Up

At ERSL, we are currently focusing on the following projects:

bulletGeneralized Dimensional Reduction
bullet Exploitation of Repeated Spatial Patterns

Generalized Dimensional Reduction

This work is supported under NSF Grant DMI 0322134

Thin mechanical components are ubiquitous and constitute a multi-billion dollar industry; examples include outer-shell of automobiles and aircrafts and most plastic components. See example below.

  

Existing analysis techniques for thin components, for historical reasons, are based on the mid-surface of a thin component. Unfortunately, the mid-surface is often ill defined for complex objects. This has led to fragile and inaccurate algorithms.

We propose that the notion of a ‘mid-surface’ be abandoned, and instead we propose the use of the skeleton of a solid. The skeleton, unlike the mid-surface, is well defined and can be computed uniquely for a solid. Initial investigation shows that this would result in robust algorithms. If successful, the research would lead to a fundamentally new and unified theory of thin component analysis consisting of an unambiguous and complete mathematical language of skeletal reduction, and associated algebra.

The mathematical principles would also be of interest to researchers involved in the broad area of boundary value problem analysis.

We expect to articulate the findings of the research to University of Wisconsin students through a graduate course, and students and researchers elsewhere through seminars. It is also the intention to facilitate a transfer of technology of the ensuing theory and algorithms to the CAD/CAE industry. An environmental impact of the proposal is in the plastics industry where components are almost exclusively thin; better analysis techniques have the potential of significantly reducing the tonnage of plastics used (and discarded) today.

Exploitation of Repeated Spatial Patterns 

The objective is to develop a comprehensive framework for exploiting repeated spatial patterns (RSPs) in engineering analysis. Example of repeated spatial patterns include:

bullet

Classical point-group mirror and cyclic symmetries (Figure 1)

bullet

Other classical point-group symmetries (Figure 2a)

bullet

Non-classical finite translational symmetry (Figure 2b)

bullet

Partial symmetries (Figure 2c)

bullet

Assemblies containing components of the above type (Figure 2d)

The term RSP-structure is used here to refer to an assembly or component containing any of the above repeated spatial patterns.

    

Figure 1: Examples of cyclic and mirror symmetry RSP structures.

               

            Figure 2: Targeted RSP structures.

Motivation

      RSP structures are ubiquitous. In a study conducted under the supervision of the PI, over 550 random components from the NIST design repository were visually checked for cyclic and/or mirror patterns. Out of these, over 80% were found to possess some form of symmetry pattern or the other. The results are summarized in Figure 3 where non-RSP structures are marked ‘X’. (The percentage of RSP structures is unusually high; informal surveys suggest about 50%.)

Figure 3: Results from a survey.

The reason for their ubiquity is the functional, aesthetic and economic superiority of RSP structures over non-RSP structures. For example:

bullet

RSP features are often essential for functional reasons; example: the teeth profile in the bevel-gear (Figure 2b)

bullet

 RSP components are functionally superior. The symmetric propeller (Figure 1) is more efficient and less likely to fail (than an asymmetric one). Similar conclusions can be drawn about symmetric bearings, machine supports, etc

bullet

RSP-components are also easier to assemble and manufacture [Reference]. MEMS devices contain recurring patterns for functional and manufacturing reasons

bullet

RSP-structures, in general, have an inherent appeal to humans, and play an important role in industrial design

Proposed Research

      The proposed research rests on the (partially established) existence of a deep mathematical relationship between RSP structures and engineering analysis, i.e., partial differential equations governing such structures. Indeed, this relationship is exploited today, albeit in ad hoc fashion, for the particular case of cyclic/mirror symmetry. For example, ANSYS suggests a manual process for exploiting the cyclic symmetry of the turbine assembly in Figure 1. While the suggested procedures result in faster and more accurate analysis, they have two drawbacks:

  1. They are limited to cyclic or mirror symmetry (Figure 1); in fact, even a combination of cyclic and mirror symmetries is never addressed. 

  2. They are inherently manual, defeating the CAD/ CAE objective of automation and error-free analysis.

The proposed research aims to correct these limitations by

  1. Formalizing the underlying mathematics to a large class of RSP structures (Figure 2).

  2. Providing a framework for automating generalized RSP exploitation.

In particular, the proposed activities are broken down into five closely related categories:

1.       RSP detection algorithms: Lack of RSP detection algorithms is one of the fundamental reasons why they are not fully exploited today. Extant RSP detection algorithms focus largely on cyclic and mirror symmetries of 2-D polygonal and 3-D polyhedral solids. We will establish 3-D free form classical and non-classical symmetry detection algorithms. In addition, we will develop detection algorithms for near and partial-symmetry. These algorithms will be based on some new ideas recently described in Section 3.1, developed by the PI and his research group. Methods to manage RSP components in assemblies will be addressed.

2.       Codification and standardization of RSP: It is not sufficient if one recognizes spatial patterns. It is important that this information be captured unambiguously for future use, and perhaps be constantly updated during the design process. We will therefore develop algorithms for ‘monitoring’ patterns during the design process. Closely related is the ‘standardization’ of pattern information. This activity will therefore include developing a ‘standard’ for the unambiguous transfer of RSP information between various CAD systems. The PI will work with industrial partners to recommend a viable standard for embedding and communicating such information between CAD/CAE systems.

3.       RSP exploitation in engineering analysis: The PI recently published new results on the mathematical aspects of point-group symmetries in engineering analysis. A major thrust of this activity is to extend these results to a wider class of patterns and problems, including non-linear problems.

4.       Development of a ‘RSP tool-box’: Once the concepts and algorithms have been established, the PI will develop a ‘RSP toolbox’ that can be embedded within a commercially available CAD/CAE environment, such as SolidWorksÔ. The toolbox will be tested on a wide range of industrial products and applications.

5.       Education: As stated earlier, repeated spatial patterns (RSP) hold a special place in engineering. The PI will demonstrate to a wide audience, primarily high school and university students, the benefits of such patterns in design. Naturally occurring designs and man-made designs will be used to enforce the importance of RSP, through simple experiments and demonstrations.