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Delft Aerospace Computational Science
ENGINEERING MECHANICS
Mesh Association by Projection along
Smoothed-Normal-Vector Fields:
Association of Closed Manifolds
E.H. van Brummelen
E.H.vanBrummelen@tudelft.nl
Report DACS-06-005
June 2006
REPORT
c 2006: The copyright of this manuscript resides with the author.
Copyright All rights reserved. This work may not be translated or copied
in whole or in part without the written permission of the author.
TUD/LR/EM
Kluyverweg 1, 2629HS Delft, The Netherlands
P.O. Box 5058, 2600 GB Delft, The Netherlands
Phone: +31 15 278 5460 Fax: +31 15 261 1465
ISSN 1574-6992
Mesh Association by Projection along Smoothed-Normal-Vector
Fields: Association of Closed Manifolds
E.H. van Brummelen
Delft University of Technology, Faculty of Aerospace Engineering
P.O. Box 5058, 2600 GB Delft, The Netherlands
ABSTRACT
The necessity to associate two geometrically distinct meshes arises in many engineering applications. Current
mesh-association algorithms are generally unsuitable for the high-order geometry representations associated
with high-order finite-element discretizations. In the present work we therefore propose a mesh-association
method for high-order geometry representations. The associative map defines the image of a point on a
mesh as its projection along a so-called smoothed-normal-vector field onto the other mesh. The smoothednormal-vector field is defined by the solution of a modified Helmholtz equation with right-hand-side data
corresponding to the normal-vector field. Classical regularity theory conveys that the smoothed-normal-vector
field is continuously differentiable, which renders it well suited for a projection-based association. Moreover,
the regularity of the smoothed-normal-vector field increases with the regularity of the normal-vector field and,
hence, the smoothness of the association increases with the smoothness of the geometry representations.
The proposed association method thus accommodates the higher regularity that can be provided by high-order
geometry representations. The elementary properties of smoothed-normal-projection association are established
by analysis and by numerical experiments on closed manifolds.
Keywords and Phrases: Non-matching meshes, mesh association, mesh incompatibility, high-order finiteelement methods, multiphysics, fluid-structure interaction.
1. Introduction
The construction of an associative map between two distinct meshes is a common conundrum in many
engineering and scientific disciplines, e.g., multiphysics computational science, biomedical imaging,
and computer graphics. In multiphysics problems such as fluid-structure interaction, the solutions
pertaining to the different physical subsystems generally possess disparate regularity properties. These
disparate regularity properties prompt distinct approximation spaces, both in mesh size and in order
of approximation. However, this implies that the approximation spaces are incompatible at their
interface, and an association between the meshes is required to transfer data between the subsystems.
In biomedical imaging, system properties are frequently measured at different geometric realizations
of the system. For instance, measurements of the electric field or composition of the cardiovascular
system typically provide data at different instants of the cardiac cycle. If the data is represented on a
mesh, then an association between geometrically distinct meshes is required to enable a comparison
of the data. In computer graphics, mesh association occurs in, for instance, morphing. The emphasis
in this paper is on multiphysics applications, although many of the considerations extend to other
applications as well.
A fundamental problem of mesh incompatibility is the concomitant incompatibility of the geometric realizations, i.e., the sets represented by the meshes are noncoincident, unless both meshes
allow an exact representation of the geometry; see [8] for geometrically exact representations of nontrivial geometric objects. An elementary premise for defining a proper association between distinct
meshes, is that their geometric realizations are homeomorphic, i.e., there must exist a continuous
and continuously invertible bijection between the geometric realizations. In multiphysics applications,
this premise is generally satisfied, because both meshes represent the same geometric object, e.g.,
the interface between a fluid and a structure. A proper association between meshes must comply
2
E.H. van Brummelen
with several conditions. First, the association must be a homeomorphism, i.e., the association must
provide a continuous and continuously invertible one-to-one map. Second, the map should be close
to the identity or, equivalently, the distance between any one point on a geometric realization and
its associate in the other geometric realization should be minimal. Third, the association should be
sufficiently smooth, so that functions retain their regularity under the transformation induced by the
map. Fourth, the association must be generic. In particular, it must in principle be applicable to
geometric realization of arbitrary order and, moreover, it should be essentially dimension independent.
The latter implies that the association should not be contingent on primitives which are particular for
a certain spatial dimension, and which do not generalize to other dimensions. Finally, the association
must be constructive and must allow an efficient and robust implementation.
Currently available mesh association algorithms do not comply with the aforementioned conditions.
For instance, the conventional nearest-neighbour algorithm violates the homeomorphism property, as
it is not generally one-to-one. Denoting by A and B the geometric realizations of two meshes, the
nearest-neighbour algorithm associates with each point a on set A the point b on B which minimizes
the distance to a. However, as this does not imply that a is the nearest point on A to b, nearestneighbour association is not one-to-one. Moreover, one can simply infer that this map is only injective,
and not generally surjective. For instance, near boundaries there can be points in B which do not
have an original in A. Hence, nearest-neighbour association does not provide a homeomorphism.
The normal-projection (or orthogonal-projection) association in [15] defines the image of a point a
in A as its projection onto B along the normal vector on A at a. However, unless the set A is
continuously differentiable, the normal vector is not continuous on A and, accordingly, the normalprojection association does not provide a continuous map. Moreover, one can infer that the association
is not necessarily injective, nor surjective. Therefore, normal-projection association does not provide
a homeomorphism. On the other hand, both the nearest-neighbour association and the normalprojection association allow very efficient implementations, e.g., through the vine-search algorithm [9]
or extensions of the techniques in [14] and, moreover, they are close to the identity. This renders them
acceptable for multiphysics computations with low-order discretizations, for which the smoothness of
the map is not so relevant because the regularity of the approximation functions is low, and for which
small discontinuities and small violations of the one-to-one correspondence do not essentially interfere
with the accuracy of the approximation.
A homeomorphic mesh association for surfaces is provided by the mesh-overlay method in [10,
11]. The mesh-overlay method constructs a common refinement of the meshes, viz., a mesh which
consists of the combinatorial union of both meshes, i.e., the union in the sense of mesh topology. The
association is then made in this common refinement. The existence of such a common refinement is
ascertained by the theorem that two simplicial complexes that triangulate a surface have simplicially
isomorphic subdivisions; see, for instance, Ref. [5, Th.3.4.5]. The mesh-overlay method is based
on a nearly-orthogonal projection, viz., the projection along the vector field that is formed by the
interpolation of the average normal vectors in the mesh vertices. The inverse of the association is
defined accordingly. As the interpolated average-normal-vector field is continuous, the association
and its inverse are continuous. Moreover, under nonrestrictive conditions the association is one-toone in the interior of the domain. Near boundaries and sharp features, special treatments must be
invoked [9, 12]. The mesh-overlay method yields an association that is homeomorphic and close to
the identity, and it allows an efficient implementation. Moreover, most primitives of the association,
e.g., the nearly-orthogonal projection, are essentially dimension independent. However, the method
does not generalize to high-order geometry representations and, moreover, it is not generally smooth.
The construction of the nearly-orthogonal vector field by means of a piecewise-linear interpolation of
the average normal vectors is particular for piecewise-linear geometry representations. Moreover, the
nearly-orthogonal vector field is only continuous and, hence, the association is not generally smooth.
For multiphysics applications, mesh association forms a primitive, which in conjunction with an
appropriate interpolation method can be used to transfer data between the boundaries of the domains
pertaining to the physical subsystems. An alternative means of transferring data in multiphysics
applications, that bypasses the complications of mesh association, is by means of extension, e.g., by
Mesh Association by Projection along Smoothed-Normal-Vector Fields
3
finite-interpolation elements [3] or radial basis functions [4], or by simply evaluating finite-element
interpolation functions outside the corresponding elements. See also [14] for an extension approach.
Considering two nearly contiguous domains A and B, the section of the boundary of B that is identified
with the interface will partly intersect with the interior of A, and will partly be exterior to A. On the
intersection, data is trivially transferred from the interior of A to the boundary of B. The data on
the part that is exterior to A is obtained by extending the solution on A beyond its boundaries, for
instance, by means of radial basis functions. For finite-volume and finite-difference methods, the data
between the grid points in the interior of A can be defined similarly. It may be noted that the extension
approach obviates an association of the interface meshes. However, the extension approach has two
important disadvantages. First, the method violates trace-space relations. In brief, a trace-space
relation defines the manner in which a function is to be evaluated at the boundary. This is in general
not simply the function value as the boundary is approached, which is the underlying assumption in
the extension approach. Instead, it depends sensitively on the manner in which boundary conditions
are enforced. Violation of the trace-space relations generally results in loss of accuracy and loss of
conservation. Second, if a solution exhibits large gradients near the boundary, e.g., in the case of
boundary layers or sharp features, the extension and the evaluation away from the boundary can
result in large errors.
In the present article we present a mesh-association method that is suitable for high-order finiteelement discretizations, with corresponding high-order geometry representations. The presented mesh
association is homeomorphic, it is close to the identity, it is sufficiently smooth, it is generic, and
it admits an efficient implementation. Similar to [10, 11], we define the associates of points in one
geometric realization as their projection along a nearly-orthogonal vector field onto the other geometric realization. However, we define the vector field by the solution of a modified Helmholtz
equation on the geometric realization, with right-hand-side data corresponding to the normal-vector
field. The solution to this problem is approximated by means of finite elements of the same order
as the geometric realization. Classical regularity theory for elliptic boundary-value problems conveys
that the solution of the modified Helmholtz equation, referred to as the smoothed-normal-vector field ,
is at least continuously differentiable. This renders it well suited for a projection-based association.
Moreover, the smoothness of the smoothed-normal-vector field increases with the smoothness of the
geometric realization. The associative map therefore adapts to the higher regularity that can be provided by high-order geometry representations. In the present paper, we present the fundamentals of
the smoothed-normal-projection association. To do so, we restrict our considerations to boundaryless
manifolds. The extension to manifolds with boundaries will be treated elsewhere.
The contents of this paper are organized as follows: In section 2 we attend to a precise problem statement. Section 3 presents the fundamentals of the smoothed-normal-projection association
for boundaryless manifolds. Based on the regularity properties of the smoothed-normal-vector field,
this section establishes the suitability of the smoothed-normal-vector field for a projection-based association. In section 4 we examine several elementary properties of the smoothed-normal-projection
association, such as an upper bound on the distance between a point and its image for which uniqueness of the association can be ascertained. Section 5 illustrates the theory by means of representative
numerical experiments for 2D and 3D settings. Finally, section 6 contains concluding remarks.
2. Problem statement
In this paper we are concerned with a methodology for associating two geometrically disparate meshes.
To provide a precise specification of the mesh-association problem, this section first presents a general
setting for the problem. Next, we provide details of the Lagrange elements used in this paper. The
Lagrange basis is convenient for mesh-association problems by virtue of its nodal character. However,
the presented analyses extend to other polynomial bases without modifications. Finally, we specify
the mesh-association problem.
4
E.H. van Brummelen
2.1 Problem setting
To furnish a setting for the problem, let X represent a Euclidean space of dimension d (d ∈ {2, 3, 4}),
and let {e(i) }i=d
i=1 represent an orthonormal basis of X. We remark that the case d = 4 bears practical
relevance for problems in space/time. Any element x ∈ X can be characterized by its coordinates
(x1 , . . . , xd ) with respect to the basis according to x = x1 e(1) + · · · xd e(d) . Thus, we obtain an
isomorphism between X and Rd , which enables us to identify X and Rd .
Consider a manifold M of dimension d := d − 1 embedded in Rd . For instance, M can model
the interface between a solid and a fluid in 3D space. Below, we are concerned with approximations
of M . We restrict our considerations to simplicial approximations, i.e., approximations based on line
segments in 2D, triangles in 3D, tetrahedrons in 4D, etc., because any other polytope can be subdivided
into simplices. More precisely, we presume that the approximations are based on simplicial complexes;
see, e.g., [5]. This implies that the simplices in the complex are connected in a one-to-one manner at
their faces. The simplicial complexes generated by conventional meshing algorithms comply with this
premise.
Denoting by A a simplicial complex in Rd , the underlying space of A is the subset |A| ⊂ Rd
consisting of the union of all the simplices in A. If |A| is intended as an approximation to M , then
we assume that |A| is homeomorphic to M , i.e., that there exists a continuous and continuously
invertible one-to-one map from |A| to M . In particular, for surfaces, the simplicial complex A (in
conjunction with the homeomorphism |A| 7→ M ) is referred to as a triangulation of M . The underlying
space |A| of a simplicial complex A represents a piecewise linear approximation of M . Each simplex σ
approximates a part of M according to
Pi=d
Pi=d
σ = x ∈ Rd : x = i=1 ai Ni (λ1 , . . . , λd ), 0 ≤ λi , i=1 λi = 1 ,
(2.1)
where ai ∈ Rd are the vertices of the simplex, λ1 , . . . , λd are barycentric coordinates, and Ni = λi
are the linear base functions on the simplex. To define higher-order approximations, we conceive of
the underlying space |A| as the domain of a finite-element-approximation space A (|A|) of functions
from |A| into Rd . Accordingly, the simplices are identified with finite elements. Considering a function
u : |A| → Rd , the sets u(|A|) and |A| and, hence, M are homeomorphic only if u is continuous. Hence,
we can limit ourselves to continuous finite-element spaces. For any u ∈ A (|A|), the set u(|A|) is called
the geometric realization of u. Note that if A (|A|) contains the piecewise linear functions, then there
exists a u ∈ A (|A|) such that the geometric realization u(|A|) of u coincides with the underlying
space |A|.
2.2 Lagrange elements on simplices
In particular, in this paper we opt for finite-element spaces based on classical Lagrange elements.
Consider a generic simplex ∆ ⊂ Rd with vertices e(1) , e(2) , . . . , e(d) ; see figure 1 for an illustration.
Let λ = (λ1 , . . . , λd ) represent barycentric coordinates on ∆. To facilitate the notation, we introduce
the multi-index i = (i1 , . . . , id ) and the index set
I υ = {i ∈ [0, 1, . . . , υ]d : i1 + · · · + id = υ} ,
(2.2)
where υ is a positive integer. The Lagrange base functions of order υ+1 (degree υ) are the polynomials
ψiυ (λ) (i ∈ I υ ) defined as
ψiυ1 ,...,id (λ1 , . . . , λd ) =
ik d Y
Y
υλk + 1
k=1 j=1
j
−1 .
Introducing the nodal positions αi = υ −1 (i1 e(1) , · · · + id e(d) ), it holds that
(
1 if i = j,
ψiυ αj =
0 otherwise.
(2.3)
(2.4)
Mesh Association by Projection along Smoothed-Normal-Vector Fields
5
α004
e(3)
α013
α103
α022
α112
α202
α211
α031
∆
α121
α301
α130
e(1)
α400
e(2) α040
α220
α310
Figure 1: Illustration of a simplex ∆ in R3 with the nodal positions for υ = 4.
Hence, the base functions can be associated with the nodes. We denote by P υ (∆) the span of the
Lagrange polynomials of degree υ. This space coincides with the span of all polynomials in λ1 , . . . , λd of
degree at most υ; see, e.g., Brenner & Scott [6]. Furthermore, we indicate by [P υ (∆)]d or P υ (∆, Rd )
the vector-valued functions from ∆ into Rd which reside component-wise in P υ (∆). The global
finite-element-approximation space A υ (|A|) pertaining to the Lagrange polynomials of degree υ is
now defined as the aggregate of all continuous functions from |A| into Rd of which the restriction to
a simplex σ ∈ A belongs to [P υ (σ)]d :
A υ (|A|) = u ∈ C 0 (|A|, Rd ) : u|σ ∈ [P υ (σ)]d .
(2.5)
For simplicity we have assumed in (2.5) that υ is uniform on the entire simplicial complex A, but this
assumption is nonessential.
Any function u ∈ A υ (|A|) can be expressed as:
X σ(x)
u(x) =
ûi ψiυ (λ(x)) .
(2.6)
i∈I υ
The maps x 7→ σ(x) and x 7→ λ(x) associate with each x ∈ |A| a simplex σ such that x ∈ σ, and the
barycentric coordinate λ corresponding to the position of x in σ, respectively. If x ∈ int σ, then the
association x 7→ σ is unique; see, for instance, [5, Lemma 3.3.4]. However, if x resides on the boundary
of a simplex, then x can belong to multiple simplices. On account of the continuity of u, however,
this ambiguity can be resolved by any conclusive rule, and we can proceed under the assumption
that the relation x 7→ (σ, λ) is invertible. From (2.4) it follows that the coefficients ûσi ∈ Rd can be
conceived as nodal positions corresponding to the geometric realization of u. Moreover, continuity
of u is ascertained by identifying the nodal positions at the boundaries of contiguous simplices, viz.,
ûσi = ûσ̄ı̄ for all simplex/index pairs (σ, i) and (σ̄, ı̄) such that aσ1 i1 +· · ·+aσd id = aσ̄1 ı̄1 +· · ·+aσ̄d ı̄d , with
aσk the vertices of the simplex σ. We remark that if the degree υ is nonuniform on the simplices, then
continuity of u between contiguous simplices can be maintained similarly, by specifying appropriate
relations between the nodal positions at the mutual boundary.
In summary, an approximation of a manifold M ⊂ Rd by means of Lagrange polynomials on a
complex of d-dimensional simplices can be constructed as follows: First, an appropriate vertex set
{ai }i=N
i=1 is constructed, by positioning points ai on M . This vertex set serves as the basis of a
simplicial complex A, such that the underlying space |A| is homeomorphic to M and, moreover, |A|
is close to M , e.g., in terms of the Hausdorff distance. Second, each simplex σ ∈ A is enhanced with
(d + υ)!/(υ!d!) nodes ασi = υ −1 (aσ1 i1 + · · · + aσd id ), and coincident nodes of contiguous simplices are
identified. Finally, these nodes are appropriately repositioned relative to M . For instance, the nodes
can be positioned on M . In that case, it is to be noted that a node with ik = υ and, necessarily,
6
E.H. van Brummelen
Figure 2: Approximation of the unit sphere with piecewise Lagrange polynomials for υ = 1 (left)
and υ = 4 (right). The lines indicate the boundaries of the simplices. The nodes associated with
the Lagrange polynomials are indicated by dots.
ij = 0 for all j 6= k coincides with the vertex ak . Because the vertices are already on M , such nodes
need not be repositioned.
To illustrate the approximation of a manifold by means of piecewise Lagrange polynomials, figure 2
presents the approximation of the unit sphere in R3 on a complex A of 12 simplices in a cuboid
constellation for υ = 1 (left ) and υ = 4 (right ). The nodes have been positioned on the sphere
through a gnomonic projection (sometimes referred to as a radial projection). The underlying space√
|A|
coincides with the geometric realization in the left figure, viz., the cube with vertices (±1, ±1, ±1)/ 3.
2.3 Mesh association
The above exposition enables us to give a precise specification of the problem considered in this paper.
Let us consider a manifold M ⊂ Rd and two distinct approximations of M , viz., an approximation
based on a simplicial complex A with a finite-element space A υ (|A|) and an approximation based on
a simplicial complex B with a finite-element space A ϕ (|B|). For example, M can represent a surface
modeling the interface between a fluid and a solid subdomain in R3 , A and B are surface meshes at
the interface corresponding to finite-element partitions of the subdomains, and A υ (|A|) and A ϕ (|B|)
are the position trace spaces pertaining to the subsystems, viz., the aggregates of interface positions
in accordance with the finite-element spaces in the fluid and the solid. The approximations to M
are furnished by the geometric realizations of a function u in A υ (|A|) and a function v in A ϕ (|B|).
Disparity of the finite-element spaces A υ (|A|) and A ϕ (|B|) then generally implies that the geometric
realizations u(|A|) and v(|B|) are noncoincident, i.e., the sets u(|A|) and v(|B|) do not match. Our
objective is to construct a proper associative map u(|A|) → v(|B|) for the purpose of transferring
functions between u(|A|) and v(|B|).
3. Smoothed-normal-projection association of boundaryless manifolds
The mesh-association method presented in this paper is based on projection along a so-called smoothednormal-vector field , viz., the solution of a modified Helmholtz equation with right-hand-side data
corresponding to the normal-vector field. In section 3.1 we give a dimension-independent definition
of the normal-vector field on simplicial complexes. Section 3.2 provides the notational conventions
pertaining to the functional setting of the smoothing operation, i.e., the map between the normalvector field and the solution of the modified Helmholtz equation. The smoothing operation is specified
Mesh Association by Projection along Smoothed-Normal-Vector Fields
7
in section 3.3. In section 3.4 we attend to the smoothness properties of the smoothed-normal-vector
field on the basis of classical theory on the interior regularity of solutions of elliptic equations. Finally,
section 3.5 presents the smoothed-normal-projection association.
3.1 Normal-vector fields on geometric realizations
Let us consider a simplicial complex A with a boundaryless underlying space |A| (i.e., |A| is a closed
manifold), an approximation space A υ (|A|), and the geometric realization u(|A|) pertaining to a
particular u in the approximation space. The geometric realization u(|A|) admits a simplex-wise
parameterization:
[ u(|A|) =
x ∈ Rd : x = uσ (λ), 0 ≤ λi , λ1 + · · · + λd = 1 .
(3.1)
σ∈A
To accommodate the partition-of-unity property of the barycentric coordinates, we define the hypersurface coordinates (or coordinate chart ) ξ ∈ Rd and the map ξ 7→ λ(ξ):
ξ = (ξ1 , . . . , ξd ),
ξ 7→ λ(ξ) = (ξ, 1 − |ξ|),
(3.2)
where |ξ| = ξ1 + · · · + ξd . The vectors ǫ(i) = ∂i uσ (λ(ξ)) (∂i := ∂ξi , i = 1, . . . , d) provide a basis of the
tangent space to u(|A|). It is to be noted that
ǫ(i) = ∂i uσ (λ(ξ)) = Di uσ − Dd uσ ,
(3.3)
where Di denotes differentiation with respect to the i-th argument. Indicating the unit normal vector
d
(i) i=d
to u(|A|) by ǫ(d) , the vectors {ǫ(i) }i=d
i=1 provide a basis of R . The reciprocal basis, {ǫ }i=1 , is
j
j
defined by hǫ(i) , ǫ(j) i = δi , where δi stands for the Kronecker delta and h·, ·i generically represents
the contraction of tensors. The vectors ǫ(i) and ǫ(i) (i = 1, . . . ,) are referred to as covariant and
contravariant base vectors, respectively. On account of hǫ(i) , ǫ(d) i = δid and |ǫ(d) | = 1 with | · | the
vector norm in Rd , it holds that ǫ(d) = ǫ(d) . Upon collecting the components of {ǫ(i) }i=d
i=1 with respect
i=d
σ
to the basis {e(i) }i=1 in a matrix Tij = ǫ(j)i = hǫ(j) , e(i) i, the components of the unit normal vector
follow from Tijσ ǫdi = δjd . Throughout, the summation convention applies to repeated indices, unless
explicitly stated otherwise. Application of Cramer’s rule yields
σ
ǫdi = (−1)i+d gσ−1/2 Mid
=: nσi (ξ),
(3.4a)
σ 2
σ 2
) + · · · + (Mdd
) ,
gσ = (M1d
(3.4b)
with
Mijσ
σ
and
the minors of T , i.e., the determinant of the matrix that is obtained by removing the i-th
row and j-th column of T σ . The normal-vector field n on u(|A|) is now defined as the piecewisecontinuous vector-valued function from u(|A|) into Rd of which the restriction to any simplex σ
complies with (3.4). We assume throughout that n is directed outward.
3.2 Functional setting
To provide an appropriate functional setting for the smoothing operation, we regard a generic manifold M . Let C ∞ (M ) represent the space of infinitely-differentiable functions on M . For all nonnegative
integers m and all real p ≥ 1, the Sobolev space H k,p (M ) is defined as the closure of C ∞ (M ) with
respect to the norm
( R
1/p
k
p
X
if 1 ≤ p < ∞,
l
M |U | dµM
k∇s ukLp (M) , kU kLp(M) =
kukH k,p (M) =
(3.5)
sup
|U
(x)|
if
p = ∞,
x∈M
l=0
with µM the measure on M and |U | the local norm of the tensor U , i.e., |U |2 = hU, U i. The operator
∇s designates the gradient on the manifold M . If M admits a (local) parameterization,
M = {x ∈ Rd : x = u(ξ1 , . . . , ξd },
(3.6)
8
E.H. van Brummelen
then ∇s w = ǫ(1) ∂1 w + · · · + ǫ(d) ∂d w, where ǫ(i) denote the reciprocal-base vectors corresponding to
ǫ(i) = ∂i u and the normal vector to M . For p = 2, we use the condensed notation H k (M ). The
space H k (M ) is a Hilbert space under the inner product
(u, v)H k (M) =
k Z
X
l=0
M
h∇ls u, ∇ls vi dµM .
(3.7)
The definition of the Sobolev spaces is extended to negative integer numbers by identifying H −k (M )
with the dual of H k (M ) with L2 as pivot space; see, e.g., [13, 17]. The above definitions are extended
to vector-valued functions in the usual manner. The vector-valued functions M → Rd that reside
component-wise in H k,p (M ) are indicated by H k,p (M, Rd ) or [H k,p (M )]d .
3.3 Smoothing of normal-vector fields
For a general geometric realization, the normal-vector field is only piecewise smooth, i.e., the normalvector field is smooth on each simplex, but it can be discontinuous across edges. On account of this
irregularity, the normal-vector field is unsuitable for a projection-based association. Therefore, we
propose to use instead a smoothed-normal-vector field , viz., the H 1 representation of the normalvector field.
We define the smoothed-normal-vector field corresponding to the normal-vector field n as the
vector field m ∈ H 1 (u(|A|), Rd ) according to
Z
Z
hn, wi dµM
∀w ∈ H 1 (M, Rd ),
(3.8)
hm, wi + γ h∇s m, ∇s wi dµM =
M
M
with M = u(|A|). The positive parameter γ is referred to as the smoothing parameter. Let us remark
that equation (3.8) represents a weak formulation of the modified Helmholtz equation,
m − γ ∇2s m = n ,
(3.9)
with ∇2s the Laplace-Beltrami operator on the manifold u(|A|).
The integrals in (3.8) can be separated into sums of contributions of the simplices. Upon introducing simplex-wise coordinate charts ξ 7→ x = uσ (λ(ξ)), the identity (3.8) can be cast into the
form
√
XZ
XZ √
gσ dξ =
(3.10)
hnσ , wi gσ dξ,
hm, wi + γ gσjk h∂j m, ∂k wi
σ∈A
Σ
σ∈A
Σ
where gσij = hǫ(i) , ǫ(j) i (i, j = 1, . . . , d) is the contravariant metric tensor pertaining to the simd
plex σ and
√ Σ represents the simplex Σ = {ξ ∈ R : 0 ≤ ξi , ξ1 + · · · + ξd ≤ 1}. One can infer that gσ in accordance with (3.4b) represents the ratio of the measures dµu(σ) /dξ. In tensor analysis, gσ is conventionally defined as the determinant of the covariant metric tensor, viz.,
gσ = det(gσ ij ) with gσ ij = hǫ(i) , ǫ(j) i (i, j = 1, . . . , d). However, gσ according to (3.4b) is identical
to det2 (ǫ(j)i ) (i, j = 1, . . . , d). Moreover, by virtue of hǫ(i) , ǫ(d) i = δid and |ǫ(d) | = 1, it holds that
dethǫ(i) , ǫ(j) i = dethǫ(k) , ǫ(l) i (i, j = 1, . . . , d but k, l = 1, . . . , d). The product rule for determinants
yields dethǫ(i) , ǫ(j) i = det(ǫ(i)k ) det(ǫ(j)k ) = det2 (ǫ(j)i ) (i, j, k = 1, . . . , d). Hence, the definition of gσ
according to (3.4b) is equivalent to the conventional definition.
Of course, in an actual computation the smoothed-normal-vector field according to (3.8) is not
determined explicitly. Instead, it is approximated by finite elements. We typically opt to use the same
finite-element-approximation space for m as for the geometric realization, i.e., we replace the Sobolev
spaces H 1 (u(|A|), Rd ) that furnish the setting of (3.8) by Aυ (|A|).
3.4 Regularity of the smoothed-normal-vector field
An important property of the smoothed-normal-vector field m pertains to its regularity in relation to
the regularity of the normal-vector field n. Equation (3.8) or, equivalently, (3.9) or (3.10) defines m
Mesh Association by Projection along Smoothed-Normal-Vector Fields
9
as the solution of a second-order elliptic equation with right-hand-side data n. Hence, to establish the
relation between the regularity of n and the regularity of m we can appeal to classical theory on the
regularity of solutions of elliptic equations; see, for instance, [2, 7, 13, 16]. Here, we shall be concerned
with interior regularity only, because we restrict our considerations to boundaryless manifolds. The
case with boundaries, which is profoundly more complicated, will be treated elsewhere.
The most useful expression of the interior-regularity theorem for elliptic equations for our purposes
is that according to [2, Theorem 3.54]: Let Ω be an open subset of Rn and A = al ∇l an elliptic linear
operator of order 2m with C ∞ coefficients. If f ∈ H k,p (Ω) and u is the distributional solution of
Au = f , then u belongs locally to H k+2m,p (Ω). The latter implies that u ∈ H k+2m,p (ω) for all open
subsets ω ⊂ Ω with compact closure in Ω.
The interior-regularity theorem conveys that under the stated premises, the solution is in fact
more regular than is to be expected from the general setting. For a boundaryless manifold associated
with a geometric realization u(|A|), the theorem implies that the smoothed-normal-vector field resides
in H 2,p (u(|A|), Rd ) for any p ≥ 1. The derivation of the regularity of the smoothed-normal-vector
field is elaborated by the possible non-smoothness of manifolds pertaining to geometric realizations
and, therefore, it is deferred to Appendix A. An essential attribute of the smoothed-normal-vector
field is its continuous differentiability: By the Sobolev embedding theorem for compact manifolds [2,
Theorems 2.10 and 2.20], for a d-dimensional manifold it holds that functions in H k,p (M ) are equivl
l
alent to functions in CB
(M ) if (k − l)/d > 1/p, where CB
(M ) is the space of l-times continuously
l
differentiable functions that are bounded in the k · kH l,∞ (M) -norm. This implies that m ∈ CB
(M, Rd )
for all l < 2 − d/p, p ≥ 1 and, in particular, that the smoothed-normal-vector field is bounded continuously differentiable. Therefore, the smoothed-normal-vector field is appropriate for a projection-based
association.
The interior-regularity theorem moreover yields the important corollary that an increase in the
smoothness of the geometric realization leads to higher regularity of the corresponding smoothednormal-vector field. Hence, the smoothness of the projection-based association between geometric
realizations increases with the smoothness of the geometric realizations.
3.5 Smoothed-normal-projection association
On account of its continuity, the smoothed-normal-vector field is suitable for a projection-based association. We denote by ℓ(x) the line through x ∈ u(|A|) in the direction of the smoothed-normal
vector at x. Denoting by v(|B|) a geometric realization homeomorphic to u(|A|), the smoothednormal-projection associates which each x ∈ u(|A|) the intersection of ℓ(x) and v(|B|). If v(|B|)
is sufficiently close to u(|A|), then the intersection is nonempty. However, the intersection generally
contains multiple elements, as the line ℓ(x) can have multiple intersections with the manifold v(|A|).
The associate y ∈ v(|B|) of x ∈ u(|A|) is therefore isolated as the element of the intersection set that
is closest to x:
u(|A|) ∋ x 7→ y = arg inf |y − x| ∈ v(|B|) .
(3.11)
y∈v(|B|)∩ℓ(x)
The smoothed-normal-projection association according to (3.11) is indicated by κ.
To retain the necessary one-to-one correspondence, the inverse κ −1 : v(|B|) → u(|A|) is based on
the smoothed-normal-vector field on u(|A|), i.e., x = κ −1 (y) is the closest point on u(|A|) for which y
is located on the line ℓ(x):
κ −1 (y) =
arg inf
|y − x|.
(3.12)
x∈{x∈u(|A|):y∈ℓ(x)}
This definition of the inverse is similar to that in [10, 11].
Because both κ and κ −1 utilize the smoothed-normal-vector field on u(|A|), the pair of associations (κ, κ −1 ) possesses a so-called master/slave structure. If the role of u(|A|) and v(|B|) is
reversed, then the association is based on the smoothed-normal-vector field on v(|B|). In general, the
corresponding association v(|B|) ∋ y 7→ x ∈ u(|A|) is different from κ −1 .
10
E.H. van Brummelen
4. Properties of smoothed-normal-projection association
In this section we consider elementary properties of the smoothed-normal-vector field and of the
the smoothed-normal-projection association. In section 4.1 it is shown that the smoothed-normalvector field is strictly outward, under some nonrestrictive conditions on the geometry. Based on this
externality of the smoothed-normal-vector field, section 4.2 establishes a local-uniqueness condition,
viz., a lower bound on the distance between a point and its image such that no nearby point possesses
the same image.
4.1 Externality of the smoothed-normal-vector field
An elementary property of the smoothed-normal-vector field is that under the standing assumption
that n represents the outward normal-vector field, and some quite nonrestrictive conditions on the
geometry which are elaborated below, it holds that m is strictly outward, i.e., hm, ni > 0. A precise
verification of this property and the corresponding prerequisites is beyond the scope of this paper. In
the analysis below we restrict ourselves to a rudimentary analysis. The externality of m is further
considered in section 5.
Let us consider a point x ∈ M and a local normal coordinate system (see [2, p.7] and also
Appendix A), i.e., a system of coordinates (ξ1 , . . . , ξd ) such that the components of the metric tensor
at x satisfy gij = δij and ∂k gij (x) = 0. With respect to the normal coordinates, (3.9) reduces to:
1
√
m − γ ∇2s m = m − γ √ ∂i ( g g ij ∂j m) = m − γ ∇2ξ m = n,
g
(4.1)
where ∇2ξ is a condensed notation for the Laplace operator in ξ-coordinates, ∇2ξ = ∂12 + · · · + ∂d2 .
By means of the Green’s functions for the modified Helmholtz equation, the solution to (4.1) can be
expressed as:
Z
1
|ξ − ξ′ |
n(ξ ′ ) dξ ′ ,
(4.2a)
m(ξ) =
√ K
√
γ
γ
where the kernel K(·) is given by
K(r) =
e−|r|
,
2
K(r) =
K0 (|r|)
,
2π
K(r) =
e−|r|
,
4π |r|
(4.2b)
for d = 1, 2, 3, respectively. In (4.2b), K0 (·) represents the modified Bessel function of the second kind
of order 0; see [1]. Equation (4.2) implies that m is outward at ξ unless
Z
1
|ξ − ξ′ |
hm(ξ), n(ξ)i =
hn(ξ ′ ), n(ξ)i dξ ′ ≤ 0 .
(4.3)
√ K
√
γ
γ
The kernels (4.2b) have the following essential properties: First, they are strictly decreasing and
vanish exponentially for large |r|, i.e., K(r) > K(r′ ) for all r > r′ ≥ 0 and 0 < K(r) ≤ exp (−|r|)
as r → ±∞. Second, they are symmetric, i.e., K(−r) = K(r). Third, they are strictly positive,
i.e., K(r) > 0 for all r ∈ R. On account of the strict decrease and strict positivity of the kernels, hn(ξ ′ ), n(ξ)i must be negative for ξ ′ near ξ for (4.3) to hold. This can only occur if u(|A|) is
strongly curved, for instance, near kinks. Let us now consider a point x(ξ) in the vicinity of an edge e
where u(|A|) displays a strong kink; see figure 3 for an illustration. The inner product hn(ξ ′ ), n(ξ)i
is positive if x(ξ ′ ) resides in the simplex σ + and negative if it resides in σ − . However, for sufficiently
small fixed |ξ − ξ′ | the angle between n(ξ) and n(ξ ′ ) is smaller on σ + than on σ − . Hence, the negative
contribution from σ ′ is subordinate to the positive contribution from σ, so that (4.3) cannot hold.
The above argument fails near cusp-like features and certain types of reentrant corners. In section 5
it will be shown that in the neighborhood of such features the smoothed-normal-vector field can
indeed become non-outward. Near reentrant corners, however, the externality of the smoothed-normalvector field can be restored by an appropriate choice of the smoothing parameter γ: The asymptotic
11
Mesh Association by Projection along Smoothed-Normal-Vector Fields
e
n(ξ)
n−
x(ξ)
n+
σ
+
σ−
Figure 3: Illustration of the normal vector field near a kink: |hn+ , n(ξ)i| > |hn − , n(ξ)i|
behavior of the kernels (4.2b) implies that the contribution of points at distance |ξ −ξ ′ | is proportional
√
to exp (−|ξ − ξ ′ |/ γ). Therefore, the smoothed-normal-vector field approaches the normal-vector
field as γ → 0 and, hence, for sufficiently small γ the smoothed-normal-vector field will be directed
outwards. It is to be noted, however, that in this case the externality comes at the expense of
smoothness, as the smoothness of m, e.g., in the sense of kmkL2 (M) /kmkH 1 (M) , decreases with γ.
4.2 Local-uniqueness of the association
The smoothed-normal-projection association enables a unique association between geometric realizations, provided that these realizations are sufficiently close. To elaborate the notion of closeness,
let us consider a parameterized manifold M ⊂ Rd in conformity with (3.6) and an arbitrary vector field m ∈ C 1 (M, Rd ). Consider two arbitrary distinct points u(ξ 0 ) and u(ξ 1 ) on M . If the
lines through u(ξ 0 ) and u(ξ 1 ) in the directions of m(ξ 0 ) and m(ξ 1 ) intersect, then the association through projection along m is nonunique at the intersection. In particular, if u(ξ 0 ) + θ0 m(ξ 0 )
and u(ξ 1 ) + θ1 m(ξ 1 ) are the images of u(ξ 0 ) and u(ξ 1 ) under the associative map, respectively, then
the association is nonunique if the images coincide:
u(ξ 0 ) + θ0 m(ξ 0 ) = u(ξ 1 ) + θ1 m(ξ 1 ).
(4.4)
Upon inserting ξ 1 = ξ0 + εη and θ1 = θ0 + εω into (4.4) and taking the limit ε → 0, we obtain
u(ξ 0 ) + θ0 m(ξ 0 ) = u(ξ 0 ) + εηi ∂i u(ξ 0 ) + θ0 m(ξ 0 ) + εθ0 ηi ∂i m(ξ 0 ) + εωm(ξ 0 ) + o(ε),
(4.5)
as ε → 0, where the Landau symbols o and O (used below) represent terms such that o(ε)/ε = 0 and
O(ε)/ε ≤ const as ε → 0. On account of the fact that terms of O(ε) must vanish separately as ε → 0,
it holds that θ(:= θ0 ) is determined by the generalized eigenvalue problem: Find (θ, η, ω) such that
(∂i u + θ ∂i m)ηi + ωm = 0.
(4.6)
To elucidate the structure of the generalized eigenvalue problem, we define the matrices E, H ∈ Rd×d
and the vector h ∈ Rd according to
 




η1
∂1 m1 · · · ∂d m1 0
∂1 u1 · · · ∂d u1 m1
 .. 

..
..  , H =  ..
..
..  , h =  .  .
(4.7)
E =  ...
 
 .
.
. 
.
.
ηd 
∂1 ud · · · ∂d ud md
∂1 md · · · ∂d md 0
ω
With these definitions, equation (4.6) can be expressed in the form (Eij + θHij )hj = 0.
It is important to note that as ε → 0 in (4.5), we restrict ourselves to local considerations.
Specifically, the association of u(ξ) to u(ξ)+θm(ξ) is locally unique if θ is distinct from the generalized
eigenvalues according to (4.6). This does, however, not prevent a point on M at finite distance
from u(ξ) from having the same image as u(ξ). Such a global violation of uniqueness is not included
in the analysis.
12
E.H. van Brummelen
The association of u(ξ) to u(ξ) + θm(ξ) is locally unique if |θ| < |θmin |, where θmin represent
the generalized eigenvalue of smallest modulus. This implies that a sufficient condition for local
uniqueness of the association through projection along m is that the distance between a point on M
and its associate is less than |θmin ||m|. We refer to this condition as the local-uniqueness condition
on the association distance. Let us remark that if, specifically, m represents the unit-normal-vector
field on M , then the eigenvalues θ can be identified as the principal radii of curvature and, indeed,
the local-uniqueness condition stipulates that the association distance is subordinate to the smallest
principal radius of curvature.
To corroborate that the local-uniqueness condition depends on intrinsic properties of M and m
and that it is independent of the selected coordinate chart ξ1 , . . . , ξd , we recall the definition of the
tangential base vectors ǫ(i) = ∂i u and, accordingly, we identify ηi ǫ(i) with a vector η in the tangential
hyperplane {n}⊥ . Thus, problem (4.6) can be reformulated as: Find (θ, η, ω) ∈ R × {n}⊥ × R such
that
η + θhη, ∇s im + ωm = 0 .
(4.8)
From (4.8) it is apparent that θ depends only on M via the vector in the tangent space η ∈ {n}⊥ and
on the vector field m via m and ∇s m.
To construct a lower bound for the association distance |θmin ||m| corresponding to an arbitrary
vector field m ∈ C 1 (M, Rd ), we cast the generalized eigenvalue problem (4.6) into the variational
form: Find (θ, h) ∈ R × Rd such that
wi Eij hj + θ wi Hij hj = 0
∀w ∈ Rd .
(4.9)
Let us now consider an eigenvector h in accordance with (4.9). Observing that the identity in (4.9)
must hold for all w ∈ Rd , it follows for the corresponding eigenvalue θ(h) that
|θ(h)| ≥ sup
w∈Rd
|wi Eij hj |
.
|wi Hij hj |
(4.10)
By virtue of the fact that the minimum of |θ(h)| over the subset of eigenvectors is at least equal to
the minimum over all vectors in Rd , it holds that
|θmin | ≥ inf sup
h∈Rd
w∈Rd
|wi Eij hj |
.
|wi Hij hj |
(4.11)
Upon expanding E, H and h in accordance with (4.7) and identifying ηi ∂i u with a vector η ∈ {n}⊥ ,
we obtain the following inequalities:
|θmin | ≥
inf
sup
(η,ω)∈Rd ×R
w∈Rd
|hw, ηi ∂i ui + ωhw, mi|
|hw, ηi + ωhw, mi|
≥
inf
sup
. (4.12a)
|hw, ηi ∂i mi|
|hηw, ∇s mi|
d
⊥
(η,ω)∈{n} ×R w∈R
To continue the sequence of inequalities, we first restrict w to {m}⊥ , thus eliminating the term
multiplied by ω in the numerator. Next, we judiciously set w = P{m}⊥ η, i.e., we specify w as the
orthogonal projection of η onto the hyperplane {m}⊥ :
|θmin | ≥
inf
sup
η∈{n}⊥
w∈{m}⊥
|hP{m}⊥ η, ηi|
|hw, ηi|
≥ inf
.
|hηw, ∇s mi| η∈{n}⊥ |hηP{m}⊥ η, ∇s mi|
(4.12b)
As w = P{m}⊥ η minimizes the angle between w ∈ {m}⊥ and η ∈ {n}⊥ , it is anticipated that the
second inequality in (4.12b) does not severely degrade the sharpness of the lower bound. Geometric
considerations impart that for all η ∈ {n}⊥ it holds that
|hP{m}⊥ η, ηi|
|hm, ni|
|hm, ni|
≥
=
;
|P{m}⊥ η| |η|
|m| |n|
|m|
(4.13)
Mesh Association by Projection along Smoothed-Normal-Vector Fields
13
see Appendix B for proof. From (4.12), (4.13) it follows that
|θmin ||m| ≥
inf
η∈{n}⊥
|P{m}⊥ η| |η|
|hm, ni|
|hm, ni| ≥
,
|hη P{m}⊥ η, ∇s mi|
|∇s m|
(4.14)
with |∇s m| := h∇s m, ∇s mi1/2 .
The lower bound for the association distance (4.14) implies that under the condition that the
smoothed-normal-vector field m is strictly outward, the smoothed-normal-projection association according to (3.11) admits a nonzero association distance, provided that |∇s m| is pointwise bounded.
To establish that the latter stipulation is satisfied, we note that according to the second part of the
l
Sobolev embedding theorem [2, Theorem 2.10], if (k − l)/d ≥ 1/p then H k,p (M ) ⊂ CB
(M ) and the
k,p
l
injection from H (M ) into CB (M ) is continuous, i.e., there exists a constant C such that for all
u ∈ H k,p (M ) it holds that kukH l,∞ (M) ≤ C kukH k,p (M) . Therefore, in particular, if m ∈ H 2,p (M ) for
some p > 1/d then it holds for all x ∈ M that
|∇s m(x)| ≤ kmkH 1,∞ (M) ≤ C kmkH 2,p (M) .
(4.15)
The results from section 3.4 in turn convey that |∇s m| is indeed bounded.
5. Numerical Experiments
To illustrate the properties of the smoothed-normal-projection association, we present numerical experiments in 2D and 3D. Section 5.1 examines the properties of the smoothed-normal-vector field near
kinks and cusps in 2D. In section 5.2 we consider smoothed-normal-projection association of high-order
representations of the unit sphere in 3D. In this section, we also investigate the convergence behavior
of the associative map under mesh refinement. Finally, section 5.3 presents numerical results for a
complex geometry in 3D corresponding to a scanned object.
5.1 Properties of the smoothed-normal-vector field: 2D examples
To illustrate the properties of the smoothed-normal-vector field, we first consider m according to (4.2)
in a 2D setting (d = 1) for an infinite domain with a kink, viz., M = M − ∪ M + with
M − = {x ∈ R2 : x = (−ξ, 0), −∞ < ξ ≤ 0},
M + = {x ∈ R2 : x = (ξ cos(a), ξ sin(a)), 0 ≤ ξ < ∞}.
(5.1)
Figure 4 displays the smoothed-normal-vector field m for smoothing parameters γ = 1 (top) and γ =
50 (bottom) and for kink angles of a = π/2 (left ) and a = π/8 (right ). The smoothed-normal-vector
field has been extended to the interior, to illustrate the effects pertaining to inward association. The
figure conveys that m is indeed outward, even near sharp kinks. In particular, it follows from (4.2) that
m(0) = (n+ +n− )/2 with n+ = (− sin(a), cos(a)) and n− = (0, −1). This implies that hm(ξ), n(ξ)i is
continuous at the kink and that hm(0), n(0)i = sin2 (a/2). Moreover, it holds that |m(0)| = sin(a/2).
Hence, the local norm |m(0)| at the kink decreases with the kink angle a, but |m(0)| remains strictly
positive for all nonzero kink angles.
The positions marked by the symbol ◦ indicate the points closest to M where local uniqueness
of the association by projection along m is violated; cf. section 4.2. Hence, the association distance
is determined by the distance between ◦ and M along m. Figure 4 (top, right ) illustrates that
the association distance can become excessively small if the smoothing is weak and the manifold M
contains a strong kink. However, figure 4 (bottom, right ) conveys that the association distance can
be effectively increased by increasing the smoothing parameter. It is to be noted that this result is in
accordance with the lower bound (4.14), as an increase in the smoothing
√parameter leads to a reduction
in |∇s m|.√Specifically, equation (4.2) yields ∇s m(0) = (n+ − n− )/(2 γ) and, therefore, |∇s m(0)| =
cos(a/2)/ γ. This leads to the following lower bound for the association distance:
|θmin ||m| ≥
|hm, ni| √ sin2 (a/2)
= γ
;
|∇s m|
cos(a/2)
(5.2)
14
E.H. van Brummelen
1
1
0.5
0.5
0
0
−0.5
−0.5
−1
−1
−0.5
0
0.5
1
−1
−1
1
1
0.5
0.5
0
0
−0.5
−0.5
−1
−1
−0.5
0
0.5
1
−1
−1
−0.5
0
0.5
1
−0.5
0
0.5
1
Figure 4: Illustration of the smoothed-normal-vector field m in 2D (d = 1) according to (4.2)
for a kinked domain with smoothing parameters γ = 1 (top) and γ = 50 (bottom) and kink angles
a = π/2 (left) and a = π/8 (right). The positions marked by ◦ indicate the points closest to M
where local uniqueness of the association by projection along m is violated.
15
Mesh Association by Projection along Smoothed-Normal-Vector Fields
a = π/2, γ = 1 a = π/2, γ = 50 a = π/8, γ = 1 a = π/8, γ = 50
exact
7.071 × 10−1
5.000 × 100
3.881 × 10−2 2.744 × 10−1
−1
0
lower bound 7.071 × 10
5.000 × 10
3.881 × 10−2 2.744 × 10−1
Table 1: Comparison of the exact association distance and the lower bound (4.14) or, equivalently, (5.2) for the test cases in figure 4.
0.5
0
−0.5
−0.5
0
0.5
Figure 5: Illustration of the non-outwardness of the smoothed-normal-vector field near a cusp.
cf. equation (4.14). Table 1 compares the exact association distance to the lower bound according
to (5.2). The table conveys that in the present case the lower bound in fact coincides with the actual
association distance. This indicates that the simplifications that lead from (4.8) to (4.14) do not
severely degrade the sharpness of the lower bound (4.14).
Next, we consider the effect of cusp-like features of the manifold on the smoothed-normal-vector
field. To this end, we consider the smoothed-normal-vector field according to (4.2) associated with
the manifold M = M − ∪ M + with
M − = {x ∈ R2 : x = (−ξ, 0), −∞ < ξ ≤ 0},
M + = {x ∈ R2 : x = (sin(a), 1 − cos(a)), 0 ≤ a < π/2} ∪ {x ∈ R2 : x = (1, ξ), 1 ≤ ξ < ∞}.
(5.3)
The manifold (5.3) contains a cusp at x = (0, 0). Figure 5 plots the smoothed-normal-vector field m
for γ = 1 near the cusp. The figure clearly illustrates that m is non-outward in the neighborhood of
the cusp. The non-outwardness of m is caused by fact that the normal vectors n+ on M + and n−
on M − are oppositely directed near the cusp. Hence, their contributions to m according to (4.2)
cancel on account of the symmetry of the kernel. In the vicinity of the cusp, m is then essentially
determined by the normal-vector field away from the cusp, and this can cause m to be non-outward.
In particular, in figure 5 the infinite horizontal branch M − yields a dominant contribution in the
direction n− = (0, −1) which causes m to be inward on M + . In view of the restricted importance of
cusp-like features in practical applications, however, we will not pursue this issue further.
5.2 Association between approximations of the unit sphere
As an elementary example of smoothed-normal-projection association in 3D (d = 2), we consider the
association between two distinct approximations of the unit sphere. One of the simplicial complexes
consists of 4 simplices in a tetrahedral constellation, the other of 12 simplices in a cuboid constellation.
16
E.H. van Brummelen
The underlying spaces pertaining to these simplicial complexes coincide with the tetrahedral and
hexahedral geometric realization in figure 6 (top, left ) and (top, right ), respectively. The geometric
realizations have been obtained by positioning the vertices on the unit sphere. Let us note that the
geometric realizations in figure 6 (top) can be conceived as piecewise linear (υ = 1) approximations
of the unit sphere. In addition, figures 6 (bottom) depict the geometric realizations corresponding
to cubic (υ = 3) approximations of the sphere. These geometric realizations have been obtained by
positioning the nodes of the Lagrange elements on the sphere through a gnomonic projection; see also
section 2.2.
To exemplify the smoothed-normal-projection association, figure 6 displays the associates of edges
(faces) of the simplices of the cuboid configuration in the tetrahedral configuration (left ) under the
map κ according to (3.11) and the associates of the edges of the tetrahedral configuration in the
cuboid configuration (right ) under the inverse map κ −1 according to (3.12). In particular, the dotted
lines in figure 6 (left ) represent the edges of simplices of the tetrahedron itself, whereas the solid
lines represent the images of the edges of the hexahedron under the smoothed-normal-projection
association. The same legend applies to figure 6 (right ). For completeness, we mention that the
smoothing parameter in (3.8) has been set to γ = 1 and that the cuboid configuration acts as master
and the tetrahedral configuration as slave, i.e., the association between the geometric realizations is
based on the smoothed-normal-vector field on the hexahedron; cf. section 3.5. Furthermore, it is to
be remarked that the smoothed-normal-vector field according to (3.8) is approximated by means of
finite-elements of the same order as the geometric realizations. As a digression, we note that this
underlies the asymmetry of the association in figure 6 (top): Although both the tetrahedron and the
regular hexahedron are symmetric with respect to the plane x2 = 0, the association is not, on account
of the fact that the finite-element configuration in the hexahedron does not possess this symmetry. For
υ = 3, this asymmetry is less pronounced by virtue of the improvement of the approximation of the
smoothed-normal-vector field corresponding to the increase in the polynomial degree. Figure 6 (top)
illustrates that the smoothed-normal-projection association yields a meaningful association, even if
the discrepancy between the geometric realizations is large. For υ = 3 (figure 6 (bottom)), the two
geometric realizations are closer and, accordingly, also the distance between a point and its image
under κ or κ −1 is smaller. This illustrates that the associative map converges to the identity.
To examine the approximation-to-the-identity properties of the association in more detail, we consider the convergence of κ(·) − (·) under refinement of the simplicial complexes for distinct polynomial
degrees. The refinement consists in subdividing each simplex in the original simplicial complexes
into n2 (n ∈ N) uniform simplices. Each of the simplices in the refined complex acts as a finite
element of degree υ. The geometric realizations are again obtained by positioning the nodes of the
elements on the unit sphere via a gnomonic projection. Figure 7 displays the norms kκn (·)−(·)kL2 (Mn )
and kκn (·) − (·)kH 1 (Mn ) for n = 1, 2, . . . , 12 and υ = 1, 2, 3, where {Mn } represents the sequence of
refinements of the cuboid constellation and {κn } is the sequence of associative maps corresponding
to the sequence of refinements. For completeness, we mention that the integrals on Mn that underly
the norms have been computed element-wise by means of a 24-point Gauss-quadrature rule. Because both geometric realizations represent distinct polynomial approximations of degree υ of a C ∞
manifold, interpolation theory (see, e.g., [17, §8.5]) conveys that the optimal convergence behavior is
bounded by:
kκn (·) − (·)kH s (Mn ) ≤ const hυ+1−s
,
(5.4)
n
where hn designates the maximum of the diameters of the elements. Noting that on both geometric
realizations hn is proportional to n−1 , it follows from figure 7 that the smoothed-normal-projection
association indeed yields optimal convergence behavior in the L2 and H 1 norms.
5.3 Association between approximations of a scanned object
To demonstrate the versatility of smoothed-normal-projection association, we finally consider the association of two distinct representations of a complex geometry associated with a scanned object, viz.,
a hippic figure; see figure 8. The fine and coarse representations comprise 6668 and 668 triangular ele-
Mesh Association by Projection along Smoothed-Normal-Vector Fields
Figure 6: Smoothed-normal-projection association between approximations of the unit sphere in
3D by means of 4 elements in a tetrahedral constellation (left) and 12 elements in a hexahedral
constellation (right) for polynomial degrees υ = 1 (top) and υ = 3 (bottom). The solid lines in
the left figures represent the images of edges of the hexahedral constellation in the tetrahedral
constellation under κ according to (3.11). The solid lines in the right figures are the images of edges
of the tetrahedral constellation in the hexahedral constellation under κ −1 according to (3.12).
17
18
E.H. van Brummelen
2
1
1
log kκn (·) − (·)kV
−2
1
2
1
−6
2
3
−10
1
4
−14
0
1
2
2
log n
3
4
Figure 7: kκn (·) − (·)kVn versus n for Vn = H 1 (Mn ) (△) and Vn = L2 (Mn ) (◦) and υ = 1 (−−−),
υ = 2 (− −), υ = 3 (−·). The triangles below the plot illustrate the refinement. The triangles
adjacent to the curves indicate the slopes of the curves.
Mesh Association by Projection along Smoothed-Normal-Vector Fields
19
ments, respectively.1 Figure 8 (top) displays the coarse representation and the associates of the edges
of the fine representation under the smoothed-normal-projection association κ. The bottom image
depicts the fine representation and the associates of the edges of the coarse representation under κ −1 .
The smoothing parameter in (3.8) has been set to γ = 5 × 10−4 . This value has been selected on the
basis of the size of the object (w1.8, d0.8, h1.5) and its geometric complexity. Furthermore, the fine
representation acts as master, i.e., the association is based on the smoothed-normal-vector field on the
fine representation. In addition, figure 9 zooms in on the head of the horse. The figures illustrate that
the smoothed-normal-projection association yields a proper one-to-one association, even near sharp
features such as the ears.
6. Conclusion
Motivated by the inappropriateness of concurrent mesh-association methods for high-order geometry
representations, we developed a new mesh-association technique based on projection along a so-called
smoothed-normal-vector field. The smoothed-normal-vector field consists of the solution of a modified
Helmholtz equation on the geometric realization of a mesh, with right-hand-side data provided by
the normal-vector field. The image of a point on one mesh is then defined as its projection along the
smoothed-normal-vector field onto the other mesh. To retain the necessary one-to-one correspondence,
the association possesses a master/slave structure, i.e., the association and its inverse are both based
on the same smoothed-normal-vector field.
By means of regularity theory for elliptic partial-differential equations we established that the
smoothed-normal-vector field is bounded continuously differentiable, which supports its suitability for
a projection-based association. Furthermore, we showed that the smoothed-normal-vector field is in
general strictly outward, under quite nonrestrictive conditions on the geometry, such as the absence
of cusp-like features. We then derived a lower bound for the distance between a point and its image
for which non-uniqueness of the association can occur, and we showed that this distance is bounded
from below, provided that the smoothed-normal-vector field is strictly outward.
We illustrated the properties of the smoothed-normal-vector field in 2D by means of numerical
experiments. The numerical experiments indicate that the lower bound on the association distance
is sharp. Moreover, the results corroborate that the smoothed-normal-vector field can indeed become
non-outward near cusps. Numerical experiments on distinct approximations of the unit sphere in
3D convey that the smoothed-normal-projection association displays optimal convergence in the L2
and H 1 norms under mesh refinement and under increasement of the polynomial degree of the geometry
representations. Finally, we tested the smoothed-normal-projection association on a complicated
geometry in 3D. The numerical results confirm that the smoothed-normal-projection association yields
a meaningful one-to-one correspondence, even near sharp features.
A. Regularity of smoothed-normal-vector fields on geometric realizations
Let us consider a simplicial complex A in Rd with underlying space |A| in conjunction with a geometric
realization u(|A|), where u is a member of a continuous, simplex-wise-polynomial space A υ (|A|). The
normal-vector field, simplex-wise defined by (3.4), serves as right member for the smoothed-normalvector field m according to (3.8). To facilitate the presentation, we use the notation M := u(|A|) and
[H s (M )]d := H s (M, Rd ). Our objective is to show that if n ∈ [H k,p (M )]d then m ∈ [H k+2,p (M )]d .
To enable application of the interior-regularity theorem for elliptic equations (see §3.4), we cover
u(|A|) with an atlas (see, e.g., [2]), viz.,Pa collection of open charts (Ωi , χi )i∈I such that ∪i∈I Ωi = M .
Clearly, it holds that kmk2[H s (M)]d ≤ i∈I kmk2[H s (Ωi )]d . By definition, Ωi is homeomorphic to the
subset Ωi := χi Ωi ⊂ Rd . On account of (3.8), we have on each subset Ωi :
Z
Z
√
√
jk
(A.1)
hn, wi g dξ
hm, wi + g h∂j m, ∂k wi g dξ =
∀w ∈ [H01 (Ωi )]d ,
Ωi
Ωi
1 We acknowledge the use of the gts library (gts.sourceforge.net) for constructing the meshes of the hippic figure,
and of the original geometry file of the horse from the gts sample-file repository.
20
E.H. van Brummelen
Figure 8: Smoothed-normal-projection association between nontrivial geometries: Coarse representation of a hippic figure with the image of edges of the fine representation under κ according
to (3.11) (top) and fine representation of the figure with the image of edges of the coarse representation under κ −1 according to (3.12).
21
Mesh Association by Projection along Smoothed-Normal-Vector Fields
Figure 9: Zoom of the regions near the head of figure 8.
where g jk is the contravariant metric tensor on Ωi and h·, ·i stands for the vector inner product in Rd .
Moreover, [H01 (Ωi )]d represents the subspace of functions in √
[H 1 (Ωi )]d that vanish on the boundary,
∂Ωi , in the appropriate sense. For the derivatives of gij and g with g := det(gij ) it holds that
√
g ji
√
1
lm
∂k gij = −gil gjm ∂k g ,
∂k g = √ ∂k det(gij ) =
g ∂k gij .
(A.2)
2 g
2
√
The identities (A.2) imply that g ij ∈ C s (ω) ⇔ gij ∈ C s (ω) ⇒ g ∈ C s (ω) for any positive integer s
and any open bounded subset ω ⊂ Rd . Therefore, it follows from the interior-regularity theorem
that if the covariant or contravariant metric tensor resides in C ∞ (Ωi ) and n ∈ H k,p (Ωi , Rd ), then
m ∈ H k+2,p (ω, Rd ) for any subset ω with compact closure in Ωi .
Conversely, it holds for the norm of m on any ω = χ−1
i ω, ω ⊂ Ωi that
kmk2[H s (ω)]d =
s Z
X
l=0
ω
√
g j1 k1 · · · g jl kl hm,j1 ···jl , m,k1 ···kl i g dξ ,
(A.3)
with m,j1 ···jl the covariant derivatives of order l. In particular, for orders 1, 2 and 3 it holds that
m,i = ∂i m,
m,ij = ∂ij m − Γkij ∂k m,
l
m
l
m
m,ijk = ∂i (∂jk m − Γm
jk ∂m m) − Γij (∂lk m − Γlk ∂m m) − Γik (∂jl m − Γjl ∂m m), (A.4)
where Γlij = 21 (∂i gkj + ∂j gki − ∂k gij )g kl is the Christoffel symbol of the second kind. In general,
m,i1 ···il (l ≥ 1) contains derivatives of the covariant metric tensor gij upto order l − 1. Hence, under
the standing hypotheses on the metric tensor, there exists a constant C such that kmk[H s (ω)]d ≤
Ckmk[H s (ω)]d .
It follows from the preceeding theory that if there exists an atlas (Ωi , χi )i∈I of M with the
following properties:
(1 ) there exists a finite cover ∪j∈J ωj = M such that each ωj has compact closure in some Ωi ,
22
E.H. van Brummelen
Ωσ
e
Ωe
v
Ωv
Figure 10: Illustration of the covering of the geometric realization with simplex, edge, and vertex
domains.
(2 ) in each subset Ωi = χi Ωi ⊂ Rd the covariant metric tensor gkl resides in C ∞ (Ωi ).
and, moreover, n ∈ H k,p (M, Rd ) then m ∈ H k+2,p (M, Rd ). Below, we prove the existence of an atlas
with the aforementioned properties for d = 2 by construction. The proof generalizes mutatis mutandis
to d > 2.
Let us consider a cover of the geometric realization M conforming to figure 10. More precisely,
if A, E and V denote the sets of simplices, edges and vertices, respectively, then the atlas is defined
by (Ωσ , χσ )σ∈A ∪ (Ωe , χe )e∈E ∪ (Ωv , χv )v∈V , where Ωσ = u(σ), Ωv is a topological disc covering a
neighborhood of M around a vertex v and Ωe is defined as an open subset of the union of the closure of
the simplices adjacent to edge e such that u(e) ⊂ Ωv1 ∪ Ωe ∪ Ωv2 for certain vertex neighborhoods Ωv1
and Ωv2 .
The existence of a homeomorphism χσ : x ∈ Ωσ 7→ ξ ∈ Ωσ ⊂ R2 such that the covariant metric
tensor gij = h∂i x, ∂j xi ∈ C ∞ (Ωσ ) follows straightforwardly from the smoothness of u on the simplices.
For the edge sets Ωe , a homeomorphism with covariant-metric-tensor components in C ∞ can be
constructed as follows: Let σe arbitrarily represent a simplex adjacent to edge e and let us consider
¯ 7→ x ∈ u(σe ) with ∆
¯ = {ξ̄ ∈ R2 : 0 ≤ ξ¯i , ξ̄ 1 + ξ̄ 2 ≤ 1}. The covariant
a homeomorphism ξ̄ ∈ ∆
metric tensor corresponding to this map can be expanded as ḡ ij = ∂¯i xk ∂¯j xk , where ∂¯i := ∂/∂ ξ̄i
and summation on the repeated index k is implied. Being the gradient of a scalar field, ∂¯i xk forms
¯ into R2 . There exist 2 linear combination of these vector fields,
an irrotational vector field from ∆
qki = hkl ∂¯i xl , such that ḡij = qki qkj . In particular, the covariant metric tensor can be identified with
a symmetric positive-definite matrix, again denoted by ḡij , which admits a Cholesky factorization
¯ → R such
according to ḡ ij = qki qkj . The vectors qki being irrotational, there exist scalar fields ξk : ∆
that ∂¯i ξk = qki . In particular, ξk can be defined as the classical solution of the Poisson problem
1 ∂ √ ij
1 ∂ √ ij ∂ξk
= √
(A.5)
√
ḡ ḡ
ḡ ḡ qkj ,
ḡ ∂ ξ̄ i
ḡ ∂ ξ̄ i
∂ ξ̄ j
¯ Denoting the covariant-metricfor k = 1, 2, subject to certain Dirichlet boundary conditions on ∂ ∆.
tensor components with respect to (ξ1 , ξ2 ) by gij , it holds that
ḡij =
∂ξk ∂ξl
g kl = qki qlj gkl ,
∂ ξ̄ i ∂ ξ̄ j
(A.6)
which implies that gij = δij . Let us mentioned that the coordinates (ξ1 , ξ2 ) define a so-called normal
coordinate system; see [2, p.7]. The covariant-metric-tensor components gij with respect to the normal
¯ and, by (A.2), so do g ij and √g. Analogously, we can define
coordinates (ξ1 , ξ2 ) reside in C ∞ (∆)
coordinates in the other simplex adjacent to edge e. Moreover, we can select the boundary conditions
Mesh Association by Projection along Smoothed-Normal-Vector Fields
23
in the Poisson problems such that the simplex-wise mappings are continuous across the edge e. The
composite map yields a homeomorphism χe : x ∈ Ωe 7→ ξ ∈ Ωe ⊂ R2 with the desired properties.
For the vertex sets, a homeomorphism χv : x ∈ Ωv 7→ ξ ∈ Ωv ⊂ R2 such the covariant metric
tensor with respect to the coordinate chart (ξ1 , ξ2 ) is of class C ∞ can be constructed by separating the
covering disc into nv sectors, where nv is the cardinality of set of simplices connected to the vertex v.
On each sector we can define coordinates in a similar manner as for the edge sets above, and continuity
of the composite map can again be accomplished through a suitable choice of the boundary conditions
on the sector-wise Poisson problems.
B. Proof of lower bound (4.13)
The objective here is to prove that for all η ∈ {n}⊥ ⊂ Rd , it holds that the orthogonal projection
P{m}⊥ η onto {m}⊥ ⊂ Rd satisfies (4.13). The orthogonal projection π := P{m}⊥ η is defined by the
variational problem: Find π ∈ {m}⊥ such that
hv, πi = hv, ηi
∀v ∈ {m}⊥
(B.1)
where hv, wi := v T · w denotes the scalar product in Rd . By means of the Lagrange-multiplier
formalism, π according to equation (B.1) can be equivalently defined as: Find (π, λ) ∈ {m}⊥ × R
such that
hv, πi + λhv, mi = hv, ηi
∀v ∈ Rd .
(B.2)
Upon inserting v = π in (B.2), we obtain kπk2 = hπ, ηi. Hence, by the Cauchy-Schwarz inequality,
|hπ, ηi|
kπk
kπkknk
|hπ, ni|
=
=
≥
.
kπkkηk
kηk
kηkknk
kηkknk
(B.3)
Moreover, by inserting v = n and v = m we can extract from (B.2) the identities
hn, πi + µhn, mi = 0,
µkmk2 = hm, ηi .
(B.4)
The first identity in (B.4) yields |hn, πi| = |µ||hn, mi|. Furthermore, by the second identity and the
Cauchy-Schwarz inequality we obtain |µ|kmk ≤ kηk. Therefore, in continuation of (B.3),
|µ||hn, mi|
|µ||hn, mi|
|hn, mi|
|hπ, ni|
=
≥
=
,
kηkknk
kηkknk
|µ|kmkknk
kmkknk
(B.5)
and thus we have established the lower bound (4.13).
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