b UPS-IUT Tarbes
1, rue Lautréamont, F-65016 Tarbes, France
ABSTRACT: Resin Transfer Molding is widely used to produce fiber-reinforced materials. In the process, the resin enters a close mold containing the dry fiber preform. For mold designer, **numerical** simulation is a useful tool to optimize the mold filling, in particular to identify the best positions of the ports and the vents. An issue in mold filling simulation is the front tracking, because the shape of the resin front changes during the flow. In particular, topological changes can appear resulting from internal obstacles dividing the front or multi-injection. A previous approach [1] using the Boundary Element **Method** (BEM) in a moving mesh framework shows the capability of the **method** to compute accuratlely the front propagation at low CPU time. The present paper describes a **method** developed to handle complex shapes, using BEM together with a **Level** **Set** approach. **Numerical** results in two dimensions are presented, assuming a Newtonian non-reactive fluid, and an homogeneous and not-deformable reinforcement. The resin flow in the fibrous reinforcement is modeled using Darcy’s law and mass conservation. The resulting equation reduces to Laplace’s equation considering an isotropic equivalent mold. Laplaces equation is solved at each time step using a constant Boundary Element **Method** to compute the normal velocity at the flow front. It is extended to the fixed grid and next used to feed a **Level** **Set** solver computing the signed distance to the front. Our model includes a boundary element mesher and a Narrow Band **method** to speed up CPU time. The **numerical** model is compared with an analytical solution, a FEM/VOF-based simulation and experimental measurements for more realistic cases involving multiple injection ports and internal obstacles.

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Two numerical models for studying the dynamics of formation and rise of single bubbles in high‐viscosity ionic liquids were implemented using the level‐set method. The models describe [r]

b UPS-IUT Tarbes
1, rue Lautréamont, F-65016 Tarbes, France
ABSTRACT: Resin Transfer Molding is widely used to produce fiber-reinforced materials. In the process, the resin enters a close mold containing the dry fiber preform. For mold designer, **numerical** simulation is a useful tool to optimize the mold filling, in particular to identify the best positions of the ports and the vents. An issue in mold filling simulation is the front tracking, because the shape of the resin front changes during the flow. In particular, topological changes can appear resulting from internal obstacles dividing the front or multi-injection. A previous approach [1] using the Boundary Element **Method** (BEM) in a moving mesh framework shows the capability of the **method** to compute accuratlely the front propagation at low CPU time. The present paper describes a **method** developed to handle complex shapes, using BEM together with a **Level** **Set** approach. **Numerical** results in two dimensions are presented, assuming a Newtonian non-reactive fluid, and an homogeneous and not-deformable reinforcement. The resin flow in the fibrous reinforcement is modeled using Darcy’s law and mass conservation. The resulting equation reduces to Laplace’s equation considering an isotropic equivalent mold. Laplaces equation is solved at each time step using a constant Boundary Element **Method** to compute the normal velocity at the flow front. It is extended to the fixed grid and next used to feed a **Level** **Set** solver computing the signed distance to the front. Our model includes a boundary element mesher and a Narrow Band **method** to speed up CPU time. The **numerical** model is compared with an analytical solution, a FEM/VOF-based simulation and experimental measurements for more realistic cases involving multiple injection ports and internal obstacles.

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Instead of using the approaches mentioned above, an attempt to employ an alternative **method**, i.e. XFEM combined with **level** **set** **method**, is made in the present work. By contrast with the conventional **numerical** methods, XFEM exists to cope with discontinuous problems. Since it was rst developed in 1999 within the eld of fracture mechanics [ 4 , 10 ], XFEM has been extensively and successfully applied in many elds [ 14 ]. The **level** **set** **method** has also proven to be an accurate and robust way to capture various free boundaries (even topologically changed) based on xed grids [ 25 ]. Combining XFEM with **level** **set** **method** is natural, since **level** **set** **method** not only is able to determine where ought to be enriched but also itself is utilized to construct the enrichment function in XFEM [ 34 ]. XFEM combined with **level** **set** **method** was rst applied to solve the classical Stefan problem in 2002 [ 7 , 18 ]. Based on the work done in [ 7 , 18 ], this combined approach has been further investigated and developed on some detailed aspects but still conned within the framework of classical Stefan problem [ 27 , 5 , 9 , 32 , 22 ]. Beyond that, very few research works are found taking the inuence of melt ow into account [ 38 , 33 , 23 ]. In [ 38 ], an externally forced ow described by volume-averaged momentum and continuity equations [ 39 ] was introduced to transport the heat, while the density dierence between phases and buoyancy eect were regarded negligible. The sharp interface was smeared over a certain thickness, so that FEM was utilized to solve the hydrodynamic portion. In [ 33 ], the melt ow driven either by the density dierence between the two phases or by an arbitrary ux at the interface was considered. But the velocity eld is simply constructed by extending the liquid normal speed at the interface over the whole domain, so no hydrodynamic calculation was involved. In [ 23 ], the ow induced by the density dierence between the two phases was modelled by solving Stokes equations. A FEM-void scheme (without any enrichment) was used in the approximation of velocity and pressure elds.

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Fig. 1. Self-assembled colloidal clusters. a) Electron micrograph of a suspension of triplet clusters. Scale bar, 30 μ m. b–e) Close up of doublet, triplet, quadruplet, and quintuplet clusters. Scale bars, 10 μ m. Further details are available in [7] , photograph courtesy of Dr. Joshua Ricouvier.
the use of boundary integral methods (BIM) [8] , e.g. most recently the GGEM-based BIM [9,10] solving the Stokes equations in general geometries. However, it is also possible to use the conventional unsteady, fractional-step/projection-**method** NS solver at low Reynolds number, combined with an interface description **method** [11,12] . The latter approach is more versa- tile, probably less diﬃcult to implement, and enjoys a rich literature of standard **numerical** techniques. Here, in view of a rich range of possible applications and considering also the rapid development of inertial microﬂuidics (where inertial effects are used to better control the ﬂow behavior) we take the approach of simulating the incompressible, two-ﬂuid NS as out- lined in [13] . The splitting procedure proposed in [13] enables the use of fast solvers for the pressure Poisson equation also for large density and viscosity contrasts. The remaining choice then is to be made among the available interface-description methods.

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second-order moments), probabilistic approaches may be considered for the minimization of the average value or the standard deviation of the performance criterion; see [ 48 ] for an overview. These approaches generally rely on very costly sampling strategies, such as Monte-Carlo, or collocation methods, involving a large number of evaluations of the considered cost function and its derivative; see for instance [ 47 ] and the references therein. Linearized approximations of such problems have been proposed [ 44 , 3 ]. In this article, we rely on a simple sampling strategy for the robust worst-case optimization when small uncertainties are expected; this **method** is particularly well-suited in situations where the uncertain data lie in a low-dimensional space. Our approach is guided by the large CPU cost of the **numerical** resolution of systems of the form (2.23) , which makes methods involving a large number of evaluations of the objective function and its derivative totally impractical in our context. The general principle of the **method** is presented in Section 4.1 in an abstract and formal way. Its particular application to deal with robustness with respect to the incoming wavelength and to the geometry of shapes themselves are discussed in Sections 4.2 and 4.3 , respectively.

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tion [8, 9], geometry processing [22], virtual surgery [14] and flow visualization [6, 35].
Pure **level** **set** **method** suffers from **numerical** diffusion because all the computation is performed on finite Eule- rian grids. Sharp geometry features are gradually smoothed and distorted during the evolution, and the interface keeps shrinking due to the large mass loss. This is a severe limita- tion for **level**-**set** based applications such as fluid simulation and flow visualization. Several techniques have been pro- posed to address the problem, such as the Coupled **Level** **Set** and Volume of Fluid **method** (CLSVOF) [33], the Par- ticle **Level** **Set** **method** (PLS) [7, 8], the semi-Lagrangian contouring **method** [1] and the recent Marker **Level** **Set** **method** (MLS) [19, 20]. However, these methods are com- putationally expensive for two reasons: First, the **level** **set** itself is embedded into a grid with higher dimensions, and many iterations over the grid at each time step can be costly. Second, significant computation cost has to be spent on the **level** **set** correction process. Although adaptive representa- tions such as the local **level** **set** [26], the octree decompo- sition [18] and the Hierarchical RLE **level** **set** [15] help to alleviate the computation load, the performance is still lim- ited for real-time applications.

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Figure 2: A 2000-grain 2D equiaxed polycrystal described using five **level** **set** functions, as shown by the 5
different colors.
The above technique however presents the limitation that when two grains belonging to the same container **level** **set** function start touching each other, they coalesce. In [7], a **method** is proposed to avoid this problem: when two different grains belonging to the same container **level** **set** function get closer than a critical distance, one of the two grains is removed from the container **level** **set** function, and placed into another one. However, if the proposed methodology is applicable in the context of regular grids where the connected components (the individual grains) of each container **level** **set** function can easily be extracted, the problem becomes much more complex when dealing with non-uniform FE meshes. Therefore, in this work, it was chosen to delay the onset of grains coalescence by introducing a constraint in the

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k∇ψ(x, t)k = 1, x ∈ Ω,
ψ(x, t) = 0, x ∈ Γ(t). (2) There exist different approaches to solve this equation including the well-known Fast Marching **Method** introduced by Sethian [9] which propagates a front from the interface and ensures directly a gradient equal to unity. Though this approach has been later extended to unstructured meshes [10], its implementation becomes extremely complicated when it comes to consider anisotropic (i.e. obtuse) triangulations [11]. The latter relies on the insertion of **numerical** supports for obtuse triangle and is mentioned as ”cumbersome” in [11]. To our knowledge, this variant is not used in the recent literature. Another major drawback of the Fast Marching **Method** lies in the parallel implementation. More specifically, the algorithm has to be performed several time on each partition to synchronize the values between the processors, which requires significant implementation effort and poor parallel efficiency.

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problem (3.26).
Discussion on topological changes
The optimized shape of Figure 3.22(b) urges to discuss a very important topic, with severe **numerical** implications for our **method**. We can see in this shape the existence of very small holes, the absence of which would result in a severe violation of the thickness constraint. The question that comes naturally in mind, is whether a solution of problem (3.26) could be obtained by perforating the optimized shape with- out the thickness constraint with inﬁnitesimally small holes. This perforation would have an inﬁnitesimal impact on the compliance and it would satisfy the maximum thickness constraint, in the way it has been mathematically formulated. However, from an engineering point of view, this would not be a satisfying solution both because the size of the holes would violate some tooling limitations and also because the ”modulo” ratio used in casting (see Chapter 2) would remain unchanged. If one wants to avoid such tiny holes, one should impose at the same time a constraint on their size. Else, it is inevitable that such holes can appear in the optimized shape, since they are preferable for solving the problem (3.26).

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Motivated by this need, **numerical** modeling has become an extensive field of investigation for the last three decade. Several models based on continuum mechanics and heat transfer has been studied at different scales. Simulations at fibre or tow scale are generally conducted using Stokes’ flow. Due to the complexity of the fibrous network, Representative Elementary Volumes are usually considered. Simulations at part scale are frequently adopted in industry considering an average flow modeled using Darcy’s law.

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Keywords: Convective reactive **level**-**set**, Scaled heaviside function, Anisotropic multi criteria mesh adaptation, Conservative interpolation
1. Introduction
Two-fluid flows is the term given for two fluids with different properties. Two-fluid flows appear in a wide variety of natural processes and industrial applications such as geophysical flows, water waves, drop impacts, micro-fluidics, bio mechanics and many others. These applications typically involve immiscible fluids that are separated by thin layers known as the interface. The interface is the region across which the fluids properties as well as some of the flow variables are subjected to variations and deformations. It is a sharp front where density and viscosity change abruptly. The challenge in the **numerical** simulation of two-fluid flows is how to represent the interface and to model its kinematics.

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1. The size of the elements;
2. The maximum thickness of the part;
3. The number of sub-grids that have to be generated.
This last parameter is linked to the fact that contrary to Moumnassi et al. [17], the different surfaces of the model are not decomposed, which means that the edges and angles of the geometry cannot be represented on coarse meshes by the **Level**-**Set** (which tends to smooth sharp geometrical features). This is illustrated in a simple 2D case in figure 2(a-b). In order to overcome this issue, the finite element mesh is refined near sharp features such as edges or angles, see figure 2(c). A multilevel approach is considered, and the elements containing such features are recursively refined. This approach differs from an octree [20], as no database is created to navigate through the element hierarchy (it avoids the overhead due to the classical octree database). The construction of the **Level**-**Set** mesh is now detailed, the objective is to make it as efficient as possible. The general description of the algorithm is as follows (see figure 3 for an illustration of the different steps): a) Create the STL discretization of the part (to obtain, from a prescribed geometrical accuracy a

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Motivated by this need, **numerical** modeling has become an extensive field of investigation for the last three decade. Several models based on continuum mechanics and heat transfer has been studied at different scales. Simulations at fibre or tow scale are generally conducted using Stokes’ flow. Due to the complexity of the fibrous network, Representative Elementary Volumes are usually considered. Simulations at part scale are frequently adopted in industry considering an average flow modeled using Darcy’s law.

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terms of both the distribution as well as its topology. In section 2 we specify the problem setting, define our objective function that needs to be optimized and describe the notion of a Hadamard shape derivative. In section 3 we introduce the **level** **set** that is going to implicitly characterize our domain and give a brief description of the smoothed interface ap- proach. Moreover, we compute the shape derivatives and describe the steps of the **numerical** algorithm. Furthermore, in Section 4 we compute several examples of multi-layer auxetic material that exhibit negative apparent Poisson ratio in 2D. For full 3D systems the steps are exactly the same, albeit with a bigger computational cost.

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usually done in Continuum Surface Force methods, the full expression of the divergence of the viscous stress tensor ∇.( 2µ D )
is discretized, the viscous stress tensor being defined by 2µ D for a Newtonian fluid, with D the rate-of-deformation tensor. In this approach, contrary to previous work [10] , it seems that the contribution of the normal viscous stresses has not to be
added explicitly to the pressure jump condition at the interface. Valuable **numerical** validations of this **method** are proposed in [26] . However, whether these two approaches are formally equivalent or not remained an open question. In particular,

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The evaporation of droplets is also a topic of interest which is involved in many practical situations. For instance, it is an important step in the description of the combustion in automotive and aircraft engines. The prediction of this complex phenomenon requires an accurate description of the interaction between a cloud of moving and vaporizing droplets and ﬂame fronts. As the dynamics of droplets can be strongly affected by thermal, chemical and collective effects, performing direct **numerical** simulations of such ﬂows with a well-resolved description of the droplets would be a step forward in the description and the understanding of these ﬂames. The evaporation of a droplet spray is also a signiﬁcant phenomenon in the steel industry for cooling systems using liquid jet impingements. This latter involves the description of the interactions of vaporizing droplets with a very hot steel plate. This situation seems even more diﬃcult than spray combustion. Indeed, many complex phenomena arise during the impact of a droplet on a hot wall. For example, in the so-called “Leidenfrost regime”, which occurs when the wall superheat is high, the formation of a thin layer of saturated vapor between the impinging droplet and the hot plate leads to the droplet levitation during its spreading; no contact line is formed during the impact. This regime is quite similar to the ﬁlm boiling regime.

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On use of the thick **level** **set** **method** in 3D quasi-static crack simulation of quasi-brittle material
Alexis Salzman · Nicolas Moës · Nicolas Chevaugeon
Abstract This work demonstrates the 3D capabil- ity of the thick **level** **set** (TLS) **method**, first intro- duced by Moës et al. (Int J Numer Methods Eng 86:358–380, 2011 . doi: 10.1002/nme.3069 ) and later in Stolz and Moës (Int J Fract 174(1):49–60, 2012 . doi: 10.1007/s10704-012-9693-3 ). The thick **level** **set** approach is a non-local damage **method** embedding fracture mechanics discontinuity. Enhanced numer- ical implementation for elastic quasi-brittle materi- als in 2D under quasi-static loading conditions was presented in Bernard et al. (Comput Methods Appl Mech Eng 233–236:11–27, 2012 . doi: 10.1016/j.cma. 2012.02.020 ). The present work focuses on using this enhanced **numerical** implementation in a 3D context. This work adds a new way to construct the crack faces by use of a “double cut algorithm”. The regulariza- tion computation, part of the non-local feature of the model, is also reviewed to improve its accuracy. As 3D models are computationally intensive, CPU aspects are discussed. Five test cases are presented. The first one illustrates the capability of the **method** to deal with crack coalescence, which is quite unique for this kind of simulation. Three other cases point out a comparison with literature examples (**numerical** and experimental) and good agreement is observed. One is a more com- plex example, which deals with an engineering oriented application. This work confirms good performance of A. Salzman ( B ) · N. Moës · N. Chevaugeon

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The evolution of a metallic material’s microstructure along its manufacture influences almost all of its final properties (yield strength, conductivity, elastic limit, corrosion resis- tance, etc.). However, both the exact property/microstructure and microstructure/process relationships are, in themselves, active fields of research with few definite answers and many unanswered questions. For example, while it is known that materials with finer mi- crostructures tend to last longer under fatigue loading, the exact role of secondary phases in both crack initiation and propagation is still the subject of many studies [2]. Here, computer models of crystal plasticity are already contributing elements to try and answer these questions in real-world applications [3–5]. The total effects of deformation and heat- ing on metallic microstructures remain also relatively unknown even though certain results are reproducible. Here too, HPC technologies can have a great impact on the prediction of microstructures and subsequent parametrization of manufacturing chains or in testing working hypotheses of the mechanisms at play [6–10]. However, even if one chooses to use the tools of HPC in order to improve design cycles and gain both time and energy, a **numerical** framework is only as good as the predictive power of the physical laws it is based on.

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Finally, the third benchmark focuses on the rise of a non-spherical bubble in a very viscous liquid as in Refs. [2,6,26] . These three benchmarks consist of axisymmetric computations.
3.1. A spherical bubble rising under gravity
Currently, performing well-resolved simulations of bubbly flows or droplet sprays is a challenge for the direct **numerical** simulation of two-phase flows. Many issues must be overcome to achieve such ambitious targets. One can expect that the development of High Performance Computing hardware and software will allow performing simulations involving a growing number of bubbles or drops. However well-resolved 3D simulations remain costly and a particular attention must be paid to the development and the assessment of accurate **numerical** methods to solve the fluid dynamics equations. In particular, a computation which is efficient to describe accurately the dynamics with half the number of grid cells on a bubble diameter makes possible to compute eight times more bubbles with the same computational resources. Actually, the gain would even be significantly greater since, for explicit methods, a larger time step can be used on a coarser grid. The test-case proposed in this section allows us to assess the different discretizations of the viscous terms. Simulations are performed with an axisymmetric coordinate system. The radial and axial sizes of the computational domain are defined relatively to the bubble radius R bubble : l r = 8R bubble and l z = 4l r . Considering a spherical bubble, reference results on the drag coefficient are provided

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