scholarly journals Stress analysis of elastic bi-materials by using the localized method of fundamental solutions

2021 ◽  
Vol 7 (1) ◽  
pp. 1257-1272
Author(s):  
Juan Wang ◽  
◽  
Wenzhen Qu ◽  
Xiao Wang ◽  
Rui-Ping Xu ◽  
...  
Keyword(s):  
Stress Analysis ◽  
Large Scale ◽  
Classical Method ◽  
Fundamental Solutions ◽  
Least Square ◽  
Collocation Methods ◽  
Method Of Fundamental Solutions ◽  
Computational Domain ◽  
Desktop Computer ◽  
Moving Least Square

<abstract> <p>The localized method of fundamental solutions belongs to the family of meshless collocation methods and now has been successfully tried for many kinds of engineering problems. In the method, the whole computational domain is divided into a set of overlapping local subdomains where the classical method of fundamental solutions and the moving least square method are applied. The method produces sparse and banded stiffness matrix which makes it possible to perform large-scale simulations on a desktop computer. In this paper, we document the first attempt to apply the method for the stress analysis of two-dimensional elastic bi-materials. The multi-domain technique is employed to handle the non-homogeneity of the bi-materials. Along the interface of the bi-material, the displacement continuity and traction equilibrium conditions are applied. Several representative numerical examples are presented and discussed to illustrate the accuracy and efficiency of the present approach.</p> </abstract>

The Method of Fundamental Solutions for Solving Exterior Axisymmetric Helmholtz Problems with High Wave-Number

2013 ◽  
Vol 5 (04) ◽  
pp. 477-493 ◽  
Author(s):  
Wen Chen ◽  
Ji Lin ◽  
C.S. Chen
Keyword(s):  
Large Scale ◽  
Three Dimensional ◽  
Matrix Decomposition ◽  
Coefficient Matrix ◽  
Fundamental Solutions ◽  
Method Of Fundamental Solutions ◽  
Wave Number ◽  
High Wave Number ◽  
Memory Space ◽  
High Wave

AbstractIn this paper, we investigate the method of fundamental solutions (MFS) for solving exterior Helmholtz problems with high wave-number in axisymmetric domains. Since the coefficient matrix in the linear system resulting from the MFS approximation has a block circulant structure, it can be solved by the matrix decomposition algorithm and fast Fourier transform for the fast computation of large-scale problems and meanwhile saving computer memory space. Several numerical examples are provided to demonstrate its applicability and efficacy in two and three dimensional domains.

Fast Solution for Solving the Modified Helmholtz Equation withthe Method of Fundamental Solutions

2015 ◽  
Vol 17 (3) ◽  
pp. 867-886 ◽  
Author(s):  
C. S. Chen ◽  
Xinrong Jiang ◽  
Wen Chen ◽  
Guangming Yao
Keyword(s):  
Helmholtz Equation ◽  
Large Scale ◽  
Solution Process ◽  
Homogeneous Solution ◽  
Fundamental Solutions ◽  
Method Of Fundamental Solutions ◽  
Dense Matrix ◽  
Modified Helmholtz Equation ◽  
Large Scale Problems ◽  
Particular Solution

AbstractThe method of fundamentalsolutions (MFS)is known as aneffective boundary meshless method. However, the formulation of the MFS results in a dense and extremely ill-conditioned matrix. In this paper we investigate the MFS for solving large-scale problems for the nonhomogeneous modified Helmholtz equation. The key idea is to exploit the exponential decay of the fundamental solution of the modified Helmholtz equation, and consider a sparse or diagonal matrix instead of the original dense matrix. Hence, the homogeneous solution can be obtained efficiently and accurately. A standard two-step solution process which consists of evaluating the particular solution and the homogeneous solution is applied. Polyharmonic spline radial basis functions are employed to evaluate the particular solution. Five numerical examples in irregular domains and a large number of boundary collocation points are presented to show the simplicity and effectiveness of our approach for solving large-scale problems.

scholarly journals Free surface groundwater flow solution using boundary collocation methods

2018 ◽  
Vol 196 ◽  
pp. 03026 ◽  
Author(s):  
Juraj Mužík ◽  
Roman Bulko
Keyword(s):  
Diffusion Equation ◽  
Groundwater Flow ◽  
Moving Boundary ◽  
Numerical Algorithms ◽  
Type Equation ◽  
Fundamental Solutions ◽  
Collocation Methods ◽  
Method Of Fundamental Solutions ◽  
Singular Boundary ◽  
State Diffusion

In this paper, two meshless numerical algorithms are developed for the solution of two-dimensional steady-state diffusion equation that describes the stationary groundwater flow. The proposed numerical methods, which are truly meshless, quadrature-free and boundary only, are based on the method of fundamental solutions and singular boundary method respectively. The diffusion equation is transformed into a Poisson-type equation with a known fundamental solution. Numerical examples with moving boundary are presented and compared to the solutions obtained by the finite element method.

Solution of Grad–Shafranov equation by the method of fundamental solutions

2014 ◽  
Vol 80 (3) ◽  
pp. 477-494 ◽  
Author(s):  
D. Nath ◽  
M. S. Kalra
Keyword(s):  
Singular Integrals ◽  
Fundamental Solutions ◽  
Method Of Fundamental Solutions ◽  
Computational Domain ◽  
Tokamak Plasmas ◽  
Shafranov Equation ◽  
The Individual ◽  
The Right ◽  
Physical Boundary ◽  
Good Agreement

In this paper we have used the Method of Fundamental Solutions (MFS) to solve the Grad–Shafranov (GS) equation for the axisymmetric equilibria of tokamak plasmas with monomial sources. These monomials are the individual terms appearing on the right-hand side of the GS equation if one expands the nonlinear terms into polynomials. Unlike the Boundary Element Method (BEM), the MFS does not involve any singular integrals and is a meshless boundary-alone method. Its basic idea is to create a fictitious boundary around the actual physical boundary of the computational domain. This automatically removes the involvement of singular integrals. The results obtained by the MFS match well with the earlier results obtained using the BEM. The method is also applied to Solov'ev profiles and it is found that the results are in good agreement with analytical results.

Fast sparse image reconstruction method in through-the-wall radars using limited memory Broyden–Fletcher–Goldfarb–Shanno algorithm

2021 ◽  
pp. 1-11
Author(s):  
Candida Mwisomba ◽  
Abdi T. Abdalla ◽  
Idrissa Amour ◽  
Florian Mkemwa ◽  
Baraka Maiseli
Keyword(s):  
Image Reconstruction ◽  
Large Scale ◽  
Optimization Problems ◽  
Classical Method ◽  
Computational Cost ◽  
Least Square ◽  
Computational Time ◽  
Reconstruction Method ◽  
Nonlinear Methods ◽  
Limited Memory

Abstract Compressed sensing allows recovery of image signals using a portion of data – a technique that has drastically revolutionized the field of through-the-wall radar imaging (TWRI). This technique can be accomplished through nonlinear methods, including convex programming and greedy iterative algorithms. However, such (nonlinear) methods increase the computational cost at the sensing and reconstruction stages, thus limiting the application of TWRI in delicate practical tasks (e.g. military operations and rescue missions) that demand fast response times. Motivated by this limitation, the current work introduces the use of a numerical optimization algorithm, called Limited Memory Broyden–Fletcher–Goldfarb–Shanno (LBFGS), to the TWRI framework to lower image reconstruction time. LBFGS, a well-known Quasi-Newton algorithm, has traditionally been applied to solve large scale optimization problems. Despite its potential applications, this algorithm has not been extensively applied in TWRI. Therefore, guided by LBFGS and using the Euclidean norm, we employed the regularized least square method to solve the cost function of the TWRI problem. Simulation results show that our method reduces the computational time by 87% relative to the classical method, even under situations of increased number of targets or large data volume. Moreover, the results show that the proposed method remains robust when applied to noisy environment.

APPLICATION OF THE METHOD OF FUNDAMENTAL SOLUTIONS FOR THE INVERSE PROBLEM OF DETERMINATION OF THE BIOT NUMBER

2013 ◽  
Vol 10 (02) ◽  
pp. 1341002 ◽  
Author(s):  
J. A. KOŁODZIEJ ◽  
M. MIERZWICZAK
Keyword(s):  
Heat Conduction ◽  
Biot Number ◽  
Heat Conduction Problem ◽  
Fundamental Solutions ◽  
Least Square ◽  
Method Of Fundamental Solutions ◽  
Nonlinear System Of Equations ◽  
Marquardt Algorithm ◽  
Steady State Heat

This paper deals with the iterative inverse determination of the Biot number in the 2D steady-state heat conduction problem. The identification of the value of the Biot number is obtained by using the boundary data and additionally from the knowledge of the temperature inside the domain. The method of fundamental solutions is used to solve the 2D heat conduction problem. The nonlinear system of equations obtained from collocation is solved in a least square sense by using the Levenberg–Marquardt algorithm.

A FAST METHOD OF FUNDAMENTAL SOLUTIONS FOR SOLVING HELMHOLTZ-TYPE EQUATIONS

2013 ◽  
Vol 10 (02) ◽  
pp. 1341008 ◽  
Author(s):  
XINRONG JIANG ◽  
WEN CHEN ◽  
C. S. CHEN
Keyword(s):  
Large Scale ◽  
Fast Multipole Method ◽  
Coefficient Matrix ◽  
Fundamental Solutions ◽  
Memory Storage ◽  
Method Of Fundamental Solutions ◽  
Computational Time ◽  
Fast Method ◽  
Dense Matrix ◽  
Fast Multipole

The method of fundamental solution (MFS) has been known as a simple and effective boundary meshless method. However, the MFS generates dense square coefficient matrix and thus requires a large amount of computational time and memory storage for solving large-scale problems. The fast multipole method (FMM) is a technique that reduces computational operations and storage requirements for solving such dense matrix equations. This study makes the first attempt to apply the FMM to the MFS calculation of acoustic problems, where the operations are reduced to O(N log N) while O(N2) operations are required for the traditional MFS using the standard iterative methods. Numerical examples with up to 100,000 points are successfully tested on a desktop personal computer. Our results clearly demonstrate efficiency and accuracy of the fast multipole MFS for solving large-scale Helmholtz-type problems.

Studying the deployment of high-lift devices by using dynamic immersed boundaries

2021 ◽  
Vol ahead-of-print (ahead-of-print) ◽  
Author(s):  
Francesco Capizzano ◽  
Triyantono Sucipto
Keyword(s):  
Research Effort ◽  
Logical Consequence ◽  
Least Square ◽  
Immersed Boundary ◽  
Computational Domain ◽  
Time Step ◽  
High Lift ◽  
Content Type ◽  
Moving Least Square ◽  
Ib Method

Purpose This paper aims to describe a research effort towards the comprehension of the unsteady phenomena due to the deployment of high-lift devices at approach/landing conditions. Design/methodology/approach The work starts from a preexisting platform based on an immersed boundary (IB) method whose capabilities are extended to study compressible and viscous flows around moving/deforming objects. A hybrid Lagrangian-Eulerian approach is designed to consider the motion of multiple bodies through a fixed Cartesian mesh. That is, the cells’ volumes do not move in space but rather they observe the solid walls crossing themselves. A dynamic discrete forcing makes use of a moving least-square procedure which has been validated by simulating well-known benchmarks available for rigid body motions. Partitioned fluid-structure interactions (FSI) strategies are explored to consider aeroelastic phenomena. A shared platform, between the aerodynamic and the structural solvers, fulfils the loads’ transfer and drives the sequence of the operating steps. Findings The first part of the results is devoted to a basic two-dimensional study aiming at evaluating the accuracy of the method when simple rigid motions are prescribed. Afterwards, the paper discusses the solution obtained when applying the dynamic IB method to the rigid deployment of a Krueger-flap. The final section discusses the aeroelastic behaviour of a three-element airfoil during its deployment phase. A loose FSI coupling is applied for estimating the possible loads’ downgrade. Research limitations/implications The IB surfaces are allowed to move less than one IB-cell size at each time-step de-facto restricting the Courant-Friedrichs-Lewy (CFL) based on the wall velocity to be smaller than unity. The violation of this constraint would impair the explicit character of the method. Practical implications The proposed method improves automation in FSI numerical analysis and relaxes the human expertise/effort for meshing the computational domain around complex three-dimensional geometries. The logical consequence is an overall speed-up of the simulation process. Originality/value The value of the paper consists in demonstrating the applicability of dynamic IB techniques for studying high-lift devices. In particular, the proposed Cartesian method does not want to compete with body-conforming ones whose accuracy remains generally superior. Rather, the merit of this research is to propose a fast and automatic simulation system as a viable alternative to classic multi-block structured, chimaera or unstructured tools.

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