Integration of CAD and boundary element analysis through subdivision methods

doi:10.1016/j.cie.2009.01.009

Computers & Industrial Engineering

Volume 57, Issue 3, October 2009, Pages 691-698

https://doi.org/10.1016/j.cie.2009.01.009 Get rights and content

Abstract

Subdivision methods have been mainly used in computer graphics. This paper extends their applications to mechanical design and boundary element analysis (BEA), and fulfills the seamless integration of CAD and BEA in the model and representation.

Traditionally, geometric design and BEA are treated as separate modules requiring different representations and models, which include continuous parametric models and discrete models. Due to the incompatibility of the involved representations and models, the post-processing in geometric design or the pre-processing in BEA is essential. The transition from geometric design to BEA requires substantial effort and errors are inevitably introduced during the transition. In this paper, a framework of realizing the integration of CAD and BEA was first presented based on subdivision methods. A common model or a unified representation for geometric design and BEA was created with subdivision surfaces. For general 3D structures, automatic mesh generation for geometric design and BEA was fulfilled through subdivision methods. The seamless integration improves the accuracy of numerical analysis and shortens the cycle of geometric design and BEA.

Introduction

Traditionally, geometric design and boundary element analysis are treated as separate modulus requiring different methods and representations, which include continuous parametric models and discrete models. NURBS (Non-Uniform Rational B-splines) (Piegl & Tiller, 1997), which are continuous parametric models, are often used for geometric design in CAD systems; while meshes, which are discrete models, are used in BEA. Due to the incompatibility of the involved different representations and models, the post-processing in geometric design or the pre-processing in BEA is essential. Therefore, conversion and remodeling are required for iterations between geometric design and BEA. Errors are inevitably introduced during the conversion and remodeling. The integration of geometric design and BEA has become more and more important.

One of ways in the integration of CAD and CAE components is direct using the CAD model for CAE downstream applications. Therefore, directly usable and accurate CAD models and data are highly desirable for the cycle of geometric design and BEA. However, CAD/CAE environments are generally heterogeneous due to highly task-dependent components with corresponding mathematical models. The requirements for the properties of the objects and mathematical models are different in areas such as grid generation and boundary element analysis (Andrey & Thomas, 1999), which are different from geometric design. Therefore, trimming operations for continuous parametric CAD models to generate trimmed patch boundary elements (Kane, Maier, Tosaka, & Atluri, 1993) and modifications of CAD models are often a necessity as a precursor to effective BEA mesh generation (Dan, 1998). Furthermore, CAD models with boundary representation can contain errors, such as gaps, incorrect topology of trimming curves. In most cases, the problems of these CAD errors do not affect the efficiency of graphical applications because these errors are too small to be observed visually. However, major problems are encountered in the creation of the downstream CAE mathematical models such as finite/boundary element meshes, which require the global continuity of the object boundary. Therefore, a CAD mathematical model of the object should be pre-processed to meet specific requirements of the downstream BEA application. The problem of CAD geometric model pre-processing is known as CAD repair (Andrey & Thomas, 1999). CAD model repair is defined as the process of fixing geometric and topological definition errors in a design model so that it can be used for the efficient creation of its computational model in a given downstream process, e.g. finite/boundary element simulation. There are various errors, which can be detected in CAD models. These errors include: inverted faces, gaps between surfaces in a volume, folded geometry, surface geometry with no bounding face, faces with no finite area, self-intersecting edges and faces, face/edge sloppiness, boundary edges that do not lie on the faces, overlapping faces, etc. Editing and fixing the geometry directly is cumbersome, tedious, and expensive (David, Sunil, & Steven, 2003). Informal studies reveal that engineering analysts are spending more than half of their time on re-working CAD files before analysis can begin. This situation gets worse with the growing usage and complexity of these models (Dan, 1998).

Subdivision methods can provide a common model or a unified representation for geometric design and BEA, avoiding the above problems inherent in traditional spline patch based approaches. Geometric discretization (mesh generation) in BEA is one of the major sources in the BEA error (Zhao & Wang, 1999). The accuracy and the convergence of the BEA solutions are strongly related to the quality of the BEA meshes (Liapis, 1994). The adaptive BEA can significantly improve the accuracy of BEA (Zhao & Wang, 1999). Therefore, automatic and adaptive BEA mesh generation is an important issue. Adaptive schemes in subdivision methods can generate adaptive BEA mesh automatically. Subdivision methods have multuresolution capability, and can generate different levels (fine or coarse) of mesh according to the requirement in accuracy. The goal of the research in this paper is to realize the advantageous features resulting from the integration of CAD and BEA based on subdivision methods. The subdivision-based integration can lead to at least the following advantageous features.

•
A common model, unified framework and representation for geometric design and BEA.
•
No post-processing in geometric design or pre-processing in BEA.
•
No error due to the conversion from a geometric design model to a BEA model.
•
Automatic and adaptive BEA mesh generation.
•
Capability of mesh generation at different levels due to the multuresolution property of subdivision methods.
•
Providing a new approach to studying the shape optimization for complex shapes.
•
Allowing vigorous consideration of BEA issues at early design stage.
•
Seamless integration, reduction in the trial-and-error, and remarkable shortening of the lead-time at the geometric design and BEA stages.

Section snippets

Subdivision methods and related research

Subdivision methods (Dyn et al., 2007, Kobbelt, 1996, Levin, 1999, Levin, 2000, Litke et al., 2001, Sabin, 2005, Zorin et al., 1996a, Zorin et al., 1996b, Zulti et al., 2006) generate a sequence of recursively-refined meshes (polyhedral surfaces) starting from an initial coarse control mesh. At each step of the subdivision, a finer polyhedral surface with more vertices and faces is constructed from the previous one via an iterative refinement process. After a few steps, the geometric design

Advantages of subdivision methods in geometric design

In current mechanical CAD/CAM systems, the commonly used representation models for free-form surfaces are NURBS (Non-Uniform Rational B-splines) (Piegl & Tiller, 1997). However, NURBS have some limitations (Levin, 1999, Mandal et al., 2000).

(1)
NURBS do not allow modelers and designers to arrange control vertices in a natural way. The control vertices must be maintained in a homomorphic rectangular structure in their parameter domains. Therefore, NURBS cannot represent arbitrary topology.
(2)
A

Integration of geometric design and BEA in product development through the subdivision method

The purpose of the integration of geometric design and BEA in this paper is creating a common model or a unified representation for geometric design and BEA. The integration is implemented through the subdivision method. The created common model or a unified representation is fine mesh, which represents the geometric shape for geometric design after rendering and can be directly used as boundary element mesh in BEA. This paper focuses on studying the common issue: creating a common model or a

BEM formulation for 3D subdivision models

The method of boundary element (Brebbia et al., 1984) solves boundary value problems by transforming a differential equation with boundary conditions into an integral equation. If body forces are omitted for simplicity, the partial differential equation of static equilibrium with boundary conditions in the elastic state can be transformed into the following integral equation specialized for boundary points: $C_{ij} (ξ) u_{j} (ξ) = \int_{Γ} u_{ij}^{*} (ξ, x) p_{j} (x) d Γ (x) - \int_{Γ} p_{ij}^{*} (ξ, x) u_{j} (x) d Γ (x)$ where $u_{i} (ξ)$ is the displacement

Steps for the implementation of the subdivision-based geometric design and BEA in product development

The subdivision method can be used to create a common model or a unified representation for CAD, BEA and NC machining (if required) in product development. The following steps are summarized for the implementation of the approach:

1.
Creating an initial control mesh for a mechanical part according to its features in geometry.
2.
Creating fine mesh through an iterative refinement process using the subdivision method. The number of subdivision is set according the required accuracy or the tolerance of

Conclusions

Based on subdivision methods, a common model or a unified representation for geometric design and BEA is developed in order to shorten the cycle of geometric design and BEA, and allow vigorous consideration of analysis issues at early design stage. It is shown that subdivision methods can offer the ideal common platform or model to realize the concept and needs of geometric design and BEA integration.

Appropriate geometric models created by subdivision methods can be used as the mesh of BEA.

References (52)

E. Catmull et al.
Recursively generated B-spline surfaces on arbitrary topological meshes
Computer Aided Design
(1978)
G.M. Chaikin
An algorithm for high speed curve generation
Computer Graphics Image Processing
(1974)
K.M. Cheang et al.
Post-processing of compressed 3D graphic data
Journal of Visual Communication and Image Representation
(2000)
C.K. Chui et al.
From extension of Loop’s approximation scheme to interpolatory subdivisions
Computer Aided Geometric Design
(2008)
D. Doo et al.
Behavior of recursive division surfaces near extraordinary points
Computer Aided Design
(1978)
K. Hormann et al.
A family of subdivision schemes with cubic precision
Computer Aided Geometric Design
(2008)
Z. Huang et al.
A bound on the approximation of a Catmull–Clark subdivision surface by its limit mesh
Computer Aided Geometric Design
(2008)
K.C. Hui et al.
Generating subdivision surfaces from profile curves
Computer Aided Design
(2007)
J. Ivanova et al.
Design sensitivity analysis for shape optimization by the BEM
Engineering Analysis with Boundary Elements
(1991)
J. Ivanova et al.
Design sensitivity analysis for shape optimization by the BEM
Engineering Analysis with Boundary Elements
(1991)

L. Kobbelt

A variational approach to subdivision

Computer Aided Geometric Design

(1996)

G. Li et al.

Composite sqrt (2) subdivision surfaces

Computer Aided Geometric Design

(2007)

S. Liapis

A review of error estimation and adaptivity in the boundary element method

Engineering Analysis with Boundary Elements

(1994)

N. Litke et al.

Trimming for subdivision surfaces

Computer Aided Geometric Design

(2001)

C. Mandal et al.

A novel FEM-based framework for subdivision surfaces

Computer Aided Design

(2000)

S. Schaefer et al.

Nonlinear subdivision through nonlinear averaging

Computer Aided Geometric Design

(2008)

S. Schaefer et al.

Exact evaluation of limits and tangents for non-polynomial subdivision schemes

Computer Aided Geometric Design

(2008)

A. Tafreshi et al.

General-purpose computer program for shape optimization of engineering structures using the boundary element method

Computers & Structures

(1995)

H. Wang et al.

Biorthogonal wavelets based on gradual subdivision of quadrilateral meshes

Computer Aided Geometric Design

(2008)

Z. Zhao et al.

Error estimation and h-adaptive boundary elements

Engineering Analysis with Boundary Elements

(1999)

A. Zulti et al.

C^2 subdivision over triangulations with one extraordinary point

Computer Aided Geometric Design

(2006)

Andrey, A. M., & Thomas, W. (1999). Methods and algorithms of automated CAD repair for incremental surface meshing. In...

C.A. Brebbia et al.

Boundary element techniques: Theory and applications in engineering

(1984)

C.S. Chen et al.

Boundary element technology XIII: Incorporating computational methods and testing for engineering integrity

(1999)

Dan, M. (1998). Model quality: The key to CAD/CAM/CAE interoperability. In The conference proceedings for the 1998 MSC...

David, R. W., Sunil, S., Steven, J. O. (2003). Meshing complexity of single CAD models. In Proceeding of twelfth...

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Integration of CAD and boundary element analysis through subdivision methods

Abstract

Introduction

Section snippets

Subdivision methods and related research

Advantages of subdivision methods in geometric design

Integration of geometric design and BEA in product development through the subdivision method

BEM formulation for 3D subdivision models

Steps for the implementation of the subdivision-based geometric design and BEA in product development

Conclusions

Computer Aided Design

Computer Graphics Image Processing

Journal of Visual Communication and Image Representation

Computer Aided Geometric Design

Computer Aided Design

Computer Aided Geometric Design

Computer Aided Geometric Design

Computer Aided Design

Engineering Analysis with Boundary Elements

Engineering Analysis with Boundary Elements

Computer Aided Geometric Design

Computer Aided Geometric Design

Engineering Analysis with Boundary Elements

Computer Aided Geometric Design

Computer Aided Design

Computer Aided Geometric Design

Computer Aided Geometric Design

Computers & Structures

Computer Aided Geometric Design

Engineering Analysis with Boundary Elements

Computer Aided Geometric Design

Boundary element techniques: Theory and applications in engineering

Boundary element technology XIII: Incorporating computational methods and testing for engineering integrity