Solving Boundary Value Problems Via the Nyström Method Using Spline Gauss Rules

We propose to use spline Gauss quadrature rules for solving boundary value problems (BVPs) using the Nyström method. When solving BVPs, one converts the corresponding partial differential equation inside a domain into the Fredholm integral equation of the second kind on the boundary in the sense of boundary integral equation (BIE). The Fredholm integral equation is then solved using the Nyström method, which involves a use of a particular quadrature rule, thus, converting the BIE problem to a linear system. We demonstrate this concept on the 2D Laplace problem over domains with smooth boundary as well as domains containing corners. We validate our approach on benchmark examples and the results indicate that, for a fixed number of quadrature points (i.e., the same computational effort), the spline Gauss quadratures return an approximation that is by one to two orders of magnitude more accurate compared to the solution obtained by traditional polynomial Gauss counterparts

Geometry and tool motion planning for curvature adapted CNC machining

CNC machining is the leading subtractive manufacturing technology. Although it is in use since decades, it is far from fully solved and still a rich source for challenging problems in geometric computing. We demonstrate this at hand of 5-axis machining of freeform surfaces, where the degrees of freedom in selecting and moving the cutting tool allow one to adapt the tool motion optimally to the surface to be produced. We aim at a high-quality surface finish, thereby reducing the need for hard-to-control post-machining processes such as grinding and polishing. Our work is based on a careful geometric analysis of curvature-adapted machining via so-called second order line contact between tool and target surface. On the geometric side, this leads to a new continuous transition between “dual” classical results in surface theory concerning osculating circles of surface curves and oscu- lating cones of tangentially circumscribed developable surfaces. Practically, it serves as an effective basis for tool motion planning. Unlike previous approaches to curvature-adapted machining, we solve locally optimal tool positioning and motion planning within a single optimization framework and achieve curvature adaptation even for convex surfaces. This is possible with a toroidal cutter that contains a negatively curved cutting area. The effectiveness of our approach is verified at hand of digital models, simulations and machined parts, including a comparison to results generated with commercial software

Solving boundary value problems via the Nyström method using spline Gauss rules

We propose to use spline Gauss quadrature rules for solving boundary value problems (BVPs) using the Nyström method. When solving BVPs, one converts the corresponding partial differential equation inside a domain into the Fredholm integral equation of the second kind on the boundary in the sense of boundary integral equation (BIE). The Fredholm integral equation is then solved using the Nyström method, which involves the use of a particular quadrature rule, thus, converting the BIE problem to a linear system. We demonstrate this concept on the 2D Laplace problem over domains with smooth boundary as well as domains containing corners. We validate our approach on benchmark examples and the results indicate that, for a fixed number of quadrature points (i.e., the same computational effort), the spline Gauss quadratures return an approximation that is by one to two orders of magnitude more accurate compared to the solution obtained by traditional polynomial Gauss counterparts

The Construction of Conforming-to-shape Truss Lattice Structures via 3D Sphere Packing

Truss lattices are common in a wide variety of engineering applications, due to their high ratio of strength versus relative density. They are used both as the interior support for other structures, and as structures on their own. Using 3D sphere packing, we propose a set of methods for generating truss lattices that fill the interior of B-rep models, polygonal or (trimmed) NURBS based, of arbitrary shape. Once the packing of the spheres has been established, beams between the centers of adjacent spheres are constructed, as spline based B-rep geometry. We also demonstrate additional capabilities of our methods, including connecting the truss lattice to (a shell of) the B-rep model, as well as constructing a tensor-product trivariate volumetric representation of the truss lattice - an important step towards direct compatibility for analysis.RYC-2017-2264

Constant probe orientation for fast contact-based inspection of 3D free-form surfaces using (3+2)-axis inspection machines

A new probe optimization method for contact based (3+2)-axis inspection machines is proposed. Given an inspection path of a stylus on a free-form surface, an optimal orientation of the stylus is computed such that (i) the inclination angle of the stylus is within a given angular range with respect to the surface normal, (ii) the motion of the stylus is globally collision free, and (iii) the stylus remains constant in the coordinate system of the measuring machine. The last condition guarantees that the inspection motion requires only the involvement of the three translational axes of the measuring machine. The numerical simulations were validated through physical experiments on a testcase of a tooth of a bevel gear due to the surface complexity and probe accessibility. This optimized method was compared to 3-axis and 5-axis inspection strategies, showing that the fixed (3+2)-axis stylus returns more accurate inspection results compared to the traditional 3-axis approach and similar to 5-axis approach

Constant probe orientation for fast contact-based inspection of 3D free-form surfaces using (3+2)-axis inspection machines

A new probe optimization method for contact based (3+2)-axis inspection machines is proposed. Given an inspection path of a stylus on a free-form surface, an optimal orientation of the stylus is computed such that (i) the inclination angle of the stylus is within a given angular range with respect to the surface normal, (ii) the motion of the stylus is globally collision free, and (iii) the stylus remains constant in the coordinate system of the measuring machine. The last condition guarantees that the inspection motion requires only the involvement of the three translational axes of the measuring machine. The numerical simulations were validated through physical experiments on a testcase of a tooth of a bevel gear due to the surface complexity and probe accessibility. This optimized method was compared to 3-axis and 5-axis inspection strategies, showing that the fixed (3+2)-axis stylus returns more accurate inspection results compared to the traditional 3-axis approach and similar to 5-axis approach.RYC-2017-2264

Kramers escape driven by fractional Brownian motion

We investigate the Kramers escape from a potential well of a test particle driven by fractional Gaussian noise with Hurst exponent 0<H<1. From a numerical analysis we demonstrate the exponential distribution of escape times from the well and analyze in detail the dependence of the mean escape time as function of H and the particle diffusivity D. We observe different behavior for the subdiffusive (antipersistent) and superdiffusive (persistent) domains. In particular we find that the escape becomes increasingly faster for decreasing values of H, consistent with previous findings on the first passage behavior. Approximate analytical calculations are shown to support the numerically observed dependencies.Comment: 14 pages, 16 figures, RevTeX

Phenotypic Signatures Arising from Unbalanced Bacterial Growth

Fluctuations in the growth rate of a bacterial culture during unbalanced growth are generally considered undesirable in quantitative studies of bacterial physiology. Under well-controlled experimental conditions, however, these fluctuations are not random but instead reflect the interplay between intra-cellular networks underlying bacterial growth and the growth environment. Therefore, these fluctuations could be considered quantitative phenotypes of the bacteria under a specific growth condition. Here, we present a method to identify “phenotypic signatures” by time-frequency analysis of unbalanced growth curves measured with high temporal resolution. The signatures are then applied to differentiate amongst different bacterial strains or the same strain under different growth conditions, and to identify the essential architecture of the gene network underlying the observed growth dynamics. Our method has implications for both basic understanding of bacterial physiology and for the classification of bacterial strains

A Microscope Automated Fluidic System to Study Bacterial Processes in Real Time

Most time lapse microscopy experiments studying bacterial processes ie growth, progression through the cell cycle and motility have been performed on thin nutrient agar pads. An important limitation of this approach is that dynamic perturbations of the experimental conditions cannot be easily performed. In eukaryotic cell biology, fluidic approaches have been largely used to study the impact of rapid environmental perturbations on live cells and in real time. However, all these approaches are not easily applicable to bacterial cells because the substrata are in all cases specific and also because microfluidics nanotechnology requires a complex lithography for the study of micrometer sized bacterial cells. In fact, in many cases agar is the experimental solid substratum on which bacteria can move or even grow. For these reasons, we designed a novel hybrid micro fluidic device that combines a thin agar pad and a custom flow chamber. By studying several examples, we show that this system allows real time analysis of a broad array of biological processes such as growth, development and motility. Thus, the flow chamber system will be an essential tool to study any process that take place on an agar surface at the single cell level

Experimentally Guided Computational Model Discovers Important Elements for Social Behavior in Myxobacteria

Identifying essential factors in cellular interactions and organized movement of cells is important in predicting behavioral phenotypes exhibited by many bacterial cells. We chose to study Myxococcus xanthus, a soil bacterium whose individual cell behavior changes while in groups, leading to spontaneous formation of aggregation center during the early stage of fruiting body development. In this paper, we develop a cell-based computational model that solely relies on experimentally determined parameters to investigate minimal elements required to produce the observed social behaviors in M. xanthus. The model verifies previously known essential parameters and identifies one novel parameter, the active turning, which we define as the ability and tendency of a cell to turn to a certain angle without the presence of any obvious external factors. The simulation is able to produce both gliding pattern and spontaneous aggregation center formation as observed in experiments. The model is tested against several known M. xanthus mutants and our modification of parameter values relevant for the individual mutants produces good phenotypic agreements. This outcome indicates the strong predictive potential of our model for the social behaviors of uncharacterized mutants and their expected phenotypes during development