Search Results (15)

Transfinite interpolation, originally proposed in the early 1970s as a global interpolation method, was first implemented using Lagrange polynomials and cubic Hermite splines. While initially developed for computer-aided geometric design (CAGD), the method also found application in global finite element analysis. With the advent of isogeometric analysis (IGA), Bernstein–Bézier polynomials have increasingly replaced Lagrange polynomials, particularly in conjunction with tensor product B-splines and non-uniform rational B-splines (NURBSs). Despite its early promise, transfinite interpolation has seen limited adoption in modern CAD/CAE workflows, primarily due to its mathematical complexity—especially when blending polynomials of different degrees. In this context, the present study revisits transfinite interpolation and demonstrates that, in four broad classes, Lagrange polynomials can be systematically replaced by Bernstein polynomials in a one-to-one manner, thus giving the same accuracy. In a fifth class, this replacement yields a robust dual set of basis functions with improved numerical properties. A key advantage of Bernstein polynomials lies in their natural compatibility with weighted formulations, enabling the accurate representation of conic sections and quadrics—scenarios where IGA methods are particularly effective. The proposed methodology is validated through its application to a boundary-value problem governed by the Laplace equation, as well as to the eigenvalue analysis of an acoustic cavity, thereby confirming its feasibility and accuracy. Full article

This paper extends the well-known transfinite interpolation formula, which was developed in the late 1960s by the applied mathematician William Gordon at the premises of General Motors as an extension of the pre-existing Coons interpolation formula. Here, a conjecture is formulated, which claims that the meaning of the involved blending functions can be enhanced, such that it includes any linear independent and complete set of functions, including piecewise-linear, trigonometric functions, Bernstein polynomials, B-splines, and NURBS, among others. In this sense, NURBS-based isogeometric analysis and aspects of T-splines may be considered as special cases. Applications are provided to illustrate the accuracy in the interpolation through the L₂ error norm of closed-formed functions prescribed at the nodal points of the transfinite patch, which represent the solution of partial differential equations under boundary conditions of the Dirichlet type. Full article

This paper is devoted to numerical algorithms based on harmonic transformations with two goals: (1) face boundary formulation by blending techniques based on the known characteristic nodes and (2) some challenging examples of face resembling. The formulation of the face boundary is imperative for face recognition, transformation, and combination. Mapping between the source and target face boundaries with constituent pixels is explored by two approaches: cubic spline interpolation and ordinary differential equation (ODE) using Hermite interpolation. The ODE approach is more flexible and suitable for handling different boundary conditions, such as the clamped and simple support conditions. The intrinsic relations between the cubic spline and ODE methods are explored for different face boundaries, and their combinations are developed. Face combination and resembling are performed by employing blending curves for generating the face boundary, and face images are converted by numerical methods for harmonic models, such as the finite difference method (FDM), the finite element method (FEM) and the finite volume method (FVM) for harmonic models, and the splitting–integrating method (SIM) for the resampling of constituent pixels. For the second goal, the age effects of facial appearance are explored to discover that different ages of face images can be produced by integrating the photos and images of the old and the young. Then, the following challenging task is targeted. Based on the photos and images of parents and their children, can we obtain an integrated image to resemble his/her current image as closely as possible? Amazing examples of face combination and resembling are reported in this paper to give a positive answer. Furthermore, an optimal combination of face images of parents and their children in the least-squares sense is introduced to greatly facilitate face resembling. Face combination and resembling may also be used for plastic surgery, finding missing children, and identifying criminals. The boundary and numerical techniques of face images in this paper can be used not only for pattern recognition but also for face morphing, morphing attack detection (MAD), and computer animation as Sora to greatly enhance further developments in AI. Full article

This paper presents the advanced classes of linear symmetric subdivision schemes for the fitting of data and the creation of geometric shapes. These schemes are derived from the B-spline and Lagrange’s blending functions. The important characteristics of the derived schemes, including continuity, support, and the impact of parameters on the magnitude of the artifact and Gibbs oscillations are discussed. Schemes additionally generalize various subdivision schemes. Linear symmetric subdivision schemes can produce Gibbs oscillations when the initial data is taken from discontinuous functions. Additionally, these schemes may generate unwanted artifacts in the limit curve that do not exist in the original polygon. One solution is to use non-linear schemes, but this approach increases the computational complexity of the scheme. An alternative approach is proposed that involves modifying the linear symmetric schemes by introducing parameters into the linear rules. The suitable values of these parameters reduce or eliminate Gibbs oscillations and artifacts while still using linear symmetric schemes. Our approach provides a balance between reducing or eliminating Gibbs oscillations and artifacts while maintaining computational efficiency. Full article

A class of zipper fractal functions is more versatile than corresponding classes of traditional and fractal interpolants due to a binary vector called a signature. A zipper fractal function constructed through a zipper iterated function system (IFS) allows one to use negative and positive horizontal scalings. In contrast, a fractal function constructed with an IFS uses positive horizontal scalings only. This article introduces some novel classes of continuously differentiable convexity-preserving zipper fractal interpolation curves and surfaces. First, we construct zipper fractal interpolation curves for the given univariate Hermite interpolation data. Then, we generate zipper fractal interpolation surfaces over a rectangular grid without using any additional knots. These surface interpolants converge uniformly to a continuously differentiable bivariate data-generating function. For a given Hermite bivariate dataset and a fixed choice of scaling and shape parameters, one can obtain a wide variety of zipper fractal surfaces by varying signature vectors in both the x direction and y direction. Some numerical illustrations are given to verify the theoretical convexity results. Full article

Recently, biomass has become an increasingly widely used energy resource. The problem with the use of biomass is its variable composition. The most important property that determines the energy content and thus the performance of fuels such as biomass is the heating value (HHV). This paper focuses on selecting the optimal number of input variables using linear regression (LR) and the multivariate adaptive regression splines approach (MARS) to create an artificial neural network model for predicting the heating value of selected biomass. The MARS model selected the input data better than the LR model. The best modeling results were obtained for a network with three input neurons and nine neurons in the hidden layer. This was confirmed by a high correlation coefficient of 0.98. The obtained results show that artificial neural network (ANN) models are effective in predicting the calorific value of woody and field biomass, and can be considered a worthy simulation model for use in selecting biomass feedstocks and their blends for renewable fuel applications. Full article

The emergence of Machine learning (ML) algorithms has shown competency in a variety of fields and are growing in popularity in their application to geospatial science issues. Most recently, and notably, ML algorithms have been applied to flood susceptibility (FS) mapping. Leveraging high-power computing systems and existing ML algorithms with national datasets of Canada, this project has explored methods to create a national FS layer across a geographically large and diverse country with limited training data. First, approaches were considered on how to generate a map of FS for Canada at two different levels, (i) national, which combined all training data into one model, and (ii) regional, where multiple models were created, based on regional similarities, and the results were mosaicked to generate a FS map. The second experiment explored the predictive capability of several ML algorithms across the geographically large and diverse landscape. Results indicate that the national approach provides a better prediction of FS, with 95.7% of the test points, 91.5% of the pixels in the training sites, and 89.6% of the pixels across the country correctly predicted as flooded, compared to 65.5%, 80.6% and 75.6%, respectively, in the regional approach. ML models applied across the country found that support vector machine (svmRadial) and Neural Network (nnet) performed poorly in areas away from the training sites, while random forest (parRF) and Multivariate Adaptive Regression Spline (earth) performed better. A national ensemble model was ultimately selected as this blend of models compensated for the biases found in the individual models in geographic areas far removed from training sites. Full article

The isogeometric collocation method (IGA-C), which is a promising branch of isogeometric analysis (IGA), can be considered fitting the load function with the combination of the numerical solution and its derivatives. In this study, we develop an iterative method, isogeometric least-squares progressive-iterative approximation (IG-LSPIA), to solve the fitting problem in the collocation method. IG-LSPIA starts with an initial blending function, where the control coefficients are combined with the B-spline basis functions and their derivatives. A new blending function is generated by constructing the differences for collocation points (DCP) and control coefficients (DCC), and then adding the DCC to the corresponding control coefficients. The procedure is performed iteratively until the stop criterion is reached. We prove the convergence of IG-LSPIA and show that the computation complexity in each iteration of IG-LSPIA is related only to the number of collocation points and unrelated to the number of control coefficients. Moreover, an incremental algorithm is designed; it alternates with knot refinement until the desired precision is achieved. After each knot refinement, the result of the last round of IG-LSPIA iterations is used to generate the initial blending function of the new round of iteration, thereby saving great computation. Experiments show that the proposed method is stable and efficient. In the three-dimensional case, the total computation time is saved twice compared to the traditional method. Full article

Compared with wheeled and tracked robots, legged robots have better movement ability and are more suitable for the exploration of unknown environments. In order to further improve the adaptability of legged robots to complex terrains such as slopes, obstacle environments, and so on, this paper makes a new design of the legged robot’s foot sensing structure that can successfully provide accurate feedback of the landing information. Based on this information, a new foot trajectory planning method named three-element trajectory determination method is proposed. For each leg in one movement period, the three elements are the start point in the support phase, the end point in the support phase, and the joint angle changes in the transfer phase where the first two elements are used to control the height, distance, and direction of the movement, and the third element is used make decisions during the lifting process of the leg. For the support phase, the trajectory is described in Cartesian space, and a spline of linear function with parabolic blends is used. For the transfer phase, the trajectory is described in joint-space, and the joint angle function is designed as the superposition of the joint angle reverse-chronological function and the interpolation function which is obtained based on joint angle changes. As an important legged robot, a hexapod robot that we designed by ourselves with triangle gait is chosen to test the proposed foot trajectory planning method. Experiments show that, while the foot’s landing information can be read and based on the three-element trajectory planning method, the hexapod robot can achieve stable movement even in very complex scenes. Although the experiments are performed on a hexapod robot, our method is applicable to all forms of legged robots. Full article

Linear independence of the blending functions is a necessary requirement for T-spline in isogeometric analysis. The main work in this paper focuses on the analysis about T-splines of degree one, we demonstrate that all the blending functions of such T-spline of degree one are linearly independent. The advantage owned by one degree T-spline is that it can avoid the problem of judging whether the model is analysis-suitable or not, especially for occasions that need a quick response from the analysis results. This may provide a new way of using T-spline for a CAD and CAE integrating scenario, since one degree T-spline still guarantees the topology flexibility and is compatible with the spline-based modeling system. In addition, we compare the numerical approximations of isogeometric analysis and finite element analysis, and the experiment indicates that isogeometric analysis using T-spline of degree one can reach a comparable result with classical method. Full article

This work deals with heat conduction problems formulation in the framework of a CAD-compatible topology optimization method based on a pseudo-density field as a topology descriptor. In particular, the proposed strategy relies, on the one hand, on the use of CAD-compatible Non-Uniform Rational Basis Spline (NURBS) hyper-surfaces to represent the pseudo-density field and, on the other hand, on the well-known Solid Isotropic Material with Penalization (SIMP) approach. The resulting method is then referred to as NURBS-based SIMP method. In this background, heat conduction problems have been reformulated by taking advantage of the properties of the NURBS entities. The influence of the integer parameters, involved in the definition of the NURBS hyper-surface, on the optimized topology is investigated. Furthermore, symmetry constraints, as well as a manufacturing requirement related to the minimum allowable size, are also integrated into the problem formulation without introducing explicit constraint functions, thanks to the NURBS blending functions properties. Finally, since the topological variable is represented by means of a NURBS entity, the geometrical representation of the boundary of the topology is available at each iteration of the optimization process and its reconstruction becomes a straightforward task. The effectiveness of the NURBS-based SIMP method is shown on 2D and 3D benchmark problems taken from the literature. Full article

This paper discusses the construction of a type-2 fuzzy B-spline model to model complex uncertainty of surface data. To construct this model, the type-2 fuzzy set theory, which includes type-2 fuzzy number concepts and type-2 fuzzy relation, is used to define the complex uncertainty of surface data in type-2 fuzzy data/control points. These type-2 fuzzy data/control points are blended with the B-spline surface function to produce the proposed model, which can be visualized and analyzed further. Various processes, namely fuzzification, type-reduction and defuzzification are defined to achieve a crisp, type-2 fuzzy B-spline surface, representing uncertainty complex surface data. This paper ends with a numerical example of terrain modeling, which shows the effectiveness of handling the uncertainty complex data. Full article

A seamless mosaic has been constructed including a 3D terrain model at 50 m grid-spacing and a corresponding terrain-corrected orthoimage at 12.5 m using a novel approach applied to ESA Mars Express High Resolution Stereo Camera orbital (HRSC) images of Mars. This method consists of blending and harmonising 3D models and normalising reflectance to a global albedo map. Eleven HRSC image sets were processed to Digital Terrain Models (DTM) based on an opensource stereo photogrammetric package called CASP-GO and merged with 71 published DTMs from the HRSC team. In order to achieve high quality and complete DTM coverage, a new method was developed to combine data derived from different stereo matching approaches to achieve a uniform outcome. This new approach was developed for high-accuracy data fusion of different DTMs at dissimilar grid-spacing and provenance which employs joint 3D and image co-registration, and B-spline fitting against the global Mars Orbiter Laser Altimeter (MOLA) standard reference. Each HRSC strip is normalised against a global albedo map to ensure that the very different lighting conditions could be corrected and resulting in a tiled set of seamless mosaics. The final 3D terrain model is compared against the MOLA height reference and the results shown of this intercomparison both in altitude and planum. Visualisation and access mechanisms to the final open access products are described. Full article

Track geometry is a fundamental subject in railway construction. With the demand for increased capacity in terms of load and speed, the need for suitable transitions between consecutive track sections is highly relevant. Properly constructed transition curves lead to improved travel comfort, increased safety, and reduced wear. The well known clothoid curve is widely used as a transition curve; however, the linear curvature is not sufficiently smooth to meet the requirements for railways carrying high speed trains or heavy hauls. Blending spline curves are flexible spline constructions possessing favourable smoothness properties at the end points, which makes them considerable for use as transition curves. This paper demonstrates some selected blending splines applied as transition curves between two existing circular arc segments selected from the Ofotbanen railway. The main results in this paper are related to the smoothness at the end points and the behaviour of the curvature of the curves, where the new transition curves were shown to be smoother than the original clothoid. Another new result is the observation that the proposed method allows for the improvement of existing railways without forcing extensive changes to the original track. Some representative examples are included to highlight the flexibility of this first instance of blending splines as transition curves. Full article

The functional properties of multi-component materials are often determined by a rearrangement of their different phases and by chemical reactions of their components. In this contribution, a material model is presented which enables computational simulations and structural optimization of solid multi-component systems. Typical Systems of this kind are anodes in batteries, reactive polymer blends and propellants. The physical processes which are assumed to contribute to the microstructural evolution are: (i) particle exchange and mechanical deformation; (ii) spinodal decomposition and phase coarsening; (iii) chemical reactions between the components; and (iv) energetic forces associated with the elastic field of the solid. To illustrate the capability of the deduced coupled field model, three-dimensional Non-Uniform Rational Basis Spline (NURBS) based finite element simulations of such multi-component structures are presented. Full article