Improved Representation of Dependencies in Feature-based

Parametric CAD Models using Acyclic Digraphs

Jorge D. Camba

and Manuel Contero

Dwight Look College of Engineering, Texas A&M University, College Station, TX, U.S.A.

I3BH, Universitat Politècnica de València, València, Spain

Keywords: Parametric CAD Modeling, Graph Theory, 3D Model Visualization, Mechanical Design.

Abstract: In an engineering design context, the process of modeling complex 3D parts with feature-based parametric

software systems often produces a large number of dependencies between geometric features that are

difficult to manage. The resulting network of feature interdependencies can be understood as an alternative

representation of the CAD model that can help identify the most important aspects of the geometry, its

critical features, and understand the overall complexity and interconnectedness of the model. Being able to

visualize and process this information efficiently can significantly enhance design activities and facilitate

model reuse, which can ultimately lead to cost and time savings, and better quality models. In this paper, we

identify some of the simplifications and elements that are overlooked by current representation techniques

and describe the negative consequences of not taking these elements into consideration. We describe a new

method and a software solution to generate a simplified, intuitive, and more accurate visualization of a

parametric model as an acyclic digraph. Finally, to validate our approach, the proposed representation is

compared to existing techniques using a group of CAD models.

1 INTRODUCTION

Over the years, feature-based parametric Computer-

Aided Design (CAD) technology has steadily

progressed to become a mature and commonly

deployed technology for the creation of 3D CAD

models and assemblies. In a parametric model, a

series of variable parameters and geometric relations

control the geometry of the object so it can be

modified easily to create different design variants.

These elements can be defined by dimensional,

geometric, and algebraic constraints (Shah, 1991).

Feature-based parametric modeling systems rely

on data structures that maintain geometric

information of specific aspects of the model

(features) in an associative manner, specifically in

the form of parent/child relationships. Therefore, all

individual features in the CAD model are connected

hierarchically, creating a network structure where

every node represents a feature and every connection

represents a dependency between two features

(Hanratty, 1995). This structure is commonly known

as design tree, feature tree, or history tree.

Because of the adaptable nature of the design

tree, parametric 3D CAD systems allow the

incorporation of design semantics to the model,

which facilitates the modification of the geometry by

simply changing the values of the parameters and

dependencies. In this regard, a parametric model can

be considered an intelligent representation of a part.

When dependencies are properly defined,

alterations to a parent node will automatically

propagate downstream to all its child nodes and the

CAD model will adjust and react to changes in a

predictable manner (Bodein, Rose, and Caillaud,

2014). Therefore, from a designer’s perspective, it is

important to plan the modeling procedure

beforehand to determine the most efficient sequence

of features, as poor modeling strategies often result

in parts that take longer to create and are more

difficult or impossible to modify (Hartman, 2005).

Unfortunately, parent/child interdependencies are

also the root of many regeneration problems in

parametric modeling. As the size and complexity of

a parametric model grows, so do the number of

dependencies and the degree of interconnectedness

of its design tree, which can severely impact

maintainability and model reuse (Salehi and

McMahon, 2009).

Camba J. and Contero M..

Improved Representation of Dependencies in Feature-based Parametric CAD Models using Acyclic Digraphs.

DOI: 10.5220/0005261500160025

In Proceedings of the 10th International Conference on Computer Graphics Theory and Applications (GRAPP-2015), pages 16-25

ISBN: 978-989-758-087-1

 2015 SCITEPRESS (Science and Technology Publications, Lda.)

From a user’s standpoint, modeling complex

parts requires using and managing large amounts of

parametric information and dependencies efficiently,

which can be overwhelming and confusing, even for

the most experienced designers (Baxter et al., 2007;

Mohammed, May, and Alavi, 2008). For this reason,

many design firms claim they have difficulties

visualizing parametric information, which often has

a negative impact on the documentation of design

knowledge (Myung and Han, 2001).

Although some tools have been developed to

improve the visualization of a CAD model structure

(some of which will be discussed in the next

section), there are currently no tools that can

automatically generate a simple and intuitive

representation of all the dependencies in a

parametric model for visualization and

documentation purposes (Marchenko et al., 2011). In

fact, most commercial CAD tools only provide tree

views to navigate the model features in a linear

manner, which often hinders the understanding of

complex relations. Furthermore, the simplifications

and assumptions made by these representations are

frequently inaccurate and ambiguous, making them

impractical and unreliable in production

environments.

In this paper, the problems with current

representations of parametric model structures are

identified and discussed. An improved

representation method based on directed acyclic

graphs (DAG) is proposed that eliminates ambiguity

and simplifies inherited dependencies between

features at multiple levels. Additionally, a prototype

of our method is presented along with examples of

achieved results.

2 RELEVANT LITERATURE

A general description of the hierarchical structure of

a parametric CAD model was provided by

(Marchenko et al., 2011). According to the authors,

a parametric model can be understood as a mono-

hierarchical, tree-like structure consisting of general

assemblies, subassemblies, parts, features,

dependencies, and parameters, where all elements

stand in relation to exactly one parent.

If individual CAD elements are isolated and

studied independently, additional dependencies can

be added on separate hierarchical layers, resulting in

a directed acyclic graph structure, also called poly-

hierarchy, in which multiple parents are possible for

any given node. Consequently, a more complex

network structure is produced (Marchenko et al.,

2011).

Visualizing the CAD model structure is

problematic, particularly for complex models.

Efforts include the use of modeling languages such

as UML (Wang and Li, 2012) and SysML (Peak et

al., 2007; Wölkl and Shea, 2009) to help designers

define structures that can be implemented in a CAD

model, or the application of entity-relationship

diagrams borrowed from database technologies

(Zhou, 2011). These methods, however, were

developed to describe models during the early stages

of the design process, not as tools to generate

visualizations of existing structures.

The standard method to visualize the model

structure implemented by most commercial systems

such as SolidWorks®, Catia®, or PTC Creo®, is the

design tree (see Figure 1).

Figure 1: Sample design tree in SolidWorks®.

The design tree is not a tool itself, but a

chronological representation of all the steps and

operations performed to create a specific model. As

new features are created, they are sequentially

inserted at the bottom of the model’s design tree.

The design tree allows the user to go back to any

specific point in the design and edit a particular

feature or sketch. Nevertheless, it is difficult to

visualize and analyze feature interdependencies.

Although CAD packages allow designers to select

node from the design tree and query their

dependencies (the result is typically two lists of

nodes with parent and child features), an overall

view of the model’s structure is often not available.

Various methods for modeling dependencies in

engineering processes have been applied to the

visualization of parametric models. For example,

feature dependencies can be represented as design

structure matrices. A Design Structure Matrix, or

Dependency Structure Matrix, (DSM), is a method

originally introduced by Steward (1981) for

representing and analyzing interdependencies

between elements and has become a common

modeling tool in a variety of application areas

(Eppinger, 1991).

ImprovedRepresentationofDependenciesinFeature-basedParametricCADModelsusingAcyclicDigraphs

A DSM is a square matrix (i.e., it has an equal

number of rows and columns) that shows

relationships between elements in a system. In the

context of parametric models, a binary matrix is

used because it can represent the presence or

absence of a relationship between pairs of features in

a model. This matrix is described as follows:

 Features of the model are placed down the left

side of the matrix as row headings and across

the top as column headings in the same order.

 If there exists a parent-child relation from

node i to node j, then the value of element i,j

(row i, column j) is 1. Otherwise, the value of

the element is zero.

The diagonal elements of the matrix do not have

any interpretation in describing the system, so they

are usually either left empty or blacked out. An

example of a DSM is shown in Figure 2.

Figure 2: DSM showing model dependencies.

Authors Tang et al. (2010) used DSM structures to

capture and reuse past design knowledge.

Specifically, they use DSM to record information

such as interaction levels and design parameters.

Researchers Lai and Gershenson (2008) applied

DSM techniques based on design features to the

representation of dependencies for assembly

modularity. Karniel, Belsky, and Reich (2005) used

DSM to decompose complex 3D-surface fitting

reengineering problems from geometry constraints.

In terms of CAD model visualization, Bhaskara

(2011) suggested a novel approach to analyze and

restructure complex CAD models using DSM

techniques. In his work, once all the dependencies

were identified and represented as a DSM, he

applied partitioning and clustering algorithms to

restructure and optimize the final matrix (and, thus,

the original model structure). For instance, heavily

used features and their closely related dependencies

were automatically moved to the bottom of the

DSM, and closely connected features were grouped

together to form clusters. For his study, he

developed the DSMs manually, but he recognized

the need for software systems that can automatically

generate accurate DSMs for large and complex CAD

models. He also acknowledged the need to represent

dependencies from auxiliary features and sketches

(Bhaskara, 2011).

A more visual, intuitive, and common method to

represent dependencies in parametric models is

provided by graph-based tools. Generally speaking,

a graph G can be defined as a pair G = (V, E) where

V is a set of nodes (or vertices) and E, the links (or

edges) between two connected nodes. When the

edges have a direction associated with them, then the

graph is called a directed graph, or digraph.

Graphs have been used to model a variety of

problems in many disciplines such as mathematics

(Bondy and Murty, 1976), engineering and computer

science (Deo, 2004), and economics (Michael and

Battiston, 2009). In the context of parametric CAD,

models can be understood as directed graph

structures where every feature is represented as a

node, and every parent-child relation is represented

with a directed edge from the parent to the child

node, as shown in Figure 3. Furthermore, because

the nature of parent-child relations prevents the

appearance of directed cycles of loops in the

structure, the directed graph is always acyclic.

Figure 3: Directed graph showing model dependencies.

Graph-based tools have been used to define the

relationships and dependencies among the geometric

features of parts in mechanical assemblies (Srikanth

and Turner, 1990) and more recently, to check

tolerance specifications assigned to a CAD model

(Franciosa, Patalano, and Riviere, 2010).

Researchers Marchenko et al. (2011) proposed a

method that uses graphs to visualize and document

parametric information of 3D CAD models. Their

system is limited to assemblies in the area of sheet

metal forming tools, although the authors claim that

GRAPP2015-InternationalConferenceonComputerGraphicsTheoryandApplications

their visualizations and tools can be generalized for

design tasks in other branches.

More recently, Patalano, Vitolo, and Lanzotti

(2013) presented a software tool that uses graph

theory to generate the geometric modeling of

mechanical assemblies. Similarly to the previous

authors, Patalano, Vitolo, and Lanzotti (2013) also

limited their study to assemblies and did not

consider the representation of individual parts.

Graphs are also used by Owensby and Summers

(2014) to estimate assembly times of products. In

their work, the authors generate the connectivity

graph of an assembly from assembly constraint

information and analyze the structural complexity of

the graph using a variety of metrics. Once again,

only assemblies are considered.

Some high-end commercial CAD packages such

as Catia® and UG-NX® provide tools to explore the

structure of CAD models and the links between

assembled parts in the form of graphs (Tickoo,

2014).

In the next section, some of the visualization

problems with current representation tools are

discussed, focusing on the ambiguity of the

representations when certain simplifications are

made.

3 PROBLEMS WITH CURRENT

GRAPH-BASED

VISUALIZATION METHODS

Some of the problems with current graph

visualization techniques in the area of parametric

modeling were identified by Marchenko et al.

(2011). The problems described by the authors,

however, are mainly concerned with the

descriptiveness, readability, and clarity of the graph,

i.e., the challenges that need to be overcome to make

the graph representation more expressive and

understandable. The problems they identified as well

as some of their recommendations can be

summarized as the following:

 In many models, especially those with a large

number of relations, some dependencies

cannot be drawn without intersections, which

can result in a confusing visualization. Objects

and lines in the graph should be drawn

without overlapping.

 In large graphs, it is often difficult to track

down the hierarchy of individual CAD

elements, particularly if the graph is not

clearly organized by levels. Connections

should be easy to follow, with minimum

crossings and bends.

 The designation of certain parameters is not

unique. In some cases, to reduce the number

of intersections, some graph representations

show the same feature as multiple nodes.

 The representation of specific input and output

parameters related to a dependency is not

shown, which hinders the tracking of

information flows. In fact, these specific

elements cannot be efficiently displayed by a

static visualization, as the large number of

dependencies would overwhelm the viewer.

Instead, an interactive approach is needed to

control the number of elements that are

displayed at any given time.

 There is a lack of standardization in terms of

the graphical language (icons) used to

represent individual CAD elements, which can

be confusing due to the abundance of CAD

elements and the variety of CAD systems.

Although the previous recommendations can

certainly improve the graph representation of the

parametric CAD model, they do not consider some

of the specific relations that may result in ambiguous

representations or incorrect graphs. In the following

subsections, two of these specific problems are

described.

3.1 Multi-level Relations

An ambiguous graph can be generated if multi-level

relations (grandparent-parent-child) are ignored,

simplified, or not carefully analyzed. As part of this

research, the authors of this paper evaluated this

situation using two popular parametric modeling

packages, SolidWorks® and Autodesk Inventor®.

To illustrate the problem, two versions (A and B)

of a CAD model were used. Both versions are

identical in terms of geometry, but different in terms

of the parameters used to control the features. The

geometry of the models is shown in Figure 4.

The two models differ slightly in the way step 6

(“cut hole”) was performed. This 3D operation takes

the two-dimensional profile of the circle as input and

removes material up to a specified depth, which can

be parameterized. Since the design of the part

requires the hole to pierce through the top horizontal

block, which is 0.5” thick, the depth of the hole must

also be 0.5” deep.

In version A of the model, the depth of the cut

operation was indicated as a fixed value of 0.5”

which goes through the entire top block. In version

B, however, the depth of the cut was indicated by

ImprovedRepresentationofDependenciesinFeature-basedParametricCADModelsusingAcyclicDigraphs

Figure 4. Modeling steps to create sample part.

using an existing point as a reference (one of the

corners on the base block). This way, if the thickness

of the horizontal block needs to be increased, the

depth of the cut does not have to be manually

adjusted. The two approaches are shown in Figure 5.

Both of them will successfully perform the cut and

deliver the desired result.

Figure 5. Versions of CAD model based on different

parameters for the depth of the cut.

When asked to display the feature dependencies for

both models, the CAD packages return the exact

same information. The dependency graph for both

versions of the model is shown in Figure 6. The

node “Extrude1” represents the first extrusion of the

original sketch (step 2), node “Extrude2” represents

the extrusion of the horizontal block (step 4), and

node “Cut” represents the circular hole (step 6).

The dependency between the first node

(“Extrude1”) and the third (“Cut”) is problematic.

Even though the dependency is displayed in the

graph in both versions of the model, its meaning

differs greatly.

In the first case (version A), the link between

“Extrude1” and “Cut” represents an inherited

dependency. “Cut” depends on “Extrude2” and

“Extrude2” depends on “Extrude1”, therefore “Cut”

depends on “Extrude 1.” In other words, “Extrude1”

is the grandparent node of “Cut.” This relation

indicates that the node “Cut” will be affected by

changes in either “Extrude1” or “Extrude2.”

In the second case (version B), the same link

represents a direct dependency between “Extrude1”

and “Cut.” The geometry of the cut is partially

dictated by a geometric element (point P

) that

belongs to the feature “Extrude1.”

Figure 6: Dependency graph for both versions of the CAD

model according to the parent/child relationships

displayed by SolidWorks®.

This inherited dependency occurs naturally in single

body parts as a by-product of the Boolean operations

that are performed internally to manipulate

geometric bodies. As new features are created and

added to the model, they are automatically merged

or combined with existing features to create one

single block of geometry. As a result, all new

features will depend, one way or another, on

previous geometry. However, from a practical

modeling standpoint, these dependencies do not add

value to the visualization. The fact that all features

are merged together in one single body implies that

these types of dependencies exist and that the chain

of dependencies always flows from the feature that

is created first to the feature that is created last.

To illustrate the situations discussed earlier, we

can analyze how the two models react to changes

when the geometry is modified by adding a new

feature (a third extrusion that passes through point

) before the hole is cut. While version A will

GRAPP2015-InternationalConferenceonComputerGraphicsTheoryandApplications

Figure 7: Model behavior after geometry modifications.

update correctly, version B will return a rebuild error

because point P

will no longer available after the

third extrusion is created. The sequences of changes

are shown in Figure 7.

This example demonstrates how current graph-

based representation methods for parametric models,

even those implemented by commercial systems, are

imprecise when dealing with multi-level or inherited

dependencies in single body parts. When such

relations are displayed, the resulting graph becomes

ambiguous, as it allows for multiple interpretations,

each of which represents a different CAD model.

As a solution to this problem, we suggest

labelling or marking the inherited dependencies, or

using a different color, line type, or symbol for the

connector. In certain cases, inherited dependencies

can even be eliminated, especially in multi-level

relations (such as great-grandparent, grandparent,

parent, child) that involve single-body parts with

features that already share a different type of

dependency. These simplifications can significantly

reduce the visual complexity of the graph and help

users easily determine the most relevant parametric

constraints. This statement reinforces the position of

authors Marchenko et al. (2011), whose research

pointed out the problem of visually overwhelming

users with unnecessary dependencies.

3.2 The Role of Sketches

Although certain 3D features do not require the use

of two-dimensional sketches (fillets and chamfers,

for example, only consume existing 3D edges, and

mirror creates a symmetrical copy of a set of

features about a face or a plane), most 3D operations

that are used to define the building blocks of a

feature-based parametric 3D model require prior

definition of one or more sketches as input to

generate the desired volume. From a purely

functional point of view, however, the vast majority

of 2D sketches are used only once (and almost

exclusively by one feature). Only occasionally are

sketches shared among multiple features to define

different geometric elements (a situation that

translates into a graph with one parent sketch node

and multiple child feature nodes) or used as

connecting nodes of a sketch-to-sketch dependency

(which is represented as a parent sketch node with a

child sketch node).

In this regard, most sketches are shown as

intermediate elements in the dependency graph

linking two feature nodes, which results in an extra

level of connectors that add unnecessary clutter to

the graph but no added value in terms of

visualization of the parametric relations. To simplify

the overall graph, these sketch nodes can be

represented with distinctive styles or even

eliminated from the graph.

ImprovedRepresentationofDependenciesinFeature-basedParametricCADModelsusingAcyclicDigraphs

Figure 8: Modeling steps (top) and three visualizations of the model structure (bottom). The classic design tree

representation (bottom left) does not provide information on interdependencies. Notice the reduction of nodes in the

simplified graph (bottom right) when compared to the full graph (middle) when connecting sketches in single body parts are

not shown.

An example of a CAD model (displayed as the

sequence of sketching and modeling steps require to

build the geometry) along with its corresponding full

graph (including all sketch nodes) and the simplified

version of the graph according to the criteria

discussed in this section are shown in Figure 8.

4 IMPROVED GRAPH

VISUALIZATION

Based on the discussion from previous sections, we

propose a simplified representation of a dependency

graph for parametric CAD models that is more

readable, less cluttered, and reduces ambiguity. Our

representation is built by reducing the number of

nodes in the graph according to the following

criteria:

 Connecting sketches used to create a feature

that will be merged with previous geometry do

not provide value to the user in terms of

visualizing and understanding the graph

complexity. Therefore, sketch nodes are not

displayed, unless they are explicitly used as

part of a constraint. Connecting sketch nodes

can be identified easily as sketch elements

with only one child node (the 3D feature they

generate).

 Inherited dependencies are not displayed.

Multi-level relations (grandparent-parent-

child) are only displayed if there is a specific

GRAPP2015-InternationalConferenceonComputerGraphicsTheoryandApplications

constraint that relates the grandparent node to

the child node. Otherwise, the dependencies

can be implied and therefore, there is no need

to make them visible.

The pseudo code for a simplification algorithm

to reduce the node and dependency count in the

model graph can be written as:

TraverseModel()

For each node n

{

If n is a Sketch and has

only one child node then

{

//Reconnect dependencies

For each Parent(n)

Reconnect to Child(n);

Remove(n);

}

Else If n is a Feature then

{

For each Child(n)

Check constraints;

If there is only one

constraint and constraint is

‘Merge geometry’ then

//multi-level

delete dependency;

}

Using two CAD models, a side-by-side

comparison of the full graph and our simplified

version of the graph according to the criteria

discussed in this paper is provided in Figure 9. Node

names have been omitted for clarity.

5 CONCLUSIONS

The constant need for tools and methods to support

design activities and reuse increasingly complex

CAD models naturally demands the development of

efficient visualization systems to manage parametric

model geometry.

In this paper, we have reviewed some of the most

common representation and visualization methods

for parametric 3D CAD models, particularly

focusing on dependency graphs. Our work describes

a series of problems with current graph

representation techniques and analyzes the

consequences of certain dependencies and CAD

elements, and the effects of representing relations

that have not been rigorously examined. In both

cases, a considerable level of ambiguity is

introduced to the graph, resulting in an imprecise or

cluttered representation of the model.

Figure 9: Comparison of the full graph and simplified version for two CAD models (A and B). Node names have been

omitted for clarity.

ImprovedRepresentationofDependenciesinFeature-basedParametricCADModelsusingAcyclicDigraphs

We have proposed a simplified representation of

the dependency graph by eliminating sketch nodes

that do not add value to the overall visualization and

only showing the multilevel dependencies

(grandparent-child) that are explicitly defined by

geometric and dimensional constraints. The result is

a clearer and more intuitive representation of the

internal structure of the parametric model. While the

system need to be tested on a large, "real-world"

example with a more extensive group of models, in

many cases the number of nodes can be reduced by

at least half, which contributes to a more effective

visualization.

Although the algorithm presented in this paper

was implemented as an add-in for the CAD package

SolidWorks, further developments are planned for

the near future. At the present time, the software

prototype can extract model information and export

it to a graph structure (which is visualized by an

external application), but a complete interactive

solution that is fully integrated within the CAD

environment would be beneficial. In addition, the

simplified model representation can be analyzed,

restructured, and optimized by applying clustering

and partitioning algorithms, which have the potential

to make models more flexible and reusable.

Furthermore, comparative studies of the graph

structures can be performed and assessed using

complexity metrics.

ACKNOWLEDGEMENTS

The Spanish Ministry of Economy and

Competitiveness and the FEDER Funds, through the

ANNOTA project (Ref. TIN2013-46036-C3-1-R)

supported this work.

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ImprovedRepresentationofDependenciesinFeature-basedParametricCADModelsusingAcyclicDigraphs