This paper presents a compact and efficient 88-line MATLAB code for the parameterized level set method based topology optimization using radial basis functions (RBFs), which is applied to minimize ...the compliance of a two-dimensional linear elastic structure. This parameterized level set method using radial basis functions can maintain a relatively smooth level set function with an approximate re-initialization scheme during the optimization process. It also has less dependency on initial designs due to its capability in nucleation of new holes inside the material domain. The MATLAB code and simple modifications are explained in detail with numerical examples. The 88-line code included in the
appendix
is intended for educational purposes.
Polynomials are used together with polyharmonic spline (PHS) radial basis functions (RBFs) to create local RBF-finite-difference (RBF-FD) weights on different node layouts for spatial discretizations ...that can be viewed as enhancements of the classical finite differences (FD). The presented method replicates the convergence properties of FD but for arbitrary node layouts. It is tested on the 2D compressible Navier–Stokes equations at low Mach number, relevant to atmospheric flows. Test cases are taken from the numerical weather prediction community and solved on bounded domains. Thus, attention is given on how to handle boundaries with the RBF-FD method, as well as a novel implementation for hyperviscosity. Comparisons are done on Cartesian, hexagonal, and quasi-uniform node layouts. Consideration and guidelines are given on PHS order, polynomial degree and stencil size. The main advantages of the present method are: 1) capturing the basic physics of the problem surprisingly well, even at very coarse resolutions, 2) high-order accuracy without the need of tuning a shape parameter, and 3) the inclusion of polynomials eliminates stagnation (saturation) errors. A MATLAB code is given to calculate the differentiation weights for this novel approach.
The aim of the present study is to develop a series of artificial neural networks (ANN) and to determine, by comparison to experiments, which type of neural network is able to predict the measured ...structural deformations most accurately. For this approach, three different ANNs are proposed. Firstly, the classical form of an ANN in the form of a feedforward neural network (FFNN). In the second approach a new modular radial basis function neural network (RBFNN) is proposed and the third network consists of a deep convolutional neural network (DCNN). By means of comparative calculations between neural network enhanced numerical predictions and measurements, the applicability of each type of network is studied.
•Three types of artificial neural networks are proposed for structural dynamics.•All calculated results are verified by means of shock tube experiments.•A new modular radial basis function network is developed.•The convolutional neural networks capture the loading and deformation history.
We propose a set of novel radial basis functions with adaptive input and composite trend representation (AICTR) for portfolio selection (PS). Trend representation of asset price is one of the main ...information to be exploited in PS. However, most state-of-the-art trend representation-based systems exploit only one kind of trend information and lack effective mechanisms to construct a composite trend representation. The proposed system exploits a set of RBFs with multiple trend representations, which improves the effectiveness and robustness in price prediction. Moreover, the input of the RBFs automatically switches to the best trend representation according to the recent investing performance of different price predictions. We also propose a novel objective to combine these RBFs and select the portfolio. Extensive experiments on six benchmark data sets (including a new challenging data set that we propose) from different real-world stock markets indicate that the proposed RBFs effectively combine different trend representations and AICTR achieves state-of-the-art investing performance and risk control. Besides, AICTR withstands the reasonable transaction costs and runs fast; hence, it is applicable to real-world financial environments.
•Large PHS+poly based RBF-FD stencils can lead to high orders of accuracy without numerical ill-conditioning.•It can also combine high orders of accuracy near boundaries with an absence of ...Runge-phenomenon-type boundary errors.•Numerical explanations to this behavior are provided based on a closed-form expression for the RBF+poly cardinal functions.•It explains the role of polynomials and RBFs in RBF+poly approximations.
Radial basis function generated finite difference (RBF-FD) approximations generalize grid-based regular finite differences to scattered node sets. These become particularly effective when they are based on polyharmonic splines (PHS) augmented with multi-variate polynomials (PHS+poly). One key feature is that high orders of accuracy can be achieved without having to choose an optimal shape parameter and without having to deal with issues related to numerical ill-conditioning. The strengths of this approach were previously shown to be especially striking for approximations near domain boundaries, where the stencils become highly one-sided. Due to the polynomial Runge phenomenon, regular FD approximations of high accuracy will in such cases have very large weights well into the domain. The inclusion of PHS-type RBFs in the process of generating weights makes it possible to avoid this adverse effect. With that as motivation, this study aims at gaining a better understanding of the behavior of PHS+poly generated RBF-FD approximations near boundaries, illustrating it in 1-D, 2-D and 3-D.
In this paper we present a new adaptive two-stage algorithm for solving elliptic partial differential equations via a radial basis function collocation method. Our adaptive meshless scheme is based ...at first on the use of a leave-one-out cross validation technique, and then on a residual subsampling method. Each of phases is characterized by different error indicators and refinement strategies. The combination of these computational approaches allows us to detect the areas that need to be refined, also including the chance to further add or remove adaptively any points. The resulting algorithm turns out to be flexible and effective through a good interaction between error indicators and refinement procedures. Several numerical experiments support our study by illustrating the performance of our two-stage scheme. Finally, the latter is also compared with an efficient adaptive finite element method.
•Novel regularization method based on radial basis function discretization for calculating the DRT.•Improved estimation of DRT only when data collection range is truncated.•MATLAB GUI for computing ...the DRT.
The distribution of relaxation times (DRT) is an approach that can extract time characteristics of an electrochemical system from electrochemical impedance spectroscopy (EIS) measurements. Computing the DRT is difficult because it is an intrinsically ill-posed problem often requiring regularization. In order to improve the estimation of the DRT and to better control its error, a suitable discretization basis for the regularized regression needs to be chosen. However, this aspect has been invariably overlooked in the specialized literature. Pseudo-spectral methods using radial basis functions (RBFs) are, in principle, a better choice in comparison to other discretization basis, such as piecewise linear (PWL) functions, because they may achieve fast convergence. Furthermore, they can yield improved estimation by extending the estimated DRT to the entire frequency spectrum, if the underlying DRT decays to zero sufficiently fast outside the measured frequency range. Additionally, their implementation is relatively easier than other types of pseudo-spectral methods since they do not require ad hoc collocation point distributions. The as-developed novel RBF-based DRT framework was tested against controlled synthetic EIS spectra and real experimental data. Our results indicate that the RBF discretization performance is comparable with that of the PWL discretization at normal data collection range, and with improvement when the EIS acquisition is incomplete. In addition, we also show that applying RBF discretization for deconvolving the DRT problem can lead to faster numerical convergence rate as compared with that of PWL discretization only at error free situation. As a companion to this work we have developed a MATLAB GUI toolbox, which can be used to solve DRT regularization problems.
This paper develops a surrogate-assisted evolutionary programming (EP) algorithm for constrained expensive black-box optimization that can be used for high-dimensional problems with many black-box ...inequality constraints. The proposed method does not use a penalty function and it builds surrogates for the objective and constraint functions. Each parent generates a large number of trial offspring in each generation. Then, the surrogate functions are used to identify the trial offspring that are predicted to be feasible with the best predicted objective function values or those with the minimum number of predicted constraint violations. The objective and constraint functions are then evaluated only on the most promising trial offspring from each parent, and the method proceeds in the same way as in a standard EP. In the numerical experiments, the type of surrogate used to model the objective and each of the constraint functions is a cubic radial basis function (RBF) augmented by a linear polynomial. The resulting RBF-assisted EP is applied to 18 benchmark problems and to an automotive problem with 124 decision variables and 68 black-box inequality constraints. The proposed method is much better than a traditional EP, a surrogate-assisted penalty-based EP, stochastic ranking evolution strategy, scatter search, and CMODE, and it is competitive with ConstrLMSRBF on the problems used.
Partial differential equations (PDEs) on surfaces appear in many applications throughout the natural and applied sciences. The classical closest point method (Ruuth and Merriman (2008) 17) is an ...embedding method for solving PDEs on surfaces using standard finite difference schemes. In this paper, we formulate an explicit closest point method using finite difference schemes derived from radial basis functions (RBF-FD). Unlike the orthogonal gradients method (Piret (2012) 22), our proposed method uses RBF centers on regular grid nodes. This formulation not only reduces the computational cost but also avoids the ill-conditioning from point clustering on the surface and is more natural to couple with a grid based manifold evolution algorithm (Leung and Zhao (2009) 26). When compared to the standard finite difference discretization of the closest point method, the proposed method requires a smaller computational domain surrounding the surface, resulting in a decrease in the number of sampling points on the surface. In addition, higher-order schemes can easily be constructed by increasing the number of points in the RBF-FD stencil. Applications to a variety of examples are provided to illustrate the numerical convergence of the method.
•In this paper, a new method for the numerical approximation of PDEs on surfaces is proposed. Our method has the advantage of being comprised of standard computational components, such as the closest point representation of the surface, and RBF finite difference methods.•Our approach uses a narrow computational tube around the surface and avoids the need for a quasi-uniform distribution of surface points. This makes the method a natural candidate for coupling with grid-based methods such as the grid-based particle method for moving interface problems (Leung and Zhao, J. Comput. Phys. 228 (8) (2009) 2993–3024).•The method is also efficient: it exploits repeated patterns in computational geometry, it uses small computational tubes, and it avoids an explicit interpolation step. Further-more, a change in the order of the method is carried out simply by changing the number of points in the finite difference stencil. See our novelty statement for details on how the method compares with the original closest point method (Ruuth and Merriman, J. Comput. Phys. 227 (3) (2008) 1943–1961) and recent RBF methods (e.g., Piret, J. Comput. Phys. 231 (14) (2012) 4662–4675).•We conduct convergence studies in two and three dimensions and apply the method to a variety of problems, including reaction–diffusion systems and image denoising. Second order accurate results are observed in our experiments.
A weighted strong form collocation method using radial basis functions and explicit time integration is proposed to solve the incompressible Navier-Stokes equations. The velocities and pressure are ...solved directly at the same time step and the continuity equation is satisfied at each time step which improve the solution accuracy and stability. No artificial compressibility coefficient needs to be introduced for modeling the incompressible flows and no pressure oscillation arises in the numerical solutions. Optimal convergence can be achieved by imposing the derived proper weights on the boundaries and the continuity equation. Radial basis collocation method in a Lagrangian form is quite easy to capture the moving boundary or free surface in flow problems. Moreover, solid boundary conditions can be enforced directly and no special treatments are required. Further, critical time step for the explicit time integration is predicted in the stability analysis and the influences on the stability are evaluated. Numerical studies validate the high accuracy as well as good stability of the presented method.
•A weighted collocation method is raised for incompressible Navier-Stokes equations.•Optimal convergence can be gained by imposing the proper weights on the boundaries.•No artificial compressibility coefficient is used to model the incompressible flow.•No pressure oscillation arises in the numerical solutions.•It's quite easy to capture the moving boundary or free surface in flow problems.
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