GEOMETRIC MODELING OF COMPLEX OBJECTS USING R-FUNCTIONS

АННОТАЦИЯ

В этой статье представлена комплексная структура для геометрического моделирования сложных крепежных деталей, таких как гайки, болты и винты, с использованием R-функций. R-функции предоставляют мощный математический инструмент для построения неявных поверхностей и выполнения булевых операций, что позволяет эффективно представлять сложные геометрии. Мы предлагаем новые математические модели для распространенных крепежных деталей, используя теорию R-функций для создания параметризованных функциональных представлений. Эти модели особенно полезны в автоматизированном проектировании (САПР) и аддитивном производстве, где точные геометрические описания имеют важное значение. Предлагаемая структура проверена с помощью нескольких тематических исследований, демонстрирующих ее эффективность в реальных приложениях. Кроме того, мы исследуем интеграцию этих моделей с алгоритмами машинного обучения для автоматизированной оптимизации формы и генеративного проектирования.

ABSTRACT

This paper presents a comprehensive framework for the geometric modeling of complex fasteners, such as nuts, bolts, and screws, using R-functions. R-functions provide a powerful mathematical tool for constructing implicit surfaces and performing Boolean operations, enabling the efficient representation of intricate geometries. We propose new mathematical models for common fasteners, leveraging the theory of R-functions to create parameterized functional representations. These models are particularly useful in computer-aided design (CAD) and additive manufacturing, where precise geometric descriptions are essential. The proposed framework is validated through several case studies, demonstrating its effectiveness in real-world applications. Additionally, we explore the integration of these models with machine learning algorithms for automated shape optimization and generative design [7].  

Ключевые слова: R-функции, геометрическое моделирование, неявные поверхности, САПР, аддитивное производство, крепеж, машинное обучение.

Keywords: R-functions, geometric modeling, implicit surfaces, CAD, additive manufacturing, fasteners, machine learning.

Introduction

Geometric modeling of complex objects is a critical aspect of modern engineering, particularly in fields such as computer-aided design (CAD) and additive manufacturing [4].  Fasteners, including nuts, bolts, and screws, are ubiquitous in mechanical systems, and their accurate representation is essential for the design and analysis of these systems. Traditional modeling techniques, such as boundary representation and constructive solid geometry (CSG), have limitations in handling complex geometries, especially when it comes to intricate features like threads and chamfers.

R-functions offer a robust alternative for geometric modeling by providing a mathematical framework for constructing implicit surfaces and performing Boolean operations [1]. This paper builds on the foundational work of Rvachev and others, extending the theory of R-functions to create detailed models of fasteners. We propose new mathematical models for common fasteners, parameterized to allow for flexibility in design and manufacturing.

Theoretical Foundations

R-functions are real-valued functions that can represent logical operations (negation, conjunction, disjunction) on implicit functions. This allows for the construction of complex geometries by combining simpler primitives. The basic R-functions are defined as follows:

• Negation:  f(x)
• Conjunction:  f(x)∧g(x)
• Disjunction:  f(x)∨g(x)

These functions can be used to construct implicit surfaces that represent complex geometries [6,8]. For example, a cylinder can be represented by the implicit function f(x,y,z)=r²−x²−y², where r is the radius of the cylinder.

Modeling of Fasteners

We propose new mathematical models for common fasteners, including nuts, bolts, and screws, using R-functions. These models are parameterized to allow for flexibility in design and manufacturing.

1. Nut Model: The nut is modeled as a hexagonal prism with a cylindrical hole. The implicit function for the nut is constructed using R-functions to combine the hexagonal prism and the cylindrical hole. The parameters include the size of the nut (S), the diameter of the hole (d), and the height of the nut (H).

F_nut​(x,y,z)=F_hexagon​(x,y,z)∧−F_cylinder​(x,y,z)

2. Bolt Model: The bolt is modeled as a combination of a hexagonal head and a cylindrical shaft. The implicit function for the bolt is constructed using R-functions to combine the hexagonal head and the cylindrical shaft. The parameters include the size of the bolt (S), the diameter of the shaft (d), and the length of the bolt (L).

F_bolt​(x,y,z)=F_hexagon​(x,y,z)∨F_cylinder​(x,y,z)

3. Screw Model: The screw is modeled as a combination of a cylindrical head and a threaded shaft. The implicit function for the screw is constructed using R-functions to combine the cylindrical head and the threaded shaft. The parameters include the diameter of the head (D), the diameter of the shaft (d), and the length of the screw (L).

F_screw​(x,y,z)=F_cylinder​(x,y,z)∨F_thread​(x,y,z)

Figure 1. R functions for modeling nuts, bolts and screws

4. Regular Polygon (Hexagon): A regular hexagon is used as a primitive in the construction of the nut and bolt models [5,10].  The implicit function for a regular hexagon is given by:

Mathematical Formula:

F_{regular polygon}​(x,y,r,x₀​,y₀​,n)=_i=1⋀^n​F_HS​(x,y,ξ_i​,η_i​,ξ_i+1​,η_i+1​)

where F_HS​ defines a half-plane:

F_HS​(x,y,x₁​,y₁​,x₂​,y₂​)=(x−x₁​)(y₂​−y_1​)−(y−y₁​)(x₂​−x₁​)

5. Chamfered Rectangle: A chamfered rectangle is used to model the edges of fasteners. The implicit function for a chamfered rectangle is given by:

Mathematical Formula:

Fchamfered rectangle​(x,y,w,h,d,α)=_i=1⋀⁴​FHS​(x,y,ξi​,ηi​,ξi+1​,ηi+1​)

where w and h are the width and height of the rectangle, d is the distance to the chamfer, and α is the chamfer angle.

Figure 2. R functions for modeling regular polygon (hexagon) and chamfered rectangle

Integration with Machine Learning

We explore the integration of these models with machine learning algorithms for automated shape optimization and generative design. Neural networks can be trained to learn the R-function representations from data, enabling the reconstruction of complex shapes from partial or noisy inputs. This approach can significantly reduce the time and effort required for designing and optimizing fasteners.

Applications

The proposed models have several applications in CAD and additive manufacturing. They can be used to create precise geometric descriptions of fasteners, which are essential for the design and analysis of mechanical systems. Additionally, these models can be integrated with finite element analysis (FEA) tools to simulate the mechanical behavior of fasteners under various loading conditions.

Results

We validated the proposed models using case studies in CAD and additive manufacturing environments [2]. Our approach allowed for the generation of parameterized fastener designs, demonstrating high precision in geometric representation. The implicit modeling method proved efficient in computational simulations and manufacturing prototypes.

Discussion

The proposed R-function-based modeling technique offers several advantages over conventional geometric modeling approaches [3].  These include:

• Efficiency in Boolean operations for complex shapes
• Better integration with automated design and machine learning-based optimization
• Flexibility in adjusting parameters for different fastener types

Future work will focus on extending these models to include more complex geometries and integrating them with advanced manufacturing techniques [9]. 

Conclusion

This paper presents a comprehensive framework for the geometric modeling of complex fasteners using R-functions. We propose new mathematical models for common fasteners, parameterized to allow for flexibility in design and manufacturing. The proposed framework is validated through several case studies, demonstrating its effectiveness in real-world applications. Additionally, we explore the integration of these models with machine learning algorithms for automated shape optimization and generative design. Future work will focus on extending these models to include more complex geometries and integrating them with advanced manufacturing techniques.

References:

1. Rvachev, V. L. (1982). Theory of R-functions and some applications. Kiev: Naukova Dumka.
2. Farin, G., Hoschek, J., & Kim, M. S. (2002). Handbook of computer-aided geometric design. Amsterdam: Elsevier.
3. Agoston, M. K. (2005). Computer graphics and geometric modeling: Implementation and algorithms. London: Springer.
4. Golovanov, N. N. (2002). Geometric modeling. Moscow: Fizmatlit.
5. Potishko, A. V., & Krushevskaya, D. P. (1983). Handbook of engineering graphics. Kiev: Budivelnyk.
6. Rvachev, V. L., & Sheyko, T. I. (2001). Introduction to the theory of R-functions. Problems of Mechanical Engineering, Vol. 4, No. 1–2, pp. 46–58.
7. Kravchenko, V. F., & Rvachev, V. L. (2006). Algebra of logic, atomic functions, and wavelets in physical applications. Moscow: Fizmatlit.
8. Ryachev, V. L., Sheyko, T. I., & Shapiro, V. (1998). Method of R-functions in boundary value problems with geometric and physical symmetry. Mathematical Methods and Physico-Mechanical Fields, 41(1), pp. 146–159.
9. Ryachev, V. L., Tolok, A. V., Uvarov, R. A., & Sheyko, T. I. (2000). New approaches to constructing equations of three-dimensional loci using R-functions. Bulletin of Zaporizhzhya State University, No. 2, pp. 119–130.
10.  Nuraliev, F.M., Morozov, M.N., Giyosov, U.E., Yorkulov, J. (2023). About the application of the R-function for geometric modeling of 3D objects of complex shapes in a virtual educational environment. Software systems and computational methods, No. 3, pp. 18–28.