Periodic B-spline Research Articles

A unified optimization design method for multi-stage non-circular gear transmission based on periodic B-spline interpolation

This research introduces a novel and unified optimization design method for multi-stage non-circular gear transmission (MNCGT) to address the challenges in designing MNCGT for complex motion requirements. The method optimizes non-circular gears comprehensively, reducing design and manufacturing difficulties while ensuring the realization of specified transmission requirements. A unified parameterization method, grounded on periodic B-spline interpolation, is introduced to establish the transmission function of non-circular gears and map it to a finite unified variable space. This innovative approach effectively reduces the difficulty and constraints of MNCGT optimization design. The proposed method takes into account crucial factors such as non-circularity, smoothness, processing conditions, and contact ratio, which significantly impact the transmission performance and manufacturing feasibility of non-circular gears. The effectiveness and superiority of this method are demonstrated through two practical examples and a real-world application in a planetary gear transplant mechanism, highlighting its potential for solving complex engineering problems.

Stochastic occupancy modeling for spaces with irregular occupancy patterns using adaptive B-Spline-based inhomogeneous Markov Chains

This paper presents a discrete time, discrete state-space in-homogeneous Markov Chains model for stochastic occupancy modeling in spaces with irregular occupancy patterns. The goal of the model is to provide accurate predictions of occupancy numbers, enabling appropriate actions to be taken for HVAC system to maintain optimal indoor environment. The proposed Markov Chain model incorporates time in-homogeneity by coupling the time-varying model parameters using a Periodic B-Spline expansion with adaptive knots, which effectively captures patterns in occupancy activity. This method optimizes the distribution of knots based on specific occupancy characteristics observed in different types of rooms. To evaluate the effectiveness of the proposed method, six months of occupancy data collected from a meeting room are utilized. A comprehensive comparison is conducted between the proposed adaptive B-Spline method and other approaches, including the counting method and uniform B-Spline method. The comparison considers both model accuracy and complexity, using metrics such as the Akaike Information Criterion and Bayesian Information Criterion. Results indicate that the proposed model achieves more accurate predictions with fewer model parameters compared to other methods. These forecasts are particularly useful in optimizing the control of HVAC systems, where accurate predictions of future occupancy numbers are essential.

Periodic implicit representation, design and optimization of porous structures using periodic B-splines

Porous structures are intricate solid materials with numerous small pores, extensively used in fields like medicine, chemical engineering, and aerospace. However, the design of such structures using computer-aided tools is a time-consuming and tedious process. In this study, we propose a novel representation method and design approach for porous units that can be infinitely spliced to form a porous structure. We use periodic B-spline functions to represent periodic or symmetric porous units. Starting from a voxel representation of a porous sample, the discrete distance field is computed. To fit the discrete distance field with a periodic B-spline, we introduce the constrained least squares progressive-iterative approximation algorithm, which results in an implicit porous unit. This unit can be subject to optimization to enhance connectivity and utilized for topology optimization, thereby improving the model’s stiffness while maintaining periodicity or symmetry. The experimental results demonstrate the potential of the designed complex porous units in enhancing the mechanical performance of the model. Consequently, this study has the potential to incorporate remarkable structures derived from artificial design or nature into the design of high-performing models, showing the promise for biomimetic applications.

Camera Path Generation for Triangular Mesh Using Toroidal Patches

Triangular mesh data structures are principal in computer graphics, serving as the foundation for many 3D models. To effectively utilize these 3D models across diverse industries, it is important to understand the model’s overall shape and geometric features thoroughly. In this work, we introduce a novel method for generating camera paths that emphasize the model’s local geometric characteristics. This method uses a toroidal patch-based spatial data structure, approximating the mesh’s faces within a predetermined tolerance ϵ, encapsulating their geometric intricacies. This facilitates the determination of the camera position and gaze path, ensuring the mesh’s key characteristics are captured. During the path construction, we create a bounding cylinder for the mesh, project the mesh’s faces and associated toroidal patches onto the cylinder’s lateral surface, and sequentially select grids of the cylinder containing the highest number of toroidal patches as we traverse the lateral surface. The centers of the selected grids are used as control points for a periodic B-spline curve, which serves as our foundational path. After initial curve generation, we generated camera position and gaze path from the curve by multiplying factors to ensure a uniform camera amplitude. We applied our method to ten triangular mesh models, demonstrating its effectiveness and adaptability across various mesh configurations.

Fourier analysis of membrane locking and unlocking

Despite four decades of research membrane locking remains an important issue hindering the development of effective beam and shell finite elements. In this article, we utilize Fourier analysis of the complete spectrum of natural vibrations and propose a criterion to identify and evaluate the severity of membrane locking. To demonstrate our approach, we utilize primal and mixed Galerkin formulations applied to a circular Euler–Bernoulli ring discretized using uniform, periodic B-splines. By analytically computing the discrete Fourier operators, we obtain an exact representation of the normalized error across the entire spectrum of eigenvalues. Our investigation addresses key questions related to membrane locking, including mode susceptibility, the influence of polynomial order, and the impact of shell/beam thickness and radius of curvature. Furthermore, we compare the effectiveness of mixed and primal Galerkin methods in mitigating locking. By providing insights into the parameters affecting locking and introducing a criterion to evaluate its severity, this research contributes to the development of improved numerical methods for thin beams and shells.

Symbolic Parametric Representation of the Area and the Second Moments of Area of Periodic B-Spline Cross-Sections

The calculation of moments of area is one of the most fundamental aspects of engineering mechanics for calculating the properties of beams or for the determination of invariants in different kind of geometries. While a variety of shapes, such as circles, rectangles, ellipses, or their combinations, can be described symbolically, such symbolic expressions are missing for freeform cross-sections. In particular, periodic B-spline cross-sections are suitable for an alternative beam cross-section, e.g., for the representation of topology optimization results. In this work, therefore, a symbolic description of the moments of area of various parametric representations of such B-splines is computed. The expressions found are then compared with alternative computational methods and checked for validity. The resulting equations show a simple method that can be used for the fast conceptual computation of such moments of area of periodic B-splines.

Vectorized instance segmentation using periodic B-splines based on cascade architecture

Instance segmentation is an essential part of image semantic understanding. In this paper we propose a novel cascade framework for instance segmentation. Unlike some existing methods that only output contour discrete coordinates, our approach obtains a vectorized representation of contours using periodic B-splines. In order to make better use of geometry and appearance information, we consider the global and local features of objects and introduce two types of graph structures, the star graph and circular graph, for feature extraction. Thereby, we develop a neural network, termed the mix network, to better exploit extracted features. Specifically, we first regress the spline control points to an object boundary via the mix network, then perform spline sampling to obtain the initial predictions of contours, and finally deform the predicted contours to the real contours of the objects. In addition, we add a regularization to further constrain the fairness of contour splines. Experiments show that our approach achieves 34.6% in mask mAP, Mean Average Precision, with a ResNet-101-FPN-DCN backbone on the challenging COCO benchmark, which significantly improves the performance of contour-based methods.

Open-loop optimal control of a flapping wing using an adjoint Lattice Boltzmann method

We present the usage of an adjoint Lattice Boltzmann Method (LBM) for open-loop control of two-dimensional flapping wing motion. Movement of the wing is parametrised with periodic B-Splines, while the wing interacts with the surrounding flow via an Immersed Boundary (IB) method. Multi-objective optimisation is performed using a gradient optimisation algorithm, for which sensitivities are calculated with an adjoint method. The objectives selected were the mean lift force and mechanical power. To achieve performance suitable for optimisation, we also present an efficient GPU implementation of the LBM and adjoint LBM. The Immersed Boundary approach employed for the LBM is verified against results from the literature, while for the flapping case it is compared with two different Finite Volume Method (FVM) approaches. The obtained Pareto front of optimal designs shows a clear discrepancy between the power consumption and the mean lift force. A significant improvement of the basic wing design is demonstrated, and highlights the applicability of adjoint LBM simulations in complex open-loop control problems.

Understanding narwhal diving behaviour using Hidden Markov Models with dependent state distributions and long range dependence.

Diving behaviour of narwhals is still largely unknown. We use Hidden Markov models (HMMs) to describe the diving behaviour of a narwhal and fit the models to a three-dimensional response vector of maximum dive depth, duration of dives and post-dive surface time of 8,609 dives measured in East Greenland over 83 days, an extraordinarily long and rich data set. Narwhal diving patterns have not been analysed like this before, but in studies of other whale species, response variables have been assumed independent. We extend the existing models to allow for dependence between state distributions, and show that the dependence has an impact on the conclusions drawn about the diving behaviour. We try several HMMs with 2, 3 or 4 states, and with independent and dependent log-normal and gamma distributions, respectively, and different covariates to characterize dive patterns. In particular, diurnal patterns in diving behaviour is inferred, by using periodic B-splines with boundary knots in 0 and 24 hours.

A two-scale stochastic framework for predicting failure strength probability of heterogeneous materials

In the present study, a two-scale stochastic framework has been proposed for predicting the failure strength probability of heterogeneous materials. The analysis at both scales (meso and macro) is performed under plane stress condition. The meso-scale analysis is performed by XFEM whereas the macro-scale analysis is performed by FEM. The heterogeneities (pores and reinforced particles) are considered at meso-scale. The effect of shape, size, clustering and volume fraction of heterogeneities is analyzed at meso-scale. A new scheme is developed for modeling the arbitrary shape heterogeneities using periodic B-splines. An adaptive hanging node mesh refinement technique is employed to reduce the computational cost. Maximum principal stress failure criterion has been implemented for modeling both tensile and compressive behaviors at meso-scale. The volume fraction of pores and reinforcement particles is distributed stochastically to the elements at macro-scale. The average volume fraction of the pores is taken as 8%, 10%, 12% and 14% whereas the average volume fraction of the reinforced particles is kept constant at 20%. The statistical analysis of numerical data is performed through normal and Weibull distribution fits. K-S goodness of fit predicts that the numerical data is better fitted by the normal distribution.

Forecasting wind power – Modeling periodic and non-linear effects under conditional heteroscedasticity

In this article we present an approach that enables joint wind speed and wind power forecasts for a wind park. We combine a multivariate seasonal time varying threshold autoregressive moving average (TVARMA) model with a power threshold generalized autoregressive conditional heteroscedastic (power-TGARCH) model. The modeling framework incorporates diurnal and annual periodicity modeling by periodic B-splines, conditional heteroscedasticity and a complex autoregressive structure with non-linear impacts. In contrast to usually time-consuming estimation approaches as likelihood estimation, we apply a high-dimensional shrinkage technique. We utilize an iteratively re-weighted least absolute shrinkage and selection operator (lasso) technique. It allows for conditional heteroscedasticity, provides fast computing times and guarantees a parsimonious and regularized specification, even though the parameter space may be vast. We are able to show that our approach provides accurate forecasts of wind power at a turbine-specific level for forecasting horizons of up to 48h (short- to medium-term forecasts).

Boundary-to-Marker Evidence-Controlled Segmentation and MDL-Based Contour Inference for Overlapping Nuclei.

This paper presents a novel method for automated morphology delineation and analysis of cell nuclei in histopathology images. Combining the initial segmentation information and concavity measurement, the proposed method first segments clusters of nuclei into individual pieces, avoiding segmentation errors introduced by the scale-constrained Laplacian-of-Gaussian filtering. After that a nuclear boundary-to-marker evidence computing is introduced to delineate individual objects after the refined segmentation process. The obtained evidence set is then modeled by the periodic B-splines with the minimum description length principle, which achieves a practical compromise between the complexity of the nuclear structure and its coverage of the fluorescence signal to avoid the underfitting and overfitting results. The algorithm is computationally efficient and has been tested on the synthetic database as well as 45 real histopathology images. By comparing the proposed method with several state-of-the-art methods, experimental results show the superior recognition performance of our method and indicate the potential applications of analyzing the intrinsic features of nuclei morphology.

基于周期B 样条曲面的环状<italic>N</italic> 边洞<italic>G<sup>n</sup></italic> 连续性过渡

The orbicular <italic>N</italic>-sided hole filling problem is usually introduced by filleting an end-point of a part with large radius. The existing methods based on quadrilateral partition or constrained-optimization can rarely generate high-order continuous blending surfaces under these circumstances. This paper first reparameterizes the boundary of the specified orbicular <italic>N</italic>-sided hole to ensure the compatibility of neighboring cross-boundary derivatives on the connecting points, preserving their <italic>Gⁿ</italic> continuity. Then we compute the control points of the periodic B-spline surface using the sufficient <italic>Gⁿ</italic> continuity condition on the pole and the algorithm of extending parametric surfaces. This method generates single blending surface, which can be converted into standard B-spline surface by adding knots without introducing errors. It only elevates the degree of the boundary by <italic>n</italic>. The construction method is simple and efficient, without iteration nor large-scale matrix solving. It achieves <italic>Gⁿ</italic> continuity under compatible conditions. The blending examples underline its feasibility and practicability.

G n filling orbicular N-sided holes using periodic B-spline surfaces

The orbicular N-sided hole filling problem is usually introduced by filleting an end-point of a part with large radius. The existing methods based on quadrilateral partition or constrained-optimization can rarely generate high-order continuous blending surfaces under these circumstances. This paper first reparameterizes the boundary of the specified orbicular N-sided hole to ensure the compatibility of neighboring cross-boundary derivatives on the connecting points, preserving their Gn continuity. Then we compute the control points of the periodic B-spline surface using the sufficient Gn continuity condition on the pole and the algorithm of extending parametric surfaces. This method generates single blending surface, which can be converted into standard Bspline surface by adding knots without introducing errors. It only elevates the degree of the boundary by n. The construction method is simple and efficient, without iteration nor large-scale matrix solving. It achieves Gn continuity under compatible conditions. The blending examples underline its feasibility and practicability.

The numerical solution of dynamic response of SDOF systems using cubic B-spline polynomial functions

In this paper, we present a new explicit procedure using periodic cubic B-spline interpolation polynomials to solve linear and nonlinear dynamic equation of motion governing single degree of freedom (SDOF) systems. In the proposed approach, a straightforward formulation was derived from the approximation of displacement with B-spline basis in a fluent manner. In this way, there is no need to use a special pre-starting procedure to commence solving the problem. Actually, this method lies in the case of conditionally stable methods. A simple step-by-step algorithm is implemented and presented to calculate dynamic response of SDOF systems. The validity and effectiveness of the proposed method is demonstrated with four examples. The results were compared with those from the numerical methods such as Duhamel integration, Linear Acceleration and also Exact method. The comparison shows that the proposed method is a fast and simple procedure with trivial computational effort and acceptable accuracy exactly like the Linear Acceleration method. But its power point is that its time consumption is notably less than the Linear Acceleration method especially in the nonlinear analysis.

B-spline quasi-interpolation on sparse grids

We propose a periodic B-spline quasi-interpolation for multivariate functions on sparse grids and develop a fast scheme for the evaluation of a linear combination of B-splines on sparse grids. We prove that both of these operations require only O ( n log d − 1 n ) number of multiplications, where n is the number of univariate B-spline basis functions used in each coordinate direction and d is the number of variables of the functions. We also establish the optimal approximation order of the periodic B-spline quasi-interpolation. Numerical examples are presented to confirm the theoretical estimates.

Automated process planning method to machine A B-Spline free-form feature on a mill–turn center

In this paper, we present a methodology for automating the process planning and NC code generation for a widely encountered class of free-form features that can be machined on a 3-axis mill–turn center. The free-form feature family that is considered is that of extruded protrusions whose cross-section is a closed, periodic B-Spline curve. In this methodology, for machining a part with B-Spline protrusion located at the free end, the part is first rough turned to the maximum profile diameter of the B-Spline, followed by rough profile cutting and finish profiling with axially mounted end mill tools. The identification and sequencing of machining volumes is completely automated, as is the generation of actual NC code. The approach supports both convex and non-convex profiles. In the case of non-convex profiles, the process planning algorithm ensures that there is no gouging of the work piece by the tool. The algorithm also identifies when sections of the tool path lie outside the work piece and utilizes rapid traverses in these regions to reduce cutting time. This methodology presents an integrated turn–mill process planning where by making the process fully automated from design with no user intervention making the overall process planning efficient. The algorithm was tested on several examples and test parts using the unmodified NC code obtained from the implementation were run on a Moriseiki mill–turn center. The parts that were produced met the dimensional specifications of the desired part.

Dominant point detecting based non-uniform B-spline approximation for grain contour

Three-dimension reconstruction from serial sections has been used in the last decade to obtain information concerning three-dimensional microstructural geometry. One of the crucial steps of three-dimension reconstruction is getting compact and fairing grain contours. Based on the achievement of closed raw contours of ceramic composite grains by using wavelet and level set, an adaptive method is adopted for the polygonal approximation of the digitized raw contours. Instead of setting a fixed length of support region in advance, the novel method computes the suitable length of support region for each point to find the best estimated curvature. The dominant points are identified as the points with local maximum estimated curvatures. Periodic closed B-spline approximation is used to find the most compact B-spline grain boundary contours within the given tolerance. A flexible distance selection approach is adopted to obtain the common knot vector of serial contours consisting of less knots that contain enough degrees of freedom to guarantee the existence of a B-spline curve interpolating each contour. Finally, a B-spline surface interpolating the serial contours is generated via B-spline surface skinning.

Producing Follower Motions Through Their Digitized Cam Contours

A computing procedure that can be used to generate follower motions with a higher degree of accuracy by using their digitized 2-D cam surfaces is presented. Based on the geometric relationships at the contact point between a planar cam and its follower, the reverse functions that can be solved to find the follower motion of the cam-follower mechanism are established. To approximate the digitized cam surfaces and to create their derivatives required in determining the follower motions, a least squares periodic B-spline approximation is also introduced. In comparison with analytical results obtained from a theoretical design, the procedure is verified for its accuracy and reliability. Furthermore, by using the cam surface data points taken from a CMM, a practical example is given to show the effectiveness and usefulness of the proposed techniques.

B-spline signal processing using harmonic basis functions

This paper deals with the concept of orthogonal transformation and proposes an orthogonal discrete spline transform (DSPLT) for processing and representation of signals in terms of modified spline basis functions. This work reinterprets the B-spline digital filtering techniques proposed by Unser et al. by introducing harmonic spline basis functions. The principle of periodic B-spline interpolation is explored. Based on the orthonormal properties of the eigenvectors of the periodic B-spline interpolation matrix, a complete set of orthonormal modified spline basis functions which constitute the basis set for the DSPLT is developed. A 16-point, not-in-place, radix-2, decimation-in-time fast DSPLT algorithm is presented. An efficient VLSI algorithm for computing the discrete spline transform coefficients of variable length is proposed. The spectral properties of the modified spline basis functions are explored. A multiresolution analysis (MRA) technique using wave-packets in terms of the modified spline basis is also developed. Finally, the present approach of B-spline coefficient extraction followed by interpolative signal reconstruction is compared with previous filtering methods.