Offset Surface Research Articles

Reconstruction approximating method by biquadratic splines of offset surfaces holes

Standard Offset surfaces are defined as locus of the points which are at constant distance along the unit normal direction from the generator surfaces. Offset are widely used in various practical applications, such as tolerance analysis, geometric optics and robot path-planning. In some of the engineering applications, we need to extend the concept of standard offset to the generalized offset where distance offset is not necessarily constant and offset direction are not necessarily along the normal direction. Normally, a generalized offset is functionally more complex than its progenitor because of the square root appears in the expression of the unit normal vector. For this, an approximation method of its construction is necessary. In many situation it is necessary to fill or reconstruct certain function defined in a domain in which there is a lack of information inside one or several sub-domains (holes). In some practical cases, we may have some specific geometrical constrains, of industrial or design type, for example, the case of a specified volume inside each one of these holes. The problem of filling holes or completing a 3D surface arises in all sorts of computational graphics areas, like CAGD, CAD-CAM, Earth Sciences, computer vision in robotics, image reconstruction from satellite and radar information, etc. In this work we present an approximation method of filling holes of the generalized offset of a surface when there is a lack information in a sub-domain of the function that define it. We prove the existence and uniqueness of solution of this problem, we show how to compute it and we establish a convergence result of this approximation method. Finally, we give some graphical and numerical examples.

Footvector Representation of Curves and Surfaces

This paper proposes a foot mapping-based representation of curves and surfaces which is a geometric generalization of signed distance functions. We present a first-order characterization of the footvector mapping in terms of the differential geometric invariants of the represented shape and quantify the dependence of the spatial partial derivatives of the footvector mapping with respect to the principal curvatures at the footpoint. The practical applicability of foot mapping representations is highlighted by several fast iterative methods to compute the exact footvector mapping of the offset surface of CSG trees. The set operations for footpoint mappings are higher-order functions that map a tuple of functions to a single function, which poses a challenge for GPU implementations. We propose a code generation framework to overcome this that transforms CSG trees to the GLSL shader code.

Geometric modeling of complex knitting stitches using a bicontinuous surface and its offsets

Helicoids have been utilized as a scaffold on which to define the topology and geometry of yarns in a weft-knitted fabric. The centerline of a yarn in the fabric is specified as a geodesic path, with constrained boundary conditions, running along a helicoid at a fixed distance. The properties and constraints of the yarn are formulated into a single “energy” function, which is then minimized to produce the desired resulting models. We present improvements to this approach that address the deficiencies of the original work and extend its capabilities to more complex stitches, such as transfer, tuck and miss. A single bicontinuous surface is described, which replaces discrete helicoids and produces higher quality, continuous yarn models. A new computational method is employed that significantly speeds up the optimization computations. Including offset surfaces with the scaffold, as well as removing sections of the scaffold, allow for the modeling of complex stitches. The improved approach produces superior geometric results, consisting of complex knitting stitches, at a fraction of the computational cost of the previous method.

Exchange variations of generalized dual parallel curves and surfaces

Parallel curves (or offset curves) and parallel surfaces (or offset surfaces) have a big importance for CAD/CAM, robotics, cam design and many industrial applications, especially for mathematical modelling of cutting paths milling machines. Any vector space has a corresponding dual vector space that consists of all linear functions on vector space. Dual spaces are used in mathematics such as describing measures, distributions, and Hilbert spaces. Consequently, the dual space is an important concept in functional analysis. This paper proposes a novel definition of generalized and standard dual parallel curves and surfaces. Additionally, we give some properties of generalized dual parallel curves and surfaces using this novel definition. We also express the variation of the generalized dual parallel curves, the first and second variation of area change of the standard dual parallel surfaces and the first variation of area change of the generalized dual parallel surfaces.

Non-Parametric Shape Design of Free-Form Shells Using Fairness Measures and Discrete Differential Geometry

A non-parametric approach is proposed for shape design of free-form shells discretized into triangular mesh. The discretized forms of curvatures are used for computing the fairness measures of the surface. The measures are defined as the area of the offset surface and the generalized form of the Gauss map. Gaussian curvature and mean curvature are computed using the angle defect and the cotangent formula, respectively, defined in the field of discrete differential geometry. Optimization problems are formulated for minimizing various fairness measures for shells with specified boundary conditions. A piecewise developable surface can be obtained without a priori assignment of the internal boundary. Effectiveness of the proposed method for generating various surface shapes is demonstrated in the numerical examples.

Research on Curved Layer Slicing and Spatial Path Generation Method in Five-axis Material Extrusion Process

The traditional additive manufacturing technology uses a plane-cutting method, and the printing path of each layer is limited to a single plane. When printing thin-walled parts, there are problems such as high surface displacement, poor mechanical properties, and the need to add support structure. The paper proposes a five-axis additive manufacturing technology using curved surface slice method. Specifically, it includes a triangular mesh vertex offset surface slicing algorithm based on half-edge data structure and a path planning algorithm based on triangular mesh adjacent topology and geodesic distance field; a series-parallel hybrid five-axis material extrusion printing device has been developed, which fully combines the advantages of the small cumulative error of the parallel mechanism and the large working space of the series mechanism. The proposed surface cutting and path planning method is used to print thin-walled spherical parts and skull substitutes. The surface roughness of the printed parts is significantly improved. The results show the correctness and feasibility of the algorithm.

Hydrodynamic Bearing Performance Trade-Off Study and Fuzzy-Based Multi-Objective Optimization on an Offset Surface Textured Journal Bearing

The main purpose of this research is to determine the behavior of bearing performance having various operating conditions. Experimental studies are performed to analyze the dynamic characteristics of phosphorous bronze hydrodynamic journal bearing with a surface textured under various loading conditions with multi-grade oil as lubricating at different constant rotational speeds (i.e., 1000 to 4000 rpm). The results indicated that the experimentally obtained oil film pressure and temperature distribution profile were in good agreement with the theoretical results. In this experimental study, Fuzzy-based Taguchi optimization was used to predict the optimal input parameters that result in the optimal temperature combination and stated that the theoretical findings are in good agreement with the oil film pressure and temperature distribution profile obtained experimentally. To maximize the output parameters, this analysis also aims to explain the implementation of the fuzzy logic-based Taguchi analysis.

Hydrodynamic Bearing Performance Trade-off Study and Fuzzy Based Multi-objective Optimisation on a Offset Surface Textured Journal Bearing

ABSTRACT The primary objective of this research is to study the bearing performance characteristics having different operating conditions. Experimental investigations are carried out on phosphorous bronze hydrodynamic journal bearing with a surface textured at different loading conditions with multi-grade oil as lubricating at different constant rotational speeds (i.e.,1000 to 4000 rpm) for studying the dynamic characteristics. Results reported that oil film pressure and temperature distribution profile obtained experimentally are in good agreement with the theoretical results. Fuzzy-based Taguchi optimisation has been used in this experimental analysis to predict the optimal input parameters, which result in the optimal combination of temperature and reported that the oil film pressure and temperature distribution profile obtained experimentally are in good agreement the theoretical results. This analysis also attempts to describe the application of the fuzzy logic-based Taguchi analysis to optimise the performance parameters.

The sealing of fixed joints

The article is devoted to the study of the seals of fixed joints. The mechanisms of leakage occurrence at different values of roughness and pressure on the joint surfaces are considered taking into account the real contact area. The estimation of deformations and forces in the contact zone was carried out taking into account the profile of surfaces, limit deviations of shape and location of surfaces, as well as physical and mechanical properties of materials of the sealed joint. The obtained theoretical dependence allows to evaluate the tightness of the fixed connection depending on the main parameters of the seal roughness, manufacturing errors and the location of the interfaces, their physical and mechanical properties and characteristics of sealed liquid, including differential pressure and the nominal pressure in the joint. A new law of probability density distribution is proposed, which more adequately describes the considered boundary conditions of the experiment. Relations for the probabilistic evaluation of the contact area and surface deformations taking into account the characteristics of the material are obtained. Simple engineering relations for the estimation of probability of contact of the projections and offset surfaces for the two profiles are obtained. The simulation of the process of leakage depending on these conditions is carried out. It is shown that the complete tightness of rigid joints is insufficient even with low roughness provided by the finishing methods of processing and in the absence of deviations in the shape of the joint surfaces. The comparison of experimental data with the obtained theoretical results showed that the proposed engineering dependences provide a good match with the experimental data.

Oblique Right‐Lateral Faulting Along the Northern Margin of the Ili Basin in the Northern Tian Shan, Northwest China

AbstractThe quantification of the geometry and kinematics of the boundary faults of an active orogenic belt is key to exploring its deformation pattern. The Kashihe fault (KSHF), which bounds the northern margin of the Ili Basin and is more than 400 km long, is a WNW‐striking fault in the northern Tian Shan. Although the KSHF displays active faulting and is the seismogenic fault of the 1812 Ms 8 Nilka earthquake, the quantitative slip rate and its role in regional strain accommodation have not been reported. Here, we quantify the late Quaternary activity of the KSHF based on high‐resolution remote sensing image interpretations and detailed field investigations. Six field observation sites along a ~270 km active fault in the Chinese Tian Shan indicate that the KSHF is characterized by dextral strike‐slip faulting and N‐S thrusting. Based on the surveying of offset geomorphic surfaces with an unmanned drone and the dating of late Quaternary sediments using optically stimulated luminescence (OSL) and radiocarbon methods, we estimated a late Quaternary right‐lateral strike‐slip rate across the fault of 2.39 ± 0.55 mm/yr and a shortening rate of 0.62 ± 0.30 mm/yr. Analysis of field investigations and fault section exposing indicated that a young paleoearthquake event occurred on the western segment of the KSHF. Active tectonics around the Ili Basin imply that the rigid Ili block extrudes westward along its boundary strike‐slip faults and rotates counterclockwise. We propose that this block rotation driven by dextral faulting accommodates regional N‐S convergence and sinistral shearing.

Late Pleistocene slip rate of the Chaman fault based on 10Be exposure dating of offset geomorphic surfaces near Kabul, Afghanistan

We report the first late Pleistocene left-lateral slip rate of the northern Chaman fault in Afghanistan based on 10Be dating of offset alluvial fan surfaces. Field mapping complemented with the analysis of CORONA and ALOS PRISM satellite images and shaded relief and topographic data allows us to identify piercing points across the fault zone at two displaced alluvial fans (Ghat Bandakul and Azadkhail) at Wardak Province south of Kabul. We constrain the ages of offset alluvial fans based on 10Be terrestrial cosmogenic nuclides (TCN) dating of well-embedded quartzite sandstone boulders and cobbles. The Ghat Bandakul alluvial fans are offset at 165 ± 15 m and 235 ± 24 m, for which 10Be TCN model ages average 46.9 ± 3.5 kyr and 66.6 ± 4.9 kyr, respectively, yielding a slip rate of 3.5 ± 0.4 mm/yr. At the Azadkhail site, we identified two offset terraces. Based on 500 ± 50 m offset of a geomorphic surface with 111.2 ± 10.5 kyr age, we obtain a slip rate of 4.5 ± 0.6 mm/yr. Combining the 800 ± 70 m left-lateral offset matching with an average age of 176.8 ± 17.1 kyr for the fan yields the same geomorphic slip rate of 4.5 ± 0.6 mm/yr. These slip rates are considerably lower than geologically, geomorphologically, and geodetically defined slip rates previously obtained along the southern Chaman fault. These results show a clear trend of decreasing fault slip rate with increasing distance to the northeast. The observations suggest that the southern portion of the fault accommodates most of the fault-parallel component of relative motion between the Indian and Eurasian plates. The lower slip rate of the Chaman fault in Afghanistan may be explained by the transfer of the left-lateral shear onto the Gardez and Mokur faults that splay northward from the Chaman fault.

The Sizing of Plate-Fin Exchangers to Fixed Dimensions Within a Volume Design Region

This paper shows the development of a design approach for plate and fin heat exchangers to meet fixed dimensions. This approach adopts the concept of volume design region that establishes the limits within which the physical dimensions (length, width and height) of a specific design problem can be set. The design region is determined by minimum and maximum dimensions. The heat exchanger volume is dictated by the problem specifications and the type of secondary surface used on each of the fluids. High density surfaces tend to produce small volumes, while the opposite applies to low density surfaces. In principle, if the heat transfer and friction factor correlations for secondary surfaces are expressed as a function of the geometrical parameters that define the fin density, then it possible to fix the surface density boundaries that give the smallest and largest exchanger volume. The design methodology presented in this work enables to include exchanger dimensions as a design objective along with the heat load and the pressure drop. To achieve these objectives, surface design is a central strategy. In this work, triangular, louvered, rectangular and offset surfaces are used to demonstrate the methodology.

Quantitative Constraints on Faulting and Fault Slip Rates in the Northern Main Ethiopian Rift

AbstractThe Boset magmatic segment (BMS) of the northern Main Ethiopian Rift (MER) is an ideal natural laboratory to investigate the kinematics, interaction, and rates of activity within a fault network in a magma‐rich rift. In this paper we take advantage of the availability of (1) high‐resolution remote sensing data (LiDAR, ASTER); (2) absolute age chronology on offset reference surfaces; and (3) well‐exposed active normal fault arrays to place new constraints on rift kinematics and strain distribution, and to quantify the architecture and fault slip rates at different temporal scales within a magmatic segment. We found that the rift border faults strike approximately NE, while the younger faults in the rift segments strike NNE. Analyses of geometric rift parameters show that the axial active part of the rift is transtensional with an increase of the shear component northward. The fault displacement analyses and displacement:length ratios increase toward the segment tips, suggesting a significant contribution of fault growth by linkage. In contrast, magmatism is focused on the segment center and localized to a narrow zone. Estimated fault slip rates vary, with rates of up to ~0.37 mm/year in ~0.3 Ma old rift floor deposits, whereas higher rates of up to ~4.4 mm/year are observed for faults cutting through ~6 Ka lavas. The difference in slip rates indicates short‐term variability or a very active recent episode compared to long‐term low average slip rates.

Surface roughness control of root analogue dental implants fabricated using selective laser melting

Selective laser melting (SLM) is a promising technology for use in “immediate implantation” to quickly fabricate customized dental implants. However, the implant surface produced using SLM has a high and inconsistent surface roughness, which greatly affects early cell behaviors and osseointegration. In this work, samples were produced with different border process parameters and inclination angles. The surface roughness and morphology of the side surfaces were measured and studied. The results indicate that a large offset value increases surface roughness due to an insufficient energy input, while a small offset increases surface roughness due to an intensified Marangoni convection. Different inclination angles affect surface roughness due to stair effects and the heat-affected zone. Based on the above results, a dental implant was fabricated using gradient processing. Compared with the implant fabricated with a single parameter process, the implant processed with gradient parameters had a low and consistent surface roughness.

Stable Rate of Slip Along the Karakax Section of the Altyn Tagh Fault from Observation of Interglacial and Postglacial Offset Morphology and Surface Dating

AbstractDigital elevation maps obtained using TanDEM‐X and Pleiades data combined with newly obtained surface age estimates using cosmogenic radionuclide (CRN) and optically simulated luminescence (OSL) methods are used to quantify the slip rate along the western section of the Altyn Tagh Fault in southern Xinjiang. The reconstruction of the conical shape of massive alluvial fans inferred to be from the Eemian (115±7 ka) from CRN dating shows consistent left‐lateral offsets of 300±20 m, yielding a slip rate of 2.6±0.3 mm/year. Successive episodes of incision have left cut terraces inset in wide canyons, 10–25 m below the fans' surface. The incision was followed by the deposition of a broad terrace of early Holocene age, which is reincised by modern stream channels. Near the village of Shanxili, a 200 m wide valley is partially dammed by a shutter ridge displaced by the fault. A fill terrace deposited upstream from the ridge has an OSL age of 8.8±0.6 ka. The 23±2 m offset of the riser incising the terrace indicates a minimum postdepositional movement on the fault, yielding a Holocene rate of 2.6±0.5 mm/year, consistent with the 115‐ka average slip rate. Scarp degradation analysis using mass diffusion reveals a nonlinear relationship between fault displacement and degradation coefficient along the progressively exposed fault scarp, a pattern suggesting either seismic clustering or variable diffusion rate since the Eemian. Together with the Gozha Co‐Longmu Co fault to the south, the Karakax section of the Altyn Tagh Fault contributes to the eastward movement of the western corner of Tibet.

Local Modified Becke-Johnson Exchange-Correlation Potential for Interfaces, Surfaces, and Two-Dimensional Materials

The modified Becke-Johnson meta-GGA potential of density functional theory has been shown to be the best exchange-correlation potential to determine band gaps of crystalline solids. However, it cannot be consistently used for the electronic structure of nonperiodic or nanostructured systems. We propose an extension of this potential that enables its use to study heterogeneous, finite, and low-dimensional systems. This is achieved by using a coordinate-dependent expression for the parameter c that weights the Becke-Russel exchange, in contrast to the original global formulation, where c is just a fitted number. Our potential takes advantage of the excellent description of band gaps provided by the modified Becke-Johnson potential and preserves its modest computational effort. Furthermore, it yields with one single calculation band diagrams and band offsets of heterostructures and surfaces. We exemplify the usefulness and efficiency of our local meta-GGA potential by testing it for a series of interfaces (Si/SiO2, AlAs/GaAs, AlP/GaP, and GaP/Si), a Si surface, and boron nitride monolayer.

Multi-material topology optimization of lattice structures using geometry projection

This work presents a computational method for the design of architected truss lattice materials where each strut can be made of one of a set of available materials. We design the lattices to extremize effective properties. As customary in topology optimization, we design a periodic unit cell of the lattice and obtain the effective properties via numerical homogenization. Each bar is represented as a cylindrical offset surface of a medial axis parameterized by the positions of the endpoints of the medial axis. These parameters are smoothly mapped onto a continuous density field for the primal and sensitivity analysis via the geometry projection method. A size variable per material is ascribed to each bar and penalized as in density-based topology optimization to facilitate the entire removal of bars from the design. During the optimization, we allow bars to be made of a mixture of the available materials. However, to ensure each bar is either exclusively made of one material or removed altogether from the optimal design, we impose optimization constraints that ensure each size variable is 0 or 1, and that at most one material size variable is 1. The proposed material interpolation scheme readily accommodates any number of materials. To obtain lattices with desired material symmetries, we design only a reference region of the unit cell and reflect its geometry projection with respect to the appropriate planes of symmetry. Also, to ensure bars remain whole upon reflection inside the unit cell or with respect to the periodic boundaries, we impose a no-cut constraint on the bars. We demonstrate the efficacy of our method via numerical examples of bulk and shear moduli maximization and Poisson’s ratio minimization for two- and three-material lattices with cubic symmetry.

Ансамбль нейромереж GRNN на підставі зміщених поверхонь відгуку для задач електронної комерції

Розв'язок задач електронної комерції, які здебільшого характеризуються нелінійними поверхнями відгуку, є важливим завданням. Застосування сучасних засобів обчислювального інтелекту не завжди є доречним зважаючи на складність реалізації процедур навчання і налагодження. Неітеративні засоби та нейронні мережі без навчання також не забезпечують задовільної точності результату. З огляду на це у роботі описано новий ансамбль на підставі нейронних мереж узагальненої регресії. Основна ідея розробленого ансамблю полягає в лінеаризації поверхні відгуку, що задається даними наявної вибірки. Для цього отримана за допомогою мережі GRNN поверхня подається на вхід лінійної нейроподібної структури. Така комбінація забезпечує підвищення точності роботи ансамблю під час розв'язання поставленої задачі. Описаний ансамбль застосовано для розв'язання задачі прогнозування ціни на вживані автомобілі. Експериментальним способом підібрано оптимальні параметри його роботи. Шляхом порівняння із відомими методами встановлено найвищу точність його роботи. Результати експериментальних досліджень порівняно з теоретичними оцінками на підставі висновків теореми Кондорсе про журі присяжних. Розроблений ансамбль нейронних мереж узагальненої регресії на підставі зміщення поверхонь відгуку та з додатковим використанням нейроподібних структур Моделі послідовних геометричних перетворень варто застосовувати під час розв'язання різноманітних задач електронної комерції підвищеної точності.

Approximation of generalized offset surfaces by bicubic splines

We present an approximation problem of surfaces of a generalized offset surface with offset variable distances and directions. Such approximating surface fits some given data points and minimizes a Sobolev’s semi-norm of order 3. The study of the new results, from a mathematical point of view, carefully establishing the proof of the convergence between the generalized offset surface and its approximating spline in an adequate parametric bicubic spline space. Moreover, the approximating spline function is computed and an estimation of the relative error is introduced. Finally, some numerical and graphic examples are shown in order to prove the useful and the effectiveness of our method.

Geometric deviation reduction method for interpolated toolpath in five-axis flank milling of the S-shaped test piece

Geometric deviation, defined as the distance between the designed surface and the machined surface, is an important component of machining errors in five-axis flank milling of the S-shaped test piece. Since the interpolated toolpath in practical machining process is the approximation of the theoretical toolpath, the geometric deviation caused by the interpolated toolpath appears. To overcome this problem, a novel geometric deviation reduction method is suggested in this study. First, the features of the S-shaped test piece are analyzed. Second, the theoretical toolpath is generated according to the designed surface and the cutter location data is obtained by discretizing the theoretical toolpath. The linear interpolation of the cutter location data is carried out to obtain the interpolated toolpath. Then, the geometric deviation is modeled by calculating the Hausdorff distance between the tool axis trajectory surface based on the interpolated toolpath and the offset surface of the designed surface. Finally, the geometric deviation is reduced by optimizing the cutter location data without inserting more cutter location points. The machining experiment is conducted to verify the effectiveness of the proposed method. The experimental results agree with the simulation results, and both of them indicate the geometric deviation on the machined surface reduces after optimization.