COMPUTATIONAL MECHANICS WCCM VI in conjunction with APCOM’04, 6HSW%HLMLQJ&KLQD ¸ 2004 Tsinghua University Press & Springer-Verlag
0HWULF PDS DQG DQLVRWURSLF PHVK DGDSWDWLRQ IRU VWDWLF DQG PRYLQJVXUIDFHV T. Coupez1*, C. Gruau1, Ch. Pequet1, J. Bruchon2
&(0()(FROHGHV0LQHVGH3DULV805&156%3 6RSKLD$QWLSROLV)UDQFH *,5()8QLYHUVLWpGH/DYDO4XpEHF*.3&DQDGD e-mail:
[email protected],
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[email protected] 1 2
$EVWUDFW Mesh adaptation of anisotropic type is shown as a possible technique to capture and calculate accurately the interfaces between different domains of various nature : fluid-fluid, fluid-gas, fluidstructure. Moving free surfaces within complex fluid flow calculation are also investigated. 3D mesh generation of mixed thin/thick geometry is performed by the way of the natural metric driving an anisotropic mesher. The metric contains the distribution of length in any direction. Multidomain mesh generation is revisited through a mesh adaptation process, thin layer of elements being automatically generated at the interfaces between domain. Dynamic is investigated by considering complex moving free surfaces and moving interfaces in continuation of previous works. Interfaces motion are assumed to be solution of a transport equation. The numerical diffusion is controlled by using an adaptive procedure. The proposed approach does not rely on a particular mesh adaptation technique and example will be run using a genuine 3D anisotropic mesh generator as well as in conjunction with a simple moving node technique. .H\ZRUGV anisotropic mesh, adaptativity, interface refinement, fluid-fluid, fluid-structure ,1752'8&7,21 Unstructured anisotropic meshing consists in building stretched elements rather than isotropic triangles or tetrahedra. It is well established that this kind of mesh may be used both in fluid dynamic and in solid and more particularly when using the finite element method. In continuation of previous work [2], this paper intends to show that anisotropic adaptation of mesh is a possible technique to capture and calculate accurately the interfaces between different domains: solid/solid, solid/fluid, fluid/fluid. Muldidomain mesh generation as well as moving free surfaces within complex fluid flow calculation can be investigated with the help of this technique. We focus first on the 3D mesh generation of mixed thin/thick geometry. Our underlying objective is to produce 3D meshes of thin parts with enough elements in the thickness, without producing a huge number of elements. This goal can be achieved by building anisotropic meshes which are clearly allowed when the solution is locally a shear flow or a poiseuille flow. Under this condition the 3D modelling can be competitive in comparison to 2D1/2 modelling (thin layer approximation), although allowing much more flexibility and generality when simple shear flows combine with complex 3D flow in different regions of industrial relevant geometries. The “natural” metric [3] is introduced in order to drive an anisotropic mesher. The metric construction is done by calculating a conformation tensor which contains the statistical distribution of length in any direction. From any mesh respecting a given boundary, an adaptive procedure leads to a more useful mesh (enough elements in the thickness). Multidomain meshing is another difficult task because of the constraint of inner domain boundaries. It must be adressed in fluid-structure calculation for instance and plays a role on the development of the numerical approach. Using a diffuse interface approach it is possible to relax the boundary constraint and the multidomain mesh generation can be revisited as a mesh adaptation problem. Here again we focus on
the information to be passed to an anisotropic mesher rather than on the mesher methodology which is based on local modification [16,3]. The mesh size map (scalar) that classically drives the isotropic refinement must be generalized by a metric map (tensor) to pilot the anisotropic mesh adaptation. The metric map contains the geometrical information such as the interfaces between subdomains gave rise to a thin layer of stretched elements which favor in return the capture of interfaces. After several iterations between remeshing and “remapping” the multidomain mesh is completed. Dynamic is investigated by considering complex moving free surfaces and moving interfaces in continuation of our previous work [1,2]. Interface motions are assumed to be solution of a transport equation (common underlying approach of VOF, level set, pseudo concentration methods). The approximation error leads to numerical diffusion which depends on the local element size crossed by interfaces. It is well reduced by using a adaptive procedure leading again to anisotropic mesh properties. The proposed approaches do not rely on a particular mesh adaptation technique and example will be run using a genuine 3D anisotropic mesh generator as well as in conjunction with a simple moving node technique. Different kind of 3D applications will be presented including 3D mould filling, fluid structure interaction, interacting bubble growth in a liquid [4]. $35,25,5(),1(0(17$5281'67$7,&685)$&(6
Meshing a calculation domain W consists in three major steps: designing W' s boundary by some CAD tool, generating a first (coarse) mesh of W from the mesh of its boundary and improving this mesh until suitability for the simulation. Focusing on the third step, this section only relies on a priori information, since a posteriori error estimation cannot be used before the mesh enables the simulation.
,QWURGXFLQJ VHYHUDO HOHPHQW OD\HUV WKURXJK WKH WKLFNQHVV In many applications, the first improvement to be performed is refining the mesh in order to get more than one element through the domain thickness. This is commonly adressed by prescribing an element size to the mesh generator. Since the geometry of W may be of a great complexity, such element size could not be uniform over W. Besides, an isotropic element size would lead to a mesh containing too many nodes. Thus, recent developments have been devoted to anisotropic mesh refinement. 3D anisotropy means three different sizes in the thickness direction, in the width direction and in the length direction. These directions and sizes can change from one point of W to another. Such anisotropic refinement can be performed without using any metric map: uni-directional refinement [5] or extrusion from boundary [6] or median surface [7]. But metric driven adaptation is now widely used, relying either on geometrical considerations as in hybrid Delaunay / frontal method [8] or on topological optimizations [9, 3]. In previous works [3], we have introduced a well-suited metric map, called natural metric and denoted by 0 . This metric automatically detects the local anisotropy of W, and especially the thickness direction. In this paragraph, we define the natural metric when the whole calculation domain W is composed of several subdomains W . We denote by ½W the median surface of W [10]. We define the natural metric on [ in ½W by the largest metric 0 such that the unit ball ( 0[ {\such that \ - [ 1 } lies inside W (figure 1 on the left), the metrics 0 being ordered by their shortest eigenvalues. Then, we define the natural metric on x in W by the natural metric of the closest point [ in ½W (if [ is not unique, the averaged natural metric is taken).
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This definition cannot be directly implemented, because it would involve a infinite number of operations. According to [3], an approximation of the natural metric can be constructed in three steps: first, detecting the local anisotropy by aggregation of W ’s elements around each node of W , then, computing elliptic interpolation of these element patches and finally, using the interpolation error as stoping condition. Among all techniques to compute elliptic interpolation 0 of an element patch &, we retain a method based on statistical distribution of length in all directions (figure 1 on the left), which turns out to be particularly robust. In this method, the inverse of 0 corresponds to the 2nd order tensorial approximation of this distribution : 0
Ë Û = m ÌÌ × ( [ - F ) © ( [ - F ) d[ ÜÜ Í Ý
1
(1)
where F is the center of & and is a scaling constant than can be easily optimized. With 0 , Q elements layers can be obtained through the thickness of each subdomain W by multiplying its largest eigenvalue (corresponding to the shortest size) by Q . Figure 2 shows the natural mesh of a simple geometry with two subdomains: a tube (in which Q= 4) completed by a cube (in which Q= 3).
&DSWXULQJ LQWHUIDFHV When an application involves several objects W with boundaries W , the geometry of these objects must be correctly represented by the mesh or the meshes on which the simulation is carried out. The first idea to fullfil this requirement is to generate one mesh of each subdomain W . In this case, the boundary W is correctly designed by the CAD step and well respected by the mesh generation and adaptation steps. However, the main drawback is that many localizations have to be performed between two subdomains in contact during the simulation, which is highly CPU time consuming. In forging process simulation for instance, this technique needs the development of a relevant contact management algorithm [11]. The second idea is to generate one mesh of the global domain W and to impose the surface triangulations of W in this mesh. In this configuration, the sub-meshes of W are perfectly connected (figure 2 in the middle) and no localization has to be performed during the simulation. Nevertheless, the triangulations of W have to be blocked during the CAD step and during the mesh generation and adaptation steps, which represents a great constraint that is not easy to respect.
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In this paragraph, we introduce a third alternative: generating one mesh of the global geometry W and refining this mesh around boundaries W . Again, this technique leads to no localization. Besides, the refinement around W is driven by a metric computed without much human intervention, namely the multidomain metric 0 . Furthermore, whereas interfaces between subdomains is no more exact but fuzzy, the mesh is accurately tighten around these interfaces. To tighten the mesh of W around W , we represent W by a function J such that J = 1 inside, J = 0 outside and 0 < J < 1 around W . For instance, J can be defined by J # ( [) =
1+ H
1
& " (! , % $
(2)
)
with » 1 and G([ W ) is the signed distance between [ and W (i.e. G([ W ) > 0 if [ is outside W
and 0 otherwise). Then, it can be proved [3] that the multidomain metric is 0-
= 0 '() *+ (, + P 2¶J © ¶J
*, ) . /0 - (. '
where P is the number of element layers around W and ¶J = max ¶J 1 . 1
(3)
Since the additional tensor ¶J © ¶J can be anisotropic, the refinement around W is also anisotropic. '
