Similarity measures for mid-surface quality evaluation
Introduction
Mid-surface models have received a great deal of interest in recent years because they can reduce the computational cost of performing engineering analyses on thin-walled parts. The mid-surface representation of a part is a dimensionally reduced abstraction in which each wall is represented by a surface of zero thickness; for thin-walled objects the mid-surfaces provide a simplified description which retains the main shape characteristics of the original design. Mid-surface models are used extensively for the analysis of thin-walled parts in engineering analysis applications such as finite element analysis and mould filling/cooling analysis [1], [2], [3], [4]. Mid-surface models have also been used as a basis for feature recognition from thin-walled moulded parts [5].
There are a number of automated mid-surface generation techniques that can create a mid-surface abstraction from a CAD solid model, but there are currently no applications that can guarantee to generate a representative mid-surface model for any arbitrary solid part. This is partly due to limitations in the current algorithms, but also because there are many part shapes for which it is not possible to generate a representative mid-surface model [6]. In practice it is often necessary for the analyst to make manual adjustments to an automatically generated mid-surface model before it can be used for engineering analysis. Fig. 1. shows an example of good and poor quality mid-surface models generated using a commercial mid-surface generation tool (UGS I-DEAS NX). In Fig. 1(b) it can be seen that all of the features of the solid part are represented in the mid-surface model, whereas in Fig. 1(c) the boss has not been captured and the main walls have not been correctly connected together.
The motivation for the research presented in this paper is to assist users of mid-surface models who need to judge how well a mid-surface model represents the solid shape from which it has been generated. It should be emphasised that mid-surface representations can only accurately represent parts where the wall thickness is small compared to the other dimensions, and the methods presented in this paper are oriented towards practical applications of mid-surface models. For example in the finite element analysis two-dimensional meshes constructed on the mid-surface geometry are appropriate when “the length in one of the spatial dimensions, for example the material thickness, is much less than the lengths in the other two dimensions” [7]. Similarly parts designed for injection moulding must have a thin and relatively uniform wall thickness. Bralla [8] states that “uniform wall sections will help to produce warp-free and strain-free molded parts…Dimensional variations…are accentuated by uneven or abrupt wall-thickness changes”.
The objective of the research is to develop techniques that can be used to evaluate the accuracy of a mid-surface model by measuring its similarity to the solid model from which it was generated. Two techniques are presented, one to compare the geometric similarity of the two models and the second to compare their topological similarity. The remaining sections of the paper are structured as follows: Section 2 provides a review of the relevant literature relating to the mid-surface model generation and shape similarity measures and Section 3 presents the proposed similarity measures. In Section 4 the demonstrator is described, and in Section 5 case studies are presented. Section 6 discusses the results and conclusions drawn.
Section snippets
Literature review
The automatic generation of mid-surface abstractions from solid models has been a subject of research for many years and there has been significant progress in the field, however there is still no method that can generate an accurate mid-surface model for any arbitrary shape. The following sections provide a brief overview of existing mid-surface generation techniques, and a review of shape similarity measures that have been used to compare computer based geometry models.
Mid-surface model similarity evaluation
An effective mid-surface model similarity evaluation technique must be able to measure the similarity of a mid-surface model to a solid model with respect to both geometry and topology. The geometric evaluation should be sensitive enough to determine whether all the design features of the solid model are represented on the mid-surface model and the topological evaluation must evaluate whether the connectivity between solid model faces is correctly represented on the mid-surface model. The
Geometric similarity
A demonstrator for the geometric similarity evaluation has been implemented using C++ and the CAD system UGS I-DEAS NX. A flow chart of the evaluation process is shown in Fig. 6.
For the demonstrator the grids of points on the mid-surface and solid models have been generated using the automatic finite element mesh generation function in UGS I-DEAS, however a wide range of other techniques could be used. The Hausdorff comparison has been performed using a simple algorithm similar to that
Test cases
The mid-surface quality evaluation techniques have been tested on a range of thin-walled parts. The analysis for one part is presented in detail in Section 5.1, and the results for six other parts are included in Section 5.1.1.
Summary and conclusions
This paper has described two similarity measures for comparing mid-surface and solid models. The literature review highlighted that geometric similarity is not sufficient for evaluating mid-surface model quality, and that in order to obtain useful results it is necessary to use a topological measure of similarity in combination with the geometric similarity evaluation.
The geometric similarity measure proposed in this research is based on the Hausdorff distance and requires the models to be
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