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Simplification of CAD Models by Automatic Recognition and Suppression of Blend Chains

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Abstract

This paper presents a CAD model simplification procedure that consists in recognizing and suppressing blend chains of certain types. The proposed method involves Euler operators (KEV, KEF, and KFMV) developed on top of an open-source geometric modeling kernel. The simplification process consists of two stages: recognition and suppression. The suppression stage ensures the geometric and topological validity of the simplification result. The proposed approach is targeted for use in batch mode, which poses strict requirements to the robustness of the suppression algorithm. The essential properties of the approach are its sustainability, predictability of the result, and extensible architecture, which allows new topological cases to be added without modifying the algorithm's core. At the recognition stage, the algorithm constructs an attributed adjacency graph, which is then enriched with the information about types of edges, their properties, and assumed kinds of blend faces. At the suppression stage, the algorithm iterates through the adjacency graph and composes candidate blend faces into chains. For each face in a chain, local topology analysis is carried out to determine the corresponding sequence of Euler operators that are supposed to eliminate that face. The algorithm can be extended by adding descriptors of new topological cases into the processing. Upon applying the Euler operators, the affected edges are reconstructed to obtain a geometrically correct boundary representation of the model.
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ISSN 0361-7688, Programming and Computer Software, 2020, Vol. 46, No. 3, pp. 233–243. © Pleiades Publishing, Ltd., 2020.
Russian Text © The Author(s), 2020, published in Programmirovanie, 2020, Vol. 46, No. 3.
Simplification of CAD Models by Automatic Recognition
and Suppression of Blend Chains
S. E. Slyadneva,* and V. E. Turlapova,**
aLobachevsky State University of Nizhny Novgorod, pr. Gagarina 23, Nizhny Novgorod, 603950 Russia
*e-mail: sergey.slyadnev@gmail.com
**e-mail: vadim.turlapov@itmm.unn.com
Received December 20, 2019; revised January 9, 2020; accepted January 19, 2020
Abstract—This paper presents a CAD model simplification procedure that consists in recognizing and sup-
pressing blend chains of certain types. The proposed method involves Euler operators (KEV, KEF, and
KFMV) developed on top of an open-source geometric modeling kernel. The simplification process consists
of two stages: recognition and suppression. The suppression stage ensures the geometric and topological
validity of the simplification result. The proposed approach is targeted for use in batch mode, which poses
strict requirements to the robustness of the suppression algorithm. The essential properties of the approach
are its sustainability, predictability of the result, and extensible architecture, which allows new topological
cases to be added without modifying the algorithm’s core. At the recognition stage, the algorithm constructs
an attributed adjacency graph, which is then enriched with the information about types of edges, their prop-
erties, and assumed kinds of blend faces. At the suppression stage, the algorithm iterates through the adja-
cency graph and composes candidate blend faces into chains. For each face in a chain, local topology analysis
is carried out to determine the corresponding sequence of Euler operators that are supposed to eliminate that
face. The algorithm can be extended by adding descriptors of new topological cases into the processing. Upon
applying the Euler operators, the affected edges are reconstructed to obtain a geometrically correct boundary
representation of the model.
DOI: 10.1134/S0361768820030081
1. INTRODUCTION
Simplification of CAD models is carried out to solve
a variety of engineering problems. Among them, we can
mention the preparation of geometries for computation
(numerical simulation), data compression for efficient
visualization, intellectual property protection during
data exchange, performance improvement of CAD sys-
tems, and elimination of secondary structural elements
to ensure recognition of primary ones. In history-
based parametric design systems, a CAD model can be
simplified by eliminating a certain set of structural ele-
ments (also called features) from the tree of opera-
tions. However, if the design history is not available or
does not contain desired structural elements, this
approach is not applicable. Thus, a need in direct edit-
ing operators arises, which are currently absent in
open-source libraries for geometric modeling.
In this paper, we describe an operator for blend
chain suppression that is based on the recognition of
structural elements with the subsequent use of Euler
operators. The contribution of this work is as follows:
(a) the implementation of some Euler operators (KEV,
KEF, and KFMV) on top of an open-source geomet-
ric modeling kernel; (b) the development of an exten-
sible architecture for the recognition and suppression
of blend chains; and (c) the implementation of an
algorithm for suppressing a number of structurally sig-
nificant types of blend chains.
The proposed algorithm can operate in global and
local modes. In the former case, incremental suppres-
sion is carried out, i.e., blend chains are eliminated
one after another with the intermediate modification
of auxiliary data structures. In the latter (local) case,
the recognition of a blend chain starts with a candidate
face selected by the user, the other chains being unaf-
fected. The suppression algorithm accumulates the
continuous history of model modifications, which
allows the resulting boundary representation to be
matched with the original one, thus ensuring the pres-
ervation of the associated data (colors, names, anno-
tations, tolerances, etc.).
The paper is organized as follows. Section 2 over-
views the most significant works on the recognition
and suppression of blend chains. Section 3 describes
our approach to the blend face recognition based on
an attributed adjacency graph. Section 4 presents a
suppression algorithm and an incremental procedure
for running this algorithm in the batch mode. Section 5
contains some examples that illustrate the application
of the algorithm to several CAD models from our
234
PROGRAMMING AND COMPUTER SOFTWARE Vol. 46 No. 3 2020
SLYADNEV, TURLAPOV
everyday practice. That section also compares the
developed method with a more general face elimina-
tion algorithm. The directions for further research are
outlined in Section 6.
2. STATE OF THE PROBLEM
In [1], we analyzed some publications devoted to the
problem of CAD model simplification and described
the corresponding software packages. Let us dwell on
the results related mainly to the suppression of blend
chains.
The simplification of a CAD model consists in its
geometric transformation in accordance with certain
functional and technological requirements. With the
engineering semantics of the model being reflected in
its structural elements (SEs), to simplify its geometry,
the SEs need to be represented in an explicit form.
When the design history of a parametric model is
available, we can dispense with computationally com-
plex procedures, e.g., SE recognition and morpholog-
ical analysis [2]. Unfortunately, such parametric mod-
els are not always available, and even if they are, their
design history may not contain desired SEs. For
instance, a blind hole defined in a parametric model
can be transformed into a through hole when a new
volume element is removed from the model. Similarly,
when extending a prismatic body, the blend elements
defined on a f lat sketch are not automatically trans-
formed into the SEs of edge-blend faces. Thus, for
geometry simplification, direct editing tools need to be
employed even when a parametric model is available.
In [3, 4], a procedure for blend face suppression
implemented in Rhinoceros, a popular CAD system,
was described. The algorithm performs a preliminarily
classification of faces based on their types: edge-blend
face (EBF) and vertex-blend face (VBF) (see Fig. 1).
Then, the algorithm computes new geometric carriers
of boundary elements (an edge for the EBF and a ver-
tex for the VBF). The topology of the model is modi-
fied at the f inal stage of the procedure without any
preliminary prediction.
In [5, 6], a topological procedure for blend face sup-
pression based on Euler operators with subsequent
geometry stitching was described. In contrast to heuris-
tic “geometric” approaches, e.g., [4], the algorithms
described in [5, 6] first generate a topology transforma-
tion plan as a sequence of Euler operators that guaran-
tee the validity of the resulting boundary representation.
The predictability of the result achieved through the
local analysis of the blend face topology, in our opin-
ion, makes the algorithms [5, 6] suitable for automatic
simplification of CAD models in the batch mode.
However, complex blend configurations that cannot be
processed due to the limitations of the algorithm are
misrecognized already at the initial stage. Thus, the
algorithm processes only topological cases for which
the corresponding sequence of Euler operators is a pri-
ori known. On the other hand, the implementation of
such algorithms requires a geometric kernel with Euler
operators (e.g., Parasolid or ACIS). Since OpenCas-
cade, the only open-source geometric kernel, does not
provide these operators, their implementation was one
of our goals in the context of this research.
The recognition of blend elements on a boundary
representation is considered a relatively simple problem
(in contrast to the problem of their suppression).
Indeed, any face is defined by an exact equation of a
parametric surface , which generally has an
explicit specifier of its type at the level of data structures
(plane, cylinder, sphere, torus, spline, etc.). Based on
this specifier and differential properties of the surface, it
can be concluded whether the face is a blend element
or not.
The recognition of blend chains requires local topol-
ogy analysis in the neighborhood of several candidate
faces. A sufficiently general (from a practical perspec-
tive) chain recognition algorithm, which also extracts
the order of blend chain construction, was proposed in
[7]. The work [6] enriched the result obtained in [7]
with a suppression algorithm, which uses a face removal
operator [8] developed by the same authors.
The series of works [6–8], in our opinion, provides
a fundamental solution to the problem of blend face
recognition and suppression. As mentioned above, the
basic idea of this topological approach is to guess the
structure of a resulting model with the subsequent
transformation of the initial boundary representation
to an a priori known result. Thus, our goal is to formal-
ize and implement this principle by using open-source
geometric modeling tools.
3. BLEND FACE RECOGNITION
To suppress blend chains, they should be automat-
ically recognized in the original model. At this stage,
v
(,)
su
Fig. 1. Recognizable types of blend faces: edge-blend face
(EBF) and vertex-blend face (VBF).
Vertex-Blend Face
Edge-Blend Face
PROGRAMMING AND COMPUTER SOFTWARE Vol. 46 No. 3 2020
SIMPLIFICATION OF CAD MODELS BY AUTOMATIC RECOGNITION 235
the problem consists not only in finding the faces, but
also in classifying the edges that belong to them (see
Fig. 2). The classification is carried out based on the
following types: (a) spring edges; (b) cross edges; and
(c) terminating edges. The recognition result is repre-
sented as attributes of an adjacency graph. This stage
does not form chains. A recognition technique that
transforms the adjacency graph to determine the
sequence of Euler operators is published for the first
time.
Below is a brief description of the recognition pro-
cess.
1. Construction of the attributed adjacency graph
(AAG).
2. EBF recognition.
3. VBF recognition.
4. Refinement of terminating edges (a terminating
edge cannot be part of two faces recognized as blend
elements).
5. Extraction of the result.
For EBF and VBF recognition, a set of geometric
and topological heuristics are employed, which are
described below.
The recognition procedure transforms the initial
adjacency graph G into an equivalent graph G*, which
is enriched with additional attributes associated with
its vertices (Fig. 3). The main types of attributes are a
blend element (attribute value is ) and a carry-
ing face (attribute value is ).
3.1. Attributed Adjacency Graph
The AAG is a key element of the proposed recogni-
tion architecture. This data structure can be used to
solve a wide range of recognition problems. The AAG,
which contains data on the convexity and concavity of
edges (see Fig. 4), allows us to recognize certain basic
SEs, e.g., cylindrical holes, pockets, and protrusions
[14]. The AAG is also a convenient data storage that
makes it possible to emphasize the adjacency relation
extracted by analyzing the topology of the model’s
boundary representation. Let us list the main advan-
tages of the AAG in solving SE recognition problems.
First, the AAG enables the transition from the geo-
metric description of a model to the corresponding
formalism of graph theory. Model classification pro-
cedures (recognition of sheet bodies, tubes, 2.5D
models, etc.) also use operations on graphs, e.g., find-
ing connected components, subgraphs, and shortest
paths, analyzing degrees of vertices, etc.
Second, the AAG acts as a short-term storage
(cache) for identifiers of edges, vertices and faces. The
boundary representation of a model in the OpenCas-
cade library does not include unique identifiers of
boundary elements in an explicit form. These identifi-
ers must be extracted when analyzing topological
structures, which can adversely affect the performance
(
)
1
A=
(
)
2
A=
Fig. 2. Types of edges that bound blend faces.
Spring
Cross
Terminating
Fig. 3. Enrichment of the adjacency graph G with attri-
butes as a result of blends recognition.
GG*
Attr
Attr
Fig. 4. Attributed adjacency graph.
G1
G1
G2
G2
G3
G3
G4
G4
F
F
Convex
Concave
G5
G5
G6
G6
236
PROGRAMMING AND COMPUTER SOFTWARE Vol. 46 No. 3 2020
SLYADNEV, TURLAPOV
of the corresponding software. The presence of the
identifier cache as part of the AAG allows it to be used
as an “accelerating data structure.”
Third, the AAG is a carrier of boundary represen-
tation semantics and supplements it with the informa-
tion about the SEs extracted in the process of recogni-
tion.
3.2. EBF Recognition
EBF recognition relies on the classification of all
model edges based on the following types (Fig. 2).
1. Spring edges are smooth guide edges for EBFs.
2. Cross edges can be smooth and non-smooth;
they implement the adjacency of two blend faces
(EBFs or VBFs).
3. Terminating edges complete blend chains; a
closed chain does not contain terminating edges.
At the first stage, all smooth edges are found. With
dihedral angles being classified at the AAG construc-
tion stage, this stage is reduced to the attribute-based
selection of the corresponding edges from the AAG.
From the set of smooth edges, spring edges for
EBFs are selected. For this purpose, the principal cur-
vatures on a candidate edge are analyzed. The rela-
tionships between the absolute values of the curvatures
allow us to make an assumption about whether a face
A with the curvatures a1 and a2 is a blend face or not
(Fig. 5). Thus, the following conditions must be satis-
fied.
1. The absolute value a2 exceeds the absolute value
a1; i.e., the curvature in a transverse direction domi-
nates.
2. The absolute value a2 exceeds the absolute value
b2 by at least a factor of c (c > 1). This heuristic indi-
cates that the given EBF rests on a less curved surface,
e.g., on a plane.
The radius of the candidate blend face is assumed
to be .
Cross edges can be found from the condition that
the curvature of an EBF along a cross edge coincides
2
1/
a
with the principal curvature a2 found earlier. The cur-
vature of a surface along a curve is computed by the
following well-known differential geometry formula,
which involves the coefficients of the first and second
quadratic forms:
Here, and are parameters
of the carrying surface (see Fig. 6).
Terminating edges are not smooth and are not
included in the sets of spring and cross edges. A closed
chain has no terminating edges. An open chain is
bounded by terminating edges. Some terminating
edges are overridden to non-smooth cross edges (see
Section 3.4).
3.3. VBF Recognition
The next recognition stage is VBF extraction. At this
stage, a set of topological heuristics prompted by typical
blend co nfigu rations on real-world mo dels is empl oyed .
First, the VBF cannot be adjacent to another face of this
type (this heuristic ref lects the restriction of the algo-
rithm). Second, for the VBF, there must be at least
three adjacent EBFs with the adjacency being imple-
mented through cross (non-terminating) edges. Rec-
ognized VBFs receive the corresponding attribute in
the AAG.
3.4. Refinement of Terminating Edges
Finally, when all blend faces are marked, terminating
edges are refined. For each terminating edge e, a pair of
faces with the indices such that = 1
(i.e., both the faces are elements of a blend chain) is
found. Since the terminating edge must not occur
internally in a blend chain, its type is overridden to the
cross edge. A specific feature of the overridden cross
edges is that they do not form smooth connections
between blend elements (see Fig. 7).
()
+λ
λ=
+λ
2
2
2
.
LM N
k
EFG
λα
v
/tan
=d du=
v
(,)
u
(
)
,
ee
fg
() ()
ee
Af =Ag
Fig. 5. Curvature analysis to determine the blend face.
A
b2b1
a1
a2
p
B
Fig. 6. To the measurement of the curvature along the edge.
du
dv
v
α
u
x0
PROGRAMMING AND COMPUTER SOFTWARE Vol. 46 No. 3 2020
SIMPLIFICATION OF CAD MODELS BY AUTOMATIC RECOGNITION 237
Fig. 7. Typical example of a terminating edge overridden to
a cross edge.
Overridden
terminating edge
4. BLEND FACE SUPPRESSION
Suppression of blend faces and their chains consists
in removing the corresponding faces and stitching the
adjacent ones to obtain a waterproof shell. The
removal can be implemented in several ways: (a) by
eliminating faces from the topological structure of a
model with subsequent collapsing of the gaps and (b)
by applying a sequence of Euler operators. In the first
case, the removal is carried out somewhat blindly
because there is no guarantee that the topological con-
figuration in the neighborhood of the chain allows for
a correct collapse of the gap. To control the collapsing
process, an additional analysis was carried out in [10]
to match the local topological configuration of a blend
chain with an abstract domain homeomorphic to it. If
Euler operators are employed [11], then topological
flaws are impossible by definition. On the other hand,
the Euler operator performs only the syntactic transfor-
mation of the model while preserving its semantics
(geometry). Therefore, upon applying Euler operators,
a local reconstruction of the geometry is required. We
refer to this stage as the geometry normalization pro-
cess.
Formally, the process of blend face removal
involves the following sequence of operations.
1. By traversing the graph , blend chains are
formed as sets of faces . Here,
(where K is the number of chains), while Nk
and Mk are the starting and ending indices of the faces
in the kth chain. Only the AAG vertices to which the
subgraph (where ) corre-
sponds are extracted. The function returns a
graph vertex with the number i = Nk, Mk. The func-
tion returns an attribute value for this vertex (1 for
a blend element, 2 for a spring face, or 0 for the other).
2. The faces of the chain are removed from the
model. The graph plays a supplementary role in the
process of face removal, which is why its reconstruc-
tion (or modification) is necessary only if the suppres-
sion is continued to the next chain, i.e., .
3. The affected edges of the model are recon-
structed to merge the support faces.
4.1. Basic Algorithm
The basic suppression algorithm processes the fol-
lowing topological cases of a unit face.
1. The face is an isolated blend face.
2. Th e face is part of a closed or o pen cha in of EBFs
and VBFs.
At the preliminary stage, the possibility of sup-
pressing the entire chain is checked. For this purpose,
additional topological heuristics are employed. In par-
ticular, any cross edge of the chain must be part of
exactly two faces of this chain. Otherwise, the chain is
*
G
{
}
==:,
kj kk
FfjNM
1,
k= K
*
k
CG
(( )) 1,
ik
AV C =
(
)
i
V
i
f
(
)
A
k
F
*
G
kK
not completely recognized and is considered insup-
pressible.
The next steps of the suppression algorithm are as
follows.
1. Using the Euler operators for the topological
preparation of the result.
2. Reconstructing the edges to adapt the geometry
to a new topological structure (normalization).
3. Posterior validation of the faces affected.
Step 1 consists in the sequential application of the
kill-edge-vertex (KEV), kill-edge-face (KEF), and
kill-face-make-vertex (KFMV) operators in the order
determined by the type of a particular topological con-
figuration.
Step 2 performs purely geometric constructions on
a predetermined topological structure of the model.
At this step, the edges, vertices, and faces affected by
the topological operation are reconstructed. For
instance, to obtain a new supporting curve of an edge,
the corresponding supporting surfaces are intersected.
Step 3 protects the algorithm from possible errors
in geometric constructions (intersection of surfaces,
projection of a curve onto a surface, etc.) that are due
to the imperfect reliability of the geometric kernel. In
addition, at this step, for each affected face, the
boundary of the domain D is checked for self-intersec-
tion. This is required because the suppression of blend
faces can, generally speaking, change the number of
connected components in the domain D (see Figs. 8
and 9). The number of connected components in this
domain is assumed to be an invariant of the blend sup-
pression operator. The analysis of alternative invari-
ants, e.g., a bounding box for the domain D, will make
it possible to introduce new operating modes of the
suppression algorithm.
238
PROGRAMMING AND COMPUTER SOFTWARE Vol. 46 No. 3 2020
SLYADNEV, TURLAPOV
4.2. Incremental Procedure
The basic algorithm described above simplifies a
CAD model by suppressing a single blend chain
selected. However, to simplify models in the batch
mode (without user involvement), this is not enough.
To remove all blend chains, we implemented an auxiliary
procedure of the so-called incremental suppression.
The incremental procedure begins with construct-
ing the AAG (G). The graph G must be reconstructed
whenever the topology of a model is modified,
because the graph vertices address the boundary ele-
ments of the model through their integer indices,
which are not robust to modification operators [12].
The procedure implements two nested data process-
ing loops (see Fig. 10). At each iteration of the outer
loop, the blend chains of the entire model are recog-
nized. Upon selecting an arbitrary face , an
attempt is made to suppress the corresponding chain
. If successful, the graph is reconstructed
for the updated model and the local history of modi-
fications is concatenated into the global history H. If
the chain cannot be deleted (inner loop), the
corresponding faces are removed from the suppression
queue and a new chain is selected without recon-
structing the graph. In addition, the algorithm stores
the set of addresses T for the insuppressible faces,
which generally remain relevant even after the modifi-
cation of the topology (these are the so-called tran-
sient indices for the boundary elements of the model).
In the inner contour, the algorithm iterates over the
chains until the suppression is complete or the queue
F is empty.
4.3. History of Modifications
Like any shape editing operator, the suppression
operator must preserve associativity between the
boundary elements of the resulting and original mod-
els. For this purpose, the suppression algorithm is sup-
plemented with a data structure that represents the
modification history of topological elements as a
graph. With the above-described incremental proce-
dure consisting in multiple calls of the suppression
F
fF
(
)
chain f
G
s
h
(
)
chain f
F
operator, the corresponding data structures must be
concatenated (glued) to obtain a continuous history of
each boundary element.
5. COMPARISON
To suppress blend faces and their chains, the face
removal (FR) operator can be employed. The FR
algorithm removes a target face or a set of faces from
the structure of a model with subsequently merging
the adjacent faces to restore adjacency [8]. In the
OpenCascade library, this approach is implemented in
Fig. 8. Violation of the topological validity of the model
when removing the inner blend.
Fig. 9. Self-intersecting contour of the support face.
Self-intersecting
contour
Fig. 10. Flowchart of the incremental suppression procedure.
Outer contour
F empty
T empty
G get or initialize AAG
F
Ok?
|F|=0
|F|=0
Ok?
Failure
Success
recognize(r, G)
F F \{f}
F F \chain(f)\{f}
F F \chain(f)\{f}
TT chain(f) {f}h, s suppress(f)
G build AAG
H H+h
f
f T
take any face from F
H get or initialize history
Inner contour
PROGRAMMING AND COMPUTER SOFTWARE Vol. 46 No. 3 2020
SIMPLIFICATION OF CAD MODELS BY AUTOMATIC RECOGNITION 239
the BRepAlgoAPI package. The blends recognition
and suppression (BRS) algorithm proposed in this
paper is an alternative to the FR algorithm.
Table 1 compares the two algorithms based on
three criteria: runtime in seconds ( and ),
validity of the result ( {0, 1}), and efficiency
in terms of the number of removed faces ( and
). The experiment was conducted on a computer
with Intel(R) Core(TM) i5-9500 @ 3.00 GHz and 32
GB RAM.
The parameter sets a limit on the maximum
radius of the blend faces to be recognized. The symbol
“i” (“interactive”) marks the time measurements in
the cases where user involvement was required. The
algorithms were run on a pre-selected public dataset
[16]. To run the FR algorithm, we used the prelimi-
nary stage of blend chain recognition described in Sec-
tion 3. The time measurements for the FR algorithm do
not include the recognition stage and do not take user
actions into account. The time measurements for the
BRS algorithm include both recognition and suppression
because, in the BRS algorithm, these stages are insepara-
ble. The symbol “d” (“design intent”) marks the simpli-
fication results that are formally valid but violate the
design intent of a model at the level of its structural ele-
ments. The symbol “?” means that the corresponding
result cannot be obtained (the algorithm did not termi-
nate in an acceptable time of 5–10 min).
A key feature of the automatic simplification algo-
rithm is its robustness. The shaded cells of Table 1
contain the results that demonstrate the insufficient
robustness of the corresponding algorithm.
The BRS algorithm runs 14.6 times faster on aver-
age (see Fig. 11) than the FR algorithm in the cases
where both algorithms perform well, i.e., and
. The higher performance of the incremental
BRS procedure is owing primarily to the fact that the
proposed algorithm is a local operation (see p. 287 in
[17]; see also [18]), while the FR algorithm is based on
the apparatus of global Boolean operations. The local-
ity of the BRS algorithm also guarantees that the
FR
T
BRS
T
,
FR BRS
VV
FR
N
BRS
N
max
r
0
FR
N
0
BRS
N
design intent of a CAD model is preserved, because
the boundary elements outside the neighborhood of a
suppressed chain are not affected.
Having lower overall productivity ( in
the majority of the cases considered), the BRS algo-
rithm is more robust, exhibiting aberrant behavior
only in 1.05% of cases versus 23.15% for the FR algo-
rithm. We should also point to a non-obvious fact that
the FR algorithm does not guarantee model simplifi-
cation, as evidenced by the result on the test set #73
(see Table 1). Indeed, with the FR algorithm, the
topology of a model is modified twice: when a face is
removed and when its neighbor faces are merged up to
intersection. The second topological modification, in
some cases, can cause the formation of new faces. This
is generally due to a poor selection of faces to be
removed (see Fig. 12).
The results of applying the incremental BRS pro-
cedure are shown in Fig. 13.
6. CONCLUSIONS
This work completes the development of our new
approach to CAD model simplification by means of
recognizing and suppressing blend chains. In the
BRS FR
N<N
Fig. 11. Performance comparison between the BRS algorithm and the FR algorithm.
30
25
20
15
10
5
0468151721333740
Rate
BRS performance compared to FR
Average
42 45 47 50 62 66 69 72 74 80 86 92 95
Fig. 12. Ab normal res ult o f th e FR alg ori thm o n th e tes t se t
#73 (see Table 1) for .
max
r=
240
PROGRAMMING AND COMPUTER SOFTWARE Vol. 46 No. 3 2020
SLYADNEV, TURLAPOV
Table 1. Comparison between the FR and BRS (incremental procedure) on the test data [16]; the cells that correspond to
the unsatisfactory results of the algorithms are shaded.
Case rmax TFR VFR NFR TBRS VBRS NBRS
1 0.05 1 1 0.01 1 0
2 0.025 1 1 0.003 1 0
3 0.07 1 2 0.008 1 0
4 0.08 1 3 0.012 1 3
5 ∞ 0.13 1 12 0.016 1 4
6 4.98 0 9 0.08 1 4
7∞0.0181 10.0051 1
8 0.02 1 2 0.004 1 2
9 0.02 1 2 0.003 1 0
10 0.025 1 4 0.005 1 4
11 0.035 1 7 0.006 1 0
12 0.0005 1 0 0.003 1 0
13 0.033 1 4 0.006 1 0
14 0.01 1 4 0.007 1 0
15 0.08 1 18 0 .0 2 1 18
16 0.067 1 4 0.006 1 1
17 0.026 1 4 0.008 1 4
18 0.075 1 5 0.02 1 5
19 0.025 (i) 1 4 0.004 1 0
20 0.02 (i) 1 20 0.017 1 0
21 0.085 1 8 0.01 1 8
22 0.045 1 2 0.005 1 2
23 0.028 1 2 0.003 1 0
24 0.96 1 (d) 26 0.005 1 0
25 0.0004 1 0 0.004 1 0
26 0.085 1 (d) 20 0.01 1 0
27 0.15 1 (d) 34 0.04 1 0
28 0.45 16.7 (i) 1 44 0.64 1 0
29 1.28 (i) 1 13 0.004 1 0
30 0.0004 1 0 0.012 1 6
31 4 11.1 0 111 0 .2 5 1 0
32 0.0005 1 0 0.01 1 3
33 2.07 1 6 0.2 1 5
34 ∞ 0.51 1 11 0.05 1 2
35 0.0004 1 0 0.015 1 3
36 ∞ 44.5 1 0 8.92 1 80
37 ∞ 111.7 1 210 19.5 0 105
38 ∞ 8.14 1 47 0.64 1 27
39 ∞ 4.26 1 16 0.06 1 0
40 13.67 1 (d) 39 0.48 1 13
41 10.73 1 (d) 39 0.35 1 13
42 1.7 1 0 0.1 1 8
43 ∞ ? ? ? 2.58 1 4
44 ∞ 0.08 1 18 0.02 1 18
PROGRAMMING AND COMPUTER SOFTWARE Vol. 46 No. 3 2020
SIMPLIFICATION OF CAD MODELS BY AUTOMATIC RECOGNITION 241
45 0.15 1 35 0.04 1 35
46 0.017 1 1 0.003 1 1
47 0.04 1 2 0.006 1 2
48 ∞ 11.8 1 97 0.447 1 75
49 2.22 (i) 1 3 0.096 1 0
50 ∞ 8.11 1 42 0.19 1 8
51 ∞ ? ? ? 0.036 1 0
52 41 ? ? ? 0.17 1 1
53 3.46 1 (d) 80 0.062 1 0
54 0.0005 1 0 0.01 1 3
55 ∞ ? ? ? 2.56 1 0
56 66 1 4 0.17 1 0
57 1.58 1 (d) 15 0.15 1 6
58 ∞ 0.5 1 11 0.003 1 0
59 ∞ ? ? ? 1.76 1 11
60 0.0004 1 0 1.22 1 11
61 ? ? ? 49. 62 1 90
62 0.15 1 5 0.013 1 3
63 0.14 1 5 0.009 1 3
64 0.09 1 3 0.004 1 0
65 1.01 1 0 0.07 1 8
66 ∞ 143.5 1 39 1.69 1 4
67 ∞ 1.48 1 27 0.133 1 6
68 0.0004 1 0 0.14 1 12
69 0.327 1 4 0.02 1 4
70 11.25 1 32 3.98 1 24
71 1.08 (i) 1 6 0.017 1 0
72 0.591 1 (d) 12 0.041 1 4
73 21.2 1 (d) 42 + 0.789 1 6
74 0.82 1 4 0.07 1 4
75 ∞ ? ? ? 9.48 1 23
76 0.15 1 4 0.018 1 4
77 0.39 1 2 0.44 1 0
78 ∞ ? ? ? 0.21 1 0
79 5 3.1 1 (d) 32 0.022 1 0
80 0.29 1 46 0.068 1 46
81 ∞ 28.52 1 51 0.14 1 0
82 5 ? ? ? 3.39 1 21
83 ∞ 8.29 1 47 0.64 1 27
84 6.5 2.08 1 30 0.071 1 0
85 0.38 1 9 0.011 1 0
86 0.16 1 3 0.015 1 3
87 6.5 2.14 1 34 0.004 1 0
88 4.55 1 (d) 87 0.94 1 0
89 ∞ 9.77 1 47 0.56 1 6
90 79.26 1 0 0.66 1 0
Case rmax TFR VFR NFR TBRS VBRS NBRS
Table 1. (Contd.)
242
PROGRAMMING AND COMPUTER SOFTWARE Vol. 46 No. 3 2020
SLYADNEV, TURLAPOV
framework of this research, for the first time, the Euler
operators KEV, KEF, and KFMV have been imple-
mented on top of the open-source geometric kernel
(OpenCascade) and the architecture for feature recog-
nition on the attributed adjacency graph (AAG) has
been proposed. The recognition procedure yields the
graph that is isomorphic to the graph and
includes some additional attributes used at the stage of
blend element suppression. The suppression algorithm
involves the incremental procedure that allows recog-
nized blend chains to be removed in the batch mode.
The algorithm can be extended by adding descriptors
of new topological cases. Due to the accumulation of
the modification history for all affected boundary ele-
ments, the semantically important information (col-
ors, names, and tolerances) associated with the origi-
nal model can be transferred to a simplified model
without losses.
*
G
G
Simplification of CAD models with absent design
histories remains an important engineering problem,
as was shown in [19, 20] and some other works. In the
framework of this research, an experimental software
environment [13] was proposed, available on a non-
profit basis with open-source code. It includes the
implementations of the AAG, Euler operators, BRS
algorithm, and incremental suppression procedure. As
input and output data, it uses STEP files (ISO 10303).
Once constructed, the AAG can be transferred to
third-party systems, e.g., MATLAB, for subsequent
analysis. In [20], the JSON format was used to transfer
AAG data.
The proposed approach to blend chain recognition
and suppression was implemented in C++ by using
the OpenCascade geometric kernel and Analysis Situs
platform [13], which was employed for prototyping.
The latest version of the algorithm and a test dataset
are available in open access [9]. The algorithm was
also integrated into CAD Processor [15].
Based on the results of the experiment, we can outline
some directions for further development of the proposed
algorithm. First, the “catalog” of supported blend types
is currently incomplete and can be extended. Second, it
seems reasonable to optimize the performance of the
incremental procedure, which carries out the global
reconstruction of the AAG, even though each iteration
modifies the topology locally. The reconstruction of
the AAG can be replaced by its modification, whereby
the faces of a deleted chain are removed and existing
attributes are copied at each iteration.
The comparison with the FR algorithm has showed
that, in some cases, the BRS algorithm does not cope
well with the blend chains correctly processed by the
FR algorithm. For these cases, the BRS algorithm
should be further improved. In addition, the list of
types of topological configurations supported by the
BRS algorithm should be extended and tested.
An obvious direction for further improvement of
the proposed approach is its adaptation to chamfer
suppression as this requires only the adaptation of the
recognition algorithm, while the set of Euler operators
remains the same due to the topological equivalence of
blend chains and chamfers.
91 1.25 (i) 1 0 0.053 1 0
92 13.32 1 232 2.55 1 161
93 0.194 1 3 0.015 1 3
94 0.0004 1 0 0.01 1 3
95 ∞ 87.67 1 38 4.95 1 61
Case rmax TFR VFR NFR TBRS VBRS NBRS
Table 1. (Contd.)
Fig. 13. Some results of the BRS algorithm (top-down):
test sets #92, #80, and #48.
PROGRAMMING AND COMPUTER SOFTWARE Vol. 46 No. 3 2020
SIMPLIFICATION OF CAD MODELS BY AUTOMATIC RECOGNITION 243
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Translated by Yu. Kornienko
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