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In this paper, a meshless Chebyshev collocation method (CCM) for the numerical solution of the eigenvalue problems of the Helmholtz equation is presented. The Chebyshev polynomials are employed for the efficient and accurate approximation of the eigenfunctions to ensure the pseudo-spectral convergence of the CCM. Two different approaches, namely the generalized eigenvalue approach and the standard eigenvalue approach, which convert the original eigenvalue problem into the generalized eigenvalue problem and the standard eigenvalue problem respectively, are implemented. Five benchmark numerical examples are presented and discussed to demonstrate the accuracy and efficiency of the proposed CCM.

Evolutionary computation has shown great performance in solving many multi-objective optimization problems; in many such algorithms, non-dominated sorting plays an important role in determining the relative quality of solutions in a population. However, the implementation of non-dominated sorting can be computationally expensive, especially for populations with a large number of solutions and many objectives. The main reason is that most existing non-dominated sorting algorithms need to compare one solution with almost all others to determine its front, and many of these comparisons are redundant or unnecessary. Another reason is that as the number of objectives increases, more and more candidate solutions become non-dominated solutions, and most existing time-saving approaches cannot work effectively. In this paper, we present a novel non-dominated sorting strategy, called Hierarchical Non-Dominated Sorting (HNDS). HNDS first sorts all candidate solutions in ascending order by their first objective. Then it compares the first solution with all others one by one to make a rapid distinction between different quality solutions, thereby avoiding many unnecessary comparisons. Experiments on populations with different numbers of solutions, different numbers of objectives and different problems have been done. The results show that HNDS has better computational efficiency than fast non-dominated sort, Arena’s principle and deductive sort.

A Guiding Evolutionary Algorithm (GEA) with greedy strategy for global optimization problems is proposed. Inspired by Particle Swarm Optimization, the Genetic Algorithm, and the Bat Algorithm, the GEA was designed to retain some advantages of each method while avoiding some disadvantages. In contrast to the usual Genetic Algorithm, each individual in GEA is crossed with the current global best one instead of a randomly selected individual. The current best individual served as a guide to attract offspring to its region of genotype space. Mutation was added to offspring according to a dynamic mutation probability. To increase the capability of exploitation, a local search mechanism was applied to new individuals according to a dynamic probability of local search. Experimental results show that GEA outperformed the other three typical global optimization algorithms with which it was compared.

A novel numerical model is developed for evaluating the temperature distribution on a two- dimensional skin tissue. The method is based on a combination of Laplace transform and meshless method as well as on the Pennes bio-heat equation. It is the first time to combine Laplace transform and RBF-MFS to study the performance of heat transfer in skin tissue governed by Pennes equation. First, Laplace transform of Pennes bio- heat equation is applied to handle the derivative of time variable. Then, the particular solution is approximated by a linear combination of radial basis functions, and the homogeneous solution is approximately determined by the method of fundamental solution. The multi-subdomain RBF-MFS technique is implemented for analyzing problems containing different materials and/or multi- connected regions. The last step of the algorithm is the inverse Laplace transforms from Laplace domain to time domain. The efficiency and availability of the proposed method is assessed by several examples including healthy tissue, tissue with tumour and burned tissue.

A hybrid graded element model is developed in this article for solving the heat conduction problem of nonlinear functionally graded materials (FGMs), whose material properties not only vary spatially but also are temperature dependent. In the proposed approach, both Kirchhoff transformation and iterative method are introduced to deal with the nonlinear term in the heat conduction equation of nonlinear FGMs. Then, the graded element is formulated based on two sets of independent temperature fields. One is the intra-element temperature field, which is defined within the element domain and constructed by a linear combination of fundamental solutions; the other is the frame field, which is defined on the element boundary only and used as the boundary interpolation functions of the element to ensure the field continuity over the inter-element boundary. This model can simulate the graded material properties naturally due to the inherent properties of fundamental solutions, which are employed in constructing the...

A novel hybrid graded element model is developed in this paper for investigating thermal behavior of functionally graded materials (FGMs). The model can handle a spatially varying material property field of FGMs. In the proposed approach, a new variational functional is first constructed for generating corresponding finite element model. Then, a graded element is formulated based on two sets of independent temperature fields. One is known as intra-element temperature field defined within the element domain; the other is the so-called frame field defined on the element boundary only. The intra-element temperature field is constructed using the linear combination of fundamental solutions, while the independent frame field is separately used as the boundary interpolation functions of the element to ensure the field continuity over the inter-element boundary. Due to the properties of fundamental solutions, the domain integrals appearing in the variational functional can be converted into boundary integrals which can significantly simplify the calculation of generalized element stiffness matrix. The proposed model can simulate the graded material properties naturally due to the use of the graded element in the finite element (FE) model. Moreover, it inherits all the advantages of the hybrid Trefftz finite element method (HT-FEM) over the conventional FEM and boundary element method (BEM). Finally, several examples are presented to assess the performance of the proposed method, and the obtained numerical results show a good numerical accuracy.

This paper presents a hybrid graded element model for the transient heat conduction problem in function- ally graded materials (FGMs). First, a Laplace transform approach is used to handle the time variable. Then, a fun- damental solution in Laplace space for FGMs is constructed. Next, a hybrid graded element is formulated based on the obtained fundamental solution and a frame field. As a result, the graded properties of FGMs are naturally reflected by us- ing the fundamental solution to interpolate the intra-element field. Further, Stefest's algorithm is employed to convert the results in Laplace space back into the time-space domain. Finally, the performance of the proposed method is assessed by several benchmark examples. The results demonstrate well the efficiency and accuracy of the proposed method.

A meshless model is developed to investigate behaviors of heat conduction in skin tissues. The model is established through combined use of Laplace transform (LT), radial basis function (RBF) and method of fundamental solution (MFS). The well-known Pennes bioheat model is used to describe the process of heat conduction in skin tissues. First, Laplace transform is applied to handle the time variable in the Pennes bioheat equation, which results in a generalized Helmholtz equation. Then, a RBF-MFS scheme is constructed to solve the Helmholtz equation in Laplace space. Further, Stefest's algorithm is employed to convert the results in Laplace space back into the time space. Finally, the performance of the proposed method is assessed by several benchmark examples. The results demonstrate desirable efficiency and accuracy of the proposed method.

Based on the radial basis functions (RBF) and T-Trefftz solution, this paper presents a new meshless method for numerically solving various partial differential equation systems. First, the analog equation method (AEM) is used to convert the original patial differential equation to an equivalent Poisson's equation. Then, the radial basis functions (RBF) are employed to approxiamate the inhomogeneous term, while the homogeneous solution is obtained by linear combination of a set of T-Trefftz solutions. The present scheme, named RBF-Trefftz has the advantage over the fundamental solution (MFS) method due to the use of nonsingular T-Trefftz solution rather than singular fundamental solutions, so it does not require the artificial boundary. The application and efficiency of the proposed method are validated through several examples which include different type of differential equations, such as Laplace equation, Hellmholtz equation, convectin-diffusion equation and time-dependent equation.