Variable Shape Parameter Research Articles

An Optimal Multiquadric Variable Shape Parameter for Boundary Value Problems Using Particle Swarm Optimization

The multiquadric radial basis function method has been widely used to solve partial differential equations-based problems regarding its flexibility and meshfree characteristics. The accuracy and stability of this method are derived and based on the use of a free-shape parameter that sensibly controls the comportment of the technique. Significant improvements have already been reported and show that variable shape parameters conduct the method to handle problems with striking results compared to global-based techniques. Nevertheless, choosing a suitable set of shape parameters is still an open topic because of the complexity of the method when the number of collocation points increases. The current work proposes a variant particle swarm optimization based on local displacement with attractors to determine the multi-quadratic function's ``best'' optimal variable shape parameter in solving boundary value problems. Based on an initially random set of variable shape parameters, the proposed algorithm first performs and evaluates the errors between the expected exact solution and the approximate solution thoroughly. In the first stage, the particle swarm algorithm search for an optimal set of shape parameters that minimize the error and the conditioning number of the radial basis system matrix. In the second stage, the obtained optimal set of shape parameters is applied to solve the considered problem. In this way, when the number of collocation points increases, the first stage based on particle swarm optimization stabilizes the strategy. It proposes an ``acceptable'' set of shape parameters for the given problem. The proposed method is applied to a set of well-known boundary value problems in one and two-dimensional spaces and compared to other techniques published in the literature. The results show that the proposed method achieves more accurate solutions than recently proposed in the literature.

Formula omitted]-Curves: Extended log-aesthetic curves with variable shape parameter

This paper proposes a novel curve called the α-curve, which is specified by a variable shape parameter α and can be applied in curve reconstruction involving curve fairing, fitting, and segmentation. In contrast to classical log-aesthetic curves in which α is constant, this study contributes a novel formulation of α that varies monotonically within a specified segment. By introducing the variable α, the logarithmic curvature graph (LCG) becomes piecewise linear and parabolic, or any other function if necessary. Consequently, in comparison to conventional log-aesthetic curves that have limited field usage based on their rigidity, α-curves have various LCGs and flexible representations, thereby opening up many practical applications. Experimental results and comparative studies on curves created by CAD experts in the automotive industry demonstrate the theoretical and practical validity of α-curves.

An improved FSDT with an efficient mesh-free approach for nonlinear static analysis of FG-GOPRC beams

The aim of this paper is to enhance the First Shear Deformation Theory (FSDT) beam and apply it to investigate the linear and nonlinear static bending behavior of beams containing functionally graded graphene oxide powder-reinforced composite (FG-GOPRC). In this study, we propose an improvement to the FSDT by utilizing a high-order mesh-free approach. Unlike some existing FSDT methods, our proposed study uses a nonlinear shear deformation term coupling, which is crucial to illustrate the effect of nonlinear cases of various boundary conditions under a strong formulation, even without selective or reduced integration, resulting in lower computation costs. Using the principle of minimum potential energy, the governing equations are rewritten in a matrix–vector strong form and discretized using the Radial Point Interpolation Method with variable shape parameter (VS-RPIM) and linearized using a High Order Path Following Techniques (HOPFT). To account for the behavior of GOPRC, we utilized the modified Halpin–Tsai model to estimate the effective Young’s modulus, while the rule of mixture was employed to determine the effective Poisson’s ratio. Furthermore, the weight fractions of the graphene oxide powder (GOP) vary continuously through the thickness direction of the composite beams. To validate the proposed numerical model, we provide various comparison studies to demonstrate its robustness and accuracy. Additionally, we conduct parametric studies to examine the effects of different parameters, such as length-to-thickness ratio, GOP distribution patterns, weight fraction, and size of GOP, types of loads, and boundary conditions on the nonlinear bending behavior of FG-GOPRC beams. We also investigate the neglected term in some literature, which is of interest in our study. In summary, this paper proposes an enhanced FSDT to investigate the nonlinear static behavior of FG-GOPRC beams. The proposed model is validated through various comparison studies and parametric analyses, demonstrating its robustness and accuracy. Our study provides useful insights into the nonlinear bending behavior of FG-GOPRC beams and highlights the importance of nonlinear shear deformation term coupling.

An Automated Parametric Surface Patch-Based Construction Method for Smooth Lattice Structures with Irregular Topologies

Additive manufacturing enables the realization of complex component designs that cannot be achieved with conventional processes, such as the integration of cellular structures, such as lattice structures, for weight reduction. To include lattice structures in component designs, an automated algorithm compatible with conventional CAD that is able to handle various lattice topologies as well as variable local shape parameters such as strut radii is required. Smooth node transitions are desired due to their advantages in terms of reduced stress concentrations and improved fatigue performance. The surface patch-based algorithm developed in this work is able to solidify given lattice frames to smooth lattice structures without manual construction steps. The algorithm requires only a few seconds of sketching time for each node and favours parallelisation. Automated special-case workarounds as well as fallback mechanisms are considered for non-standard inputs. The algorithm is demonstrated on irregular lattice topologies and applied for the construction of a lattice infill of an aircraft component that was additively manufactured.

CURVATURE BASED CHARACTERIZATION OF RADIAL BASIS FUNCTIONS: APPLICATION TO INTERPOLATION

Choosing the scale or shape parameter of radial basis functions (RBFs) is a well-documented but still an open problem in kernel-based methods. It is common to tune it according to the applications, and it plays a crucial role both for the accuracy and stability of the method. In this paper, we first devise a direct relation between the shape parameter of RBFs and their curvature at each point. This leads to characterizing RBFs to scalable and unscalable ones. We prove that all scalable RBFs lie in the -class which means that their curvature at the point xj is proportional to, where cj is the corresponding spatially variable shape parameter at xj. Some of the most commonly used RBFs are then characterized and classified accordingly to their curvature. Then, the fundamental theory of plane curves helps us recover univariate functions from scattered data, by enforcing the exact and approximate solutions have the same curvature at the point where they meet. This leads to introducing curvature-based scaled RBFs with shape parameters depending on the function values and approximate curvature values of the function to be approximated. Several numerical experiments are devoted to show that the method performs better than the standard fixed-scale basis and some other shape parameter selection methods.

A novel Gaussian-cubic-based backward substitution method using symmetric variable shape parameter

In this paper, we make the first attempt to apply the Gaussian-cubic basis function in the novel backward substitution method for solving linear and nonlinear problems in irregular domains. To improve the solution accuracy, the weighted parameters of the provided hybrid kernel are varying instead of constant ones in traditional methods. The shape parameters are determined by a symmetric variable shape parameter scheme. Furthermore, a counterintuitive ghost-points method is employed as the strategy for placing the center nodes in the basis functions. To evaluate the accuracy and efficiency of the proposed method, several 2D and 3D numerical examples are considered and the numerical results are compared with the traditional backward substitution method and the results in references. And the method can be used to solve nonlinear problems with some linearization techniques. The superior accuracy and efficiency have been validated.

GOLD DISTRIBUTION Another Look on the Generalization of Lindely Distribution

In this paper, a new generalization of one parameter Lindely distribution is proposed. The new distribution is a mixture distribution of Gamma distributions with fixed scale parameter and variable shape parameter. The distribution is called 'GOLD Distribution' as it is a generalization for several distributions such as exponential, Lindely, Sujatha, Amarendra, Devya and Shambhu distributions. The probability density and cumulative density functions are derived. Also, the statistical properties of the GOLD distribution are discussed. Parameter estimation using the maximum likelihood and the method of moments are given. Moreover, an illustration of the usefulness of the GOLD distribution in survival data analysis is discussed based on a real lifetime data.

Variable RPIM and HOCM coupling for non-linear buckling and post-buckling analysis of transverse FG sandwich beams

This paper presents an analytical-based approach to analyze non-linear buckling and post-buckling of transverse functionally graded sandwich beams (TFGSB). The proposed approach combines a high order continuation method (HOCM) and a variable radial point interpolation method (RPIM). First of all, the strong form of the beam buckling governing equations are established in the framework of first order shear deformation theory (FSDT) by taking into account the non-linear tangential term. Then, the spatial approximation of the unknown variables and their partial derivatives using the RPIM with variable shape parameters is the first step of this approach. The RPIM shape parameter is considered to change according to Franke formula, modified Franke formula or reciprocal Franke formula. The non-linear resulting system of discrete equations is linearized by using the Taylor series expansion, whilst the continuation method is employed to obtain the whole solution. To demonstrate the effectiveness of the proposed approach, numerical examples on non-linear buckling and post-buckling analysis of TFGSB are presented. These numerical tests are carried out to define the optimal RPIM with variable shape parameters and to investigate the sensitivity of the accuracy regarding boundary conditions, effects of power-law index, Young’s modulus, skin-core-skin thickness and the span-depth ratios. The results obtained through the proposed approach are compared with those computed by finite element method (FEM) and the available data in the literature.

Highly efficient mesh-free approach to simulate the non-linear bending analysis of FG porous beams and sandwich beams with FG face sheets

The main objective of the present work is to propose a coupling of the Radial Point Interpolation Method (RPIM) with variable shape parameter to High Order Continuation Method (HOCM) for analyzing the geometrically non-linear behavior of porous Functionally Graded (FG) isotropic and sandwich Timoshenko beams by taking into account the non-linear shear deformation. The FG and FG Sandwich Beams (FGSB) effective material properties are assumed to vary continuously in the thickness direction according to a power-law index. The porosity is modeled as the stiffness reduction criteria and included in the mixture rule. The strong form of the non-linear equations governing porous FG and FGSB beams are obtained using the principle of virtual displacements in the framework of First-Order Shear Deformation Theory (FOSDT). The HOCM is used to compute the solution of the non-linear problem by a transformation to recursive succession of linear problems using Taylor series expansion. The continuation technique is used to obtain the complete solution path in a step by step manner. The effectiveness and accuracy of the proposed coupling is brought out through numerical examples which concern porous FG isotropic and FG sandwich beams. The optimal parameters have been defined according to the stability analysis of the proposed solver, and then used to investigate the effect of boundary conditions, showing the effect of the power-law index, effect of geometric imperfections, and face sheet-core-face sheet thickness ratios on deflections. The validation is done by comparing the obtained results to those available in the literature and/or those computed using Finite Element Method (FEM).

A novel local radial basis function collocation method for multi-dimensional piezoelectric problems

The local radial basis function collocation method (LRBFCM), a strong-form formulation of the meshless numerical method, is proposed for solving piezoelectric medium problems. The proposed numerical algorithm is based on the local Kansa method using variable shape parameter. We introduce a novel technique for the determination of shape parameter in the LRBFCM, which leads to greater accuracy, and simplicity. The implemented algorithm is first verified with a 2D Poisson equation. Then, we employed LRBFCM in a numerical simulation for 2D and 3D piezoelectric problems involving mutual coupling of the electric field and elastodynamic equations for mechanical field. The presented meshless method is verified using corresponding results obtained from the finite element method and moving least squares meshless local Petrov–Galerkin method. In particular, the 2D piezoelectric problem is verified with an exact solution.

A time–efficient variable shape parameter Kansa–radial basis function method for the solution of nonlinear boundary value problems

In this study we consider the application of a Kansa–radial basis function (RBF) collocation method for solving two– and three–dimensional nonlinear boundary value problems (BVPs) of second and fourth order. In this variable shape parameter approach, a distinct shape parameter is linked with each RBF in the approximation of the solution and the total set of unknowns in the resulting discretized nonlinear problem comprises the RBF coefficients in the approximation and the set of (distinct) shape parameters. The solution of the system of nonlinear equations is achieved using the MATLAB©optimization toolbox functions fsolve or lsqnonlin. Unlike previous applications of these routines to nonlinear BVPs, we exploit the option offered in these functions to provide the analytical expression of the Jacobian of the nonlinear systems in question and show, in several numerical applications, how this leads to spectacular savings in computational time.

Collocation method based on radial basis functions via symmetric variable shape parameter for solving a particular class of delay differential equations

In this article, we use the collocation method based on the radial basis functions with sym- metric variable shape parameter (SVSP) to obtain numerical solutions of neutral-type functional- differential equations with proportional delays. We used Gaussian radial basis functions with SVSP. Using non uniform collocation points, we achieved a system and solving this system yielded the prob- lem solutions. Several examples are given to illustrate the efficiency and accuracy of the introduced method in comparison with the same method with the constant shape parameter (CSP) as well as other analytical and numerical methods. Comparison of the obtained numerical results shows the considerable superiority of the collocation method based on RBFs with SVSP in accuracy and convergence over the collocation method based on the RBFs with CSP and other analytical and numerical methods for delay differential equations (DDEs). Finally, numerical rate of convergence analysis of the numerical approximation was also obtained. It is observed that by comparing be- tween the obtained ROC values of error norms by the SVSP and CSP method, SVSP results were considered acceptable.

Ghost-point based radial basis function collocation methods with variable shape parameters

In this study, a strategy was proposed to determine the interval of the variable shape parameter for the ghost point method using radial basis functions. The determination of a suitable interval for the variable shape parameter remains a challenge. The modified Franke formula was used as an initial predictor of the center of the interval of the variable shape parameter in this study. After extensive tests, a numerical procedure was found for the determination of a suitable interval. The improvement from the imposition of the partial differential equation on the boundary points using the ghost point method was also investigated. To demonstrate the effectiveness of the proposed approach, four numerical examples are presented, including second and fourth order partial differential equations in 2D and 3D.

A numerical study on inverse quadratic optimal shape parameter in interpolation problems

The main purpose of this work is to shed more light into the inverse quadratic radial basis function (RBF) in the application of interpolation. This RBF contains a parameter that plays a crucial role in determining the final result quality. Five strategies of variable shape parameters are numerically investigated. Both types of node distribution; normally and scattered, are considered. It is discovered that good results can be obtained when using some strategies particularly in 1D problem. Challenges become appearing when dealing with 2D and deserves further study.

Quantifying the influence of shoreface shape on coastal responses to sea level rise

The coastal zone is a very dynamic region, constantly seeking to reach a state of equilibrium due to variations in the external forces. The shallow zone along the inner continental shelf known as the shoreface, can be divided into the upper and lower shoreface. While the former responds to processes operating across time spans ranging from that of a single event, such as storm, through to seasonal variations in environmental conditions, the latter evolution's occurs in geological scale, more in response to mean trends in environmental conditions (decades to millennia). These zones are connected via sedimentary exchanges, by a transition zone that adapts to changes imposed from behaviors and displacements of the upper and lower shoreface, such as upward profile displacement with sea level rise, aeolian transport, alongshore transport gradient, cross-shore transport. To analyze the influence of shoreface morphology variations on coastal response to mean sea level rise (MSLR), computer simulations were performed using synthetic shoreface profiles with variable shoreface shape parameter (‘m’) values. The random shoreface translation model (RanSTM) was used to simulate coastal response through variations in the shape of shoreface profiles. The ‘m’ value was changed in the upper and lower shoreface separately, to check the sensitivity of coastal response (in this case, shoreline recession distance) as a function of morphological change in each of the two zones. In addition, the variations of ‘m’ were analyzed with a variable and a fixed transition zone length (in order to isolate effects of a variable ‘m’). The simulations indicated that changes in ‘m’ on the upper shoreface have minimal influence on shoreline recession distance, especially when the transition zone remained constant. Conversely, profiles with higher ‘m’ values in the lower shoreface exhibited greater shoreline recession. It was also observed that, a longer transition zone produced a higher shoreline recession. Profiles adjusted with ‘m’ values of 0.66 and 0.4 in both shoreface zones had the highest and lowest shoreline recession distances (141.4 and 203.7 m), respectively. Sensitivity tests have shown that an increase of only 0.02 units in ‘m’ in the lower shoreface had a strong positive correlation (r = 0.97, p < 0.01) with increased shoreline recession. The results obtained in this study demonstrate the control that lower shoreface shape variations have on coastal response during MSLR. Critically, even small variations in shoreface shape parameter, such as those that occur along adjacent coastal sectors, and due to climate change effects may have implications for management and adaptation on a regional scale.

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.

Influence of the PZT Sensor Array Configuration on Lamb Wave Tomography Imaging with the RAPID Algorithm for Hole and Crack Detection.

The tomography technique is an effective way to quantitatively evaluate damage from reconstruction imaging in structure health monitoring (SHM). The reconstruction algorithm for the probabilistic inspection of damage (RAPID) algorithm based on the signal difference coefficient (SDC) feature is a promising approach due to its superior performance. This paper focuses on the influence of different patterns of PZT (Lead Zirconate Titanate) sensor array configurations, i.e., the circular, square, and parallel array, on reconstruction image qualities for evaluating hole and crack damage. Variable shape parameters are applied to account for the unequal damage distances of different actuator-sensor pairs. Considering the directionality scattering fields of cracks, the angular scattering pattern of the SDC values are studied by simulation. The SDC variations for different groups of sensing paths at the same actuator are applied to predict the crack orientation. An improved RAPID algorithm is proposed by defining an additional SDC value of 1 in the path along the predicted crack orientation, which is determined by the point of the actuator causing the minimal SDC variation and the center point of the initial reconstruction image of the crack. The results show that the improved RAPID algorithm is effective for the evaluation of crack damage. Reconstruction image qualities with three PZT sensor array configurations for both holes and cracks are compared. The research is significant for selecting the PZT sensor array configuration in SHM.

One shape fits all? A comprehensive examination of cryptocurrency return distributions

ABSTRACT We perform the most comprehensive test of cryptocurrency return distributions to date. We fit 58 hypothetical distributions to 15 major cryptocurrencies to establish which of these best describes cryptocurrency returns. The answer is: ‘It depends.’ A sharp-peaked Cauchy distribution is the most likely distribution for the majority of return series. Specific distributions are definitively identified for only a handful of cryptocurrencies. The best fitting distributions are peaked and thick-tailed, with some possessing variable shape parameters. Our findings have implications for financial modelling and its applications, such as risk measurement and risk management.

A Radial Basis Function-Generated Finite Difference Method to Evaluate Real Estate Index Options

The local radial basis functions (RBF) method is becoming increasingly popular as an alternative to the global version that suffers from ill-conditioning. The purpose of this paper is to design and describe the valuation of the real estate index options by a local RBF scheme based multiquadric radial basis function-generated finite difference (RBF-FD) method. As a generalized finite differencing scheme, the RBF-FD method functions without the need for underlying meshes to structure nodes. It offers high-order accuracy approximation and removes the difficulty of the ill-conditioned conventional global collocation methods. This paper employs an optimal variable shape parameter for the multiquadric basis functions at each grid point of the domain. Meanwhile, a local mesh refinement technique is adopted to deal with non-smooth payoffs of option. These techniques are effective and stable in improving the computational accuracy of the RBF-FD method. Several numerical experiments are presented and compared with the FD and compactly supported RBF methods to demonstrate the good performances of the proposed method. Lastly, the RBF-FD method is extended to price the American option of the real estate index.

Sensitivity analysis of raindrop size distribution parameterizations in WRF rainfall simulation

Numerical weather models such as weather research and forecasting (WRF) are increasingly used in studies on water resources. However, they have suffered from relatively poor performance in rainfall estimation. Among the various influential factors, a critical parameter in the WRF model rainfall retrieval is raindrop size distribution (DSD), which has not been fully explored. The analysis of sensitivity and uncertainty of the DSD model accuracy is significant for rainfall forecasts based on mesoscale numerical weather prediction (NWP) models. A WRF-disdrometer integrated error assessment framework is developed to analyze the accuracy and sensitivity of DSD parameterizations of gamma distribution in WRF rainfall simulation. This study adopts three different microphysics parameterizations (Morrison, WDM6, and Thompson aerosol-aware) to simulate the DSD of approximately 100 rainfall events in Chilbolton, UK that are categorized into 12 scenarios based on the season, rainfall evenness, and rainfall rate. The Thompson aerosol-aware microphysics scheme shows the best performance among the three. In comparisons of WRF rainfall simulations across different scenarios of evenness and rainfall rate, a higher accuracy is obtained with more even rain and a higher rainfall rate. The sensitivity results of different DSD parameterizations indicate that the sensitivity to the intercept parameter N0 is pronouncedly higher than those to the shape parameter μ and slope parameter λ for all studied schemes. The overall WRF rainfall shows a trend of slight underestimation followed by overestimation as μ increases; further, the rainfall is overestimated when log10N0 or λ decreases and is underestimated when it increases and then remains constant. Comparisons of different scenarios reveal that variations of DSD parameters of even rain have a relatively high impact on rainfall recognizability, and the DSD parameterizations show a higher sensitivity for rainfall with a low rate. Moreover, the sensitivity discrimination is not clear among the rainfall of different seasons. The uncertainty assessment of the WRF rainfall retrieval caused by the shape parameter suggests that a gamma DSD model with a variable shape parameter should be developed according to the evenness, rainfall rate, and microphysics parameterizations by using the WRF model. Some modified algorithms of the WRF gamma DSD model for achieving better accuracy in WRF rainfall retrievals will be explored in future studies with various climatic regimes by adjusting the DSD parameterization based on the assimilation of measured data.