In this section, the convergence behavior of the iterative surface meshing process without parameter correction is investigated. The theory for cubic spline approximation with constant knot spacing shows that the error is of the order O(h4)for the approximation of an at least C4-continuous function, see for instance [74]. This also holds for tensor product surfaces. Since the B-spline patches of the wing-fuselage and the wing-winglet-fuselage configuration are C2-continuous at most, the surface of a sphere (C∞) is used for the convergence study first. With a radius of r = 1 and the origin as the center, the sphere is defined by
x2+ y2+ z2 = 1 . (4.49)
For the approximation of this sphere, four different initial polyhedra for the iterative process are used: a cube, a tetrahedron, a polyhedron composed of two pentagons and five quadrilaterals (called poly5 in the following) and a polyhedron composed of two hexagons and six quadrilaterals (called poly6 in the following). The four initial poly-hedra are depicted in Figure 4.24, whereas Figure 4.25 shows them after three approx-imations of the sphere. The cube has to be subdivided once before the Catmull-Clark limit surface can be computed. For the other three polyhedra, two prior subdivisions have to be applied. The projection of a limit point onto the sphere is given by the
(a) cube (b) tetrahedron (c) poly5 (d) poly6
Figure 4.24: Initial polyhedra for the approximation of the surface of a sphere.
(a) cube (b) tetrahedron (c) poly5 (d) poly6
Figure 4.25: Approximated sphere surfaces after three iterations.
intersection of the straight line through the origin and the limit point with the surface of the sphere: due to the radius r = 1, the projected limit point is given by
lsi = li
klik2 . (4.50)
In the following, k denotes the current subdivision level and k = 0 is the first level at which the computation of limit points is possible. For the convergence study, the maximum error
e(k)max= max
i kl(k)i − ls,(k)i k2 (4.51)
and the average error
e(k)avg = 1
|Lk|
|LXk|−1 i=0
kl(k)i − ls,(k)i k2 (4.52)
for i = 0, . . . , |Lk| − 1 with |Lk| = |Vk| + |Fk| + |Ek| are computed. The results for the approximation of the sphere are listed in Table 4.4 for the maximum error and in Table 4.5 for the average error. The CGLS tolerance for this test has been set to ε = 10−12. The errors e(0)max and e(0)avg depend on the initial positions of the polyhedron vertices since no prior projection onto the sphere has been computed. Hence, these results are omitted in the tables. The numerical order of convergence p can be estimated by
(hp)l−m = e(l)
e(m) ⇔ p =
log
e(l) e(m)
l−m1
log (h) (4.53)
for l > m, where h denotes the change of the increment from subdivision level k to the next level k + 1, i.e.,
h = h(k+1)
h(k) . (4.54)
Since the Catmull-Clark limit surface converges to uniform bicubic B-spline patches, the increment h(k) conforms to the constant interval between two knots. This interval is halved with each subdivision such that h ≈ 12. For a realistic estimation, the level m should be taken as the level from which on the reduction factor does not change largely anymore. With m = 4 and l = 8, the numerical orders of convergence regarding the maximum error result in pcube,max = 2.56, ptetra,max = 2.57, ppoly5,max = 1.73 and ppoly6,max = 1.57. For the average errors with m = 3 and l = 8, the estimations are pcube,avg = 4.08, ptetra,avg = 4.20, ppoly5,avg = 3.85 and ppoly6,avg = 3.64. The main difference between the approximation starting from a cube and the one starting from a tetrahedron is the distribution of the eight extraordinary vertices of valence n = 3, leading to the same order of convergence regarding the maximum error and a slight improvement regarding the average error for the tetrahedron. The other two examples show that the order of convergence decreases, the higher the valence of the involved extraordinary vertices is.
k e(k)cube,max e(k)tetra,max e(k)poly5,max e(k)poly6,max 1 7.91· 10−3 5.05· 10−3 2.02· 10−3 3.77· 10−3 2 4.28· 10−4 18.48 5.35· 10−4 9.44 3.93· 10−4 5.14 1.05· 10−3 3.59 3 3.38· 10−5 12.66 2.55· 10−5 20.98 1.16· 10−4 3.39 3.36· 10−4 3.13 4 7.06· 10−6 4.79 4.97· 10−6 5.13 3.45· 10−5 3.36 1.11· 10−4 3.03 5 1.21· 10−6 5.83 8.43· 10−7 5.90 1.04· 10−5 3.32 3.73· 10−5 2.98 6 2.04· 10−7 5.93 1.41· 10−7 5.98 3.15· 10−6 3.30 1.25· 10−5 2.98 7 3.43· 10−8 5.95 2.37· 10−8 5.95 9.53· 10−7 3.31 4.21· 10−6 2.97 8 5.76· 10−9 5.95 3.98· 10−9 5.95 2.88· 10−7 3.31 1.41· 10−6 2.99 Table 4.4: Maximum errors and reduction factors for the approximation of the surface
of a sphere.
k e(k)cube,avg e(k)tetra,avg e(k)poly5,avg e(k)poly6,avg
1 2.83· 10−3 9.85· 10−4 1.92· 10−4 2.40· 10−4
2 1.19· 10−4 23.78 9.81· 10−5 10.04 1.22· 10−5 15.74 1.59· 10−5 15.09 3 8.23· 10−6 14.46 4.42· 10−6 22.19 8.93· 10−7 13.66 1.35· 10−6 11.78 4 5.02· 10−7 16.39 2.24· 10−7 19.73 6.05· 10−8 14.76 1.05· 10−7 12.86 5 2.91· 10−8 17.25 1.17· 10−8 19.15 4.09· 10−9 14.79 8.25· 10−9 12.73 6 1.69· 10−9 17.22 6.41· 10−10 18.25 2.83· 10−10 14.45 6.60· 10−10 12.50 7 9.89· 10−11 17.09 3.65· 10−11 17.56 2.01· 10−11 14.08 5.38· 10−11 12.27 8 5.88· 10−12 16.82 2.14· 10−12 17.06 1.45· 10−12 13.86 4.42· 10−12 12.17 Table 4.5: Average errors and reduction factors for the approximation of the surface of
a sphere.
The results for the average error are in good agreement with the expectation arising from the theory (order O(h4), see above).
For the convergence study of the approximation of the fuselage and the wing-winglet-fuselage configuration, the improved initial polyhedra according to Figure 4.11 are used as the particular starting point k = 0. The surface points ls,(k)i are the limit points which have been projected onto the given B-spline surface, i.e., ls,(k)i = x(u(k)i , vi(k)). The convergence results of the iterative surface meshing process are given in Table 4.6. Since the given B-spline patches are C2-continuous at most, the numer-ical orders of convergence are expected to be lower than for the approximation of the surface of a sphere. Aside from that, the interventions for the improvement of the
pro-k e(k)wfc,max e(k)wwfc,max e(k)wfc,avg e(k)wwfc,avg 1 6.70· 10−3 3.98· 10−3 7.67· 10−4 9.34· 10−4
2 1.14· 10−3 5.88 2.83· 10−3 1.41 3.55· 10−5 21.61 7.26· 10−5 12.87 3 4.57· 10−4 2.49 5.55· 10−4 5.10 3.86· 10−6 9.20 1.19· 10−5 6.10 4 1.36· 10−4 3.36 2.03· 10−4 2.73 4.76· 10−7 8.11 1.59· 10−6 7.48 5 6.50· 10−5 2.09 8.72· 10−5 2.33 9.48· 10−8 5.02 2.07· 10−7 7.68 6 3.18· 10−5 2.04 5.33· 10−5 1.64 2.36· 10−8 4.02 5.34· 10−8 3.88 Table 4.6: Maximum errors, average errors and corresponding reduction factors for the approximation of the wing-fuselage configuration (wfc) and the wing-winglet-fuselage configuration (wwfc).
jection results according to (4.15) and (4.17) lead to a further loss in the convergence rate. This additional loss is highly acceptable since the significant improvement of the projection results is much more important. These circumstances make the evaluation or prediction of the convergence behavior difficult, but the numerical orders of conver-gence regarding the maximum and the average error seem to converge to the values pwfc,max ≈ pwwfc,max≈ 1 and pwfc,avg ≈ pwwfc,avg ≈ 2.
In general, both the maximum and the average error can be reduced slightly by applying parameter correction steps, especially in the beginning during the first one or two iterations. Later, the errors with and without parameter correction equalize more and more. The order of convergence stays the same.
Generation
So far, methods for the generation of a surface mesh as a control mesh for a Catmull-Clark surface well approximating a given B-spline surface have been described. Such a surface represents an object placed in a fluid flow. For the numerical simulation of the flow field, a volume mesh, which fulfills certain requirements, is needed. These require-ments are listed in Section 5.1. Sections 5.2 and 5.3 demonstrate how a volume mesh is constructed by applying fast semiautomatic algorithms which have been developed in this work. The results are again illustrated using the wing-fuselage configuration. In addition, an example of the volume mesh generation around an airplane engine docu-ments that the introduced methods can be easily applied to other geometries, too. The conversion of an obtained volume mesh into a B-spline mesh is described in Section 5.4.
5.1 Volume Mesh Requirements and Properties
The goal of the mesh generation process developed and implemented in this work is to end up with a block-structured B-spline volume mesh. Several advantages arise from using B-spline meshes for numerical flow simulations, cf. [75]:
• B-spline volume meshes are well-suited for adaptation based on nested mesh hi-erarchies. A nested mesh hierarchy is given if each cell of a coarse adaptation level is the union of the corresponding cells on the next finer level such that the volume of the coarse cell is exactly the same as the sum of volumes of the finer cells. This requirement can be a drawback of an adaptation method since the coarse cells have to define the shape of the mesh and make the accurate discrete representation of geometries such as the wing-fuselage configuration difficult or even impossible. The solution is the use of curvilinear nested meshes, e.g., para-metric mappings which are defined by trivariate B-spline tensor product volumes.
Hence, mesh refinements can be applied by function evaluation. This leads to the next advantage:
• The mesh generation process does not depend on any discretization parameters, e.g., the number of mesh points or a special point distribution obtained by using stretching functions.
• The evaluation of B-splines is fast and numerically stable such that mesh adap-tations can be calculated efficiently.
• Due to the good approximation properties of B-splines, only a few parameters have to be stored to obtain meshes of high quality.
• Moving and deforming meshes, necessary for fluid-structure interaction, can be modeled easily since only a few mesh control points have to be changed.
As the first advantage in the list above implicates, the final B-spline volume mesh has to consist of single blocks, each one being defined by its own parametric mapping which represents a coordinate system in that block. Hence, the final result will be a block-structured volume mesh.
Constructing B-spline volume meshes also fulfills the requirement that a mesh has to be suitable for finite volume flow solvers since quantities such as the cell volumes can be computed, even though the use of curvilinear edges and nonplanar faces makes the computation more complex.
Another requirement results from a physical phenomenon: Viscous flows demand the mesh to be of particular high quality close to the object because the viscous effects mainly appear in the boundary layer which can become very thin, e.g., for high Reynolds number flows (see Section 6.1 for an explanation of the Reynolds number). Hence, in this work, the volume mesh generation process is divided into two parts: at first, a body-fitted offset mesh is constructed, followed by the generation of a surrounding far-field mesh. The methods which have been developed and implemented to realize these two meshing steps are discussed in Sections 5.2 and 5.3.