Showing papers in "Engineering Analysis With Boundary Elements in 2008"
TL;DR: In this article, a meshless numerical scheme for solving the inverse heat source problem is proposed, which is developed by using the fundamental solution of the heat equation as a basis function, and a regularized solution is obtained by employing the Tikhonov regularization method, while the choice of the regularization parameter is based on generalized cross-validation criterion.
Abstract: In this paper a meshless numerical scheme for solving the inverse heat source problem is proposed. The numerical solution is developed by using the fundamental solution of the heat equation as a basis function. Since the resulting matrix equation is badly ill-conditioned, a regularized solution is obtained by employing the Tikhonov regularization method, while the choice of the regularization parameter is based on generalized cross-validation criterion. Both continuous and discontinuous numerical examples are given to verify the efficiency and accuracy of the proposed computational method.
176 citations
TL;DR: This paper proposes a simple and efficient numerical algorithm for shape and topology optimization based on the level set method coupled with the topological derivative and implements a gradient algorithm for the minimization of the objective function.
Abstract: This paper is devoted to minimum stress design in structural optimization. We propose a simple and efficient numerical algorithm for shape and topology optimization based on the level set method coupled with the topological derivative. We compute a shape derivative, as well as a topological derivative, for a stress-based objective function. Using an adjoint equation we implement a gradient algorithm for the minimization of the objective function. Several numerical examples in 2-d and 3-d are discussed.
153 citations
TL;DR: In this article, a simplified molecular dynamics (MD) simulation method is constructed and used to simulate the thermophysical properties of nanofluids: thermal conductivity and viscosity, and better agreement between present numerical results and experimental data is presented in this paper.
Abstract: For a stationary nanofluids of the volume fractions ( α ) less than 8%, a simplified molecular dynamics (MD) simulation method is constructed and used to simulate the thermophysical properties of nanofluids: thermal conductivity and viscosity. The better agreement between present numerical results and experimental data is presented in this paper. It shows the simplified dynamics simulation method to be an effective method to forecast some thermal properties of nanofluids. Many former experiments have shown that this new heat transfer fluids–nanofluids can greatly enhance the heat-transfer efficiency. This work further gives the effects of the volume fraction and the size of nanoparticles on the thermal conductivity and the viscosity of nanofluids. Numerical results show that, decreasing size of nanoparticle or increasing the volume fraction can increase thermal conductivity with increasing viscosity; for suitable volume fraction and size, increasing viscosity with improving heat transfer capability is acceptable.
146 citations
TL;DR: In this paper, an improved IEFG method for two-dimensional elasticity is derived, where the orthogonal function system with a weight function is used as the basis function.
Abstract: This paper presents an improved moving least-squares (IMLS) approximation in which the orthogonal function system with a weight function is used as the basis function. The IMLS approximation has greater computational efficiency and precision than the existing moving least-squares (MLS) approximation, and does not lead to an ill-conditioned system of equations. By combining the element-free Galerkin (EFG) method and the IMLS approximation, an improved element-free Galerkin (IEFG) method for two-dimensional elasticity is derived. There are fewer coefficients in the IMLS approximation than in the MLS approximation, and in the IEFG method that is formed with the IMLS approximation fewer nodes are selected in the entire domain than are selected in the conventional EFG method. Hence, the IEFG method should result in a higher computing speed. For two-dimensional fracture problems, the enriched basis function is used at the tip of the crack to give an enriched IEFG method. When the enriched IEFG method is used, the singularity of the stresses at the tip of the crack can be shown better than that in the IEFG method. To provide a demonstration, numerical examples are solved using the IEFG method and the enriched IEFG method.
116 citations
TL;DR: In this paper, the meshless local Petrov-Galerkin (MLPG) method is presented for numerical solution of the two-dimensional non-linear Schrodinger equation, which is based on the local weak form and the moving least squares (MLS) approximation.
Abstract: In this paper the meshless local Petrov–Galerkin (MLPG) method is presented for the numerical solution of the two-dimensional non-linear Schrodinger equation. The method is based on the local weak form and the moving least squares (MLS) approximation. For the MLS, nodal points spread over the analyzed domain are utilized to approximate the interior and boundary variables. A time stepping method is employed for the time derivative. To deal with the non-linearity, we use a predictor–corrector method. A very simple and efficient method is presented for evaluation the local domain integrals. Finally numerical results are presented for some examples to demonstrate the accuracy, efficiency and high rate of convergence of this method.
109 citations
TL;DR: In this paper, a meshless algorithm is developed to simulate the static thermal stress distribution in two-dimensional (2D) functionally graded materials (FGMs), where the analog equation method (AEM) is used to obtain the equivalent homogeneous system to the original nonhomogeneous equation.
Abstract: On the basis of analog equation theory, the method of fundamental solutions coupling with radial basis functions (MFS–RBF), a meshless algorithm is developed to simulate the static thermal stress distribution in two-dimensional (2D) functionally graded materials (FGMs). The analog equation method (AEM) is used to obtain the equivalent homogeneous system to the original nonhomogeneous equation, after which RBF and MFS are used to construct the related approximated particular part and complementary part, respectively. Finally, all unknowns are determined by satisfying the governing equations in terms of displacement components and boundary conditions. Numerical experiments are performed for different 2D structures made of FGMs, and the proposed meshless method is validated by comparing available analytical and numerical results.
97 citations
TL;DR: In this article, the natural neighbour radial point interpolation method (NNRPIM) is extended for the analysis of thick plates and laminates, where the displacement field and the strain field are defined by the Reissner-Mindlin plate theory.
Abstract: In this work the natural neighbour radial point interpolation method (NNRPIM) is extended for the analysis of thick plates and laminates. In order to define the displacement field and the strain field the Reissner–Mindlin plate theory is considered. The nodal connectivity and the node dependent integration background mesh are constructed resorting to the Voronoi tessellation and to the Delaunay triangulation. Within NNRPIM the obtained shape functions pass through all nodes inside the influence-cell providing shape functions with the delta Kronecker property. Optimization tests and examples of well-known benchmark examples are solved in order to prove the high accuracy and convergence rate of the proposed method.
96 citations
TL;DR: In this paper, the authors studied the properties of surface integral equations of the first and second kinds in electromagnetic scattering and radiation problems and found that the second-kind equations gave better conditioned matrix equation and faster converging iterative solutions but poorer solution accuracy than the first-kind equation.
Abstract: Properties of various surface integral equations of the first and second kinds are studied in electromagnetic scattering and radiation problems. The second-kind equations are found to give better conditioned matrix equation and faster converging iterative solutions but poorer solution accuracy than the first-kind equations. The solution accuracy and matrix conditioning seem to be almost opposite properties associated with the singularity of the kernel of integral operators. The more singular/smoother the kernel, the more/less diagonally dominant and the better/poorer conditioned the matrix, but the poorer/better the solution accuracy. Accuracy of the integral equations of the second kind can be improved by increasing the order of the basis and testing functions. However, the required expansion order seems to be problem dependent. The more singular the unknown, the higher the expansion order and the finer the discretization needed in order to maintain the solution accuracy of the second-kind equations.
94 citations
TL;DR: In this article, the boundary value on an inaccessible part of a circle from an overdetermined data on an accessible part of that circle is recovered by applying a modified indirect Trefftz method.
Abstract: We consider an inverse problem for Laplace equation by recovering the boundary value on an inaccessible part of a circle from an overdetermined data on an accessible part of that circle. The available data are assumed to have a Fourier expansion, and thus the finite terms truncation plays a role of regularization to perturb the ill-posedness of this inverse problem into a well-posed one. Hence, we can apply a modified indirect Trefftz method to solve this problem and then a simple collocation technique is used to determine the unknown coefficients, which is named a modified collocation Trefftz method. The results may be useful to detect the corrosion inside a pipe through the measurements on a partial boundary. Numerical examples show the effectiveness of the new method in providing an excellent estimate of unknown data from the given data under noise.
89 citations
TL;DR: In this article, the shape of the source contour in the application of the method of fundamental solutions to the elastic torsion of prismatic rods is examined. And the results show that the local and global errors of the methods are smaller when the contours are geometrically similar to the boundary of the region under consideration in comparison to the contour with a shape of a circle.
Abstract: This paper deals with numerical experiments related with the shape of the source contour in the application of the method of fundamental solutions to the elastic torsion of prismatic rods. The following five boundary-value problems (BVPs) connected with torsion are studied: L-section, [-section, +-section, -section and I-section. For all five BVPs examined, the region of cross-section of rods is concave. Both the local and global errors are examined for two basic shapes of the source contour. In the first case, the source contour is a circle and in the second case the source contour is geometrically similar to the boundary contour of the region under consideration. Furthermore, the optimal radius of the source contour, in the case of the circle, or the optimal distance of the source contour from the boundary in the case it is geometrically similar, are studied. An influence of the method parameters (radius of the circle or distance between contours) on the condition linear system of equation is examined. In all examples examined the values of the local and global errors of the method are smaller when the source contour is geometrically similar to the boundary of the region under consideration in comparison to the source contour with a shape of a circle.
85 citations
TL;DR: In this article, a unified algorithm for numerical evaluation of weakly, strongly and hyper singular boundary integrals with or without a logarithmic term is presented, where the singular boundary element is broken up into a few sub-elements.
Abstract: In this paper, a unified algorithm is presented for the numerical evaluation of weakly, strongly and hyper singular boundary integrals with or without a logarithmic term, which often appear in two-dimensional boundary element analysis equations. In this algorithm, the singular boundary element is broken up into a few sub-elements. The sub-elements involving the singular point are evaluated analytically to remove the singularities by expressing the non-singular parts of the integration kernels as polynomials of the distance r, while other sub-elements are evaluated numerically by the standard Gaussian quadrature. The number of sub-elements and their sizes are determined according to the singularity order and the position of the singular point. Numerical examples are provided to demonstrate the correctness and efficiency of the proposed algorithm.
TL;DR: In this article, the Eulerian-Lagrangian method of fundamental solutions is proposed to solve the two-dimensional unsteady Burgers' equations, which is free from mesh generation and numerical integration.
Abstract: The Eulerian–Lagrangian method of fundamental solutions is proposed to solve the two-dimensional unsteady Burgers’ equations. Through the Eulerian–Lagrangian technique, the quasi-linear Burgers’ equations can be converted to the characteristic diffusion equations. The method of fundamental solutions is then adopted to solve the diffusion equation through the diffusion fundamental solution; in the meantime the convective term in the Burgers’ equations is retrieved by the back-tracking scheme along the characteristics. The proposed numerical scheme is free from mesh generation and numerical integration and is a truly meshless method. Two-dimensional Burgers’ equations of one and two unknown variables with and without considering the disturbance of noisy data are analyzed. The numerical results are compared very well with the analytical solutions as well as the results by other numerical schemes. By observing these comparisons, the proposed meshless numerical scheme is convinced to be an accurate, stable and simple method for the solutions of the Burgers’ equations with irregular domain even using very coarse collocating points.
TL;DR: In this article, a simple classical radial basis functions (RBFs) collocation (Kansa) method was proposed for numerical solution of the coupled Korteweg-de Vries (KdV) equations, coupled Burgers' equations, and quasi-nonlinear hyperbolic equations.
Abstract: This paper formulates a simple classical radial basis functions (RBFs) collocation (Kansa) method for the numerical solution of the coupled Korteweg-de Vries (KdV) equations, coupled Burgers’ equations, and quasi-nonlinear hyperbolic equations. Contrary to the mesh oriented methods such as the finite-difference and finite element methods, the new technique does not require mesh to discretize the problem domain, and a set of scattered nodes provided by initial data is required for realization of solution of the problem. Accuracy of the method is assessed in terms of the error norms L 2 , L ∞ , number of nodes in the domain of influence, time step length, parameter free and parameter dependent RBFs. Numerical experiments are performed to demonstrate the accuracy and robustness of the method for the three classes of partial differential equations (PDEs).
TL;DR: In this paper, the IMLS approximation has greater computational efficiency and precision than the existing moving least-squares (MLS) approximation, and does not lead to an ill-conditioned system of equations.
Abstract: In this paper, the element-free Galerkin (EFG) method and improved moving least-squares (IMLS) approximation are combined. An improved FEG (IEFG) method for two-dimensional elasticity is discussed, and the coupling of the IEFG method and the boundary element method (BEM) is presented. In the IMLS approximation, an orthogonal function system with a weight function is used as the basis function. The IMLS approximation has greater computational efficiency and precision than the existing moving least-squares (MLS) approximation, and does not lead to an ill-conditioned system of equations. There are fewer coefficients in the IMLS approximation than in the MLS approximation, and in the IEFG method that is formed with the IMLS approximation fewer nodes are selected in the entire domain than are selected using the conventional EFG method. Hence, the IEFG method should result in a higher computing speed. Based on the IMLS approximation and the IEFG method, a direct coupling of the IEFG method and the BEM is discussed for two-dimensional elasticity problems, and the corresponding formulae of the coupled method are obtained. The coupled method does not need a new sub-domain between the IEFG method and the BEM sub-domains. Selected numerical examples are solved using the coupled method.
TL;DR: In this article, an application of the method of fundamental solutions (MFS) to transient heat conduction is investigated and a denseness result for this method is discussed and the method is numerically tested showing that accurate numerical results can be obtained.
Abstract: In this paper we investigate an application of the method of fundamental solutions (MFS) to transient heat conduction. In almost all of the previously proposed MFS for time-dependent heat conduction the fictitious sources are located outside the time-interval of interest. In our case, however, these sources are instead placed outside the space domain of interest in the same manner as is done for stationary heat conduction. A denseness result for this method is discussed and the method is numerically tested showing that accurate numerical results can be obtained. Furthermore, a test example with boundary singularities shows that it is advisable to remove such singularities before applying the MFS.
TL;DR: It is shown in this paper that the emerging topological gradient method is a new way for modelling and solving image restoration problems and the computational cost is reduced drastically using spectral methods.
Abstract: We show in this paper that the emerging topological gradient method is a new way for modelling and solving image restoration problems. This method is considered in the frame of variational diffusive approaches for the minimization of potential energy with respect to conductivity. The numerical experiments show the efficiency of the topological gradient approach. The image is restored at the first iteration of the optimization process. Moreover, the computational cost of this iteration is reduced drastically using spectral methods.
TL;DR: In this paper, two new concepts for the classical moving least squares (MLS) approach are presented, one is an interpolating weighting function, which leads to MLS shape functions fulfilling the interpolation condition exactly.
Abstract: In this paper two new concepts for the classical moving least squares (MLS) approach are presented. The first one is an interpolating weighting function, which leads to MLS shape functions fulfilling the interpolation condition exactly. This enables a direct application of essential boundary conditions in the element-free Galerkin method without additional numerical effort. In contrast to existing approaches using singular weighting functions, this new weighting type leads to regular values of the weights and coefficients matrices in the whole domain even at the support points. The second enhancement is an approach, where the computation of the polynomial coefficient matrices is performed only at the nodes. At the interpolation point then a simple operation leads to the final shape function values. The basis polynomial of each node can be chosen independently which enables the simple realization of a p-adaptive scheme.
TL;DR: This approach has an edge over the traditional methods such as finite-difference and finite-element methods because it does not require a mesh to discretize the problem domain, and a set of scattered nodes in the domain of influence provided by initial data is required for the realization of the method.
Abstract: This paper formulates a meshfree radial basis functions (RBFs) collocation (Kansa) method for the numerical solution of the Korteweg-de Vries (KdV) equation. The accuracy of the method is assessed in terms of the errors in L∞, L2 and root mean square (RMS), number of nodes in the domain of influence, parameter-dependent RBFs time and spatial steps length. This approach has an edge over the traditional methods such as finite-difference and finite-element methods because it does not require a mesh to discretize the problem domain, and a set of scattered nodes in the domain of influence provided by initial data is required for the realization of the method. Numerical experiments demonstrate the accuracy and robustness of the method when applied to complicated nonlinear partial differential equations. In this work, three test problems are studied.
TL;DR: In this article, a hybrid technique is presented for the hydrodynamic analysis of floating bodies in variable bathymetry regions, which is based on the coupled-mode theory for the propagation of water waves in general bottom topography.
Abstract: In this work, a hybrid technique is presented for the hydrodynamic analysis of floating bodies in variable bathymetry regions. Our method is based on the coupled-mode theory for the propagation of water waves in general bottom topography, developed by Athanassoulis and Belibassakis [Athanassoulis GA, Belibassakis KA. A consistent coupled-mode theory for the propagation of small-amplitude water waves over variable bathymetry regions. J Fluid Mech 1999;389:275–301.] and extended to 3D by Belibassakis et al. [Belibassakis KA, Athanassoulis GA, Gerostathis TP. A coupled-mode model for the refraction–diffraction of linear waves over steep three-dimensional bathymetry. Appl Ocean Res 2001;23:319–336.], which is free of any mild-slope assumption, in conjunction with a boundary integral representation of the near field in the vicinity of the floating body. Both 2D and 3D problems have been considered. In all cases the near field is represented by boundary integral representation involving simple (Rankine) sources. In the 2D case, the far-field is modelled by complete (normal-mode) series expansions derived by separation of variables in the constant-depth half-strips. In the 3D case, the far-field is modelled by an integral representation involving the appropriate Green’s function for harmonic water waves over a bottom with different depths at infinity, developed by Belibassakis and Athanassoulis [Belibassakis KA, Athanassoulis GA. Three-dimensional Green’s function for harmonic water waves over a bottom with different depths at infinity. J Fluid Mech 2004;510:267–302.]. The numerical solution is obtained by means of a low-order panel method materialising the hybrid technique. Numerical results are presented concerning floating bodies of simple geometry, lying over sloping and undulating seabeds. With the aid of systematic comparisons with benchmark solutions the convergence and accuracy of the present method in 3D has been studied, and the effects of bottom slope and curvature on the hydrodynamic characteristics (hydrodynamic coefficients and responses) of the floating bodies are illustrated and discussed.
TL;DR: In this paper, a numerical approach for the topological optimization of 2D linear elastic problems using boundary elements is presented, where models are discretized using linear elements and a periodic distribution of internal points over the domain.
Abstract: Topological optimization provides a powerful framework to obtain the optimal domain topology for several engineering problems. The topological derivative is a function which characterizes the sensitivity of a given problem to the change of its topology, like opening a small hole in a continuum or changing the connectivity of rods in a truss. A numerical approach for the topological optimization of 2D linear elastic problems using boundary elements is presented in this work. The topological derivative is computed from strain and stress results which are solved by means of a standard boundary element analysis. Models are discretized using linear elements and a periodic distribution of internal points over the domain. The total potential energy is selected as cost function. The evaluation of the topological derivative is performed as a post-processing procedure. Afterwards, material is removed from the model by deleting the internal points and boundary nodes with the lowest values of the topological derivate. The new geometry is then remeshed using a weighted Delaunay triangularization algorithm capable of detecting “holes” at those positions where internal points and boundary points have been removed. The procedure is repeated until a given stopping criterion is satisfied. The proposed strategy proved to be flexible and robust. A number of examples are solved and results are compared to those available in the literature.
TL;DR: In this paper, a new cohesive interface model is applied to characterize carbon nanotube (CNT) composites using the boundary element method (BEM) to characterize CNT composites.
Abstract: In this paper, a new cohesive interface model is applied to characterize carbon nanotube (CNT) composites using the boundary element method (BEM). In the previous BEM models of CNT composites, a rigid-inclusion model was employed to represent the CNTs in a polymer matrix due to their extremely high stiffness as compared with the polymer. Perfect bonding interface conditions between the CNT fibers and matrix were used in these earlier models. Very good BEM results for the effective moduli were obtained as compared with other multi-scale models based on molecular dynamics (MD) and continuum mechanics. However, these simulation results yield much higher estimates of the effective Young's moduli of CNT/polymer composites than those observed in experiments of such composites. This discrepancy is largely due to the interfaces in CNT composites which have been found to be weakly, rather than strongly bonded. In this work, a new cohesive interface model has been developed by using MD simulations and employed in the BEM models to replace the perfect bonding interface models. The parameters in the cohesive interface model are obtained by conducting CNT pull-out simulations with MD and these parameters are subsequently used in the BEM models of the CNT/polymer composites. Marked decreases of the estimated effective Young's moduli are observed using the new BEM models with the cohesive interface conditions. The developed BEM models combined with the MD can be a very useful tool for studying the interface effects in CNT composites and for large-scale characterizations of such nanocomposites.
TL;DR: In this paper, the authors deal with the numerical solution of structural optimization problems of an elastic body in unilateral contact with a rigid foundation, where the contact problem with a given friction is described by an elliptic inequality of the second order governing a displacement field.
Abstract: This paper deals with the numerical solution of structural optimization problems of an elastic body in unilateral contact with a rigid foundation. The contact problem with a given friction is described by an elliptic inequality of the second order governing a displacement field. The optimization problem consists in finding, in a contact region, such topology and shape of the boundary of the domain occupied by the body that the normal contact stress is minimized. Level set methods [3], [4] are numerically efficient and robust procedures for the tracking of interfaces, which allows domain boundary shape changes in the course of iteration. The evolution of the level set function is governed by the Hamilton Jacobi equation. The speed vector field driving the propagation of the level set function is given by the Eulerian derivative [2] of an appropriately defined cost functional with respect to the free boundary.
TL;DR: In this paper, a least-square radial point collocation method (LS-RPCM) is proposed to solve the instability problem observed in the conventional RPCM using local nodes.
Abstract: This paper presents a least-square radial point collocation method (LS-RPCM) that is formulated based on the strong formulation and the local approximation using radial basis functions (RBFs). Aiming to solve the instability problem observed in the conventional RPCM using local nodes, a simple and yet effective procedure that uses the well-known least-square technique in a carefully designed manner has been proposed to restore the stability. Since stable solution can now be obtained, the LS-RPCM is then extended for adaptive analysis. Attractive features of the meshfree strong-form method that facilitate the implementation of adaptive analysis are demonstrated via a number of examples in this work. A robust residual based error estimator and a simple refinement procedure using Delaunay diagram are adopted in our adaptive scheme. Stable and accurate results are obtained in all the numerical examples.
TL;DR: In this paper, a class of nonlinear transient heat conduction problems for anisotropic inhomogeneous materials is considered and the problems under consideration are reduced to a boundary integral equation which may be solved numerically using standard techniques.
Abstract: A class of nonlinear transient heat conduction problems for anisotropic inhomogeneous materials is considered. The problems under consideration are reduced to a boundary integral equation which may be solved numerically using standard techniques. Some numerical examples are considered in order to test the accuracy of the numerical procedure.
TL;DR: In this paper, numerical solutions of the equal width wave (EW) equation are obtained by using a Galerkin method with quartic B-spline finite elements, a differential quadrature method with cosine expansion basis and a meshless method with radial-basis functions.
Abstract: Numerical solutions of the equal width wave (EW) equation are obtained by using a Galerkin method with quartic B-spline finite elements, a differential quadrature method with cosine expansion basis and a meshless method with radial-basis functions. Solitary wave motion, interaction of two solitary waves and wave undulation are studied to validate the accuracy and efficiency of the proposed methods. Comparisons are made with analytical solutions and those of some earlier papers. The accuracy and efficiency are discussed by computing the numerical conserved laws and L 2 , L ∞ error norms.
TL;DR: In this article, a fast multipole boundary element method (BEM) is presented for large-scale analysis of 2D Stokes flow problems based on a dual boundary integral equation (BIE) formulation.
Abstract: A fast multipole boundary element method (BEM) is presented in this paper for large-scale analysis of two-dimensional (2-D) Stokes flow problems based on a dual boundary integral equation (BIE) formulation. In this dual BIE formulation, a linear combination of the conventional BIE for velocity and the hypersingular BIE for traction is employed to achieve better conditioning for the BEM systems of equations. In both the velocity and traction BIEs, the direct formulations are used, that is, the boundary variables involved are the velocity and traction directly. The fast multipole formulations for both the velocity BIE and traction BIE for 2-D Stokes flow problems are presented in this paper based on the complex variable representations of the fundamental solutions. Several numerical examples are presented to study the accuracy and efficiency of the proposed approach. The numerical results clearly demonstrate the potentials of the developed fast multipole BEM for solving large-scale 2-D Stokes flow problems.
TL;DR: In this paper, a numerical solution of a time-dependent diffusion equation is given in detail, based on the meshless local Petrov-Galerkin method (MLPG), which ensures a constant number of support nodes for each point.
Abstract: Derivation and implementation of a numerical solution of a time-dependent diffusion equation is given in detail, based on the meshless local Petrov–Galerkin method (MLPG). A simple method is proposed that ensures a constant number of support nodes for each point. Numerical integrations are carried out over local square domains. The implicit Crank–Nicolson scheme is used for time discretization. A detailed convergence study was performed experimentally to optimize the number of support nodes, quadrature domain size and other parameters. The accuracy of the MLPG solution is compared with that of standard methods on a unit square and on an irregularly shaped test domain. As expected, the finite difference method on a regular mesh is incompetitive on irregularly shaped domains. MLPG is significantly more accurate when using moving least square shape functions of degree two than with degree one. It is comparable to the finite element method of degree two in the H 1 error norm and about two times less accurate in the L 2 error norm.
TL;DR: In this paper, a framework to carry out the topological sensitivity analysis in this context is proposed and a numerical example concerning the treatment of ultrasonic probing data in metallic plates is presented.
Abstract: This paper deals with the use of the topological derivative in detection problems involving waves. In the first part, a framework to carry out the topological sensitivity analysis in this context is proposed. Arbitrarily shaped holes and cracks with Neumann boundary condition in 2 and 3 space dimensions are considered. In the second part, a numerical example concerning the treatment of ultrasonic probing data in metallic plates is presented. With moderate noise in the measurements, the defects (air bubbles) are detected and satisfactorily localized by means of a single sensitivity computation.
TL;DR: In this paper, a linear combination of fundamental solutions for the 3D Helmholtz equation is used to solve the problem of sound wave propagation in three-dimensional (3D) enclosed acoustic spaces, where materials coating the enclosed space surfaces can be assumed to be sound absorbent.
Abstract: The method of fundamental solutions (MFS) is formulated in the frequency domain to model the sound wave propagation in three-dimensional (3D) enclosed acoustic spaces. In this model the solution is obtained by approximation, using a linear combination of fundamental solutions for the 3D Helmholtz equation. Those solutions relate to a set of virtual sources placed over a surface placed outside the domain in order to avoid singularities. The materials coating the enclosed space surfaces can be assumed to be sound absorbent. This effect is introduced in the model by imposing impedance boundary conditions, with the impedance being defined as a function of the absorption coefficient. To impose these boundary conditions, a set of collocation points (observation points) needs to be selected along the boundary. Time domain responses are obtained by applying an inverse Fourier transform to the former frequency domain results. In order to avoid “aliasing” phenomena in the time domain results, the computations introduce damping in the imaginary part of the frequency. This effect is later removed in the time domain by rescaling the response. After corroborating the present solution against the analytical solution, known in closed form for the case of a parallelepiped room bounded by rigid walls, the model is used to solve the case of a dome.
TL;DR: In this paper, a numerical scheme based on the mesh-free plane wave method applied to inverse boundary value problems associated with Helmholtz-type equations is investigated, and the resulting ill-conditioned system of linear algebraic equations is solved in a stable manner by employing the truncated singular value decomposition, while the optimal truncation number, i.e., the regularization parameter, is determined using the L-curve criterion.
Abstract: In this paper, a numerical scheme based on the meshfree plane wave method applied to inverse boundary value problems associated with Helmholtz-type equations is investigated. The resulting ill-conditioned system of linear algebraic equations is solved in a stable manner by employing the truncated singular value decomposition, while the optimal truncation number, i.e. the regularization parameter, is determined using the L-curve criterion. Numerical results are presented for two- and three-dimensional problems in smooth and piecewise smooth geometries, with both exact and noisy data. The accuracy, convergence and stability of the numerical method are analysed and, furthermore, a comparison with other meshless methods is also performed.