Meshfree Boundary Research Articles

Steady State Heat Conduction in Exchanger Tubes by Using the Meshfree Boundary Integral Equation Method: Conduction Shape Factor and Degenerate Scale

AbstractRegarding the steady-state heat conduction problem in exchanger tubes, the meshfree boundary integral equation method is employed to determine the conduction shape factor in this paper. Different from the conventional boundary element method, the present method is free of mesh generation. After using the parametric function to represent the boundary contour and adopting the Gaussian quadrature, only collocating points on the boundary is required to obtain the linear algebraic equations. By introducing the local exact solution, the singular integral in the sense of the Cauchy principal value can be novelly determined. In addition, the boundary layer effect due to the nearly singular integral in the boundary integral equation can be dealt with. Two cases of different boundary conditions are considered. One is the isothermal condition on both the inner and outer surfaces. The other is the isothermal condition on the inner surface and the convection condition on the outer surface. Besides, numerical instability is found and the nonuniqueness solution due to the degenerate scale is examined by calculating the conduction shape factor and the temperature on the outer surface.

Aeroelastic Topology Optimization of Wing Structure Based on Moving Boundary Meshfree Method

The increasing structural flexibility of large aircraft leads to significant aeroelastic effects. More efficient topology optimization techniques are required for the design to further take advantage of aeroelasticity and obtain lightweight structures. This paper proposes a moving boundary meshfree topology optimization that combines the Galerkin method of weighted residuals and non-uniform rational B-splines (NURBS). The solution domain is described by the control points of NURBS and its property is calculated adaptively with an integration subtraction technique. The minimal compliance is searched for using the globally convergent method of moving asymptotes (GCMMA) by designing the locations of control points as subject to volume and flux constraints. The method is first applied to a typical two-dimensional design example with symmetric boundary conditions. The results show that the shape constraints can be conveniently applied, and smoother boundaries are obtained with fewer parameters. Then, a three-dimensional wing structure with asymmetric boundary conditions is optimized. A three-dimensional flight load that combines the high-order-panel and meshfree methods is employed to calculate the elastic loads and update asymmetric external loads during the optimization process. The designed wing satisfies engineering requirements and the presented method can solve the practical topology optimization problems of three-dimensional structures.

Automatic mesh-free boundary analysis: Multi-objective optimization

The paper is concerned with the numerical solution of two-dimensional potential problems, through a mesh-free boundary model in a multi-objective optimization framework that automatically generates Pareto-optimal mesh-free discretization arrangements. This robust new strategy of analysis allows for simultaneously improving the solution accuracy, the conditioning of the numerical solver, the stability and efficiency of the mesh-free analysis.The boundary mesh-free model (BMFM) is built on the boundary integral equation of the Laplace potential, with a moving least squares (MLS) approximation of variables. The model considers independent MLS approximations in each boundary segment and performs integration with standard numerical quadrature.The main novelty of the paper is the automatic generation of Pareto-optimal nodal arrangements and corresponding compact supports of the mesh-free boundary model, by means of an evolutionary multi-objective optimization process, based on genetic algorithms, which uses reliable very efficient objective functions.A benchmark problem is presented to assess the accuracy and efficiency of the modeling strategy. The remarkably accurate results obtained, in perfect agreement with those of analytical solutions, make very reliable this robust new strategy of automatic mesh-free boundary analysis in a multi-objective optimization framework.

A new boundary meshfree method for calculating the multi-domain constant coefficient heat conduction with a heat source problem

ABSTRACTA new high-precision boundary meshfree method, namely virtual boundary meshfree Galerkin method (VBMGM), for calculating the multi-domain constant coefficient heat conduction with a heat source problem is given. In the paper, the radial basis function interpolation is used to solve the virtual source function of virtual boundary and the heat source within each subdomain. Simultaneously, the equation of VBMGM for multi-domain constant coefficient heat conduction with a heat source problem is obtained by the Galerkin method. Therefore, the proposed method has common advantages of the boundary element method, meshfree method, and Galerkin method. Coefficient matrix of this specific expression is symmetrical and the specific expression of VBMGM for the multi-domain constant coefficient heat conduction with a heat source problem is given. Two numerical examples are given. The numerical results are also compared with other numerical methods. The accuracy and feasibility of the method for the multi-domain constant coefficient heat conduction with a heat source problem are proved.

Free vibration analysis of two-dimensional functionally graded coated and undercoated substrate structures

In this paper, the free vibration behaviors of the functionally graded (FG) coated and undercoated substrate structures are studied by a meshfree boundary–domain integral equation method. Based on the two-dimensional elasticity theory, the boundary–domain integral equations for each single layer of these coating–substrate structures are derived initially by using elastostatic fundamental solutions. Employ the radial integration method to transform the domain integrals into boundary integrals and achieve a meshfree scheme. By applying the multi-region boundary element method, obtain the generalized eigenvalue system of the whole structure, which involves system matrices with boundary integrals only and the complete solutions for natural frequency and vibration modes are rigidly resolved. A comparative study of FG versus homogeneous coating is conducted. The influences of material composition, material gradient, coating thickness ratio, substrate structure aspect ratio and the boundary conditions on the natural frequencies of the FG coated and undercoated substrate structure are evaluated and discussed.

Free vibration analysis of two-dimensional functionally graded structures by a meshfree boundary–domain integral equation method

Free vibration of two-dimensional functionally graded structures with an exponential material gradation is analyzed in this paper by a meshfree boundary–domain integral equation method. Based on the two-dimensional elasticity theory, boundary–domain integral equations are derived by using elastostatic fundamental solutions. Due to the material inhomogeneity and inertial effect, two domain integrals emerge in the boundary–domain integral equation formulation. Radial integration method is employed to convert the domain integrals into boundary integrals. A meshfree scheme is achieved through approximating the normalized displacements in the domain integrals by a combination of the radial basis functions and the polynomials. Thus, the free vibration problem is reduced into a generalized eigenvalue problem, which involves system matrices with boundary integrals only. By using the developed meshfree boundary–domain integral equation method, free vibration of two-dimensional exponentially graded beams and plates with various material gradients, gradation directions, boundary conditions and aspect ratios is investigated, which demonstrates the high convergence, efficiency and accuracy of the present method.

A moving Kriging interpolation-based boundary node method for two-dimensional potential problems

In this paper, a meshfree boundary integral equation (BIE) method, called the moving Kriging interpolation-based boundary node method (MKIBNM), is developed for solving two-dimensional potential problems. This study combines the BIE method with the moving Kriging interpolation to present a boundary-type meshfree method, and the corresponding formulae of the MKIBNM are derived. In the present method, the moving Kriging interpolation is applied instead of the traditional moving least-square approximation to overcome Kronecker's delta property, then the boundary conditions can be imposed directly and easily. To verify the accuracy and stability of the present formulation, three selected numerical examples are presented to demonstrate the efficiency of MKIBNM numerically.

A truly boundary-only meshfree method for inhomogeneous problems based on recursive composite multiple reciprocity technique

This study presents a novel technique called the recursive composite multiple reciprocity method (RC-MRM), to develop a truly boundary-only meshfree boundary particle method (BPM) for general inhomogeneous problems. It does not require any inner nodes to evaluate the particular solution, and thus it is a truly boundary-only numerical method. “Composite” in the RC-MRM implies that the RC-MRM employs a high-order composite differential operator rather than a high-order Laplacian operator in the standard MRM to annihilate inhomogeneous term of various types and enables the present BPM to handle a much wider variety of inhomogeneous problems, while the “recursive” algorithm in the RC-MRM significantly reduces CPU time and storage requirements of the original MRM. In addition, we also find high-order harmonic solutions of the Laplacian operator. Numerical illustrations reveals that the present BPM has rapid convergence, high accuracy and efficiency, and mathematical simplicity, through various two- and three-dimensional benchmark problems.

431 メッシュフリー境界節点法による三次元表面き裂の進展解析(OS10.メッシュフリー/粒子法とその関連技術(7),オーガナイズドセッション)

431 メッシュフリー境界節点法による三次元表面き裂の進展解析(OS10.メッシュフリー/粒子法とその関連技術(7),オーガナイズドセッション)

Quantum mushroom billiards

We report the first large-scale statistical study of very high-lying eigenmodes (quantum states) of the mushroom billiard proposed by L. A. Bunimovich [Chaos 11, 802 (2001)]. The phase space of this mixed system is unusual in that it has a single regular region and a single chaotic region, and no KAM hierarchy. We verify Percival's conjecture to high accuracy (1.7%). We propose a model for dynamical tunneling and show that it predicts well the chaotic components of predominantly regular modes. Our model explains our observed density of such superpositions dying as E(-1/3) (E is the eigenvalue). We compare eigenvalue spacing distributions against Random Matrix Theory expectations, using 16,000 odd modes (an order of magnitude more than any existing study). We outline new variants of mesh-free boundary collocation methods which enable us to achieve high accuracy and high mode numbers (approximately 10(5)) orders of magnitude faster than with competing methods.

Trefftz interpolation based multi-domain boundary point method

The paper describes meshfree boundary point solution of problems based on Trefftz interpolation functions. The solution is a collocation method, which can use any type of Trefftz functions smooth enough for approximation of domain fields. Polynomial, Kelvin and Boussinesq functions are used in our formulations. A multi-domain boundary formulation with continuity of tractions and displacements over discrete points of the inter-domain surface is shown. The method is applied to linear elastostatics. 2D and 3D examples illustrate the formulation. Discussion about advantages and drawbacks is included. As the conservation equations (force and moment equilibrium) are not satisfied in integral sense, some benchmark tests are proposed for meshless formulations to obtain a view about the accuracy.

Boundary meshfree methods based on the boundary point interpolation methods

A group of meshfree methods based on Boundary Integral Equation have been developed in order to overcome drawbacks in the conversional boundary element method (BEM) that requires boundary elements in constructing shape functions. In this paper, the radial basis point interpolation is firstly used to formulate the boundary radial point interpolation method (BRPIM). Two boundary meshfree methods: the boundary point interpolation method (BPIM) using the polynomial PIM and the BRPIM, are then examined in detail. The numerical implementations of these two methods are studied to address several technical issues, including the size of the compact support domain, the convergence, the performance, and so on. These two boundary-type meshfree methods are also compared with the Boundary Node Method and the conventional BEM in terms of both efficiency and performance. Several numerical examples of 2D elastostatics are analyzed using BPIM and BRPIM. It is found that BPIM and BRPIM are very easy to implement, and very robust for obtaining numerical solutions for problems of computational mechanics with very good accuracy. Key issues related the future development of boundary meshfree methods are also discussed.

Meshfree boundary particle method applied to Helmholtz problems

This paper is concerned with the boundary particle method (BPM), a new boundary-only radial basis function collocation schemes. The method is developed based on the multiple reciprocity principle and applying either high-order nonsingular general solutions or singular fundamental solutions as the radial basis function. Like the multiple reciprocity BEM (MR-BEM), the BPM does not require any inner nodes for inhomogeneous problems and therefore is a truly boundary-only technique. On the other hand, unlike the MR-BEM, the BPM is meshfree, integration-free, symmetric, and mathematically simple technique. In particular, the method requires much less computational effort for the discretization than the MR-BEM. In this study, the accuracy and efficiency of the BPM are numerically demonstrated in some 2D inhomogeneous Helmholtz problems under complicated geometries.

Hierarchical enrichment for bridging scales and mesh-free boundary conditions

The finite-element method, when used with a basis made up of piecewise polynomials, often requires the generation of a very fine computational mesh in order to capture localized solution phenomena such as boundary layers or near-singularities. Enrichment of the basis with additional functions, obtained through analytical or experimental means, can allow for a coarser mesh and more accurate solution. We introduce an enrichment scheme in which an interaction or ‘bridging’ scale term is used to separate the basis formed by the enrichment functions from the original set of basis functions, in effect making the enrichment hierarchical. This separation of scales allows the simple application of essential boundary conditions. It also allows a quantification of the effects of the enrichment, leading to strategies for error estimation and control of the stiffness matrix condition number. We also find that this formulation allows for the simple application of essential boundary conditions for mesh-free shape functions, which are notoriously problematic. We find that for multiple dimensions, care must be taken to derive a weak form which is truly consistent with the strong form on the essential boundary. Copyright © 2001 John Wiley & Sons, Ltd.