Computing the minimum distance between two Bézier curves

International audienceA sweeping sphere clipping method is presented for computing the minimum distance between two Bézier curves. The sweeping sphere is constructed by rolling a sphere with its center point along a curve. The initial radius of the sweeping sphere can be set as the minimum distance between an end point and the other curve. The nearest point on a curve must be contained in the sweeping sphere along the other curve, and all of the parts outside the sweeping sphere can be eliminated. A simple sufficient condition when the nearest point is one of the two end points of a curve is provided, which turns the curve/curve case into a point/curve case and leads to higher efficiency. Examples are shown to illustrate efficiency and robustness of the new method

Automation of Process Planning for Automated Fiber Placement

Process planning represents an essential stage of the Automated Fiber Placement (AFP) workflow. It develops useful and efficient machine processes based upon the working material, composite design, and manufacturing resources. The current state of process planning requires a high degree of interaction from the process planner and could greatly benefit from increased automation. Therefore, a list of key steps and functions are created to identify the more difficult and time-consuming phases of process planning. Additionally, a set of metrics must exist by which to evaluate the effectiveness of the manufactured laminate from the machine code created during the Process Planning stage. This work begins with a ranking process which was performed through a survey of the Advanced Composites Consortium (ACC) Collaborative Research Team (CRT). Members were interviewed who possessed practical process planning experience in the composites industry. The Process Planning survey collected general input on the overall importance and time requirements for each function and which functions would benefit most greatly from semi-automation or full automation. Layup strategies, in addition to dog ears, stagger shifts, steering constraints, and starting points, represented the group of functions labeled as process optimization and ranked the highest in terms of priority for automation. The laminates resulting from the selected parameters are evaluated through the occurrences of principal defect metrics such as fiber gaps, overlaps, angle deviation and steering violations. This document presents an automated software solution to the layup strategy and starting point selection phase of Process Planning. A series of ply scenarios are generated with variations of these ply parameters and evaluated according to a set of metrics entered by the Process Planner. These metrics are generated through use of the Analytical Hierarchy Process (AHP), where relative importance between each of the fiber features are defined. The ply scenarios are selected which reduce the overall fiber feature scores based on the defects the Process Planner wishes to minimize

Image Processing Techniques for Detecting Chromosome Abnormalities

With the increasing use of Fluorescence In Situ Hybridization (FISH) probes as markers for certain genetic sequences, the requirement of a proper image processing framework is becoming a necessity to accurately detect these probe signal locations in relation to the centerline of the chromosome. Automated detection and length measurements based on the centerline relative to the centromere and the telomere coordinates would highly assist in clinical diagnosis of genetic disorders and thus improve its efficiency significantly. Although many image processing techniques have been developed for chromosomal analysis such as ’’karyotype analysis” to assist in laboratory diagnosis, they fail to provide reliable results in segmenting and extracting the centerline of chromosomes due to the high variability in shape of chromosomes on microscope slides. In this thesis we propose a hybrid algorithm that utilizes Gradient Vector Flow active contours, Discrete Curve Evolution based skeleton pruning and morphological thinning to provide a robust and accurate centerline of the chromosome, which is then used for the measurement of the FISH probe signals. Then this centerline information is used to detect the centromere location of the chromosome and the probe signal location distances were measured with respective to these landmarks. The ability to accurately detect FISH probe locations with respective to its centerline and other landmarks can provide the cytogeneticists with detailed information that could lead to a faster diagnosis

Modeling Of Tow Wrinkling In Automated Fiber Placement Based On Geometrical Considerations

Automated manufacturing of fiber reinforced composite structures via numerically controlled hardware yields parts with increased accuracy and repeatability as compared to hand-layup parts. Automated fiber placement (AFP) is one such process in which structures or parts are built by adding bands of prescribed number of tows or slit-tape with prescribed width using robotic machine heads over 3D surfaces following prescribed paths. Despite the improved accuracy, different types of defects or manufacturing features arise during fabrication. These defects can be due to geometrical features, materials, and process planning parameters and are detected in the form of wrinkling, tow twist, tow folding, overlap, gaps and several others. This thesis presents a thorough investigation of wrinkling within a path on a general surface for a composite tow constructed using the AFP process. Governing equations and assumptions for the presented model are derived based on geometric considerations only, neglecting the elastic properties of the material, and formulated for an arbitrary curve on a general three-dimensional surface. A simple form of the wrinkled shape is assumed and applied to the inner edge of the tow path. A numerical solution is implemented within Mathematica to visualize the curved paths and to indicate potential regions for wrinkling on the surface. Several examples are presented to demonstrate the model, including constant angle paths on a double-curved surface and curved paths on a flat surface. The obtained deformed patterns are compared with actual data from digital image correlation (DIC) of several towpaths

Tow-Path Characterization for Automated Fiber Placement

Automated Fiber Placement (AFP) is a manufacturing process used to fabricate large composite structures for aerospace applications. During the process, the machine head deposits multiple bands of composite material named tows over a prescribed path. Temperature, speed, and compaction pressure can be varied to obtain a good layup quality. For conventional laminated plate structures manufactured using the AFP process, fibers are laid at constant angles (0°, 90°, ±45°) in straight paths. However, to manufacture complex shell structures or variable stiffness plates, curved paths are necessary in the design leading to a length mismatch between the parallel edges of the towpath. Since finite width fiber tows are originally straight, a form of tow deformation is necessary to absorb the difference in length thus ensuring good adherence to the surface substrate. In this work, several possible tow deformation mechanisms are proposed and classified as follows: (1) elastic strain deformations (tensile, compressive, shear), (2) large in-plane deformations (fiber waviness and bunching), and (3) large out-of-plane deformations (tow wrinkling and folding). Usually, large tow deformations are unfavorable and considered as defects, thus process interruption may be necessary to perform manual repairs. The aim of this proposal is to develop models able to capture tow deformations when placed on a curved path and to validate their occurrence during the AFP process. The proposed work is tackled from three different perspectives: (1) geometrical modeling, (2) physics-based modeling, and (3) experimental investigations. Understanding the geometry of a given layup is necessary to determine critical locations for defect occurrence. A worst-case scenario is considered where all other deformation mechanisms are suppressed in favor of out-of-plane wrinkling. Governing equations for a tow placed on a general surface are derived, and a simple deformation function is applied to the shorter edge of tow showing different wrinkle patterns along the length. A simplified form of the governing equations is provided for the special case of tows steered on a flat surface. Finally, examples are presented visualizing the wrinkles patterns and showing critical locations of wrinkling for a given layup for flat and general surfaces. Relationships between the wrinkles wavelength and the steering radius can be obtained from existing mechanics models and experimental data. This information is used to improve the geometrical model, thus obtaining more accurate results for wrinkles modeling. In the physics-based model, the influence of the material properties and tow geometry on the deformations is studied. The tow is modeled as multiple fiber bundles laying on a stiff foundation. In a first step, only in-plane deformations are allowed, thus capturing the elastic deformation mode (tensile/compressive strains) and the large in-plane deformations (fiber waviness and bunching). In a second step, out-of-plane deformations are allowed, enabling the modeling of additional tow deformations mentioned earlier such as tow wrinkling. In a last step, the interaction between the neighboring fiber bundles is investigated by considering the transverse and shear stiffness properties of the uncured tow. Finally, experimental investigations measuring tow deformations over steered paths are carried using the Digital Image Correlation (DIC) technique. Thermoset pre-impregnated carbon fiber tows are speckled first, then placed using an AFP machine over vi multiple paths with different radii of curvature. Shape and strain measurements of the deformed tows are acquired using a Stereo DIC setup. Quantified measurements of wavelength, width, and amplitude for tow wrinkles are obtained as a function of the steering radius. The effect of the substrate, time, and temperature on the formation of wrinkles is also studied. Other experiments using DIC are carried out to determine the deformation of neighboring tows within a course when steered along a constant curvature path. To understand the effect of AFP process parameters on tow deformations, a benchmark reference path on a flat surface with a linear increase in the curvature is proposed. Based on the location of the first visible defect along the length of the path, a critical steering radius is determined for each set of process parameters used. Finally, steering experiments are performed on a cylindrical tool with varying process parameters. The quality of the manufactured steered paths is assessed through image processing of acquired profilometry scans during manufacturing. Recommendations regarding optimal set of process parameters for future manufacturing activities are provided based on the measured defects along the path

An Improved 2DOF Elastokinematic Surrogate Model for Continuous Motion Prediction and Visualisation of Forearm Pro-and Supination for Surgical Planning

Forearm rotation (pro-/supination) involves a non-trivial combination of rotation and translation of two bones, namely, radius and ulna, relatively to each other. Early works regarded this relative motion as a rotation about a fixed (skew) axis. However, this assumption turns out not to be exact. This thesis regards a spatial-loop surrogate mechanism involving two degrees of freedom with an elastic coupling for better forearm motion prediction. In addition, the influence of the bone morphology and position of elbow on kinematics are also considered. The model parameters are not measured directly from the anatomical components, but are fitted by reducing the errors between predicted and measured values in an optimization loop. For non-invasive measurement of bone position, magnetic resonance imaging (MRI) is employed. We present a method to self-calibrate the arm position in the MRI scanning tube and to fit the model parameters from a few, coarse MRI scans. Results show a good concordance between measurement and simulation. Moreover, the minimum distance changing between bones during forearm rotation is elucidated, which is not yet proved in anatomical and clinical literatures. The minimum distance is calculated by searching for the global shortest distance between bone contours on ulna and radius by a two-level selection and a following multidimensional Newton-Raphson algorithm. To this end, the methodology is extended from healthy bones to deformed arms and an angulated forearm model is developed. The 3D angulated bone geometry is obtained by manually separating the bone structure at the broken position, and the minimum distance and the range of motion of fractured forearms are analyzed. As shown for a single case validation, simulated results show very small deviations from anatomical data. Furthermore, the simulations discussed above are visualized using interactive interfaces, which facilitates the application of the model in clinical planning.Die Unterarmrotation beinhaltet eine nicht triviale Kombination einer Rotation und Translokation zweier Knochen, Radius und Ulna relativ zu einander. Frühere Arbeiten betrachteten diese relative Bewegung als eine Rotation um eine fixierte Achse. Allerdings scheint diese Annahme ungenau zu sein. Diese Arbeit betrachtet ein Spatial-Loop Surrogat Mechanismus unter Berücksichtigung von zwei Freiheitsgraden mit einer elastischen Gelenkverbindung für eine bessere Prognose der Unterarm-Bewegung. Zusätzlich wird der Einfluss der Knochenmorphologie und die Position des Ellenbogens auf die Kinematik berücksichtig. Die Modellparameter werden nicht direkt von den anatomischen Komponenten bestimmt, sondern unter Berücksichtigung der Abweichung von Annahme und Messung. Zur nicht invasiven Messung der Knochenposition wird die Methode der Magnetresonanztomographie (MRT) angewendet. Wir stellen hier eine Methode um die Arm-Position in das MRI Scan-Rohr selbst zu kalibrieren und die Modellparameter aus einige grobe MRT-Aufnahmen zu passen. Die simulierten Ergebnisse zeigen sehr kleine Abweichungen von anatomischen Daten. Eine minimale Änderung der Distanz zwischen den Knochen während der Unterarmrotation wird beleuchte, die bisher nicht in der anatomischen und klinischen Literatur beschrieben ist. Die Berechnung der minimalen Distanz erfolgt über die Ermittlung der gesamt kürzesten Distanz. Zu diesem Zweck wird die Methodik von gesunden Knochen auf deformiere Arme und ein angewinkeltes Unterarmmodel entwickelt. Die 3D gewinkelte Knochen-Geometrie ergibt sich aus der Knochenstruktur an der gebrochener Position manuell zu trennen, und darauf werden der Mindestabstand und der Bereich der Bewegung von dem gebrochenen Unterarm analysiert. Wie dies bei einer einzelnen Fall Validierung, zeigen die simulierten Ergebnisse sehr kleine Abweichungen von anatomischen Daten. Darüber hinaus werden die oben beschrieben Simulationen mit interaktiven Benutzeroberflächen visualisiert, welche die Anwendung des Modells in der klinischen Planung erleichtert

Isogeometric Analysis for High Order Geometric Partial Differential Equations with Applications

In this thesis, we consider the numerical approximation of high order geometric Partial Differential Equations (PDEs). We first consider high order PDEs defined on surfaces in the 3D space that are represented by single-patch tensor product NURBS. Then, we spatially discretize the PDEs by means of NURBS-based Isogeometric Analysis (IGA) in the framework of the Galerkin method. With this aim, we consider the construction of periodic NURBS function spaces with high degree of global continuity, even on closed surfaces. As benchmark problems for the proposed discretization, we propose Laplace-Beltrami problems of the fourth and sixth orders, as well as the corresponding eigenvalue problems, and we analyze the impact of the continuity of the basis functions on the accuracy as well as on computational costs. The numerical solution of two high order phase field problems on both open and closed surfaces is also considered: the fourth order Cahn-Hilliard equation and the sixth order crystal equation, both discretized in time with the generalized-alpha method. We then consider the numerical approximation of geometric PDEs, derived, in particular, from the minimization of shape energy functionals by L^2-gradient flows. We analyze the mean curvature and the Willmore gradient flows, leading to second and fourth order PDEs, respectively. These nonlinear geometric PDEs are discretized in time with Backward Differentiation Formulas (BDF), with a semi-implicit formulation based on an extrapolation of the geometry, leading to a linear problem to be solved at each time step. Results about the numerical approximation of the two geometric flows on several geometries are analyzed. Then, we study how the proposed mathematical framework can be employed to numerically approximate the equilibrium shapes of lipid bilayer biomembranes, or vesicles, governed by the Canham-Helfrich curvature model. We propose two numerical schemes for enforcing the conservation of the area and volume of the vesicles, and report results on benchmark problems. Then, the approximation of the equilibrium shapes of biomembranes with different values of reduced volume is presented. Finally, we consider the dynamics of a vesicle, e.g. a red blood cell, immersed in a fluid, e.g. the plasma. In particular, we couple the curvature-driven model for the lipid membrane with the incompressible Navier-Stokes equations governing the fluid. We consider a segregated approach, with a formulation based on the Resistive Immersed Surface method applied to NURBS geometries. After analyzing benchmark fluid simulations with immersed NURBS objects, we report numerical results for the investigation of the dynamics of a vesicle under different flow conditions