Numerical Solution of Linear Volterra Integral Equation Systems of Second Kind by Radial Basis Functions

In this paper we propose an approximation method for solving second kind Volterra integral equation systems by radial basis functions. It is based on the minimization of a suitable functional in a discrete space generated by compactly supported radial basis functions of Wendland type. We prove two convergence results, and we highlight this because most recent published papers in the literature do not include any. We present some numerical examples in order to show and justify the validity of the proposed method. Our proposed technique gives an acceptable accuracy with small use of the data, resulting also in a low computational cost.

Bow shape modification through multi-objective hydrodynamic optimization: Methodology comparison between CAD-based FreeForm Deformation and Mesh-based Radial Basis Function approach

Marine industrial engineering face crucial challenges because of environmental footprint of vehicles, global recession, construction, and operation cost. Meanwhile, Shape optimization is the key feature to improve ship efficiency and ascertain better design. Accordingly, the present paper proposes an automated optimization framework for ship hullform modification to reduce total resistance at two cruise and sprint speeds. The case study is a bow shape of a wave-piercing bow trimaran hull. To this end, a multi-objective hydrodynamic problem needs to be solved. A combined optimization strategy using CFD hullform optimization is presented using the software tools STAR-CCM+ and SHERPA algorithm as optimizer. Furthermore, a comparison is made between CAD-based and Mesh-based parametrization techniques. Comparison between geometry regeneration methods is performed to present a practical and efficient parametrization tool. Design variables are control points of FreeForm Deformation (FFD) for CAD-based method and Radial Basis Function (RBF) for Mesh-based method. The optimization results show a 4.77% and 2.47% reduction in the total resistance at cruise and sprint speed, respectively.

A Novel Radial Basis Function Neural Network with High Generalization Performance for Nonlinear Process Modelling

A radial basis function neural network (RBFNN), with a strong function approximation ability, was proven to be an effective tool for nonlinear process modeling. However, in many instances, the sample set is limited and the model evaluation error is fixed, which makes it very difficult to construct an optimal network structure to ensure the generalization ability of the established nonlinear process model. To solve this problem, a novel RBFNN with a high generation performance (RBFNN-GP), is proposed in this paper. The proposed RBFNN-GP consists of three contributions. First, a local generalization error bound, introducing the sample mean and variance, is developed to acquire a small error bound to reduce the range of error. Second, the self-organizing structure method, based on a generalization error bound and network sensitivity, is established to obtain a suitable number of neurons to improve the generalization ability. Third, the convergence of this proposed RBFNN-GP is proved theoretically in the case of structure fixation and structure adjustment. Finally, the performance of the proposed RBFNN-GP is compared with some popular algorithms, using two numerical simulations and a practical application. The comparison results verified the effectiveness of RBFNN-GP.