Arc Approximation Research Articles

Fast permeability measurement for tight reservoir cores using only initial data of the one chamber pressure pulse decay test

In this study, a mathematical model for fast determination of the permeabilities of tight rocks using measurements taken from the initial period of the One Chamber Pressure Pulse Decay (OC-PPD) test is presented. The model applies to measurements taken both before and after the pressure pulse front has reached the downstream end of the specimen. The analytical solutions for the pressure decay in the upstream chamber are derived based on a parabolic arc approximation of pore pressure distribution along the test specimen. This approximation allows converting the initial–boundary value problem of fluid diffusion in the specimen, governed by partial differential equations, to a system of ordinary differential equations that can be easily solved by explicit formulae. Thus, an explicit formula for the pressure decay rate is obtained, which enables inverse analysis of the initial experimental data to estimate the rock permeability. The proposed method expedites the pulse decay test as it does not require the system to reach equilibrium. The method is validated with three sets of experimental data of the OC-PPD test using helium as the diffusing fluid, for which the relative error of the permeability is found to be less than 6%. This method is particularly useful if the equilibrium time of the pulse decay test for rock specimens with permeabilities in the range of nano-Darcy takes hours or days.

Circular arc approximation by hexic polynomial curves

Circular arc approximation by hexic polynomial curves

Road-Map Aided Gaussian Mixture Labeled Multi-Bernoulli Filter for Ground Multi- Target Tracking

Ground multi-target tracking (MTT) is one of the core tasks of airborne ground moving target indicator (GMTI) radar and automotive radar. However, ground MTT still remains a challenging issue especially in complex traffic scenarios, since it often suffers from high clutter, dense targets, low visibility, etc. To enhance the tracking performance for ground multiple targets, in this paper, we present a comprehensive solution named road-map aided Gaussian mixture labeled multi-Bernoulli filter (RA-GMLMB) filter, which incorporates road-map information into the labeled multi-Bernoulli (LMB) filter. Specifically, we first propose a hybrid circular arc and line segments approximation approach to extract road-map information, which alleviates approximation errors in the procedure of road-map approximation. Then, we deduce the RA-GMLMB filter by integrating the extracted road-map information into the LMB filter with Gaussian mixture implementation. Simulation experiments are conducted and experimental results show that the proposed RA-GMLMB filter outperforms the state-of-the-art methods.

Theoretical responses of shallow-buried circular cavity subjected to transient P wave

Shallow-buried structures are inevitably subjected to time-varying disturbances such as drilling and blasting, earthquake and tectonic activities. The work presented in this paper focuses on the transient responses of a shallow-buried cavity. Based on the wave function expansion method, the large arc approximation and the integral transform, the time-domain responses around a cylindrical cavity subjected to transient P wave in an elastic half space are investigated analytically. The analytical results reveal that the wavelength-to-diameter ratio, the direction of disturbance and the embedment depth have a significant effect on the velocity vibrations and dynamic stress concentration around the tunnel. The theoretical investigations can provide practical value for the support of shallow tunnels.

On optimal polynomial geometric interpolation of circular arcs according to the Hausdorff distance

The problem of the optimal approximation of circular arcs by parametric polynomial curves is considered. The optimality relates to the Hausdorff distance and has not been studied yet in the literature. Parametric polynomial curves of low degree are used and a geometric continuity is prescribed at the boundary points of the circular arc. A general theory about the existence and the uniqueness of the optimal approximant is presented and a rigorous analysis is done for some special cases for which the degree of the polynomial curve and the order of the geometric smoothness differ by two. This includes practically interesting cases of parabolic G0, cubic G1, quartic G2 and quintic G3 interpolation. Several numerical examples are presented which confirm theoretical results.

A circular arc approximation algorithm for cucumber classification with image analysis

Abstract The cucumber is one of the most important consumer vegetables, and due to high freshness and appearance standards, it must be graded according to quality. One key metric for grading cucumbers is curvature (arch height) relative to length. To date, this classification standard has not been implemented in a commercial automatic system because the curvature of the cucumber is related to its length. This paper presents and tests a circular arc approximation algorithm for measuring the curvature of a cucumber using image processing. By fitting the central axis of each cucumber to establish a curve equation, the approximate eigenvalue radius, R, and the radius of the section perpendicular to the curvature of cucumber, r, could be obtained quickly and efficiently. The curve of the cucumber was then transformed into a new characteristic, R' (R' = R – r), which was used as a threshold to classify each cucumber. The method was verified by theory and experiment, and the average error of 0.71 % corresponding to the coefficient R2 = 0.9958 of the similarity index between the curve and central axis of the cucumber was verified. It was found that the fitted curve and the central axis of the cucumber were coincident. Moreover, we designed an algorithm that can detect S-shaped cucumbers based on the analysis of the contour, which can avoid the influence of S-shaped cucumbers on the results. In our tests, 148 cucumbers were classified using the circular arc approximation method, the elliptic approximation method, and a manual method. The results of the classifications were compared. According to China's standard evaluation, the classification error rate of the circular arc approximation method was 10.1 % lower than that of the elliptic approximation method. The incidence of cucumbers being classified into an adjacent grade was 0.7 % using the circular approximation method, and there were no cases of cucumbers being classified into non-adjacent grades. The classification error rate of the elliptic approximation was 10.8 %, of which 2.7 % was classified into non-adjacent grades. Under the European standard, the cucumber classification accuracy using the circular arc approximation algorithm reached 100 %. Therefore, the proposed method offers a more accurate classifier than the elliptic approximation method, and the circular arc approximation algorithm can be fully applied in the commercial cucumber classification process.

Segmentation of Breast-Region in Breast Thermogram Using Arc-Approximation and Triangular-Space Search

Accurate segmentation of the breast region in a breast thermogram is a very crucial step in the diagnosis of breast cancer, and at the same time, it is also a very challenging problem. To tackle issues related to the breast region segmentation, first, we have proposed a novel method to identify the upper boundary (UB) of the breast based on the arc approximation algorithm. Then, the lower boundary curve (LBC) of the breast is identified by the segregation of the inframammary fold region (IFR) of the breast followed by an IFR isolation method. The actual LBC of the breast is traced in the respective edge image using the segmented IFR of that breast. Then, the traced LBC is extrapolated on both the sides using our proposed triangular-space search technique. Finally, the two breasts are separated based on the point-of-intersection (PoI) of LBCs of breasts. The efficacy of the proposed method is validated on the images of Database for Mastology Research with Infrared Images (DMR-IR) database both qualitatively and quantitatively based on the ground-truths. The average segmentation accuracy of 95.75% confirms the superiority of our method in segmenting the breast region over the existing state-of-the-art breast region segmentation methods.

The best quintic Chebyshev approximation of circular arcs of order ten

<p>Mathematically, circles are represented by trigonometric parametric equations and implicit equations. Both forms are not proper for computer applications and CAD systems. In this paper, a quintic polynomial approximation for a circular arc is presented. This approximation is set so that the error function is of degree $10$ rather than $6$; the Chebyshev error function equioscillates $11$ times rather than $7$; the approximation order is $10$ rather than $6$. The method approximates more than the full circle with Chebyshev uniform error of $1/2^{9}$. The examples show the competence and simplicity of the proposed approximation, and that it can not be improved.</p>

$\boldsymbol{G^2[C^1]}$ Hermite interpolation using septic PH curves

In this paper, we examine the problem of $G^2[C^1]$ Hermite interpolation using septic Pythagorean hodograph (PH) curves. PH curves are a special class of polynomial parametric curves, which have a polynomial arc length function and rational offsets, and are thus widely used in computer-aided design. According to different factorizations of their first derivative in complex form, septic PH curves are classified into three classes. The curves in the first class are all regular, and their construction under any $G^2[C^1]$ condition has already been studied. Therefore, in this paper we focus on the remaining two classes. The number of septic PH curves in the second class is even and no more than six. The existence of septic PH curves in the third class is dependent on the initial Hermite data, and users may specify a real parameter to determine the resultant curve. In addition, we provide the approximation of arcs with septic PH curves as examples demonstrating the application of our results.

Trajectory generation in 5-axis milling of freeform surfaces using circular arc approximation and its influence in surface roughness

Current methods used to generate toolpath in 5-axis machining, multiply the program lines when the precision is sought, producing a deficient approach to the points and a highly variable feed rates. These factors cause geometric imprecision and poor surface finishes. In this work, a new method for approximating trajectories through circular arcs between selected points has been developed. It has been checked through the machining of an elliptical toroid surface and its results have been compared with those produced by other CAM strategy. The part produced by the new method shows a considerable improvement in the surface roughness.

On Poletsky theory of discs in compact manifolds

We provide a direct construction of Poletsky discs via local arc approximation and a Runge-type theorem by A. Gournay.

A general framework for the optimal approximation of circular arcs by parametric polynomial curves

We propose a general framework for geometric approximation of circular arcs by parametric polynomial curves. The approach is based on constrained uniform approximation of an error function by scalar polynomials. The system of nonlinear equations for the unknown control points of the approximating polynomial given in the B\'ezier form is derived and a detailed analysis provided for some low degree cases which might be important in practice. At least for these cases the solutions can be, in principal, written in a closed form, and provide the best known approximants according to the radial distance. A general conjecture on the optimality of the solution is stated and several numerical examples conforming theoretical results are given.

A high accuracy on-line estimation algorithm of five-axis contouring errors based on three-point arc approximation

Abstract High-accuracy on-line contouring error estimation plays a significant role in contouring accuracy control of multi-axis CNC machining. This paper focuses on developing a high-accuracy on-line estimation algorithm of five-axis contouring errors based on three-point arc approximation (TPAA). The nearest reference tool tip position related to the actual one is searched at first, then the tool tip position contouring error is estimated by the distance between the actual position and the arc formed by the nearest reference tool tip position and its two adjacent reference positions. Similar to the tool tip position, the TPAA algorithm is also conducted to estimate the tool orientation contouring error in the unit spherical coordinate system, where equal proportional factors of the tool tip position and the tool orientation arc-lengths are adopted to ensure the synchronization of the tool tip position contouring error and the tool orientation contouring error. The proposed contouring error estimation algorithm is adapted to different trajectory types and has higher estimation accuracy than linear segment approximation (LSA) algorithm for trajectory segments with high curvature. Various experiments with different testing tool paths are performed on an in-house developed five-axis experiment platform to verify the effectiveness of the proposed algorithm, and the results show that the proposed TPAA algorithm can estimate contouring errors with higher accuracy than existing algorithms based on the curvature arc iterative (CAI) algorithm and the LSA algorithm.

Circular sector area preserving approximation of circular arcs by geometrically smooth parametric polynomials

The quality of the approximation of circular arcs by parametric polynomials is usually measured by the Hausdorff distance. It is sometimes important that a parametric polynomial approximant additionally preserves some particular geometric property. In this paper we study the circular sector area preserving parametric polynomial approximants of circular arcs. A general approach to this problem is considered and corresponding (nonlinear) equations are derived. For the approximants possessing the maximal order of geometric smoothness, a scalar nonlinear equation is analyzed in detail for the parabolic, the cubic and the quartic case. The existence of the admissible solution is confirmed. Moreover, the uniqueness of the solution with the optimal approximation order with respect to the radial distance is proved. Theoretical results are confirmed by numerical examples.

Prediction of tool tip dynamics for generalized milling cutters using the 3D model of the tool body

In general, chatter is the main limitation to proper material removal in milling operations. Stability lobes are good tools to determine chatter-free cutting conditions in terms of spindle speed and cutting depth, which require the frequency response function (FRF) at the tool tip to be known. There are experimental methods to measure the tool tip FRF but this may be time consuming or even impossible for each tool and tool holder combination. Receptance coupling substructure analysis (RCSA) is a widely used approach to predict tool tip dynamics. This paper proposes the use of the RCSA approach with a stereolithographic (STL) slicing algorithm to enable the exact calculation of cross sectional properties such as area and area moment of inertia of the cutting tool from its 3D model opposed to the approximation methods. So that, the effect of flutes on cutting tool structure introduced in an exact manner and the RCSA approach becomes feasible for more complicated tool geometries with varying cross-sectional properties, i.e., tapered ball end mills, end mills with variable flute geometries, and so on. The solid model of the tool can be available by either the tool manufacturer or 3D measurement. Although, at the presence of 3D models, finite element methods (FEM) offer accurate simulation of the dynamic response for solid bodies, they suffer from the compromise between accuracy and computation time, as high number of elements is needed for accuracy. Thus, the use of analytical methods where possible improves the simulation time significantly. The proposed STL slicing algorithm is integrated with a previously developed RCSA method. The experimental results show that the proposed algorithm works more accurate in calculation of the cross-sectional properties and hence free-free response of the tool compared to the existing arc approximation methods. It is also shown that the proposed approach performs better than FEM solutions in terms of the computation time and the compromise between accuracy and computation performance. Finally, the proposed approach in prediction of tool tip dynamics for a robotic machining platform.

Circular arc approximation by quartic H-Bézier curve

The quartic H-Bezier curve is used for the approximation of circular arcs. It has five control points and one positive real free parameter. The four control points are carried out by G^1-approximation constraints and the remaining control point is dividing the line segment joining the second and fourth control points in the ratio 1:2. Optimized value of free parameter α is obtained by minimizing the maximum value of absolute radius error of the recommended approximation scheme. The developed approximation scheme is found considerably better than the existing approximation schemes for these computed values of control points and optimized value of the free parameter.

Approximation of conic section by quartic Bézier curve with endpoints continuity condition

A new method for approximation of conic section by quartic Bezier curve is presented, based on the quartic Bezier approximation of circular arcs. Here we give an upper bound of the Hausdorff distance between the conic section and the approximation curve, and show that the error bounds have the approximation order of eight. Furthermore, our method yields quartic G 2 continuous spline approximation of conic section when using the subdivision scheme, and the effectiveness of this method is demonstrated by some numerical examples.

The Best Uniform Cubic Approximation of Circular Arcs with High Accuracy

In this article, the issue of the best uniform cubic approximation of circular arcs with parametrically defined polynomial curves is considered. By a proper choice of the Bezier points, the best uniform approximation of degree 3 to a circular arc is given in explicit form. The approximation is constructed so that the error function is the Chebyshev polynomial of degree 6; the error function equioscillates 7 times; the approximation order is 6. The numerical examples demonstrate the efficiency and simplicity of the approximation method as well as satisfying the properties of the best uniform approximation and yielding the best approximation of least deviation and the highest possible accuracy.

APPROXIMATION ORDER OF C3QUARTIC B-SPLINE APPROXIMATION OF CIRCULAR ARC

In this paper, we present a <TEX>$C^3$</TEX> quartic B-spline approximation of circular arcs. The Hausdorff distance between the <TEX>$C^3$</TEX> quartic B-spline curve and the circular arc is obtained in closed form. Using this error analysis, we show that the approximation order of our approximation method is six. For a given circular arc and error tolerance we find the <TEX>$C^3$</TEX> quartic B-spline curve having the minimum number of control points within the tolerance. The algorithm yielding the <TEX>$C^3$</TEX> quartic B-spline approximation of a circular arc is also presented.

Dose calculation for hypofractionated volumetric-modulated arc therapy: approximating continuous arc delivery and tongue-and-groove modeling.

Hypofractionated treatments generally increase the complexity of a treatment plan due to the more stringent constraints of normal tissues and target coverage. As a result, treatment plans contain more modulated MLC motions that may require extra efforts for accurate dose calculation. This study explores methods to minimize the differences between in‐house dose calculation and actual delivery of hypofractionated volumetric‐modulated arc therapy (VMAT), by focusing on arc approximation and tongue‐and‐groove (TG) modeling. For dose calculation, the continuous delivery arc is typically approximated by a series of static beams with an angular spacing of 2°. This causes significant error when there is large MLC movement from one beam to the next. While increasing the number of beams will minimize the dose error, calculation time will increase significantly. We propose a solution by inserting two additional apertures at each of the beam angle for dose calculation. These additional apertures were interpolated at two‐thirds’ degree before and after each beam. Effectively, there were a total of three MLC apertures at each beam angle, and the weighted average fluence from the three apertures was used for calculation. Because the number of beams was kept the same, calculation time was only increased by about 6%‐8%. For a lung plan, areas of high local dose differences (>4%) between film measurement and calculation with one aperture were significantly reduced in calculation with three apertures. Ion chamber measurement also showed similar results, where improvements were seen with calculations using additional apertures. Dose calculation accuracy was further improved for TG modeling by developing a sampling method for beam fluence matrix. Single element point sampling for fluence transmitted through MLC was used for our fluence matrix with 1 mm resolution. For Varian HDMLC, grid alignment can cause fluence sampling error. To correct this, transmission volume averaging was applied. For three paraspinal HDMLC cases, the average dose difference was greatly reduced in film and calculation comparisons with our new approach. The gamma (3%, 3 mm) pass rates have improved significantly from 74.1%, 90.0%, and 90.4% to 99.2%, 97.9%, and 97.3% for three cases, for calculation without volume averaging and calculation with volume averaging, respectively. Our results indicate that more accurate MLC leaf position and transmission sampling can improve accuracy and agreement between calculation and measurement, and are particularly important for hypofractionated VMAT that consists of large MLC movement.PACS number(s): 87.55.kd