Free Shape Parameter Research Articles

Single-sample unmixing and parametric end-member modelling of grain-size distributions with transformed probability density functions and their performance comparison using aeolian sediments

Parametric decomposition techniques including single-sample unmixing and parametric end-member modelling are routine mathematical methods used for interpreting grain-size distributions. In this study, transformed probability density functions for the Lognormal, Weibull, Skew Normal, and Skewed Generalized Normal distributions are derived and efficient open-source numerical programs applying these functions are presented. These new functions contain two free shape parameters characterizing the peak position and magnitude of a unimodal distribution, and they can be more easily initialized and constrained compared to their original counterparts, as the two shape parameters can be estimated directly from the grain-size distribution or its derivatives and can only vary within very narrow intervals. This enables the decomposition to converge to both reproducible/stable and structurally/genetically reasonable solutions. The transformed functions are applied to grain-size distributions of aeolian sediments collected from around the Tengger Desert, using both the single-sample unmixing and parametric end-member modelling methods, and the results of different functions are compared. The implications of the results for parametric decomposition of sediment grain-size distributions using unimodal mathematical distributions are discussed.

Hermite interpolation by planar cubic-like ATPH

This paper deals with the construction of the Algebraic Trigonometric Pythagorean Hodograph (ATPH) cubic-like Hermite interpolant. A characterization of solutions according to the tangents at both ends and a global free shape parameter α is performed. Since this degree of freedom can be used for adjustments, we study how the curve evolves with respect to α. Several examples illustrating the construction process and a simple fitting method to determine the unique ATPH curve passing through a given point are proposed.

Application of C-Bézier and H-Bézier basis functions to numerical solution of convection-diffusion equations

Convection-diffusion equation is widely used to describe many engineering and physical problems. The finite element method is one of the most common tools for computing numerical solution. In 2003, Wang et al. proposed C-Bézier and H-Bézier basis functions which are not only a generalization of classical Bernstein basis functions but also have a free shape parameter bringing a lot of flexibility to geometrical modeling. In this paper, we adopt C-Bézier and H-Bézier basis functions to construct test and trial function spaces of finite element method to get numerical solution of convection-diffusion equations. Compared with Lagrange basis functions, numerical accuracy is improved by 1-3 order-of magnitudes which implies a much better approximation in simulating convection-diffusion problems. Several examples are presented to verify the feasibility and effectiveness of our method.

On the construction of polynomial minimal surfaces with Pythagorean normals

A novel approach to constructing polynomial minimal surfaces (surfaces of zero mean curvature) with isothermal parameterization from Pythagorean triples of complex polynomials is presented, and it is shown that they are Pythagorean normal (PN) surfaces, i.e., their unit normal vectors have a rational dependence on the surface parameters. This construction generalizes a prior approach based on Pythagorean triples of real polynomials, and yields more free shape parameters for surfaces of a specified degree. Moreover, when one of the complex polynomials is just a constant, the minimal surfaces have the Pythagorean–hodograph (PH) preserving property — a planar PH curve in the parameter domain is mapped to a spatial PH curve on the surface. Cubic, quartic and quintic examples of these minimal PN surfaces are presented, including examples of solutions to the Plateau problem, with boundaries generated by planar PH curve segments in the parameter domain. The construction is also generalized to the case of minimal surfaces with non–isothermal parameterizations. Finally, an application to the problem of interpolating three given points in R3 as the corners of a triangular cubic minimal surface patch, such that the three patch sides have prescribed lengths, is addressed.

A meshless technique based on the radial basis functions for solving systems of partial differential equations

The radial basis functions (RBFs) methods were first developed by Kansa to approximate partial differential equations (PDEs). The RBFs method is being truly meshfree becomes quite appealing, owing to the presence of distance function, straight-forward implementation, and ease of programming in higher dimensions. Another considerable advantage is the presence of a tunable free shape parameter, contained in most of the RBFs that control the accuracy of the RBFs method. Here, the solution of the two dimensional system of nonlinear partial differential equations is examined numerically by a Global Radial Basis Functions Collocation Method (GRBFCM). It can work on a set of random or uniform nodes with no need for element connectivity of input data. For the time-dependent partial differential equations, a system of ordinary differential equations (ODEs) is derived from this scheme. Like some other numerical methods, a comparison between numerical results with analytical solutions is implemented confirming the efficiency, accuracy, and simple performance of the suggested method.

A global RBF-QR collocation technique for solving two-dimensional elliptic problems involving arbitrary interface

In recent years, several numerical methods have been developed for differential equations with discontinuous coefficients, which are known as interface problems. Our aim is to introduce an efficient global meshless method for solving elliptic problems with arbitrary interface. Meshless methods based on infinitely smooth radial basis functions (RBFs) can achieve spectral convergence rate for solving partial differential equations. RBF-QR is a newly developed method which allows numerically stable computations with RBFs for all values of the free shape parameter $$\varepsilon$$. In this paper, an appropriate RBF-QR collocation technique is applied for two-dimensional elliptic interface problems, including Poisson, Helmholtz and elasticity problems. The Helmholtz interface problem is studied, by taking into account the fact that the solutions of such problems might be oscillatory. The elasticity interface problem is discussed as an application of elliptic interface problem in vector form. Several problems involving complex interface geometries and discontinuous variable coefficients are presented to confirm the efficiency and stability of RBF-QR method for this class of problems in the case of arbitrary small values of $$\varepsilon$$.

Kinematics of Viscous Fluid Flow during the Filling of a Pipe with a Coaxial Central Body

The flow of a viscous fluid with a free surface realized during the filling of a vertical round pipe with a coaxial central body in the gravity field is numerically simulated. The mathematical formulation of the problem includes the Navier—Stokes and continuity equations discretized using the control volume method, together with the correcting SIMPLE procedure. The natural boundary conditions on the free surface are satisfied by means of the method of invariants. A parametric investigation of the filling process is carried out. The criterial dependences of the free surface shape parameters on the main dimensionless parameters of the problem are obtained. The mass distribution pattern is studied.

On variable and random shape Gaussian interpolations

This work focuses on the invertibility of non-constant shape Gaussian asymmetric interpolation matrix, which includes the cases of both variable and random shape parameters. We prove a sufficient condition for that these interpolation matrices are invertible almost surely for the choice of shape parameters. The proof is then extended to the case of anisotropic Gaussian kernels, which is subjected to independent componentwise scalings and rotations. As a corollary of our proof, we propose a parameter free random shape parameters strategy to completely eliminate the need of users’ inputs. By studying numerical accuracy in variable precision computations, we demonstrate that the asymmetric interpolation method is not a method with faster theoretical convergence. We show empirically in double precision, however, that these spatially varying strategies have the ability to outperform constant shape parameters in double precision computations. Various random distributions were numerically examined.

On the numerical solution of nonlinear integral equations on non-rectangular domains utilizing thin plate spline collocation method

This article investigates an approximate scheme to solve nonlinear Fredholm integral equations of the second kind on non-rectangular domains. The integral equations considered in the current paper are considered together with either smooth or weakly singular kernels. The offered method utilizes thin plate splines as a basis in the discrete collocation method. We can regard thin plate splines as a type of the free shape parameter radial basis functions. These basis functions establish an accurate and stable technique to estimate an unknown function by using a set of scattered points on the solution domains. Since the thin plate splines have limited smoothness, the integrals appeared in the scheme cannot be estimated by classical integration rules. Therefore, we introduce a special precise quadrature formula on non-rectangular domains to compute these integrals. The proposed scheme does not require any mesh generations, so it is meshless and does not depend on the domain form. Error analysis is also provided for the method. The performance and convergence of the new approach are tested on four two-dimensional integral equations given on the wing, mushroom, pentagon and fish-like domains.

A numerical scheme for solving a class of logarithmic integral equations arisen from two-dimensional Helmholtz equations using local thin plate splines

This paper presents a numerical method for solving logarithmic Fredholm integral equations which occur as a reformulation of two-dimensional Helmholtz equations over the unit circle with the Robin boundary conditions. The method approximates the solution utilizing the discrete collocation method based on the locally supported thin plate splines as a type of free shape parameter radial basis functions. The local thin plate splines establish an efficient and stable technique to estimate an unknown function by a small set of nodes instead of all points over the solution domain. To compute logarithm-like singular integrals appeared in the method, we use a particular nonuniform Gauss–Legendre quadrature rule. Since the scheme does not require any mesh generations on the domain, it can be identified as a meshless method. The error estimate of the proposed method is presented. Numerical results are included to show the validity and efficiency of the new technique. These results also confirm that the proposed method uses much less computer memory in comparison with the method established on the globally supported thin plate splines. Moreover, it seems that the algorithm of the presented approach is attractive and easy to implement on computers.

Cost-efficient BLE fingerprint database construction approach via multi-quadric RBF interpolation

The demand for indoor localization is becoming urgent, but the traditional location fingerprint approach takes a lot of manpower and time to construct a fine-grained location fingerprint database. To address this problem, we propose to use the approach of combining dynamic collection of fingerprint samples with Radial Basis Function (RBF) interpolation. Specifically, the raw sparse fingerprint database is constructed from a small number of fingerprints collected on a few paths, in which the pedestrian track correction algorithm improves the validity and accuracy of the sparse fingerprint database. Then, the RBF interpolation approach is applied to enrich the sparse fingerprint database, in which the Genetic Algorithm (GA) is used to optimize the free shape parameter and the cut-off radius is determined according to the experimental results. Extensive experiments show that the proposed approach guarantees high interpolation and localization accuracy and also significantly reduces the effort of manual collection of fingerprint samples.

From rod-like to disc-like Gay–Berne biaxial nematics and back

ABSTRACTBy means of Monte Carlo simulations at constant pressure and temperature, the phase behaviour of pure systems of soft biaxial Gay–Berne ellipsoids, prototypical models for thermotropic mesogens, was determined as a function of temperature and three different pressures. To reduce the number of free shape and energy parameters to only one, the medium ellipsoid axis , particles possessing identical total volume and length were studied, with interactions parameterised in terms of products of axes lengths. The resulting phase diagrams present a variety of crystalline and liquid crystalline phases characterised by different biaxiality and preferential axis alignment, with GB particles behaving like rods for more elongated shapes, and like discs for more squashed ones. In agreement with previous simulation and theoretical studies using simpler models, the transition between rod to disc behaviour (and back) occurs approximately when the so-called dual-shape condition is reached, i.e. when . This shape ideally identifies also the position of a Landau point, in which at a specific temperature, isotropic, calamitic nematic, discotic nematic, and biaxial nematic may coexist, and hence a direct isotropic to biaxial nematic transition could in principle occur. We did not observe such event in our simulations.

G1 motion interpolation using cubic PH biarcs with prescribed length

In this paper the G1 interpolation scheme for motion data, i.e., interpolation of data points and rotations at the points, with cubic PH biarcs is presented. The rotational part of the motion is determined by the Euler–Rodrigues frame which matches the given boundary positions. In addition, the length of the biarc is prescribed. It is shown that the interpolant exists for any data and any chosen length greater than the difference between the interpolation points. The interpolant is given in a closed form and depends on some free shape parameters, which are determined so that the curve is of a nice shape and the twist of the Euler–Rodrigues frame is minimized. The spline construction is provided and numerical examples that confirm the derived theoretical results are included.

Efficiency Optimisation of Blade Shape in Steam and ORC Turbines

Abstract This paper is devoted to direct constrained optimisation of blading systems of large power and small power turbines so as to increase their internal efficiency. The optimisation is carried out using hybrid stochastic-deterministic methods such as a combination of a direct search method of Hooke-Jeeves and simulated annealing or a combination of a bat algorithm and simplex method of Nelder-Mead. Among free shape parameters are blade number and stagger angle, stacking blade line parameters and blade section (profile) parameters. One practical example of efficiency optimisation of turbine blading systems is modification of low load profiles PLK-R2 for high pressure (HP) stages of large power steam turbines. Another optimised geometry is that of an ORC radial-axial cogeneration turbine of 50 kWe. Up to 1% efficiency increase can easily be obtained from optimisation of HP blade profiles, especially by making the rotor blade more aft-loaded and reducing the intensity of endwall flows. Almost 2% efficiency rise was obtained for the optimised 50 kWe ORC turbine due to flow improvement at the suction side of the blade.

Thin plate spline Galerkin scheme for numerically solving nonlinear weakly singular Fredholm integral equations

ABSTRACTThe present work proposes a numerical method to obtain an approximate solution of non-linear weakly singular Fredholm integral equations. The discrete Galerkin method in addition to thin-plate splines established on scattered points is utilized to estimate the solution of these integral equations. The thin-plate splines can be regarded as a type of free shape parameter radial basis functions which create an efficient and stable technique to approximate a function. The discrete Galerkin method for the approximate solution of integral equations results from the numerical integration of all integrals in the method. We utilize a special accurate quadrature formula via the non-uniform composite Gauss-Legendre integration rule and employ it to compute the singular integrals appeared in the scheme. Since the approach does not need any background meshes, it can be identified as a meshless method. Error analysis is also given for the method. Illustrative examples are shown clearly the reliability and efficiency of the new scheme and confirm the theoretical error estimates.

Application of thin plate splines for solving a class of boundary integral equations arisen from Laplace's equations with nonlinear boundary conditions

This article describes a technique for numerically solving a class of nonlinear boundary integral equations of the second kind with logarithmic singular kernels. These types of integral equations occur as a reformulation of boundary value problems of Laplace's equations with nonlinear Robin boundary conditions. The method uses thin plate splines (TPSs) constructed on scattered points as a basis in the discrete collocation method. The TPSs can be seen as a type of the free shape parameter radial basis functions which establish effective and stable methods to estimate an unknown function. The proposed scheme utilizes a special accurate quadrature formula based on the non-uniform Gauss–Legendre integration rule for approximating logarithm-like singular integrals appeared in the approach. The numerical method developed in the current paper does not require any mesh generations, so it is meshless and independent of the geometry of the domain. The algorithm of the presented scheme is accurate and easy to implement on computers. The error analysis of the method is provided. The convergence validity of the new technique is examined over several boundary integral equations and obtained results confirm the theoretical error estimates.

The numerical solution of nonlinear integral equations of the second kind using thin plate spline discrete collocation method

In this article, the thin plate splines are given to obtain the numerical solution of nonlinear Fredholm integral equations of the second kind. The scheme approximates the solution using the discrete collocation method based on the shape functions of thin plate splines constructed on a set of scattered points. The thin plate splines can be seen as a type of the free shape parameter radial basis functions which are used to establish an effective technique for the interpolation of an unknown function. The numerical method developed in the current paper utilizes the non-uniform Gauss–Legendre quadrature rule to compute its integrals. Since the proposed scheme does not require any background mesh for approximations and numerical integrations, it is meshless. This approach can be easily implemented and its algorithm is simple and effective to solve nonlinear integral equations. Moreover the error estimate and the convergence rate of the method are presented. Finally, numerical examples are included to show the validity and efficiency of the new technique and confirm the theoretical error estimates.

Design of $$C^2$$ C 2 algebraic-trigonometric pythagorean hodograph splines with shape parameters

In this paper, we construct \(C^2\) Algebraic-Trigonometric Pythagorean Hodograph (ATPH) splines by solving a non-linear system of equations in complex variables. We compare these splines, which depend on several shape parameters, with their polynomial PH counterpart as well as with the well-known \(C^2\) cubic B-splines. We finally present criteria for choosing the free shape parameters based on the minimization of certain fairness functionals.

A numerical study of compact approximations based on flat integrated radial basis functions for second-order differential equations

In this paper, we propose a simple but effective preconditioning technique to improve the numerical stability of Integrated Radial Basis Function (IRBF) methods. The proposed preconditioner is simply the inverse of a well-conditioned matrix that is constructed using non-flat IRBFs. Much larger values of the free shape parameter of IRBFs can thus be employed and better accuracy for smooth solution problems can be achieved. Furthermore, to improve the accuracy of local IRBF methods, we propose a new stencil, namely Combined Compact IRBF (CCIRBF), in which (i) the starting point is the fourth-order derivative; and (ii) nodal values of first- and second-order derivatives at side nodes of the stencil are included in the computation of first- and second-order derivatives at the middle node in a natural way. The proposed stencil can be employed in uniform/nonuniform Cartesian grids. The preconditioning technique in combination with the CCIRBF scheme employed with large values of the shape parameter are tested with elliptic equations and then applied to simulate several fluid flow problems governed by Poisson, Burgers, convection–diffusion, and Navier–Stokes equations. Highly accurate and stable solutions are obtained. In some cases, the preconditioned schemes are shown to be several orders of magnitude more accurate than those without preconditioning.

Optimal variable shape parameters using genetic algorithm for radial basis function approximation

Many radial basis function (RBF) methods contain free shape parameter or parameters that play an important role for the accuracy of the method. In most papers the authors end up choosing free shape parameter by trial and error or some other ad-hoc means. However, using variable shape parameters provides a clear potential for improved accuracy and stability of the RBF method. Already some progress has been reported to select usable variable shape parameters. In this paper, we propose applying the genetic algorithm to determine good variable shape parameters of radial basis functions for the solution of ordinary differential equations. Numerical results show that the proposed algorithm based on the genetic optimization is effective and provides reasonable shape parameters along with acceptable accuracy in linear and nonlinear case compared with other strategies to determine variable shape parameters.