A meshless collocation method on nonlinear analysis of functionally graded hyperelastic plates using radial basis function

Compact RBF meshless methods for photonic crystal modelling

Radial basis function based mesh deformation applied to simulation of flow around flapping wings

The problem of numerical solution of systems of nonlinear algebraic equations with even powers is studied in this paper. Nondifferentiable (nonsmooth) unconstrained minimization problems are associated with the system under consideration, with objective functions, based on discrete l 1 - and l ∞ - norm, respectively, by using these two norms as proximity criteria, and the considered system is solved by minimizing the sum of absolute values of residuals, in the case of l 1 - norm, or by minimizing the maximal residual, in the case of l ∞ - norm, respectively. Also, a “differentiable” unconstrained minimization problem (least squares problem) is associated with the considered system of nonlinear equations, with objective function, based on the discrete l 2 -norm, and the considered system is solved by minimizing the sum of squares of the residuals. Subgradients of the objective functions of unconstrained minimization problems are calculated, and a subgradient method for solving the associated problems is presented. The iterative gradient method for solving the corresponding “differentiable” discrete least squares problem, based on l 2 - norm, is also outlined, and gradients of the objective function of the corresponding “differentiable” unconstrained minimization problems are calculated.

Multiobjective optimization for node adaptation in the analysis of composite plates using a meshless collocation method

Meshless collocation method by Delta-shaped basis functions for default barrier model

The influence of the temperature dependence of the thermal conductivity and heat transfer coefficients, as well as of the emissivity of a dielectric film and of an insulating base on formation of a stationary temperature field of the dielectric–base system exposed to irradiation by a continuous ion beam in air, has been studied theoretically. Analytical expressions have been obtained for the temperature fields of a sample and base, as well as a system of interrelated nonlinear algebraic equations for characteristic temperatures. With the aid of numerical solution of a system of nonlinear algebraic equations, the dependences of the temperature of the irradiated surface of the sample–base system and of the rear side of this system on the incident ion beam parameters have been found. It has been established that these dependences are nonlinear. It has been found that heating of a sample, including its irradiated surface, decreases substantially with increase in the thermal conductivity of the base.

Modified bounded homotopies to enable a narrow bounding zone

Orthotropic materials are particular type of anisotropic materials; In contrast with isotropic materials, their properties depend on the direction in which they are measured. Orthotropic composite materials and their structures have been extensively used in a wide range of engineering applications. Studies on their physical behaviors under in-work loading conditions are essential. In this present, we apply an extended meshfree radial point interpolation method (XRPIM) for analyzing crack behaviour in 2D orthotropic materials models. The thin plate spline (TPS) radial basis function (RBF) is used for constructing the RPIM shape functions. Typical advantages of using RBF are the satisfaction of the Kronecker’s delta property and the high-order continuity. To calculate the stress intensity factors (SIFs), Interaction integral method with orthotropic auxiliary fields are used. Numerical examples are performed to show the accuracy of the approach; the results are compared with available refered results. Our numerical experiments have shown a very good performance of the present method.

A multiple scale time domain collocation method for solving non-linear dynamical system

In this study, continuous contact problem in the functionally graded (FG) layer loaded with a FG flat punch resting on the elastic semi-infinite plane was analyzed by the finite element method (FEM). It was assumed that the shear modulus and density of the layer and punch varied according to exponentially throughout their depth. FG layer's weight was included to the problem and additionally all surfaces were considered as frictionless. Analysis of FG materials was performed with a special macro which was added to the ANSYS program. Firstly, the shear modulus of the punch was considered to be very rigid and the results of initial separation load (tcr) and distance (xcr) were compared with the analytical solution. Afterwards, results obtained from the contact analysis made according to the inhomogeneity parameters (B, r) between FG punch-FG layer which had been unprecedented in the literature were discussed. As a result, FG punch-FG layer which had been unprecedented in the literature were discussed. As a result, FG punch's stress values at the punch edges where stress accumulations occurred were found to be smaller than the rigid punch. The security of the structure, longer life of the material and ease of production are directly related to the reduction of the stress values. The results obtained in this study are important in this respect. Also this work is the first study that investigates the effect of FG punch on the FG layer.

This paper discusses the influence of different Radial Basis Function (RBF) in metamodel construction to be applied to 3D aerodynamic wing optimization problems using the Constrained Optimization with Response Surface (CORS) methodology in conjunction with a stochastic Controlled Random Search Algorithm (CRSA). CORS methodology is based on the iterative construction and optimization of response surfaces with a robust search pattern application. In the CORS methodology the response surface may be generate by at least three types of methods: (i) Classical (polynomials and parametric surfaces), (ii) Statistical (K-Nearest, Kriging and Gaussian Processes) and (iii) Advanced (RBF and Neural Network). The response surfaces used in this paper are constructed using six different types of RBF: Gaussian, Hardy’s Multiquadric and Inverse Multiquadric, which depend on a shape parameter, and linear, cubic and thin plate spline, which do not depend on a shape parameter. The RBF’s are directly applied on the response surface construction inside the CORS structure. Thus the choice of a RBF and of a shape parameter (being the case) may influence significantly the methodology efficiency. The CORS methodology is here applied for accelerating the optimization process of wing aerodynamic designs with a solver based on a first order 3D panel method and a 2D boundary layer model. Since the main objective of this paper is of prospective nature, the choice of a relatively low-fidelity flow computation solver is justified. One considers problems of minimizing the aerodynamic coefficient relation (CD/CL) and the inverse of lift coefficient (1/CL). Comparative influence of the RBF choice on the acceleration induced by CORS methodology is investigated taking into account the number of expensive objective function evaluations necessary to find the minimum value in each problem.

For the better management of available groundwater, it is essential to understand the flow behavior within the aquifer systems. The groundwater flow behavior in the complex aquifer system can be studied by solving the governing equations using either analytical or numerical methods. As the analytical solutions are available only for simple idealized cases, numerical methods such as finite difference method (FDM) and finite element method (FEM) are generally used. The Meshless method is a recently developed alternative numerical approach for the solution of existing problem. A variety of Meshless methods like point interpolation method, smooth particle hydrodynamics, Galerkin methods, etc. are under the development for the solution of many solid mechanics and fluid mechanics problems. Meshless method eliminates many of the drawbacks of grid-based methods such as difficulties in meshing and re-meshing as in FDM and FEM which require huge efforts in pre-processing and with Meshless method it translates in the lesser computational time and costs. In this paper, Meshless Local Petrov-Galerkin (MLPG) method with strong form of collocation with exponential/Gaussian radial basis function is used for solving the groundwater flow problem. A computer model in MATLAB has been developed in 1D and 2D for the solution of confined aquifer problems. The developed 1D model is verified with available analytical solution for a hypothetical problem and found to be satisfactory. Further, the 2D model is applied to two case studies and results are compared with the FEM based solutions and found to be satisfactory. The present study shows that the MLPG based Meshless method is very effective in the simulation of groundwater flow problems.

Systems of nonlinear algebraic equations have a special place in solving modeling and calculation problems in various areas of construction Sciences. These are tasks for calculating building structures, calculating distributed loads, mass and heat transfer tasks, calculating engineering networks, and many other computational and optimization tasks related to the construction industry. As is known, there are a sufficient number of fundamentally different methods for solving systems of nonlinear algebraic equations, which is related to the specifics of the systems, their degree of nonlinearity and conditionality. Non-linear ill-conditioned systems are particularly difficult; an erroneous choice of the method for solving such a system can lead to significantly distorted or simply incorrect results, which is absolutely unacceptable when performing the calculated stages of solving construction problems. In this paper, we propose to use a modification of the well – known Levenberg – Marquardt method developed by the authors, based on the regularization of the Jacobi matrix used in the classical Newton method, to solve complex, ill-conditioned systems of nonlinear equations. The modified method allows solving poorly defined systems and can significantly reduce the amount of calculations needed to ensure the required accuracy by reducing the number of iterative procedures. The paper presents a detailed description of the algorithm, given the solution of the model problem and, as an example the task of modeling and optimization of the port berthing facilities – mooring wall of massive masonry with various add-ons. When composing mathematical equations, the method of limit States was used, which is generally accepted for calculating such port hydraulic structures. The choice of the optimum scheme was carried out by minimizing the cost of structure during limit conditions the reliability of structures when using existing construction materials. The most significant geometric dimensions of the structure were selected using independent parameters to be optimized.

A novel non linear signal prediction method is presented using non linear signal analysis and deterministic chaos techniques in combination with Radial Basis Functions (RBF) Neural Networks for diode resonator chaotic circuits, used in industrial processes, as well as for Magnetic Resonance Spectroscopy (MRS) processes. The Time series analysis is performed by the method proposed by Grasberger and Procaccia, involving estimation of the correlation and minimum embedding dimension as well as of the corresponding Kolmogorov entropy. These parameters are used to construct the first stage of a one step / multistep predictor while an RBF Artificial Neural Network (ANN) is involved in the second stage to enhance prediction results. The novelty of the proposed two stage predictor lies on that the RBF ANN is employed as a second order predictor, that is, as an error predictor of the non-linear signal analysis stage application. This novel two stage predictor is evaluated through an extensive experimental study for both resonator circuits for industrial processes as well as for MRS signals in a preliminary stage of analysis. Different types of Neural Networks are compared as well.

Author index of volume 147