Volumetric Untrimming: Precise decomposition of trimmed trivariates into tensor products

3D objects, modeled using Computer Aided Geometric Design tools, are traditionally represented using a boundary representation (B-rep), and typically use spline functions to parameterize these boundary surfaces. However, recent development in physical analysis, in isogeometric analysis (IGA) in specific, necessitates a volumetric parametrization of the interior of the object. IGA is performed directly by integrating over the spline spaces of the volumetric spline representation of the object. Typically, tensor-product B-spline trivariates are used to parameterize the volumetric domain. A general 3D object, that can be modeled in contemporary B-rep CAD tools, is typically represented using trimmed B-spline surfaces. In order to capture the generality of the contemporary B-rep modeling space, while supporting IGA needs, Massarwi and Elber (2016) proposed the use of trimmed trivariates volumetric elements. However, the use of trimmed geometry makes the integration process more difficult since integration over trimmed B-spline basis functions is a highly challenging task. In this work, we propose an algorithm that precisely decomposes a trimmed B-spline trivariate into a set of (singular only on the boundary) tensor-product B-spline trivariates, that can be utilized to simplify the integration process in IGA. The trimmed B-spline trivariate is first subdivided into a set of trimmed B\'ezier trivariates, at all its internal knots. Then, each trimmed B\'ezier trivariate, is decomposed into a set of mutually exclusive tensor-product B-spline trivariates, that precisely cover the entire trimmed domain. This process, denoted untrimming, can be performed in either the Euclidean space or the parametric space of the trivariate. We present examples on complex trimmed trivariates' based geometry, and we demonstrate the effectiveness of the method by applying IGA over the (untrimmed) results.Comment: 18 pages, 32 figures. Contribution accepted in International Conference on Geometric Modeling and Processing (GMP 2019

Manufacturability analysis for non-feature-based objects

This dissertation presents a general methodology for evaluating key manufacturability indicators using an approach that does not require feature recognition, or feature-based design input. The contributions involve methods for computing three manufacturability indicators that can be applied in a hierarchical manner. The analysis begins with the computation of visibility, which determines the potential manufacturability of a part using material removal processes such as CNC machining. This manufacturability indicator is purely based on accessibility, without considering the actual machine setup and tooling. Then, the analysis becomes more specific by analyzing the complexity in setup planning for the part; i.e. how the part geometry can be oriented to a cutting tool in an accessible manner. This indicator establishes if the part geometry is accessible about an axis of rotation, namely, whether it can be manufactured on a 4th-axis indexed machining system. The third indicator is geometric machinability, which is computed for each machining operation to indicate the actual manufacturability when employing a cutting tool with specific shape and size. The three manufacturability indicators presented in this dissertation are usable as steps in a process; however they can be executed alone or hierarchically in order to render manufacturability information. At the end of this dissertation, a Multi-Layered Visibility Map is proposed, which would serve as a re-design mechanism that can guide a part design toward increased manufacturability

MATHICSE Technical Report: Volumetric Untrimming: Precise decomposition of trimmed trivariates into tensor products

3D objects, modeled using Computer Aided Geometric Design (CAGD) tools, are traditionally represented using a boundary representation (B-rep), and typically use spline functions to parameterize these boundary surfaces. However, recent development in physical analysis, in isogeometric analysis (IGA) in specic, necessitates a volumetric parametrization of the interior of the object. IGA is performed directly by integrating over the spline spaces of the volumetric spline representation of the object. Typically, tensor-product B-spline trivariates are used to parameterize the volumetric domain. A general 3D object, that can be modeled in contemporary B-rep CAD tools, is typically represented using trimmed B-spline surfaces. In order to capture the generality of the contemporary B-rep modeling space, while supporting IGA needs, Massarwi and Elber (2016) proposed the use of trimmed trivariates volumetric elements. However, the use of trimmed geometry makes the integration process more difficult since integration over trimmed B-spline basis functions is a highly challenging task Xu et al. (2017). In this work, we propose an algorithm that precisely decomposes a trimmed B-spline trivariate into a set of (singular only on the boundary) tensor-product B-spline trivariates, that can be utilized to simplify the integration process, in IGA. The trimmed B-spline trivariate is first subdivided into a set of trimmed B--spline trivariates, at all its internal knots. Then, each trimmed Bezier trivariate, is decomposed into a set of mutually exclusive tensor-product B-spline trivariates, that precisely cover the entire trimmed domain. This process, denoted untrimming, can be performed in either the Euclidean space or the parametric space of the trivariate. We present examples of the algorithm on complex trimmed trivariates' based geometry, and we demonstrate the effectiveness of the method by applying IGA over the (untrimmed) results

Feature-based hybrid inspection planning for complex mechanical parts

Globalization and emerging new powers in the manufacturing world are among many challenges, major manufacturing enterprises are facing. This resulted in increased alternatives to satisfy customers\u27 growing needs regarding products\u27 aesthetic and functional requirements. Complexity of part design and engineering specifications to satisfy such needs often require a better use of advanced and more accurate tools to achieve good quality. Inspection is a crucial manufacturing function that should be further improved to cope with such challenges. Intelligent planning for inspection of parts with complex geometric shapes and free form surfaces using contact or non-contact devices is still a major challenge. Research in segmentation and localization techniques should also enable inspection systems to utilize modern measurement technologies capable of collecting huge number of measured points. Advanced digitization tools can be classified as contact or non-contact sensors. The purpose of this thesis is to develop a hybrid inspection planning system that benefits from the advantages of both techniques. Moreover, the minimization of deviation of measured part from the original CAD model is not the only characteristic that should be considered when implementing the localization process in order to accept or reject the part; geometric tolerances must also be considered. A segmentation technique that deals directly with the individual points is a necessary step in the developed inspection system, where the output is the actual measured points, not a tessellated model as commonly implemented by current segmentation tools. The contribution of this work is three folds. First, a knowledge-based system was developed for selecting the most suitable sensor using an inspection-specific features taxonomy in form of a 3D Matrix where each cell includes the corresponding knowledge rules and generate inspection tasks. A Travel Salesperson Problem (TSP) has been applied for sequencing these hybrid inspection tasks. A novel region-based segmentation algorithm was developed which deals directly with the measured point cloud and generates sub-point clouds, each of which represents a feature to be inspected and includes the original measured points. Finally, a new tolerance-based localization algorithm was developed to verify the functional requirements and was applied and tested using form tolerance specifications. This research enhances the existing inspection planning systems for complex mechanical parts with a hybrid inspection planning model. The main benefits of the developed segmentation and tolerance-based localization algorithms are the improvement of inspection decisions in order not to reject good parts that would have otherwise been rejected due to misleading results from currently available localization techniques. The better and more accurate inspection decisions achieved will lead to less scrap, which, in turn, will reduce the product cost and improve the company potential in the market

Analysis of the inspection of mechanical parts using dense range data

More than ever, efficiency and quality are key words in modern industry. This situation enhances the importance of quality control and creates a great demand for cheap and reliable automatic inspection systems. Taking into account these facts and the demand for systems able to inspect the final shape of machined parts, we decided to investigate the viability of automatic model-based inspection of mechanical parts using the dense range data produced by laser stripers. Given a part to be inspected and a corresponding model of the part stored in the model data base, the first step of inspecting the part is the acquisition of data corresponding to the part, in our case this means the acquisition of a range image of it. In order to be able to compare the part image and its stored model, it is necessary to align the model with the range image of the part. This process, called registration, corresponds to finding the rigid transformation that superposes model and image. After the image and model are registered, the actual inspection uses the range image to verify if all the features predicted in the model are present and have the right pose and dimensions. Therefore, besides the acquisition of range images, the inspection of machined parts involves three main issues: modelling, registration and inspection diagnosis. The application, for inspection purposes, of the main representational schemes for modelling solid objects is discussed and it is suggested the use of EDT models (see [Zeid 91]). A particular implementation of EDT models is presented. A novel approach for the verification of tolerances during the inspection is proposed. The approach allows not only the inspection of the most common tolerances described in the tolerancing standards, but also the inspection of tolerances defined according to Requicha's theory of tolerancing (see [Requicha 83]). A model of the sensitivity and reliability of the inspection process based on the modelling of the errors during the inspection process is also proposed. The importance of the accuracy of the registration in different inspections tasks is discussed. A modified version of the ICP algorithm (see [Besl &; McKay 92]) for the registration of sculptured surfaces is proposed. The maximum accuracy of the ICP algorithm, as a function of the sensor errors and the number of matched points, is determined. A novel method for the measurement and reconstruction of waviness errors on sculp¬ tured surfaces is proposed. The method makes use of the 2D Discrete Fourier Transform for the detection and reconstruction of the waviness error. A model of the sensitivity and reliability of the method is proposed. The application of the methods proposed is illustrated using synthetic and real range image

Computer-Aided Geometry Modeling

Techniques in computer-aided geometry modeling and their application are addressed. Mathematical modeling, solid geometry models, management of geometric data, development of geometry standards, and interactive and graphic procedures are discussed. The applications include aeronautical and aerospace structures design, fluid flow modeling, and gas turbine design

MS

thesisGeometric constraint problems appear in many situations, including CAD systems, robotics, and computational biology. The complexity of these problems inspires the search for efficient solutions. We have developed a method to solve geometric constraint problems in the areas of geometric computation and robot path planning using configuration space subdivision. In this approach the configuration space, or parameter space, is subdivided and conservatively tested to find collision-free regions, which are then numerically searched for specific path solutions. This thesis presents a new more general approach to this last solution search step, using Monte Carlo optimization. In this new search approach, within a single subdivided area of configuration space, space is randomly sampled and then iteratively resampled based on importance weighting, until convergence to a solution with an acceptable error. We show that by using Monte Carlo optimization to extend configuration space subdivision we can solve higher dimensional problems more efficiently than configuration space subdivision by itself

Automated Digital Machining for Parallel Processors

When a process engineer creates a tool path a number of fixed decisions are made that inevitably produce sub-optimal results. This is because it is impossible to process all of the tradeoffs before generating the tool path. The research presents a methodology to support a process engineers attempt to generate optimal tool paths by performing automated digital machining and analysis. This methodology automatically generates and evaluates tool paths based on parallel processing of digital part models and generalized cutting geometry. Digital part models are created by voxelizing STL files and the resulting digital part surfaces are obtained based on casting rays into the part model. Tool paths are generated based on a general path template and updated based on generalized tool geometry and part surface information. The material removed by the generalized cutter as it follows the path is used to obtain path metrics. The paths are evaluated based on the path metrics of material removal rate, machining time, and amount of scallop. This methodology is a parallel processing accelerated framework suitable for generating tool paths in parallel enabling the process engineer to rank and select the best tool path for the job