Showing papers in "Engineering Analysis With Boundary Elements in 2010"
TL;DR: In this paper, the displacement discontinuity across crack surface is modeled by independent cover functions over different physical covers, while additional functions, extracted from the asymptotic near tip field, are incorporated into cover functions of singular physical covers to reflect the stress singularity around the crack tips.
Abstract: The numerical manifold method is a cover-based method using mathematical covers that are independent of the physical domain. As the unknowns are defined on individual physical covers, the numerical manifold method is very suitable for modeling discontinuities. This paper focuses on modeling complex crack propagation problems containing multiple or branched cracks. The displacement discontinuity across crack surface is modeled by independent cover functions over different physical covers, while additional functions, extracted from the asymptotic near tip field, are incorporated into cover functions of singular physical covers to reflect the stress singularity around the crack tips. In evaluating the element matrices, Gaussian quadrature is used over the sub-triangles of the element, replacing the simplex integration over the whole element. First, the method is validated by evaluating the fracture parameters in two examples involving stationary cracks. The results show good agreement with the reference solutions available. Next, three crack propagation problems involving multiple and branched cracks are simulated. It is found that when the crack growth increment is taken to be 0.5h≤da≤0.75h, the crack growth paths converge consistently and are satisfactory.
183 citations
TL;DR: In this paper, a numerical method based on boundary integral equation (BIE) and an application of the dual reciprocity method (DRM) was used to solve the second-order one space-dimensional hyperbolic telegraph equation.
Abstract: In this paper, we use a numerical method based on the boundary integral equation (BIE) and an application of the dual reciprocity method (DRM) to solve the second-order one space-dimensional hyperbolic telegraph equation. Also the time stepping scheme is employed to deal with the time derivative. In this study, we have used three different types of radial basis functions (cubic, thin plate spline and linear RBFs), to approximate functions in the dual reciprocity method (DRM). To confirm the accuracy of the new approach and to show the performance of each of the RBFs, several examples are presented. The convergence of the DRBIE method is studied numerically by comparison with the exact solutions of the problems.
141 citations
TL;DR: In this article, a meshless local weak-strong (MLWS) method is proposed to solve the second-order two-space-dimensional telegraph equation, which combines the advantage of local weak and strong forms to avoid their shortcomings.
Abstract: In this paper a numerical approach based on the truly meshless methods is proposed to deal with the second-order two-space-dimensional telegraph equation. In the meshless local weak–strong (MLWS) method, our aim is to remove the background quadrature domains for integration as much as possible, and yet to obtain stable and accurate solution. The MLWS method is designed to combine the advantage of local weak and strong forms to avoid their shortcomings. In this method, the local Petrov–Galerkin weak form is applied only to the nodes on the Neumann boundary of the domain of the problem. The meshless collocation method, based on the strong form equation is applied to the interior nodes and the nodes on the Dirichlet boundary. To solve the telegraph equation using the MLWS method, the conventional moving least squares (MLS) approximation is exploited in order to interpolate the solution of the equation. A time stepping scheme is employed to approximate the time derivative. Another solution is also given by the meshless local Petrov-Galerkin (MLPG) method. The validity and efficiency of the two proposed methods are investigated and verified through several examples.
131 citations
TL;DR: The singular boundary method (SBM) as mentioned in this paper employs the singular fundamental solution of the governing equation of interest as the interpolation basis function to avoid the singularity at the origin.
Abstract: This paper proposes a novel meshless boundary method called the singular boundary method (SBM). This method is mathematically simple, easy-to-program, and truly meshless. Like the method of fundamental solutions (MFS), the SBM employs the singular fundamental solution of the governing equation of interest as the interpolation basis function. However, unlike the MFS, the source and collocation points of the SBM coincide on the physical boundary without the requirement of introducing fictitious boundary. In order to avoid the singularity at the origin, this method proposes an inverse interpolation technique to evaluate the singular diagonal elements of the MFS coefficient matrix. The SBM is successfully tested on a benchmark problems, which shows that the method has a rapid convergence rate and is numerically stable.
127 citations
TL;DR: In this article, a technique is proposed for solving partial differential equations using radial basis functions, which is different from the traditional schemes and provides a closed form approximation for the solution, which can be applied to problems with nonregular geometrical domains.
Abstract: In this work a technique is proposed for solving partial differential equations using radial basis functions The approach is different from the traditional schemes The radial basis functions are very suitable instruments for solving partial differential equations of various types However, the matrices which result from the discretization of the equations are usually ill-conditioned especially in higher-dimensional problems In the current paper, a stable method will be proposed for solving the partial differential equations and will be generalized to solve higher-dimensional problems To the contrast of most existing methods, the new technique provides a closed form approximation for the solution Another advantage of the developed method is that it can be applied to problems with nonregular geometrical domains
108 citations
TL;DR: In this article, the authors apply the golden section search algorithm to determine a good shape parameter of multiquadrics (MQ) for the solution of partial differential equations, and analyze cost and accuracy for using LU decomposition and TSVD.
Abstract: In this paper we propose to apply the golden section search algorithm to determining a good shape parameter of multiquadrics (MQ) for the solution of partial differential equations. We use two radial basis function based meshless collocation methods, the method of approximate particular solutions (MAPS) and Kansa's method, to solve partial differential equations. Due to the severely ill-conditioned matrix system using MQ, we also consider the truncated singular value decomposition method (TSVD) to regularize the smoothness of the error versus shape parameter curve so that a reasonably good shape parameter can be identified. We also analyze cost and accuracy for using LU decomposition and TSVD. Numerical results show that the proposed golden section search method is effective and provides a reasonable shape parameter along with acceptable accuracy of the solution.
102 citations
TL;DR: In this paper, a boundary element analysis approach is presented for solving transient heat conduction problems based on the radial integration method, which makes the representation very simple and having no temperature gradients involved.
Abstract: In this paper, a new boundary element analysis approach is presented for solving transient heat conduction problems based on the radial integration method. The normalized temperature is introduced to formulate integral equations, which makes the representation very simple and having no temperature gradients involved. The Green's function for the Laplace equation is adopted in deriving basic integral equations for time-dependent problems with varying heat conductivities and, as a result, domain integrals are involved in the derived integral equations. The radial integration method is employed to convert the domain integrals into equivalent boundary integrals. Based on the central finite difference technique, an implicit time marching solution scheme is developed for solving the time-dependent system of equations. Numerical examples are given to demonstrate the correctness of the presented approach.
86 citations
TL;DR: The boundary distributed source (BDS) method as discussed by the authors is a boundary mesh-free method that is truly mesh free and easy to implement, which is based on the same concept in the well-known method of fundamental solutions.
Abstract: A new boundary meshfree method, to be called the boundary distributed source (BDS) method, is presented in this paper that is truly meshfree and easy to implement. The method is based on the same concept in the well-known method of fundamental solutions (MFS). However, in the BDS method the source points and collocation points coincide and both are placed on the boundary of the problem domain directly, unlike the traditional MFS that requires a fictitious boundary for placing the source points. To remove the singularities of the fundamental solutions, the concentrated point sources can be replaced by distributed sources over areas (for 2D problems) or volumes (for 3D problems) covering the source points. For Dirichlet boundary conditions, all the coefficients (either diagonal or off-diagonal) in the systems of equations can be determined analytically, leading to very simple implementation for this method. Methods to determine the diagonal coefficients for Neumann boundary conditions are discussed. Examples for 2D potential problems are presented to demonstrate the feasibility and accuracy of this new meshfree boundary-node method.
83 citations
TL;DR: In this paper, a domain integral is transformed into a boundary integral and a 1D integral, which is then utilized for the evaluation of domain integrals in meshless methods based on the weak form, such as the element-free Galerkin method and the meshless radial point interpolation method.
Abstract: A method for the evaluation of regular domain integrals without domain discretization is presented. In this method, a domain integral is transformed into a boundary integral and a 1D integral. The method is then utilized for the evaluation of domain integrals in meshless methods based on the weak form, such as the element-free Galerkin method and the meshless radial point interpolation method. The proposed technique results in truly meshless methods with better accuracy and efficiency in comparison with their original forms. Some examples, including linear and large-deformation problems, are also provided to demonstrate the usefulness of the proposed method.
73 citations
TL;DR: In this paper, a general strategy based on a nonlinear transformation technique is introduced and applied to evaluate the nearly singular integrals occurring in two dimensional (2D) thin-coated structures.
Abstract: Temperature measurement and prediction have been a major focus of machining for several decades, but now these problems become more complex due to the wider use of advanced cutting tool coatings. In all literature items cited the boundary element method (BEM) were used to find the distribution of temperature inside the uncoated tool body or along the tool–chip interface in the machining processes. The BEM-based approach proposed in this paper overcomes this limit and the temperature distribution in thin coated layers is well studied. In this study, a general strategy based on a nonlinear transformation technique is introduced and applied to evaluate the nearly singular integrals occurring in two dimensional (2D) thin-coated structures. For the test problems studied, very promising results are obtained when the thickness to length ratio is in the orders of 1.0E−6 to 1.0E−10, which is sufficient for modeling most thin-coated structures in the micro- or nano-sclaes.
59 citations
TL;DR: In this paper, a new approach based on the use of Green's function is proposed for solving specific classes of inverse source identification problems, and effective numerical algorithms are developed to recover both the intensities and locations of unknown point sources from scattered boundary measurements.
Abstract: Based on the use of Green's function, we propose in this paper a new approach for solving specific classes of inverse source identification problems. Effective numerical algorithms are developed to recover both the intensities and locations of unknown point sources from scattered boundary measurements. For numerical verification, several boundary value problems defined on both bounded and unbounded regions of regular shape are given. Due to the use of closed analytic form of Green's function, the efficiency and accuracy of the proposed method can be guaranteed.
TL;DR: A cell-based smoothed radial point interpolation method (CS-RPIM) based on the generalized gradient smoothing operation is proposed for static and free vibration analysis of solids.
Abstract: A cell-based smoothed radial point interpolation method (CS-RPIM) based on the generalized gradient smoothing operation is proposed for static and free vibration analysis of solids. In present method, the problem domain is first discretized using triangular background cells, and each cell is further divided into several smoothing cells. The displacement field function is approximated using RPIM shape functions which have Kronecker delta function property. Supporting node selection for shape function construction uses the efficient T2L-scheme associated with edges of the background cells. The system equations are derived using the generalized smoothed Galerkin (GS-Galerkin) weak form, and the essential boundary conditions are imposed directly as in the finite element method (FEM). The effects of the number of divisions smoothing cells on the solution properties of the CS-RPIM are investigated in detail, and preferable numbers of smoothing cells is recommended. To verify the accuracy and stability of the present formulation, a number of numerical examples are studied to demonstrate numerically the efficiency of the present CS-RPIM.
TL;DR: In this article, a version of meshless local Petrov-Galerkin (MLPG) method is developed to obtain three-dimensional (3D) static solutions for thick functionally graded (FG) plates.
Abstract: In this paper, a version of meshless local Petrov–Galerkin (MLPG) method is developed to obtain three-dimensional (3D) static solutions for thick functionally graded (FG) plates. The Young's modulus is considered to be graded through the thickness of plates by an exponential function while the Poisson's ratio is assumed to be constant. The local symmetric weak formulation is derived using the 3D equilibrium equations of elasticity. Moreover, the field variables are approximated using the 3D moving least squares (MLS) approximation. Brick-shaped domains are considered as the local sub-domains and support domains. In this way, the integrations in the weak form and approximation of the solution variables are done more easily and accurately. The proposed approach to construct the shape and the test functions make it possible to introduce more nodes in the direction of material variation. Consequently, more precise solutions can be obtained easily and efficiently. Several numerical examples containing the stress and deformation analysis of thick FG plates with various boundary conditions under different loading conditions are presented. The obtained results have been compared with the available analytical and numerical solutions in the literature and an excellent consensus is seen.
TL;DR: In this article, it was shown that RBF in fact performs better than polynomials, as the optimal shape parameter exists not in the limit, but at a finite value.
Abstract: For the interpolation of continuous functions and the solution of partial differential equation (PDE) by radial basis function (RBF) collocation, it has been observed that solution becomes increasingly more accurate as the shape of the RBF is flattened by the adjustment of a shape parameter. In the case of interpolation of continuous functions, it has been proven that in the limit of increasingly flat RBF, the interpolant reduces to Lagrangian polynomials. Does this limiting behavior implies that RBFs can perform no better than Lagrangian polynomials in the interpolation of a function, as well as in the solution of PDE? In this paper, arbitrary precision computation is used to test these and other conjectures. It is found that RBF in fact performs better than polynomials, as the optimal shape parameter exists not in the limit, but at a finite value.
TL;DR: An improved form of the hypersingular boundary integral equation (BIE) for acoustic problems is developed in this paper, which contains only weakly singular integrals and is directly valid for acoustic problem with arbitrary boundary conditions.
Abstract: An improved form of the hypersingular boundary integral equation (BIE) for acoustic problems is developed in this paper. One popular method for overcoming non-unique problems that occur at characteristic frequencies is the well-known Burton and Miller (1971) method [7], which consists of a linear combination of the Helmholtz equation and its normal derivative equation. The crucial part in implementing this formulation is dealing with the hypersingular integrals. This paper proposes an improved reformulation of the Burton–Miller method and is used to regularize the hypersingular integrals using a new singularity subtraction technique and properties from the associated Laplace equations. It contains only weakly singular integrals and is directly valid for acoustic problems with arbitrary boundary conditions. This work is expected to lead to considerable progress in subsequent developments of the fast multipole boundary element method (FMBEM) for acoustic problems. Numerical examples of both radiation and scattering problems clearly demonstrate that the improved BIE can provide efficient, accurate, and reliable results for 3-D acoustics.
TL;DR: In this paper, a meshless local Petrov-Galerkin (MLPG) and collocation method is proposed to treat the parabolic partial differential equations with non-classical boundary conditions.
Abstract: This paper presents a new approach based on the meshless local Petrov–Galerkin (MLPG) and collocation methods to treat the parabolic partial differential equations with non-classical boundary conditions In the presented method, the MLPG method is applied to the interior nodes while the meshless collocation method is applied to the nodes on the boundaries, and so the Dirichlet boundary condition is imposed directly To treat the complicated integral boundary condition appearing in the problem, Simpson's composite numerical integration rule is applied A time stepping scheme is employed to approximate the time derivative Finally, two numerical examples are presented showing the behavior of the solution and the efficiency of the proposed method
TL;DR: A general model of the equations of generalized thermo-microstretch for a homogeneous isotropic elastic half-space is given in this article, where the modulus of elasticity is taken as a linear function of reference temperature.
Abstract: A general model of the equations of generalized thermo–microstretch for a homogeneous isotropic elastic half–space is given The modulus of elasticity is taken as a linear function of reference temperature The formulation is applied to generalized thermoelasticity theories, the Lord–Shulman and Green–Lindsay theories, as well as the classical dynamical coupled theory The normal mode analysis is used to obtain the exact expressions for the displacement components, force stresses, temperature, couple stresses and microstress distribution The variations of the considered variables through the horizontal distance are illustrated graphically A comparison is made with the results predicted by the three theories in case of temperature–independent modulus of elasticity
TL;DR: In this article, the authors analyse flow and heat transfer characteristics of nanofluids in natural convection flows in closed cavities, and compare the performance of three types of water-based nanoparticles compared with pure water and air.
Abstract: In this paper we analyse flow and heat transfer characteristics of nanofluids in natural convection flows in closed cavities. We consider two test cases: natural convection in a three-dimensional differentially heated cavity, and flow around a hotstrip located in two positions within a cavity. Simulations were performed for Rayleigh number values ranging from 10 3 to 10 6 . Performance of three types of water based nanofluids was compared with pure water and air. Stable suspensions of Cu, Al 2 O 3 and TiO 2 solid nanoparticles in water were considered for different volume fractions ranging up to 20%. We present and compare heat flux for all cases and analyse heat transfer enhancement attributed to nanofluids in terms of their enhanced thermal properties and changed flow characteristics. Results show that, using nanofluids, the largest heat transfer enhancements can be achieved in conduction dominated flows as well as that nanofluids increase the three-dimensional character of the flow field. We additionally examine the relationship between vorticity, vorticity flux and heat transfer for flow of nanofluids. The simulations were performed using a three-dimensional boundary element method based flow solver, which has been adapted for the simulation of nanofluids. The numerical algorithm is based on the combination of single domain and subdomain boundary element method, which are used to solve the velocity–vorticity formulation of Navier–Stokes equations. In the paper we present the adaptation of the solver for simulation of nanofluids. Additionally, we developed a dynamic solver accuracy algorithm, which was used to speed up the simulations.
TL;DR: This work presents a new implementation of the boundary face method with shape functions from surface elements on the geometry directly like the boundary element method (BEM), and possesses higher accuracy and is less sensitive to the coarseness of the mesh than the BEM.
Abstract: This work presents a new implementation of the boundary face method (BFM) with shape functions from surface elements on the geometry directly like the boundary element method (BEM). The conventional BEM uses the standard elements for boundary integration and approximation of the geometry, and thus introduces errors in geometry. In this paper, the BFM is implemented directly based on the boundary representation data structure (B-rep) that is used in most CAD packages for geometry modeling. Each bounding surface of geometry model is represented as parametric form by the geometric map between the parametric space and the physical space. Both boundary integration and variable approximation are performed in the parametric space. The integrand quantities are calculated directly from the faces rather than from elements, and thus no geometric error will be introduced. The approximation scheme in the parametric space based on the surface element is discussed. In order to deal with thin and slender structures, an adaptive integration scheme has been developed. An adaptive method for generating surface elements has also been developed. We have developed an interface between BFM and UG-NX(R). Numerical examples involving complicated geometries have demonstrated that the integration of BFM and UG-NX(R) is successful. Some examples have also revealed that the BFM possesses higher accuracy and is less sensitive to the coarseness of the mesh than the BEM.
TL;DR: In this paper, a local RBF-based differential quadrature (LRBFDQ) collocation method is presented for the boundary layer problems, i.e., the singularly perturbed two-point boundary value problems.
Abstract: In this article, the local RBF-based differential quadrature (LRBFDQ) collocation method is presented for the boundary layer problems, i.e., the singularly perturbed two-point boundary value problems. This novel method has an advantage over the globally supported RBF collocation method because it approximates the derivatives by RBF interpolation using a small set of nodes in the neighborhood of any collocation node. So it needs much less computational work than the globally supported RBF collocation method. It also could easily use the nodes in local support domain on the upwind side to obtain the non-oscillatory solution of boundary layer problems. Numerical examples are made by the multiquadric (MQ) RBF. Compared with the globally supported RBF collocation method and the finite difference method, numerical results demonstrate the accuracy and easy implementation of the LRBFDQ collocation method, even for the extremely thin layers in the boundary layer problems.
TL;DR: Based on the idea of radial basis functions approximation and the method of particular solutions, a new meshless computational method to solve nonhomogeneous backward heat conduction problem was developed in this paper.
Abstract: Based on the idea of radial basis functions approximation and the method of particular solutions, we develop in this paper a new meshless computational method to solve nonhomogeneous backward heat conduction problem. To illustrate the effectiveness and accuracy of the proposed method, we solve several benchmark problems in both two- and three-dimensions. Numerical results indicate that this novel approach can achieve an efficient and accurate solution even when the final temperature data is almost undetectable or disturbed with large noises. It has also been shown that the proposed method is stable to recover the unknown initial temperature from scattered final temperature data.
TL;DR: In this article, a detailed approach highlights fully the process of applying boundary conditions, including the treatment of examples in which the assumptions made in previous work are no longer valid Verification of the coupled boundary element-scaled boundary finite element method is undertaken by means of estimating stress intensity factors and comparing them against analytical solutions.
Abstract: Issues relating to the practical implementation of the coupled boundary element–scaled boundary finite element method are addressed in this paper A detailed approach highlights fully the process of applying boundary conditions, including the treatment of examples in which the assumptions made in previous work are no longer valid Verification of the method is undertaken by means of estimating stress intensity factors and comparing them against analytical solutions The coupled algorithm shows good convergence properties Issues relating to traction scaling, the use of discontinuous boundary elements, and the greater versatility of the coupled method over its constituent methods are highlighted
TL;DR: In this paper, both the strong and weak formulations of the governing equations are discretized by using the Moving Least Squares approximations and a modified differentiation scheme is developed for approximation of higher order derivatives of displacements appearing in the discretised formulations.
Abstract: This paper is devoted to the development of advanced mesh free implementations of the governing equations and the boundary conditions for boundary value problems in elasticity. Both the strong and weak formulations are discretized by using the Moving Least Squares approximations. The weak formulation is represented by local integral equations considered on sub-domains around interior nodal points. The awkward evaluation of the shape functions and their derivatives is reduced by focusing to nodal points because of the development of analytical integrations. That results in significant saving of the computational time needed for creation of the system matrix. Furthermore, a modified differentiation scheme is developed for approximation of higher order derivatives of displacements appearing in the discretized formulations. The accuracy, convergence and computational efficiency are studied in simple numerical example.
TL;DR: A time-domain boundary element method for transient dynamic analysis of two-dimensional (2D), homogeneous, anisotropic and linear piezoelectric cracked solids is presented in this paper.
Abstract: A time-domain boundary element method (TDBEM) for transient dynamic analysis of two-dimensional (2D), homogeneous, anisotropic and linear piezoelectric cracked solids is presented in this paper. The present analysis uses a combination of the strongly singular displacement boundary integral equations (BIEs) and the hypersingular traction boundary integral equations. The spatial discretization is performed by a Galerkin-method, while a collocation method is implemented for the temporal discretization. Both temporal and spatial integrations are carried out analytically. In this way, only the line integrals over a unit circle arising in the time-domain fundamental solutions are computed numerically by standard Gaussian quadrature. An explicit time-stepping scheme is developed to compute the unknown boundary data including the generalized crack-opening-displacements (CODs) numerically. Special crack-tip elements are adopted to ensure a direct and an accurate computation of the dynamic field intensity factors (IFs) from the CODs. Several numerical examples involving stationary cracks in both infinite and finite solids under impact loading are presented to show the accuracy and the efficiency of the developed hypersingular time-domain BEM.
TL;DR: In this paper, two different integral equations are employed along the opposite sides of the crack surface to analyze the crack growth process in quasi-brittle materials are discussed, and compared with the classical approach.
Abstract: This work deals with analysis of cracked structures using BEM. Two formulations to analyse the crack growth process in quasi-brittle materials are discussed. They are based on the dual formulation of BEM where two different integral equations are employed along the opposite sides of the crack surface. The first presented formulation uses the concept of constant operator, in which the corrections of the non-linear process are made only by applying appropriate tractions along the crack surfaces. The second presented BEM formulation to analyse crack growth problems is an implicit technique based on the use of a consistent tangent operator. This formulation is accurate, stable and always requires much less iterations to reach the equilibrium within a given load increment in comparison with the classical approach. Comparison examples of classical problem of crack growth are shown to illustrate the performance of the two formulations.
TL;DR: In this article, a combination of multi-domain hybrid boundary node method (Hybrid BNM) and fracture mechanics was selected for physical representation and analysis of a pavement with a top-down crack.
Abstract: Top-down crack is a type of crack that rivals the severity and prevalence of reflective crack. It significantly reduces pavement’s quality service life. The initiation of these cracks was explained by high-contact stresses induced under radial truck tires; however, the mechanism for top-down crack propagation has not been explained. A combination of multi-domain hybrid boundary node method (Hybrid BNM) and fracture mechanics was selected for physical representation and analysis of a pavement with a top-down crack. The hybrid BNM is a boundary-only, truly meshless method, which does not require a ‘boundary element mesh’, either for the purpose of interpolation of the solution variables or for the integration of ‘energy’. In order to simulate the singularity of the stress on the crack tip, enriched basis functions are used. The factors, which influence the stress intensity factor (SIF) and the expansion path, e.g. horizontal load, thickness of asphalt concrete (AC) layer and base, AC layer and base modulus, are studied through numerical results. It can be concluded that the Hybrid BNM, which has high convergence rates and high accuracy, is able to solve top-down crack problems.
TL;DR: In this article, a new method for the boundary element analysis of unsteady heat conduction problems involving nonhomogenous and/or temperature dependent heat sources by the time-dependent fundamental solution is presented.
Abstract: A new method for the boundary element analysis of unsteady heat conduction problems involving non-homogenous and/or temperature dependent heat sources by the time-dependent fundamental solution is presented. Nonlinear terms are converted to a fictitious heat source and implemented in the present formulation. The domain integrals are efficiently treated by the recently introduced Cartesian transformation method. Similar to the dual reciprocity method, some internal grid points are considered for the treatment of the domain integrals. In the present method, unlike the dual reciprocity method, there is no need to find particular solution for the shape functions in the interpolation computations and the form of the shape functions can be arbitrary and sufficiently complicated. In the present method, at each time step the temperature at boundary nodes and some internal grid points is computed and used as pseudo-initial values for the next time step. Most of the generated matrices are constant at all time steps and computations can be carried out fast. An example with different forms of heat sources is presented to show the efficiency and accuracy of the proposed method.
TL;DR: The hierarchical matrix (H-matrix) technique combined with the adaptive cross-approximation (ACA) applied to a three-dimensional (3D) elastostatic problem using the boundary element method (BEM) is presented.
Abstract: We present the hierarchical matrix (H-matrix) technique combined with the adaptive cross-approximation (ACA) applied to a three-dimensional (3D) elastostatic problem using the boundary element method (BEM). This is used in structural geology and geomechanics for the evaluation of the deformation and perturbed stress field associated with surfaces of displacement discontinuity. Such optimization significantly reduces (i) the time and memory needed for the resolution of the system of equations, but more importantly (ii) the time needed for the post-processing at observation points where the deformation and the perturbed stress field are evaluated. Specifically, it is shown that the H-matrix structure used with the ACA, clearly captures the kernel smoothness during the post-processing stage according to the field point positions, and optimizes the computation accordingly. Combined with the parallelization on multi-core processors, this technique allows intensive computations to be done on personal desktop and laptop computers. Numerical simulations are presented, showing the advantages of such optimizations compared to the standard method.
TL;DR: In this article, a two-dimensional (2D) fast multipole boundary element analysis of magneto-electro-elastic media has been developed for smart structure applications.
Abstract: A two-dimensional (2D) fast multipole boundary element analysis of magneto-electro-elastic media has been developed in this paper. Fourier analysis is employed to derive the fundamental solution for the plane-strain magneto-electro-elasticity. The final formulations are very similar to those for the 2D potential problems, and hence it is quite easy to implement the fast multipole boundary element method. The results are verified by comparison with the analytical solutions to illustrate the accuracy and efficiency of the approach. The numerical examples of multi-inclusion magneto-electro-elastic composites are considered to show the versatility of the proposed approach in smart structure applications.
TL;DR: In this paper, the authors applied the newly developed method of particular solutions (MPS) and one-stage method of fundamental solution (MFS) for solving fourth-order partial differential equations.
Abstract: In this paper we apply the newly developed method of particular solutions (MPS) and one-stage method of fundamental solutions (MFS-MPS) for solving fourth-order partial differential equations. We also compare the numerical results of these two methods to the popular Kansa's method. Numerical results in the 2D and the 3D show that the MFS-MPS outperformed the MPS and Kansa's method. However, the MPS and Kansa's method are easier in terms of implementation.