law (Habraken and Cescotto, 1998). So, both meshes have two layers of elements (solid-shell + contact element) in thickness direction, and the spherical tool was modelled as a rigid body. Finally, Coulomb friction coefficient between the tool and sheet is set to 0.05 (Henrard et al., 2010) and the penalty coefficient is equal to 1000 [N.mm -3 ].
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Verified predictions of shape sensitivities in wall-bounded turbulent flows by an adaptive finite-element method

Verified predictions of shape sensitivities in wall-bounded turbulent flows by an adaptive finite-element method

a b s t r a c t A Continuous Sensitivity Equation (CSE) method is presented for shape parameters in tur- bulent wall-bounded flows modeled with the standard k–  turbulence model with wall functions. Differentiation of boundary conditions and their complex dependencies on shape parameters, including the two-velocity scale wall functions, is presented in details along with the appropriate methodology required for the CSE method. To ensure accuracy, grid convergence and to reduce computational time, an adaptive finite-element method driven by asymptotically exact error estimations is used. The adaptive process is controlled by error estimates on both flow and sensitivity solutions. Firstly, the proposed approach is applied on a problem with a closed-form solution, derived using the Method of the Man- ufactured Solution to perform Code Verification. Results from adaptive grid refinement studies show Verification of flow and sensitivity solvers, error estimators and the adaptive strategy. Secondly, we consider turbulent flows around a square cross-section cylinder in proximity of a solid wall. We examine the quality of the numerical solutions by performing Solution Verification and Validation. Then, Sensitivity Analysis of these turbulent flows is performed to investigate the ability of the method to deal with non-trivial geometrical changes. Sensitivity information is used to estimate uncertainties in the flow solution caused by uncertainties in the shape parameter and to perform fast evaluation of flows on nearby configurations.
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A second-order cut-cell method for the numerical simulation of 2D flows past obstacles.

A second-order cut-cell method for the numerical simulation of 2D flows past obstacles.

The numerical scheme proposed in this paper is a new cut-cell method. As in [16], the immersed boundary is geometrically represented by using the signed algebraic distance to the obstacle boundary. In fluid-cells, that is mesh cells which are far enough from the immersed boundary, classical centered, second-order finite volume schemes are used. In our approach, the location of the velocity component is, as in [16], adapted to the geometry of cut-cells. However, the discrete pressure is placed at the center of the cartesian cells for both fluid-cells and cut-cells. In the vicinity of the obstacle, second-order interpolations using boundary conditions on the solid boundaries are intro- duced to evaluate the convective fluxes. This results in a local first-order approximation of the nonlinear terms in cut-cells. A pointwise approxima- tion of the viscous terms is used in cut-cells. When boundary conditions on the immersed boundary can be used, a five-point stencil scheme for the viscous term is employed. Otherwise, a six-point first-order approximation is introduced. The resulting linear system is close to the five-point structure symmetric system obtained on cartesian mesh with the MAC scheme. A direct solver, based on a capacitance matrix method, is proposed. The effi- ciency of the solver is similar to the cartesian grid solver obtained with the MAC scheme. The incompressibility of the discrete velocity field is enforced up to the computer accuracy. While first-order truncation errors are locally introduced in the scheme in the cut-cells, a second-order global accuracy is recovered. Note that a similar superconvergence result has been proved by Yamamoto in [19] in the context of elliptic equations.
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Numerical simulation of CAD thin structures using the eXtended Finite Element Method and Level Sets

Numerical simulation of CAD thin structures using the eXtended Finite Element Method and Level Sets

Albeit these modelization issue, the use of state-of-the-art industrial simulation tools such as FEM usually imply tedious meshing steps in the case of large and complex assemblies, with complex and manual meshing steps. The idealisation and simplification of these structures into a mix of 2D and 3D Finite Elements usually takes significantly more time than the analysis itself. This is a major drawback if many complex designs are to be explored: the meshing parametrization may have to be done separately from the CAD parametrization or with complex coding of case-dependent meshing rules. This is why an alternative and pragmatic path is proposed here for the simulation of thin structures. This strategy relies on two main ingredients: (i) the eXtended Finite Element Method (X-FEM) [9] and (ii) the Level-Set method [10]. The eXtended Finite Element Method specificities are used to explore a calculation process that enables a simple automation of the meshing steps. Even though potentially computationally more expensive, the meshing automation allowed by the X-FEM may lead to drastic time reduction of the CAD to mesh process and a much tighter link between CAD and calculated assembly. This process may therefore allow easier and faster design explorations in an industrial context.
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Modelling and numerical simulation of plasma flows with two-fluid mixing

Modelling and numerical simulation of plasma flows with two-fluid mixing

where the c i are the unknown values; we have the analogous for the two-dimension framework. So at each time step, one has to solve a non linear matrix system by an iterative method. Then, using the value of the normal relative velocity −Dn.∇c at the interfaces between two neighbouring cells (n is the unit normal to the interface), one solves the mixing kinetic energy equation (28). It is done with a finite volume scheme; the only technical point is the advection term (2c − 1)ρKD∇c which is handled with an upwind technique which is explicit with respect to the time (in each cell, it depends on the sign of (2c − 1)n.∇c in the neighbouring cells). Of course, the time step has to satisfy a classical stabilty criterium for this upwind technique which is of the type |(2c − 1)Dn.∇c|δt/δx ≤ 1, but this criterium is generally less strong than the standard one related to the Wilkins scheme v ther δt/δx ≤ 1, where v ther is the sound speed.
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Progressive inductor modeling via a finite element subproblem method

Progressive inductor modeling via a finite element subproblem method

3 Université de Lyon, Ampère (CNRS UMR5005), École Centrale de Lyon, F-69134 Écully Cedex, France Abstract - The modeling of inductors is split into a sequence of progressive finite element subproblems. The source fields generated by the coil conductors alone, with a wire representation, are calculated at first via either the Biot-Savart law or finite elements. The associated reaction fields for each added or modified region, mainly the magnetic cores, and in return for the source conductor regions themselves when massive, are then calculated with finite element models. Changes of magnetic regions go from perfect magnetic properties up to volume linear and nonlinear properties. The resulting subproblem method allows efficient solving of parameterized analyses thanks to a proper mesh for each subproblem and the reuse of previous solutions to be locally corrected.
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A finite element/quaternion/asymptotic numerical method for the 3D simulation of flexible cables

A finite element/quaternion/asymptotic numerical method for the 3D simulation of flexible cables

1. Introduction During the last decades, the room available in car vehicles (e.g. in engine compartment) has plummeted because of the rapid development of on-board electronics. As a result, a need for very accurate numerical tools for design has appeared in automotive industry. In the meantime, a fast computation is necessary so that design duration remains suitable for industry. In this context, flexible pieces represent an outstanding challenge since, unlike most of car pieces, they cannot be modeled as rigid body solids in CAD software. This paper focuses on a specific type of flexible piece, namely electrical cables. Cables have a complex struc- ture. A wire is made up of copper filaments wrapped in an elastomer duct. These wires are most of the time gathered in bundles which are themselves surrounded by various protections such as tape, PVC tube … Moreover, the full cable is often constituted of several drifted cable pieces forming a system with a complex geometry.
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