Convergence of a finite difference approach for detailed deviation zone estimation in coordinate metrology

<p class="Abstract">In order to comprehend an entire surface's deviation zone, infinite measured points are required. Using the common measurement techniques through coordinate metrology, a limited number of surface actual points can be acquired. However, the obtained points would not provide sufficient information to examine the geometry thoroughly. A novel approach to predict surface behaviour via Distribution of Geometric Deviations (DGD) is examined in this paper. The methodology governs the mean value property of the harmonic functions to solve Laplace equation around each measured point. This DGD model can be used to reconstruct surface deviation values at any unmeasured point of the inspected surface based on a limited number of measured points. The convergence of the introduced approach is studied in this paper. A complete approach to implement the developed methodology is described, and the validation process is studied using actual case studies and mathematical functions. This methodology is practical in closed-loop inspection and manufacturing processes to form a scheme for compensating the surface errors during manufacturing process based on the DGD model.</p>

A Finite Element Approach to Estimate the Detailed Deviation Zone in Coordinate Metrology

The exact detailed knowledge of deviation zone in a manufactured surface needs measurement of infinite number of points when the coordinate metrology is utilized. The coordinate metrology process provides deviation of the limited number of discrete points on a measured surface, but typically the process is not capable to explore any information of the surface regions between these measured points. A Finite Element approach for Deviation Zone Evaluation (DZE) on the entire inspected surface is presented in this paper. The developed DZE solution estimates the deviation values at any unmeasured point of the inspected surface when a detailed understanding of the surface geometric deviations is required. Implementation of the developed methodology is described and case study for typical industrial parts is presented. This methodology can be used for closed-loop of inspection and manufacturing processes when a compensation scheme is available to compensate the manufacturing errors based on the DZE data.

An Approach to Estimate the Detailed Deviation Zone in Coordinate Metrology Using Harmonic Function Properties

Understanding the exact details of deviation zone related to a manufactured surface needs measurement of infinite number of points. Coordinate metrology provides deviation of the limited number of discrete points on a measured surface, but typically it is not capable to explore any information of the surface regions between these measured points. An approach to estimate the Distribution of Geometric Deviations (DGD) on the entire inspected surface is presented in this paper. The methodology is developed based on estimation of mean value property of the harmonic functions and Laplace equation. The resulting DGD model can be employed to estimate the deviation values at any unmeasured point of the inspected surface when a detailed understanding of the surface geometric deviations is required. Implementation of the developed methodology is described and case studies for typical industrial parts are presented. This methodology can be used for closed-loop of inspection and manufacturing processes when a compensation scheme is available to compensate the manufacturing errors based on the DGD model.

Profile Tolerance Allocation for Rapid Prototyping of Sculptured Surfaces in a Direct Slicing Process

Layer-based manufactured parts and surfaces are inherently subject to stair case effect which can be quantified by cusp height. Cusp height of a layer is the maximum distance measured along a surface normal between the ideal surface and the produced layer. Although calculation of local cusp high is a simple task but estimating the overall deviation zone of the produced surface is a highly nonlinear and complicated problem. This paper presents a practical approach to predict the actual profile tolerances of the surfaces. This prediction is used to allocate profile tolerances for the rapid prototyping process. Also the methodology can be used to select the optimum uniform layer thicknesses that compromise between the number of layers and the desired accuracy of the final surfaces. The unified developed methodologies are capable to analyse complex surfaces and geometries. Variety of experiments is carried out to study the effectiveness and practicality of the presented methodology. The developed methodology can be employed efficiently during design of rapid prototyping parts.

Evaluation of Deviation Zone Based on Maximum Conformance to Tolerances

The accurate estimation of the geometric deviations is not possible only by manipulating the Euclidian distances of the discrete measured points from substitute geometry. The real geometric deviations of a measured surface need to be calculated based on the desired tolerance zone of the surface. This fact is usually neglected in common practices in the coordinate metrology of surfaces. The importance of considering the desired tolerance zone in estimation of the optimum deviation zone is demonstrated in this paper. Then a best fit method is presented which complies with the tolerance requirements of the designed surface. The developed fitting methodology constructs a substitute geometry to minimizes the residual deviations corresponding to the given tolerance zone and the needs of down-stream operations that use the results of the inspection process. It is shown how the developed objective function can be adopted for a case of closed-loop manufacturing process, when the under-cut residual deviations of the manufactured part can be corrected by a down-stream operation. In order to validate the proposed methodology, experiments are conducted. The results show a significant reduction of uncertainties in coordinate metrology of geometric surfaces. Implementation of this method directly results in increasing the accuracy of the entire tolerance evaluation process, and less uncertainty in quality control of the manufactured parts.

Design for Manufacturing of Sculptured Surfaces: A Computational Platform

This paper presents a computer aided design for machining (DFMc) platform that enables designers to customize the design for the available machine tools and to estimate the effect of design decisions on the accuracy of the final machined products, particularly those containing sculptured surfaces. The platform contains two modules to model and simulate the actual machined surface and to evaluate the resulting minimum deviation zone compared to the desired geometry. In the first module, based on the configuration of the available machine tool and the limitations imposed by its inherent errors, the machined surface is simulated and presented as a nonuniform rational B-spline (NURBS) surface. In the second module, the minimum deviation zone between the actual and the nominal NURBS surfaces is evaluated when the developed method to do this task efficiently improves the convergence of the resulting optimization process. Utilizing this platform, two different applications are developed; design tolerance allocation based on the minimum deviation zone of the machined surface and adaptation of the nominal design to compensate for the effect of machining errors. Employing these applications during the design stage improves the acceptance rate of the produced parts and reduces the rate of scrap and rework. The DFMc platform and its presented applications can be implemented in any integrated computer aided design/computer aided manufacturing (CAD/CAM) system. The presented methods can be applied to any type of input geometries and are particularly efficient for design and manufacturing of precise components with complex surfaces. Products in this group, such as dies and tools, medical instruments, and biomedical implants, mostly have critical and important functionalities that demand very careful design and manufacturing decisions.