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We present a new localized boundary knot method (LBKM) for the solution of large-scale partial differential equations based on the non-singular general solutions. The proposed local algorithm only requires several discrete nodes i...
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We present a new localized boundary knot method (LBKM) for the solution of large-scale partial differential equations based on the non-singular general solutions. The proposed local algorithm only requires several discrete nodes inside the physical domain and along its boundary without any mesh. For every node, a local subdomain with a simple geometry can be firstly determined via Euclidean distance between nodes. And then the unknown variable at every node can be expressed as a linear combination of function values at nodes inside its corresponding local subdomain. Finally, a sparse linear system can be formed by using the governing equation and the corresponding boundary conditions. In our computations, two different-types of LBKM formulations have been proposed for deriving the final sparse linear system. One is based on the inverse matrix calculation and the other is based on the moving least square (MLS) technique. Unlike the traditional boundary knot method (BKM) with the "global" discretization, the present LBKM is a local discretization method and requires less computer time and storage due to its feature of sparse and banded matrix, which makes it more suitable for solving large-scale problems. Preliminary numerical experiments are carried out in the large-scale acoustics models and bimaterial problems. The comparisons of proposed two types of LBKM formulations are also investigated. The results show the accuracy, validity and great promising applications of the proposed LBKM for the simulation of large-scale problems in complicated geometries. (C) 2019 Elsevier B.V. All rights reserved.
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In this paper we propose a novel two-stage method to solve the three-dimensional Poisson equation in an arbitrary bounded domain enclosed by a smooth boundary. The solution is decomposed into a particular solution and a homogeneou...
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In this paper we propose a novel two-stage method to solve the three-dimensional Poisson equation in an arbitrary bounded domain enclosed by a smooth boundary. The solution is decomposed into a particular solution and a homogeneous solution. In the first stage a multiple-scale polynomial method (MSPM) is used to approximate the forcing term and then the formula of Tsai et al. [Tsai, Cheng, and Chen (2009)] is used to obtain the corresponding closed-form solution for each polynomial term. Then in the second stage we use a multiple/scale/direction Trefftz method (MSDTM) to find the solution of Laplace equation, of which the directions are uniformly distributed on a unit circle S-1, and the scales are determined a priori by the collocation points on boundary. Two examples of 3D data interpolation, and several numerical examples of direct and inverse Cauchy problems in complex domain confirm the efficiency of the MSPM and the MSDTM.
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We solve the inverse Cauchy problem of elliptic type partial differential equations in an arbitrary 3D closed walled shell for recovering unknown data on an inner surface, with the over-specified Cauchy boundary conditions given o...
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We solve the inverse Cauchy problem of elliptic type partial differential equations in an arbitrary 3D closed walled shell for recovering unknown data on an inner surface, with the over-specified Cauchy boundary conditions given on an outer surface. We first derive a homogenization function in the 3D domain to annihilate the Dirichlet as well as the Neumann data on the outer surface. Then, we can transform the inverse Cauchy problem to solve a direct problem inside the closed walled shell, using the homogenization technique and the domain type collocation method. The boundary functions are constructed from the 3D Pascal polynomials multiplied by an elementary boundary function, which are adopted as the bases to expand the numerical solution of the transformed elliptic equation. A simple scaling regularization is employed to reduce the condition number of the linear system to determine the expansion coefficients. Several numerical examples are presented to show that the novel method can overcome the highly ill-posed property of the inverse Cauchy problem in the 3D closed walled shell. The proposed algorithm is robust against large noise up toand very time saving to obtain accurate solution.
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In this paper we study the application of the method of fundamental solutions (MFS) to interface crack analysis in linear elastic bimaterial fracture mechanics. Such problem presents some modelling difficulties because of the osci...
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In this paper we study the application of the method of fundamental solutions (MFS) to interface crack analysis in linear elastic bimaterial fracture mechanics. Such problem presents some modelling difficulties because of the oscillatory behavior of the local asymptotic fields in the neighborhood of the interface crack-tip. The present approach is based on the combination of the classical MFS approximation for linear elasticity problems and a set of enrichment functions that take into account the asymptotic behavior of the near-tip displacement and stress fields. The enriched MFS technique automatically incorporates the oscillatory crack-tip behavior and thus can significantly improve the computational accuracy of the displacements and stresses in the vicinity of the cracktip, even with a relatively coarse MFS model. A multi-domain MFS technique and the displacement extrapolation method (DEM) are used to compute the complex stress intensity factor (SIF) of the cracked bimaterials. Several benchmark numerical examples are presented to illustrate the accuracy and efficiency of the present method. Results calculated by using the boundary element (BEM) method are also given for the purpose of comparison.
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The transient heat conduction problems in layered materials are common encountered in practical application, but bring great challenge to most numerical algorithms. In this study, the singular boundary method (SBM), a meshless bou...
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The transient heat conduction problems in layered materials are common encountered in practical application, but bring great challenge to most numerical algorithms. In this study, the singular boundary method (SBM), a meshless boundary collocation technique, is first employed to numerically simulate such problems based on the time-dependent fundamental solution of diffusion equation. Taking the boundary conditions and interface conditions into account, the computing system of the SBM for layered materials is established, and then is solved. The proposed method fully inherits the merits of conventional boundary-type methods while possessing its distinctive advantages. Furthermore, it is simple, straightforward, computationally efficient, and stable since it does not need to discretize temporal derivative term. Three numerical experiments are performed to verify the efficiency and accuracy of the proposed scheme. Numerical results clearly indicate the efficiency, accuracy and stability of the presented SBM for solving transient heat conduction problems in layered materials. (C) 2019 Elsevier Ltd. All rights reserved.
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Various non-local structural derivative diffusion models have been proposed based on different kernel functions to describe the anomalous time dependence of the mean-squared displacements. In the present study, the fundamental sol...
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Various non-local structural derivative diffusion models have been proposed based on different kernel functions to describe the anomalous time dependence of the mean-squared displacements. In the present study, the fundamental solutions for constant and variable-order structural derivative advection-dispersion models are achieved via scaling transformation and the generalized non-Euclidean Hausdorff fractal distance. Comparative numerical investigations of the structural derivative models have been conducted to reveal the influences of various kernels via the meshless method of fundamental solutions. Numerical results verify the validity of the derived fundamental solutions and the rationality of the employed numerical method for structural derivative advection-dispersion models.
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In this paper, we solve two types of inverse heat source problems: one recovers an unknown space-dependent heat source without using initial value, and another recovers both the unknown space-dependent heat source and the initial ...
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In this paper, we solve two types of inverse heat source problems: one recovers an unknown space-dependent heat source without using initial value, and another recovers both the unknown space-dependent heat source and the initial value. Upon inserting the adjoint Trefftz test functions into Green's second identity, we can retrieve the unknown space-dependent heat source by an expansion method whose expansion coefficients are derived in closed form. We assess the stability of the closed-form expansion coefficients method by using the condition numbers of coefficients matrices. Then, numerical examples are performed, which demonstrates that the closed-form expansion coefficient method is effective and stable even when it imposes a large noise on the final time data. Next, we develop a coupled iterative scheme to recover the unknown heat source and initial value simultaneously, under two over specified temperature data at two different times. A simple regularization technique is derived to overcome the highly ill-posed behavior of the second inverse problem, of which the convergence rate and stability are examined. This results in quite accurate numerical results against large noise.
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This paper presents a local non-singular knot method (LNKM) to accurately solve the large-scale acoustic problems in complicated geometries. The LNKM is a domain-type meshless collocation method, which relies only on scattered nod...
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This paper presents a local non-singular knot method (LNKM) to accurately solve the large-scale acoustic problems in complicated geometries. The LNKM is a domain-type meshless collocation method, which relies only on scattered nodes. Firstly, a series of subdomains corresponding to every nodes can be searched based on the Euclidean distance between nodes. To each subdomain, a small linear system can be yielded by using the non-singular general solutions of Helmholtz-type equations. Secondly, the unknown variables at every nodes can be explicitly expressed by the function values at their corresponding supporting nodes. Finally, a large sparse system of linear equations is formed and solved to obtain the numerical solutions of physical quantities at every nodes. The proposed LNKM is mathematically simple, numerically accurate, and more applicable to the large-scale computation. Four numerical examples conform its effectiveness and accuracy for the large-scale computation of Helmholtz-type equations in complicated geometries. (C) 2021 Elsevier Ltd. All rights reserved.
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In the paper a nonlinear inverse Cauchy problem of nonlinear elliptic type partial differential equation in an arbitrary doubly-connected plane domain is solved using a novel meshless numerical method. The unknown Dirichlet data o...
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In the paper a nonlinear inverse Cauchy problem of nonlinear elliptic type partial differential equation in an arbitrary doubly-connected plane domain is solved using a novel meshless numerical method. The unknown Dirichlet data on an inner boundary are recovered by over-specifying the Cauchy data on an outer boundary. A homogenization function is derived to annihilate the Cauchy data on the outer boundary, and then a homogenization technique generates a transformed equation in terms of a transformed variable, whose outer Cauchy boundary conditions are homogeneous. When the numerical solution is expanded by a sequence of boundary functions, which automatically satisfy the homogeneous Cauchy boundary conditions on the outer boundary, we can solve the transformed equation by the domain type meshless collocation method. For the nonlinear inverse Cauchy problems we require to iteratively solve the linear systems with the righth- and sides varying per iteration step. The accuracy and robustness of the homogenization boundary function method (HBFM) are examined through seven numerical examples, where we compare the exact Dirichlet data on the inner boundary to the ones recovered by the HBFM under a large noisy disturbance. (C) 2018 Elsevier Ltd. All rights reserved.
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In this paper, a topology optimization model is proposed for non-Fourier heat conduction design. In this model, the finite element discretization scheme and the Wilson-theta time discretization method are combined to analyze non-F...
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In this paper, a topology optimization model is proposed for non-Fourier heat conduction design. In this model, the finite element discretization scheme and the Wilson-theta time discretization method are combined to analyze non-Fourier transient heat conduction that is represented by the Cattaneo-Vernotte equation with a relaxation term. Based on the solid isotropic material with penalization (SIMP) interpolation model, the mathematical statement of the proposed optimization design is formulated by integrating the transient objective function over the time interval that considers thermal dissipation energy minimization. The adjoint variable method for sensitivity analysis and the method of moving asymptotes (MMA) for solving the optimization problem are discussed as well. Numerical examples illustrate the validity and applicability of the proposed non-Fourier heat conduction topology optimization by comparison with transient Fourier heat conduction design.
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