Mesh generation over curved surfaces
4.2 PARAMETRIC MAPPING METHOD
4.2.1 Introduction
4.2.1.1 The mapping ϕ from planar domain Ω to the surface S
Let ϕ be the mapping from a two-dimensional parametric domain 2 onto a curved surface S 3 in the three-dimensional space such that ϕ: Ω → S.
Figure 4.6 Quadrilateral mesh by merging of triangles.
Figure 4.7 Quadrilateral mesh by QMorph scheme.
2D planar domain Ω can be readily meshed by the methods, as discussed in Chapter 3, and when points on Ω are mapped by ϕ onto S in 3D space, an FE mesh is created on the surface S.
However, in general ϕ is not isometric (length-preserving), and the edges and hence the shape and the area of the triangular and quadrilateral elements would change by the mapping ϕ.
As a result, the quality of well-shaped triangles and quadrilaterals in Ω may deteriorate upon the transformation from Ω to S. The geometrical approximation may not be adequate either, as straight line segments are used in places of large curvatures on the surface.
4.2.1.2 Gap between a triangular facet and the curved surface
Let S be the curved surface of parameterisation ϕ(u,v) with parametric variables (u,v) on planar domain Ω, and T be a triangle in Ω with vertices mapped on the surface S. The gap between the triangle and the surface S, denoted by δ, is given by Filip et al. (1986)
δ= φ −
∈
Sup u v T u v
u v T
( , ) ( , ) ( , )
where T(u,v) is a point on triangle T, and ϕ(u,v) is the corresponding point on surface S.
δ ≤2 + +
Hence, the gap between the surface and a triangular facet is governed by the longest edge of the triangle and the second derivatives of the surface.
4.2.1.3 Metric for curved surface geometry
The idea to be explored in detail is the design of a mesh generated on the parametric planar domain such that the resulting mesh on surface S upon the transformation by ϕ is a close geometrical approximation to the surface. This characteristic can be achieved by controlling the mesh construction on the parametric space ensuring that the gap between an edge and the surface is within any specified tolerance.
4.2.2 Fundamental forms and the related metric
ϕ is a mapping from Ω to S and is of class C2. For curved surfaces with ridges, the C2 require-ment can be relaxed for a finite number of points, where the surface characteristics can be estimated or interpolated from the neighbouring points (Lee and Lee 2003; Clemencon et al. 2006). Two fundamental forms are defined at every point p ∈ S allowing the length of a curve on S and the curvature along the curve to be determined.
4.2.2.1 Tangent and normal vectors
p = ϕ(u,v). The tangent plane Tp at p is spanned by the basis (co-ordinate) vectors.
u p v p
=∂
∂ = ∂
∂ = =∂
∂ = ∂
∂ =
u u and
v v
u v
φ φ φ
φ
The unit normal to the surface S at p is given by
n u v
= u v×
×
Hence, (u, v, n) represents a local base at p, as shown in Figure 4.8.
As u and v are basis vectors of Tp, every vector V on Tp can be written as V=αu+βvwithα β, ∈
and the first fundamental form Φ1 is given by Φ1 = V ⋅ V = Eα2 + 2Fαβ + Gβ2
where
E = u ⋅ u, F = u ⋅ v, G = v ⋅ v
Let Λ= α = ⋅ =Λ Λ
β and M E F
F G then V V TM .
Matrix M, which measures the length of vector V, is symmetric and positive-definite called the tangent plane metric at p. By means of metric M, the length of a line segment created on the parametric space can be evaluated. As shown in Figure 4.8, a curve Γ on S is defined by
γ: (u(t), v(t)) → S t ∈ [a,b]
u v
p u
n v S
Γ
C Ω
φ
I: t [a,b]
γ
ξ
Figure 4.8 Curved surface S produced by parametric mapping ϕ.
The length of curve Γ is given by
In case C is a line segment from point A to point B, we have
ξ( )t =A tAB+ , t∈[ , ],0 1 ξ =AB and L( )Γ = AB M A tAB AB dtT ( + )
00
∫
1Geometric interpretation of M: Let A be a point on Ω and M(ϕ(A)) be the metric at point A. For arbitrary real value ε, the locus of point B on Ω such that
AB M A ABT ( ( ))φ =ε2
is an ellipse. In other words, an ellipse on Ω will map to a circle on Tp of S and vice versa. In general, an ellipse on Ω will map to an ellipse of different size and orientation on the tangent plane of S, and by controlling the shape and size of the elements on Ω, meshes of various characteristics on S can be created.
4.2.3 Principal curvatures
Principal curvatures (the maximum and minimum curvatures at a point) are of fundamen-tal importance to the curved surface under consideration, which can also be used to define natural characteristic lines on the surface. Let V = αu + βv be a vector on the tangent plane Tp of S at p and n be the unit normal at p. Set
The second fundamental form, which is the projection of n onto the second derivatives of ϕ, is given by
Φ2 = Lα2 + 2Mαβ + Nβ2
The curvature κ of S at p associated with tangent vector V = αu + βv is given by
Relative to the basis vectors u and v, the principal curvature at p is given by
∂
In case all the coefficients a, b and c of the quadratic equation are all zero, Φ2 is propor-tional to Φ1 along any direction at p, and the curvature is constant in any direction (locally spherical). In such a case, the curvature is given by
κ =L E
In general, the quadratic equation gives two distinct roots λ1 and λ2 such that the orthogo-nality relationship holds, i.e. V1⋅ V2 = 0, where V1 and V2 are tangent vectors in the cur-vatures and ρ1 and ρ2 be the corresponding principal radii of curvatures. Then
ρ κ ρ
The surface turns more rapidly at points of large curvature or small radius of curvature where finite elements of reduced size have to be used to approximate the surface geometry.
4.2.3.1 Gaussian curvature and mean curvature
The Gaussian curvature K at a point on a surface is the product of the principal curva-tures, and its value depends only on the geometrical shape of the surface but not on the way it is embedded in space. On the other hand, the mean curvature H is the arithmetic mean of the two principal curvatures. Like the principal curvatures, the Gaussian curva-ture and the mean curvacurva-ture are intrinsic properties of a curved surface at a point, and the relationships between principal curvatures, Gaussian curvature and mean curvature are given by
K= and H= +
κ κ κ κ
1 2 1 2
2
Based on these relationships, the principal curvatures can also be determined as follows:
K LN M
Classification of surface by means of Gaussian curvature K (Smith and Farouki 2001):
K > 0: elliptic, K < 0: hyperbolic, K = 0: parabolic, κ1 = κ2: umbilic point
Most surfaces consist of regions of positive Gaussian curvature of elliptical points and regions of negative Gaussian curvature of hyperbolic points separated by curves of points with zero Gaussian curvature of parabolic lines.
4.2.4 Metric and principal curvatures
Let V be a vector on the tangent plane Tp making an angle θ with ˆv1. V can be written as V= V(ˆ cosv1 θ+vˆ2sin )θ. The curvature along vector V with angle θ from ˆv1 is given by κ = κ1 cos2 θ + κ2 sin2 θ
This result also suggests that the surface curvature is a second-order tensor following the usual rule of tensor transformation with respect to angle θ. In terms of tensor notation, the surface curvature tensor κ is written as
κκ = κ1v vˆ ˆ1 1+κ2 2 2v vˆ ˆ
Curvature along vector V is given by
(κκ ⋅ ⋅ =v vˆ ˆ) κ1 1v v v vˆ (ˆ ˆ) ˆ1⋅ ⋅ +κ2 2v v v vˆ (ˆ ˆ) ˆ2⋅ ⋅ =κ1(ˆˆ ˆ) (ˆ ˆ)
v is the unit vector along direction vector V.
Let p be a point on S, vˆ1 andvˆ2 be the two principal directions and κ1 and κ2 be the respec-tive principal curvatures. If q is a point on the ellipse on the tangent plane Tp with axes κ1
and κ2, we have ‖pq‖ ≥ κ, as shown in Figure 4.9a. This result shows that the curvature at a point p in any direction is bounded by the ellipse centred at p with κ1 and κ2 as axes. Similarly, as shown in Figure 4.9b, for ellipse drawn with axes in terms of radius of curvature ρ = 1/κ, we have ‖pq‖ ≤ ρ.
As the locus of a metric is an ellipse, a metric M can be defined on the tangent plane Tp at p such that the length in any direction from p will not exceed the radius of curvature in that direction. Such a metric is given by
M = ˆ ˆ ˆ =
Let AB be a straight line segment on the tangent plane Tp. The length of edge AB with respect to M(p) is given by
Figure 4.9 Curvature at a point is bounded by an ellipse: (a) ellipse of surface curvature; (b) ellipse of radius of curvature.
If edge AB is of unit length with respect to M, i.e. L = 1, we have ρ ≥ ‖AB‖. The metric M allows us to control the edges of a mesh on S such that the unit length in any direction is always smaller than the radius of curvature at the point under consideration.
4.2.5 Geometrical control
Two common measures can be applied to control the geometrical closeness to a curved sur-face (Miranda et al. 2009). The first criterion is to specify the gap between an edge and the curved surface, as shown in Figure 4.10. Let h be the length of the edge and ρ be the radius of curvature in the direction of the edge. We have
d2 2 h d h
2
2 2
2 4
=ρ − = ρ −
δ ρ ρ ρ ρ α δ
ρ
α ε α
= − = −d 2−h2 = − − 2 or = − − 2 ≤ =h/
4 1 1
4 1 1
4 ρρ
1 1
42 42 2 2 2 2 2 2
− ≤ε −α α ≤ ε ε− α≤ ε( −ε) h≤ ε( −ε ρ)
Hence, in order that the gap δ between a line segment and the surface be bounded by the ratio ε = δ/ρ, h = αρ could not exceed the value given by
α=2 ε(2−ε)
For example, if ε is set at 1%, α = 0.282, i.e. h cannot exceed 0.282ρ; otherwise, the gap between the line segment and the curved surface will be more than 1% of ρ.
Alternatively, the ratio of the length of a chord to that of the corresponding curve can be specified, as shown in Figure 4.11. Let h be the length of an edge (chord) and s be that of the shortest curve on the surface connected to the same end points of the edge. As the curve is always longer than the chord, we can control the length of an edge by specifying a value ε such that
s h
s or h s
− ≤ε ≥ −(1 ε)
h
d δ
ρ
Figure 4.10 Gap between an edge and a curved surface.
In the limiting case,
Taking one term in the expansion, we have
ε=θ θ = ε θ= ε
2 2
3 6 6
! or
h= −(1 ε)s=2 1( −ε ρθ) =2 1( −ε) 6ερ β ρ= 1 Taking two terms in the expansion, we have
ε=θ −θ ε= θ −θ
Solving for μ and taking the smaller root, we get
ε θ ε
=10− 100 120− = = 10− 100 120−
h= −(1 ε)s=2 1( −ε ρθ) =2 1( −ε) 10− 100 120− ε ρ β ρ= 2
For instance, set ε = 1%; we have β1 = 0.485 and β2 = 0.48573, i.e. h cannot exceed 0.485ρ.
Whichever criterion we would like to apply for a close approximation of a linear FE mesh of triangular facets to an analytical curved surface, we can simply modify the metric tensor by replacing the principal curvatures with the reduced principal curvatures according to the chosen criterion, such that
Figure 4.11 Compare lengths of chord and curve.
in which the value of λ depends on the criterion adopted:
i. The gap between the edge and the surface is specified, i.e. δ
ρ≤ε, λ α= =2 ε(2−ε).
ii. The ratio of the lengths of the edge and the curve is specified, i.e. h ≥ (1 − ε)s.
λ β= 1=2 1( −ε) 6ε or λ β= 2=2 1( −ε) 10− 100 120− ε
In terms of the principal radii of curvatures, the metric tensor can be modified as
M = ˆ ˆ ˆ
In the practical implementation, once a criterion is adopted and the value of ε is specified, the value of λ can be evaluated in advance before mesh generation. By a similar argument, the modified metric M allows us to control the edges of a mesh on surface S such that edges of unit length in any direction measured by the modified metric M are smaller than the modified radius of curvature ρ at the point under consideration.
4.2.6 Metric on parametric planar domain
M is the metric defined on the tangent plane Tp at point p on surface S. Hence, M is the metric for length measure of vectors on Tp, and the length element measured along direction vector V = dϕ is given by