Non-uniform Rational Basis Spline Surface Research Articles

Spline Search for Slip Surfaces in 3D Slopes

A novel method involving the transformation of spline surfaces is introduced to search for the critical slip surface in a three-dimensional (3D) slope, which corresponds to the minimum limit equilibrium method factor of safety for overall slope stability. A slipping surface in a slope can be represented as the intersection of any continuous geometrical entity with the slope topography, over which the mass of sliding soil is discretized to solve for the factor of safety satisfying given equilibrium conditions. Traditionally, many researchers have searched for a critical ellipsoidal or spherical surface, or surfaces formed using other simple shapes. However, the critical slip surface in complicated cases, for example, a landslide, is seldom purely ellipsoidal or spherical, which leads to overestimations of the true factor of safety in a slope. To provide greater flexibility for transforming the shape of the slip surfaces during a global search, the geometry representing the slip surface is assumed to be in the form of a nonuniform rational basis spline (NURBS) surface in this paper. The proposed method involves varying the parameters of a parametric exponential function, which spawns control points within its domain to create NURBS surfaces. The parameters in the exponential function are varied to transform the slipping surface using a metaheuristic search algorithm, such as particle swarm optimization. A major advantage of the proposed method is that the final spline surface in the search is formulated such that it can then be locally optimized using surface altering optimization methods by adjusting the locations of its control points.

A Quasi-Optimal Shape Design Method for Electromagnetic Scatterers Based on NURBS Surfaces and Filter-Enhanced GWO

This paper presents a quasi-optimal and efficient method for designing the shape of electromagnetic (EM) scatterers. Optimizing the EM scatterer shape to achieve satisfactory scattering properties is a crucial, yet challenging problem due to the complex geometry involved and the high non-linearity in the optimization. In this paper, the geometric complexity is to be handled by modeling EM scatterer shapes as non-uniform rational basis spline (NURBS) surfaces, which enables an easy and intuitive shape control. Using the surfaces’ control points as optimization variables, the shape optimization problem is to be solved with the heuristic optimization strategy, which can effectively handle non-linearity and the associated local optima. Specifically, the recently developed grey wolf optimizer (GWO) is employed. Although GWO can effectively avoid local optima and achieve quasi-optimal results, it requires repeated calls to time-consuming EM simulations. To solve this efficiency issue, a filter-enhanced version of GWO is proposed to reduce the times of calling EM simulations, and an efficient, adaptive remeshing method is proposed to accelerate the numerical analysis process within each EM simulation call. Altogether, they lead to a quasi-optimal and efficient method for optimizing EM scatterers. Its effectiveness has also been validated by several examples.

High-fidelity standard model reconstruction and verification of an airliner based on point clouds.

Computational fluid dynamics (CFD) numerical simulation as the primary research tool is particularly essential for its credibility during the aerodynamic design of aircraft. To further promote CFD verification and validation on the airliner, a high-fidelity model reconstruction of the airliner is fundamental. Based on this, we put forward a novel framework, to our best knowledge, to reconstruct a high-fidelity standard model for an airliner efficiently, and the feasibility and accuracy of these reconstructed models are accessed by the CFD simulation-based validation method. First and foremost, a laser scanner was placed at each station around the airliner to scan and acquire multiview point clouds. Afterwards, the truncated least-squares-based algorithm was adopted to register these point clouds entirely. Additionally, we fitted the nonuniform rational basis spline surface based on the least-squares progressive and iterative approximation algorithm. Finally, CFD simulation results were compared and analyzed with the aerodynamic data obtained by the aircraft manufacturer under the same Mach number of the uniform model. It turns out that the coincidence between them is high, and the changing trend is basically consistent. Hence, this method is highly feasible and can establish a high-fidelity standard model of an airliner with unified high and low speeds so that its appearance, test data, and research results can be adopted as the standard data for CFD verification and validation.

Development of scalable lymphatic system in the 4D XCAT phantom: Application to quantitative evaluation of lymphoma PET segmentations.

Digital anthropomorphic phantoms, such as the 4D extended cardiac-torso (XCAT) phantom, are actively used to develop, optimize, and evaluate a variety of imaging applications, allowing for realistic patient modeling and knowledge of ground truth. The XCAT phantom defines the activity and attenuation for a simulated patient, which includes a complete set of organs, muscle, bone, and soft tissue, while also accounting for cardiac and respiratory motion. However, the XCAT phantom does not currently include the lymphatic system, critical for evaluating medical imaging tasks such as sentinel node detection, node density measurement, and radiation dosimetry. In this study, we aimed to develop a scalable lymphatic system in the XCAT phantom, to facilitate improved research of the lymphatic system in medical imaging. Using this scalable lymphatic system, we modeled the lymph node conglomerate pathology that is characteristically observed in primary mediastinal B-cell lymphoma (PMBCL). As an extended application, we evaluated positron emission tomography (PET) image quantification of metabolic tumor volume (MTV) and total lesion glycolysis (TLG) of these simulated lymphomas, though the phantoms may be applied to other imaging modalities and study design paradigms (e.g., image quality, detection). A template model for the lymphatic system was developed based on anatomical data from the Visible Human Project of the National Library of Medicine. The segmented nodes and vessels were fit with non-uniform rational basis spline surfaces, and multichannel large deformation diffeomorphic metric mapping was used to propagate the template to different XCAT anatomies. To model conglomerates observed in PMBCL, lymph nodes were enlarged, converged within the mediastinum, and tracer concentration was increased. We used the phantoms as inputs to a PET simulation tool, which generated images using ordered subsets expectation maximization reconstruction with 2-8mm Gaussian filters. Fixed thresholding (FT) and gradient segmentation were used to determine MTV and TLG. Percent bias (%Bias) and coefficient of variation (COV) were computed as measures of accuracy and precision, respectively, for each MTV and TLG measurement. Using the methodology described above, we introduced a scalable lymphatic system in the XCAT phantom, which allows for the radioactivity and attenuation ground truth to be generated in 116±2.5s using a 2.3GHz processor. Within the Rhinoceros interface, lymph node anatomy and function were modified to create a cohort of 10 phantoms with lymph node conglomerates. Using the lymphoma phantoms to evaluate PET quantification of MTV, mean %Bias values were -9.3%, -41.3%, and 20.9%, while COV values were 4.08%, 7.6%, and 3.4% using 25% FT, 40% FT, and gradient segmentations, respectively. Comparatively for TLG, mean %Bias values were -27.4%, -45.8%, and -16.0%, while COV values were 1.9%, 5.7%, and 1.4%, for the 25% FT, 40% FT, and gradient segmentations, respectively. In this work, we upgraded the XCAT phantom to include a lymphatic system, comprised of a network of 276 scalable lymph nodes and corresponding vessels. As an application, we created a cohort of phantoms with lymph node conglomerates to evaluate lymphoma quantification in PET imaging, which highlights an important application of this work.

Optimal three-dimensional design of functionally graded parts for additive manufacturing using Tamura–Tomota–Ozawa model

Although some conventional manufacturing technologies are capable of producing functionally graded materials, only a few additive manufacturing processes are able to build functionally graded materials with complex distribution of material composition. To exploit this unique advantage, we have developed a new methodology capable of optimization of material distribution for three-dimensional parts for any given conditions. Representation of material distribution was done through a new technique by extending the nonuniform rational basis spline surfaces to four-dimensional space. Mori–Tanaka, Levin, and Tamura–Tomota–Ozawa models were employed for the estimation of effective material properties of functionally graded structures. Subroutines were developed in a commercial finite element software to enable the analysis of parts made from functionally graded material. A constrained particle swarm optimization method was selected and implemented to optimize the material composition distribution taking into account the additive manufacturing limitations. As a case study, the material distribution optimization of a functionally graded femur bone plate under thermomechanical loading was considered. The objective was to maximize the safety factor; i.e. the ratio of local yield strength of the functionally graded plate over the von Mises stress. The results showed significant improvement compared to nonoptimal part and demonstrated the efficacy of the proposed methodology.

A novel NURBS surface approach to statistically monitor manufacturing processes with point cloud data

As sensor and measurement technologies advance, there is a continual need to adapt and develop new Statistical Process Control (SPC) techniques to effectively and efficiently take advantage of these new datasets. Currently high-density noncontact measurement technologies, such as 3D laser scanners, are being implemented in industry to rapidly collect point clouds consisting of millions of data points to represent a manufactured parts' surface. For their potential to be realized, SPC methods capable of handling these datasets need to be developed. This paper presents an approach for performing SPC using high-density point clouds. The proposed approach is based on transforming the high-dimensional point clouds into Non-Uniform Rational Basis Spline (NURBS) surfaces. The control parameters for these NURBS surfaces are then monitored using a surface monitoring technique. In this paper point clouds are simulated to determine the performance of the proposed approach under varying fault scenarios.

A Single-Image Super-Resolution Method Based on Progressive-Iterative Approximation

In this paper, a novel single image super-resolution (SR) method based on progressive-iterative approximation is proposed. To preserve textures and clear edges, the image SR reconstruction is treated as an image progressive-iterative fitting procedure and achieved by iterative interpolation. Due to different features in different regions, we first employ the nonsubsampled contourlet transform (NSCT) to divide the image into smooth regions, texture regions, and edges. Then, a hybrid interpolation scheme based on curves and surfaces is proposed, which differs from the traditional surface interpolation methods. Specifically, smooth regions are interpolated by the non-uniform rational basis spline (NURBS) surface geometric iteration. To retain textures, control points are increased, and the progressive-iterative approximation of the NURBS surface is employed to interpolate the texture regions. By considering edges in an image as curve segments that are connected by pixels with dramatic changes, we use NURBS curve progressive-iterative approximation to interpolate the edges, which sharpens the edges and can maintain the image edge structure without jaggy and block artifacts. The experimental results demonstrate that the proposed method significantly outperforms the state-of-the-art methods in terms of both subjective and objective measures.

Adapted Delaunay triangulation method for free-form surface generation from random point clouds for stochastic optimization applications

Free-form surfaces are defined with NURBS (non-uniform rational basis spline) for most computer-aided engineering (CAE) applications. The NURBS method requires the definition of parameters such as weights, knot vectors and degree of the curves which make the configuration of the surface computationally expensive and complex. When the control points are randomly spaced in the point cloud and the topology of the desired surface is unknown, surface configuration with NURBS method becomes a challenging task. Optimization attempts for such surfaces create enormous amounts of computing data when coupled with physics solvers such as finite element analysis (FEA) tools and computational fluid dynamics (CFD) tools. In this paper, an adapted Delaunay triangulation (ADT) method for surface generation from the random points cloud is proposed and compared with widely used implicit functions based NURBS fitting method. The surface generated from ADT method can be simultaneously used with stochastic optimization algorithms (SOA) and CFD applications to search for the optimal results with minimum computational costs. It was observed while comparing ADT with NURBS-based geometry configuration that the computation time can be reduced by 3 folds. The corresponding deviation between both geometry configuration methods has been observed as low as 5% for all optimisation scenarios during the comparison. In addition, ADT method can provide light weight CFD approach as any instance of design iteration has at least half storage footprint as compared to corresponding NURBS surface. The proposed approach provides novel methodology towards establishing light weight CFD geometry, absence of which currently isolates methodologies for optimization and CFD analysis.

Accuracy assessment of mine walls\u2019 surface models derived from terrestrial laser scanning

The monitoring of highwall slopes at open-pit mines is an important task to ensure safe mining. For this reason, several techniques such as total station, radar, terrestrial Light Detection and Ranging (LIDAR) can be employed for surface measurement. The objective of this study is to investigate mesh algorithms, which can be used to interpolate 3D models of pit walls. Experiments were carried out at Coc Sau open-pit mine at Quang Ninh province of Vietnam, and at experimental mine of Akademia Górniczo-Hutnicza University of Science and Technology in Cracow, Poland. First, 3D point cloud data for the study area was acquired by using terrestrial LIDAR, then was used to generate mesh surfaces using three algorithms—Delaunay 2.5D XY Plane, Delaunay 2.5D Best Fitting Plane, and Mesh from Points. After that, the results were rectified and optimized. Subsequently, the optimized meshes were used for generation of non-uniform rational basis spline (NURBS) surfaces. Then, the NURBS surface accuracy was assessed. The results showed that the average distance between surface and point cloud was within range of 5.6–5.8 mm with deviation of 6.2–6.8 mm, depending on the used mesh. Additionally, the quality of surfaces depends on the quality of input data set and the algorithm used to generate mesh network, and the accuracy of computed NURBS surfaces fitting into pointset was 4–5 times lower than that of optimized mesh fitting. However, the accuracy of the final product allows determining displacements on the level of centimeters.

NURBS plasticity: non-associated plastic flow

This paper extends the non-uniform rational basis spline (NURBS) plasticity framework of Coombs et al. (2016) and Coombs and Ghaffari Motlagh (2017) to include non-associated plastic flow. The NURBS plasticity approach allows any smooth isotropic yield envelope to be represented by a NURBS surface whilst the numerical algorithm (and code) remains unchanged. This paper provides the full theoretical and algorithmic basis of the non-associated NURBS plasticity approach and demonstrates the predictive capability of the plasticity framework using both small and large deformation problems. Wherever possible errors associated with the constitutive formulation are specified analytically and if not numerical analyses provide this information. The rate equations within the plasticity framework are integrated using an efficient and stable implicit stress update algorithm which allows for the derivation of the algorithmic consistent tangent which ensures optimum convergence of the global out of balance force residual when used in boundary value simulations.The important extension provided by this paper is that the evolution of plastic strain is decoupled from the yield surface normal. This allows the framework to model more realistic material behaviour, particularly in the case of frictional plasticity models where an associated flow rule is known to significantly overestimate volumetric dilation leading to spurious results. This paper therefore opens the door for the NURBS plasticity formulation to be used for a far wider class of material behaviour than is currently possible.

NURBS plasticity: Yield surface evolution and implicit stress integration for isotropic hardening

This paper extends the non-uniform rational basis spline (NURBS) plasticity framework of Coombs et al. (2016) to include isotropic hardening of the yield surfaces. The approach allows any smooth isotropic yield envelope to be represented by a NURBS surface. The key extension provided by this paper is that the yield surface can expand or contract through the movement of control points linked to the level of inelastic straining experienced by the material. The model is integrated using a fully implicit backward Euler algorithm that constrains the return path to the yield surface and allows the derivation of the algorithmic consistent tangent to ensure optimum convergence of the global equilibrium equations. This provides a powerful framework for elasto-plastic constitutive models where, unlike the majority of models presented in the literature, the underlying numerical algorithm (and implemented code) remains unchanged for different yield surfaces. The performance of the algorithm is demonstrated, and validated, using both material point and boundary values simulations including plane stress, plane strain and three-dimensional examples for different yield criteria.

Imaging freeform optical systems designed with NURBS surfaces

The designs of two imaging freeform systems using nonuniform rational basis-spline (NURBS) optical surfaces are described. The first system, a 10 deg×9 degf/2 three-mirror anastigmat has four times higher spatial resolution over the image plane compared with the equivalent conventional rotational aspheric design, and 2.5 times higher resolution compared with a 10th-order XY polynomial freeform design. The mirrors for the NURBS freeform design have more than twice the asphericity than the conventional rotational and XY polynomial designs. In the second system, a Ritchey–Chretien telescope followed by a two-mirror NURBS freeform corrector is compared to a four-mirror Korsch telescope, for imaging to a visible-infrared imaging spectrometer. The freeform corrector design had 70% smaller spot sizes over the field and eliminated the large tertiary required in Korsch type design. Both of these NURBS freeform designs are possible due to a custom optical design code for fast accurate NURBS optimization, which now has parallel raytracing for thousands of NURBS grid points.

Radiation heat transfer model using Monte Carlo ray tracing method on hierarchical ortho-Cartesian meshes and non-uniform rational basis spline surfaces for description of boundaries

Abstract The paper deals with a solution of radiation heat transfer problems in enclosures filled with nonparticipating medium using ray tracing on hierarchical ortho-Cartesian meshes. The idea behind the approach is that radiative heat transfer problems can be solved on much coarser grids than their counterparts from computational fluid dynamics (CFD). The resulting code is designed as an add-on to OpenFOAM, an open-source CFD program. Ortho-Cartesian mesh involving boundary elements is created based upon CFD mesh. Parametric non-uniform rational basis spline (NURBS) surfaces are used to define boundaries of the enclosure, allowing for dealing with domains of complex shapes. Algorithm for determining random, uniformly distributed locations of rays leaving NURBS surfaces is described. The paper presents results of test cases assuming gray diffusive walls. In the current version of the model the radiation is not absorbed within gases. However, the ultimate aim of the work is to upgrade the functionality of the model, to problems in absorbing, emitting and scattering medium projecting iteratively the results of radiative analysis on CFD mesh and CFD solution on radiative mesh.

NURBS approximation to the flank-milled surface swept by a cylindrical NC tool

In this paper, a method to approximate the flank-milled surface swept by a cylindrical cutter with a non-uniform rational basis spline (NURBS) surface is presented. The swept surface produced by the moving tool can be calculated as a collection of points organized as a series of grazing curves along the surface. The generated NURBS surface closely matches the grazing surface. The deviation between this surface and the grazing surface is calculated and is controlled by increasing the number of control points used to represent the surface.

Accurate spiral tool path generation of ultraprecision three-axis turning for non-zero rake angle using symbolic computation

An accurate spiral tool path generation method of ultraprecision three-axis turning free form surface is proposed based on symbolic computation in this paper. Many analytic optical free form surfaces often need to be machined to submicron in form error, such as optical nonaxisymmetric aspheric surfaces, but current mainstream CAM systems usually use nonuniform rational basis spline (NURBS) to describe the designed surface and generate tool path. If we want to use these systems, the analytical optical surfaces must be approximated using NURBS surfaces, but it will introduce approximation error and may be difficult to achieve the approximation error less than submicron. More importantly, there is no effective tool path generation method for the special three-axis turning machine tool in current mainstream CAM systems. In this context, we propose to calculate the tool path directly from these analytic surfaces by using symbolic math. The proposed method can be used to generate accurate spiral tool paths for zero/negative/positive rake angle in a unified way. Finally, several examples are given to prove its effectiveness.

Predictive modeling of lung motion over the entire respiratory cycle using measured pressure‐volume data, 4DCT images, and finite‐element analysis

Predicting complex patterns of respiration can benefit the management of the respiratory motion for radiation therapy of lung cancer. The purpose of the present work was to develop a patient-specific, physiologically relevant respiratory motion model which is capable of predicting lung tumor motion over a complete normal breathing cycle. Currently employed techniques for generating the lung geometry from four-dimensional computed tomography data tend to lose details of mesh topology due to excessive surface smoothening. Some of the existing models apply displacement boundary conditions instead of the intrapleural pressure as the actual motive force for respiration, while others ignore the nonlinearity of lung tissues or the mechanics of pleural sliding. An intermediate nonuniform rational basis spline surface representation is used to avoid multiple geometric smoothing procedures used in the computational mesh preparation. Measured chest pressure-volume relationships are used to simulate pressure loading on the surface of the model for a given lung volume, as in actual breathing. A hyperelastic model, developed from experimental observations, has been used to model the lung tissue material. Pleural sliding on the inside of the ribcage has also been considered. The finite-element model has been validated using landmarks from four patient CT data sets over 34 breathing phases. The average differences of end-inspiration in position between the landmarks and those predicted by the model are observed to be 0.450 +/- 0.330 cm for Patient P1, 0.387 +/- 0.169 cm for Patient P2, 0.319 +/- 0.186 cm for Patient P3, and 0.204 +/- 0.102 cm for Patient P4 in the magnitude of error vector, respectively. The average errors of prediction at landmarks over multiple breathing phases in superior-inferior direction are less than 3 mm. The prediction capability of pressure-volume curve driven nonlinear finite-element model is consistent over the entire breathing cycle. The biomechanical parameters in the model are physiologically measurable, so that the results can be extended to other patients and additional neighboring organs affected by respiratory motion.