Showing papers in "Engineering Analysis With Boundary Elements in 2017"

A simple FSDT-based meshfree method for analysis of functionally graded plates

Abstract: Modeling of mechanical behavior of plates has been accomplished in the past decades, with different numerical strategies including the finite element and meshfree methods, and with a range of plate theories including the first-order shear deformation theory (FSDT). In this paper, we propose an efficient numerical meshfree approach to analyze static bending and free vibration of functionally graded (FG) plates. The kinematics of plates is based on a novel simple FSDT, termed as S-FSDT, which is an effective four-variable refined plate theory. The S-FSDT requires C 1 -continuity that is satisfied with the basis functions based on moving Kriging interpolation. Some major features of the approach can be summarized: (a) it is less computationally expensive due to having fewer unknowns; (b) it is naturally free from shear-locking; (c) it captures the physics of shear-deformation effect present in the conventional FSDT; (d) the essential boundary conditions can straightforwardly be treated, the same as the FEM; and (e) it can deal with both thin and thick plates. All these features will be demonstrated through numerical examples, which are to confirm the accuracy and effectiveness of the proposed method. Additionally, a discussion on other possible choices of correlation functions used in the model is given.

Generalized finite difference method for two-dimensional shallow water equations

Abstract: A novel meshless numerical scheme, based on the generalized finite difference method (GFDM), is proposed to accurately analyze the two–dimensional shallow water equations (SWEs). The SWEs are a hyperbolic system of first-order nonlinear partial differential equations and can be used to describe various problems in hydraulic and ocean engineering, so it is of great importance to develop an efficient and accurate numerical model to analyze the SWEs. According to split-coefficient matrix methods, the SWEs can be transformed to a characteristic form, which can easily present information of characteristic in the correct directions. The GFDM and the second-order Runge-Kutta method are adopted for spatial and temporal discretization of the characteristic form of the SWEs, respectively. The GFDM is one of the newly-developed domain-type meshless methods, so the time-consuming tasks of mesh generation and numerical quadrature can be truly avoided. To use the moving-least squares method of the GFDM, the spatial derivatives at every node can be expressed as linear combinations of nearby function values with different weighting coefficients. In order to properly cooperate with the split-coefficient matrix methods and the characteristic of the SWEs, a new way to determine the shape of star in the GFDM is proposed in this paper to capture the wave transmission. Numerical results and comparisons from several examples are provided to verify the merits of the proposed meshless scheme. Besides, the numerical results are compared with other solutions to validate the accuracy and the consistency of the proposed meshless numerical scheme.

The meshless local collocation method for solving multi-dimensional Cahn-Hilliard, Swift-Hohenberg and phase field crystal equations

Abstract: The collocation technique based on the radial basis functions (RBFs) method is simple and efficient for solving a wide area of problems. But the mentioned technique is poor for solving problems that have shock (advection problems) or the discontinuous initial condition. The local RBFs collocation technique is a meshless method based on the strong form. The use of local collocation RBFs method overcomes the mentioned important issue. In the current paper, based on the proposed idea in Wang (2015) [54], we consider a linear combination of shape functions of local radial basis functions collocation method and moving Kriging interpolation technique. For showing the efficiency of new technique, some multi-dimensional problems such as Cahn-Hilliard, Swift-Hohenberg and phase field crystal equations have been chosen. Moreover, several test problems are given that show the acceptable accuracy and efficiency of the proposed scheme.

Iterative multi-domain BEM solution for water wave reflection by perforated caisson breakwaters

Abstract: This study develops a full solution for water wave reflection by a partially perforated caisson breakwater with a rubble mound foundation using multi-domain BEM (boundary element method). Regular and irregular waves are both considered. A quadratic pressure drop condition on caisson perforated wall is adopted, and direct iterative calculations are performed. Due to the use of quadratic pressure drop condition, the effect of wave height on the energy dissipation by the perforated wall is well considered. This study also develops an iterative analytical solution for wave reflection by a partially perforated caisson breakwater on flat bottom using matched eigenfunction expansion method. The reflection coefficients calculated by the multi-domain BEM solution and the analytical solution are in excellent agreement. The present calculated results also agree reasonably well with experimental data from different literatures. Suitable values of discharge coefficient and blockage coefficient in the quadratic pressure drop condition are recommended for perforated caissons. The effects of the wave steepness, the blockage coefficient of perforated wall and the relative wave chamber width on the reflection coefficient are clarified. The present BEM solution is simple and reliable. It may be used for predicting the reflection coefficients of perforated caisson breakwaters in preliminary engineering design.

Strong and weak formulations based on differential and integral quadrature methods for the free vibration analysis of composite plates and shells: Convergence and accuracy

Abstract: The aim of this work is to investigate and compare the accuracy and convergence behavior of two different numerical approaches based on Differential Quadrature (DQ) and Integral Quadrature (IQ) methods, respectively, when applied to the free vibration analysis of laminated plates and shells. The numerical methods at issue allow to solve the strong and the weak forms of the governing equations of these structural elements. A completely general approach is presented to evaluate numerically derivatives and integrals by using several basis functions (polynomial approximation) and grid distributions (discretization). The convergence analyses are performed for three different approaches: Strong Formulation (SF), Weak Formulation (WF) with C1 continuity conditions, and Weak Formulation (WF) with C0 continuity conditions. For each approach, a set of convergence graphs is shown, by varying both basis functions and discrete grids, in order to define the combinations that provide the best accuracy with reference to the exact solutions available in the literature.

Shape optimization of sound barrier using an isogeometric fast multipole boundary element method in two dimensions

Abstract: This study presents an isogeometric fast multipole boundary element method (IGA FMBEM) in two-dimensional (2D) acoustics and a related sensitivity-based shape optimization algorithm for sound barriers. In the isogeometric analysis, Non-Uniform Rational B-Splines (NURBS) are used to accurately represent structural geometry. The control points are set as design variables in the shape optimization procedure given that their variations can flexibly result in shape changes. Acoustic shape sensitivities with respect to control points are calculated by the sensitivity boundary integral equation (BIE) based on the direct differentiation method. The singular integrals in the sensitivity BIEs are formulated explicitly under the isogeometric discretization. The minimization of sound pressure on the reference surface is selected as design objective. The gradient-based optimization solver is finally introduced for optimization iteration after the acoustic state and sensitivity information are obtained. The fast multipole method (FMM) is applied to improve overall computational efficiency. The Burton–Miller method is adopted to conquer the fictitious eigenfrequency problem in solving exterior acoustic problems. The correctness and validity of the proposed methods are demonstrated through a number of numerical simulations, while the performance of the sensitivity-based optimization algorithm is observed in the shape optimization of a 2D Γ-shaped sound barrier.

Fundamental solutions of a multi-layered transversely isotropic saturated half-space subjected to moving point forces and pore pressure

Abstract: The steady-state dynamic response of a multi-layered transversely isotropic (TI) saturated half-space due to point forces and pore pressure moving with a constant speed is investigated in this paper. To solve this problem, the dynamic stiffness method combined with the inverse Fourier transform is employed. First, the governing equations in terms of the displacement components and pore fluid pressure are solved in the transformed domain by employing the Fourier transform. Next, the exact three-dimensional (3D) dynamic stiffness matrices for the TI saturated layer, as well as the TI saturated half-space, are constructed, and the global dynamic matrix of the problem is formulated by assembling the dynamic matrices of the discrete layers and the underlying half-space. Finally, solutions in the frequency-wavenumber domain of the displacement, pore pressure and stress are obtained through the dynamic stiffness method. The result in the time-space domain is recovered by the Fourier synthesis of the frequency responses which, in turn, are obtained by numerical integration over on one horizontal wavenumber. The accuracy of the developed formulations is confirmed by comparison with existing solutions for an isotropic and saturated medium that is a special case of the more general problem addressed. Numerical results for both low and high source velocities are presented, and the effects of moving speed, material anisotropy, permeability, surface drainage condition and TI saturated layer on the dynamic response are analyzed. It is observed that the dynamic responses reach their peak values when the source velocity is equal to or approaches the phase velocities of SH-, qP1-, qP2- and qSV- in the horizontal direction and the phase velocity of qRayleigh waves. Material anisotropy is very important for the accurate assessment of the dynamic response due to the moving point forces and pore pressure in a TI saturated medium.

Wave interaction with a submerged semicircular porous breakwater placed on a porous seabed

Abstract: In the present study, a coupled eigenfunction expansion-boundary element method is developed and used to analyze the interaction of surface gravity waves with a submerged semicircular porous breakwater placed on a porous seabed in water of finite depth. Two separate cases: (a) wave scattering by the semicircular breakwater, and (b) wave trapping by the semicircular breakwater placed near a porous sloping seawall are studied. Further, as a special case, wave trapping by a semicircular breakwater placed on a rubble mound foundation near a sloping seawall is analyzed in water of uniform depth having an impermeable bed. The wave motion through the semicircular permeable arc of the breakwater is modeled using the Darcy's law of fine pore theory, whilst the wave motion through the porous seabed, rubble mound foundation and the porous seawall are modeled using the Sollitt and Cross model. The friction coefficient defined in Sollitt and Cross model is computed by approximating the spatial dependency of the seepage velocity with the average velocity within the porous media. An algorithm for determining the friction coefficient f is provided. Various physical quantities of interests are plotted and analyzed for various values of waves and structural parameters.

Dynamic simulation of landslide dam behavior considering kinematic characteristics using a coupled DDA-SPH method

Abstract: Landslide with significant volume and considerable velocity may block the river stream in the hillslope-channel coupling system, forming the natural dam and the dammed-lake behind. Previous studies predicted the behavior of landslide dams using different dimensionless indexes derived from the geomorphological characteristics. However, the kinematic characteristics of the river and landslide also play key roles in the dam formation. To consider the kinematic characteristics, the dynamic simulation of the dam behavior (formation and failure) involves three problems: (i) the movement of the river flow, (ii) the landslide movement and (iii) the landslide-river interaction. In this study, the movement of the river flow is simulated by a particle recycling method (PRM) under the framework of smoothed particle hydrodynamics (SPH). The discontinuous deformation analysis (DDA) is used to model the landslide movement. The interaction between the solid and fluid phases is achieved by the coupled DDA-SPH method. The proposed methods have been implemented in the numerical code, and a series of examples were employed for validations. The importance of the kinematic characteristics for the dam behavior was demonstrated by a series of numerical scenarios.

On the ill-conditioned nature of C∞ RBF strong collocation

Abstract: Continuously differentiable radial basis functions ( C ∞ -RBFs) are the best method to solve numerically higher dimensional partial differential equations (PDEs). Among the reasons are: 1. An n-dimensional problem becomes a one-dimensional radial distance problem, 2. The convergence rate increases with the dimensionality, 3. Such RBFs possess spectral convergence.Finitely supported polynomial methods only converge at polynomial rates. C ∞ -RBFs have global support; the systems of equations may become computationally singular if the condition number exceeds the inverse machine epsilon, eM. The solution to computational singularity is to decrease the effective eM by either hardware or software methods. Computer scientists developed rapidly executable multi-precision packages.

The stable node-based smoothed finite element method for analyzing acoustic radiation problems

Abstract: In this paper, the stable node-based smoothed finite element method (SNS-FEM) and the well-known Dirichlet-to-Neumann (DtN) boundary condition are coupled together to reduce the dispersion error in analyzing acoustic radiation problems. An artificial boundary is introduced to truncate the infinite domain and the DtN boundary condition is imposed on the artificial boundary to guarantee the uniqueness of the solution. In the SNS-FEM formulation, a stable item which contains the gradient variance items is constructed without any uncertain parameter to strengthen the system stiffness. Through this operation, a perfect balance between the stiffness and mass matrices is established and the dispersion error is reduced significantly. Two benchmark cases and two practical engineering problems are employed to investigate the performance of the SNS-FEM. The results demonstrate that the SNS-FEM achieves super accuracy and super convergence. Additionally, the SNS-FEM is less sensitive to the wave number and high-efficiency.

Smoothed particle hydrodynamics with kernel gradient correction for modeling high velocity impact in two- and three-dimensional spaces

Abstract: High velocity impact (HVI) is associated with large deformations of structures, phase transition of materials, big craters and possible flying debris. It is thus challenging for conventional numerical methods to well capture the major physics of HVI problems. This paper presents the development of an improved smoothed particle hydrodynamics (SPH) method with kernel gradient correction (KGC) technique and the original application of the improved SPH method to modeling high velocity impact problems in two-dimensional and three-dimensional spaces. The SPH method with KGC is first validated by the 3D Taylor-Bar-Impact (TBI) test with Armco iron and OFHC copper. It is demonstrated that the obtained results agree well with experimental observations and with existing numerical results. The improved SPH method is then used to model a hypervelocity impact problem with an aluminum sphere impacting onto an aluminum plate in two- and three-dimensional spaces. It is shown that the presented SPH method can qualitatively and quantitatively model the HVI problem with main physics well captured. It is also demonstrated that the presented SPH method can reproduce experimental observations better than other numerical approaches, including the characteristic shape and evolution of the debris cloud as well the particle distribution.

An element-free based IMLS-Ritz method for buckling analysis of nanocomposite plates of polygonal planform

Abstract: The buckling behavior of nanocomposite plates of polygonal planform under in-plane loads is examined. The plate under consideration is reinforced by single-walled carbon nanotubes (CNTs). The governing eigenvalue equation to this problem is derived based on the first-order shear deformation plate theory (FSDT) with a set of element-free shape functions in approximating the two-dimensional displacement fields. To solve this eigenvalue equation, the element-free IMLS-Ritz method is employed to furnish the buckling solution. The convergence of the solution for the CNT-reinforced composite plates is examined. Comparison study is further carried out to validate the accuracy of the solution. A parametric study is performed by varying the CNT volume fraction, CNT distribution, plate thickness-to-apothem ratio and boundary conditions. This first known buckling solution may serve as benchmarks for future research.

An upwind local radial basis functions-differential quadrature (RBF-DQ) method with proper orthogonal decomposition (POD) approach for solving compressible Euler equation

Abstract: The current paper is an improvement of the developed technique in Shu et al. (2005). The proposed improvement is to reduce the used CPU time for employing the local radial basis functions-differential quadrature (LRBF-DQ) method. To this end, the proper orthogonal decomposition technique has been combined with the LRBF-DQ technique. For checking the ability of the new procedure, the compressible Euler equation is solved. This equation has been classified in category of system of advection–diffusion equations. Moreover, several test problems are given that show the acceptable accuracy and efficiency of the proposed scheme.

A hybrid improved complex variable element-free Galerkin method for three-dimensional potential problems

Abstract: Combining the dimension splitting method with the improved complex variable element-free Galerkin method, a hybrid improved complex variable element-free Galerkin (H-ICVEFG) method is presented for three-dimensional potential problems. Using the dimension splitting method, a three-dimensional potential problem is transformed into a series of two-dimensional ones which can be solved with the improved complex variable element-free Galerkin (ICVEFG) method. In the ICVEFG method for each two-dimensional problem, the improved complex variable moving least-square (ICVMLS) approximation is used to obtain the shape functions, and the penalty method is used to apply the essential boundary conditions. Finite difference method is used in the one-dimensional direction. And Galerkin weak form of three-dimensional potential problem is used to obtain the final discretized equations. Then the H-ICVEFG method for three-dimensional potential problems is presented. Four numerical examples are given to show that the new method has higher computational efficiency.

A naturally stabilized nodal integration meshfree formulation for carbon nanotube-reinforced composite plate analysis

Abstract: Naturally stabilized nodal integration (NSNI) meshfree formulations associated with the higher-order shear deformation plate theory (HSDT) are proposed to analyze bending and free vibration behaviors of carbon nanotube-reinforced composite (CNTRC) plates. An extended rule of mixture is used to compute the effective material properties of CNTRC plates. The uniform and functionally graded distributions of carbon nanotube (CNTs) via the plate thickness are studied. In the present approach, gradient strains are directly computed at nodes similar to the direct nodal integration (DNI). Outstanding features of the current approach are to alleviate instability solutions in the DNI and to significantly decrease computational cost as compared to the traditional high-order Gauss quadrature scheme. Discrete equations for bending and free vibration analyses are obtained by variational consistency in the Galerkin weak form. Enforcing essential boundary conditions is completely similar to the finite element method (FEM) due to satisfying the Kronecker delta function property of moving Kriging integration shape functions. Numerical validations with various complex geometries, stiffness ratios, volume fraction of CNTs and boundary conditions are given to show the efficiency of the present approach.

A numerical mesh-free model for elasto-plastic contact problems

Abstract: A numerical mesh-free model applied to a strong formulation for simulating elasto-plastic structures with contact is developed in the context of large deformation. This numerical mesh-free model is based on the Asymptotic Numerical Method (ANM) which is used in the meshless collocation framework to extend its application field to elasto-plastic problems with contact. The efficiency of this model is to take into account of large deformations and to avoid the meshing distortion problem. According to (ANM) techniques, the development in Taylor series is performed to obtain a sequence of linear systems to be solved. These linear systems are then discretized by a collocation meshless approach by using the Moving Least Squares (MLS) functions and a continuation method is adopted to evaluate the solution. The unilateral contact problem is identified to boundary conditions which are replaced by force-displacement relations through a regularization technique. The performance of the proposed approach is tested on several elasto-plastic bi-dimensional examples without and with contact. The obtained results are compared to those computed by the Newton–Raphson method.

Thermal shock analysis of 2D cracked solids using the numerical manifold method and precise time integration

Abstract: The numerical manifold method (NMM), combined with the precise time integration method (PTIM), is proposed for thermal shock fracture analysis. The temperature and displacement discontinuity across crack faces is naturally portrayed attributing to the cover systems in the NMM. The crack tip singularities are characterized through the use of asymptotic bases in the approximations. The discrete equations for transient thermal analysis are firstly solved with the PTIM and then the thermoelastic study is performed. With the interaction integral, the stress intensity factors are computed. Several examples are tested and the nice consistency between the present and existing results is found.

Solution of multi-dimensional Klein–Gordon–Zakharov and Schrödinger/Gross–Pitaevskii equations via local Radial Basis Functions–Differential Quadrature (RBF–DQ) technique on non-rectangular computational domains

Abstract: In the current investigation, we develop an efficient truly meshless technique for solving two models in optic and laser engineering i.e. Klein-Gordon-Zakharov and Schrodinger/Gross-Pitaevskii equations in one- two- and three-dimensional cases. The employed meshless is the upwind local radial basis functions-differential quadrature (LRBF-DQ) technique. The spacial direction is discretized using the LRBF-DQ method and also to obtain high-order numerical results, the fourth-order exponential time differencing Runge-Kutta method (ETDRK4) planned by Liang et al. [37] is applied to discrete the temporal direction. To show the efficiency of the proposed method, we solve the mentioned models on some complex shaped domains. Moreover, several examples are given and simulation results show the acceptable accuracy and efficiency of the proposed scheme.

Radial basis functions methods for boundary value problems: Performance comparison

Abstract: We present in this paper comparisons on the performances among five typical radial basis functions methods, namely radial basis collocation method (RBCM), radial basis Galerkin method (RBGM), compactly supported radial basis collocation method (CSRBCM), compactly supported radial basis Galerkin method (CSRBGM), and finite subdomain radial basis collocation method (FSRBCM), for solving problems arising from engineering industries and applied sciences. Numerical comparison results demonstrate that the RBCM and FSRBCM possess high accuracy and superior convergence rates in which the FSRBCM particularly attains higher accuracy for problems with large gradients. The FSRBCM, CSRBCM and RBCM are computationally efficient while the CSRBCM, CSRBGM and FSRBCM can greatly improve the ill-conditioning of the resultant matrix. In conclusion, its advantages on high accuracy; exponential convergence; well-conditioning; and effective computation make the FSRBCM a first-choice among the five radial basis functions methods.

Transient SH-wave scattering by the lined tunnels embedded in an elastic half-plane

Abstract: A direct half-plane time-domain boundary element method (BEM) was developed and successfully applied to analyze the transient response of ground surface in the presence of arbitrarily shaped lined tunnels, embedded in a linear elastic half-space, subjected to propagating obliquely incident plane SH-waves. To prepare the model, only the interface and inner boundary of the lining need to be discretized. The problem was decomposed into a pitted half-plane and a closed ring-shaped domain, corresponding to the substructure procedure. After computing the matrices and satisfying the compatibility as well as boundary conditions, the coupled equations were solved to obtain the boundary values. To validate the responses, a practical example was analyzed and compared with those of the published works. The results showed that the model was very simple and the accuracy was favorable. Advanced numerical results were also illustrated for single/twin circular lined tunnels as synthetic seismograms and three-dimensional frequency-domain responses. The method used in this paper is recommended to obtain the transient response of underground structures in combination with other numerical methods.

Refined shear deformation theories for laminated composite arches and beams with variable thickness: Natural frequency analysis

Abstract: A higher-order mathematical formulation with applications is presented in this paper for the free vibration analysis of arches and beams made of composite materials. Higher-order shear deformation theories are required to capture accurately the three-dimensional behavior of these structures through a one-dimensional scheme. Several orders of kinematic expansion are investigated and compared. In addition, the so-called zig–zag theories obtained through the use of the well-known Murakami's function are employed. Their effectiveness is extremely clear when laminates with inner-soft cores are analyzed. A set of numerical applications is presented to prove the accuracy of the current methodology. In particular, the comparison with exact solutions and results available in the literature or obtained through three-dimensional finite element commercial codes justifies the use of higher-order models, in which the displacement field is defined through the arbitrary choice of the maximum order of kinematic expansion. The in-plane vibrational modes of arches and beams, as well as annular ring structures, are computed and presented, discussing also the Poisson effect on the solutions.

Dynamic fracture analysis of the soil-structure interaction system using the scaled boundary finite element method

Abstract: Dynamic fracture analysis of the soil-structure interaction system by using the scaled boundary finite element method is presented in this paper. The polygon scaled boundary finite elements, which have some salient features to model any star convex polygons, are employed for modelling the near-field bounded domains. A procedure for coupling the bounded domains with an improved continued-fraction-based high-order transmitting boundary is established. The formulations of the soil-structure interaction system are coupled via the interaction force vector at the interface. The dynamic stress intensity factors and T-stress are extracted according to the definition of the generalized stress intensity factors. The dynamic stress intensity factors of the coupled system are evaluated accurately and efficiently. Two numerical examples are demonstrated to validate the developed method.

A boundary element method for a class of elliptic boundary value problems of functionally graded media

Abstract: A Boundary Element Method (BEM) is derived for obtaining solutions to a class of elliptic boundary value problems (BVPs) of functionally graded media (FGM). Some particular examples are considered to illustrate the application of the BEM.

A variational multiscale interpolating element-free Galerkin method for convection-diffusion and Stokes problems

Abstract: By combining the interpolating moving least squares (IMLS) method with the variational multiscale method, a variational multiscale interpolating element-free Galerkin (VMIEFG) method is presented in this paper for the numerical solutions of convection-diffusion and Stokes problems. In the VMIEFG, the IMLS is used to construct shape functions based on using shifted and scaled polynomial bases. Compared with the variational multiscale element-free Galerkin (VMEFG) method, the VMIEFG method can directly apply the essential boundary conditions. The VMIEFG method is an effective meshless method, especially, for convection-dominated problems. Numerical examples show that the VMIEFG method avoids the oscillation in the element-free Galerkin (EFG) method, and the computational precision of the VMIEFG method is higher than that of the EFG, the VMEFG and the finite element methods.

Why dual boundary element method is necessary

Abstract: Dual boundary integral equation (BIE) was developed for problems containing degenerate boundaries in 1988 by Hong and Chen [Journal of Engineering Mechanics-ASCE, 114, 6, 1988] and was termed the dual boundary element method (BEM) in 1992 by Portela et al. [International Journal for Numerical Methods in Engineering, 33, 6, 1992]. After near 30 years, the dual BIE/BEM for the problem containing a zero-thickness barrier was revisited mathematically to study the rank deficiency from the viewpoint of the updating term and the updating document of singular value decomposition (SVD) [Journal of Mechanics, 31, 5, 2015]. In this paper, we revisit the dual BEM from the physical point of view. Although there is no zero-thickness barrier in the real world, it is always required to simulate a finite-thickness degenerate boundary to be zero-thickness in comparison with sea, air or earth scale. For example, a sheet pile, a screen, a crack problem, a thin airfoil and a breakwater were modeled by the geometry of zero-thickness. The role of the dual BEM is evident since Lafe et al. [Journal of the Hydraulics Division-ASCE, 106, 6, 1980] used the conventional BEM to model the finite-thickness pile wall to geometrically approximate zero-thickness barrier but numerically yielding divergent solution. On the contrary, we physically model the finite-thickness breakwater as a zero-thickness barrier. The breakwater is employed as an illustrative case to demonstrate that the dual BEM simulated by a zero-thickness barrier can yield more acceptable results to match the experiment data in comparison with those of the finite thickness using the conventional BEM. Finally, a single horizontal plate and two dual horizontal plates in vertical direction and in horizontal direction are three illustrative cases to tell you why the dual BEM is necessary not only in mathematics but also in physics.

Non-Euclidean distance fundamental solution of Hausdorff derivative partial differential equations

Abstract: The Hausdorff derivative partial differential equations have in recent years been found to be capable of describing complex mechanics and physics behaviors such as anomalous diffusion, creep and relaxation in fractal media. But most research is concerned with time Hausdorff derivative models, and little has been reported on the numerical solution of spatial Hausdorff derivative partial differential equations. In this study, we derive the fundamental solutions of the Hausdorff derivative Laplace, Helmholtz, modified Hemholtz, and convection-diffusion equations via a non-Euclidean metric, called the Hausdorff fractal distance. And then the singular boundary method is used to numerically simulate the steady heat transfer governed by the Hausdorff Laplace equation in comparison with the corresponding fractional Laplacian models. Numerical experiments show the validity and applicability of the derived fundamental solution of the Hausdorff Laplace equation.

A smoothed finite element method for exterior Helmholtz equation in two dimensions

Abstract: In this work, a smoothed finite element method (SFEM), in which the gradient smoothing technique (GST) from meshfree methods is incorporated into the standard Galerkin variational equation, is proposed to handle the acoustic wave scattering by the obstacles immersed in water. In the SFEM model, only the values of shape functions, not the derivatives at the quadrature points, are required and no coordinate transformation is needed to perform the numerical integration. Due to the softening effects provided by the GST, the original “overly-stiff” FEM model has been properly softened and a more appropriate stiffness of the continuous system can be obtained, then the numerical dispersion error for the acoustic problems is decreased conspicuously and the quality of the numerical solutions can be improved significantly. To tackle the exterior Helmholtz equation in unbounded domains, we use the well-known Dirichlet-to-Neumann (DtN) map to guarantee that there are no spurious reflecting waves from the far field. Numerical tests show that the present SFEM cum DtN map (SFEM-DtN) works well for exterior Helmholtz equation and can provide better solutions than standard FEM.

An enhanced octree polyhedral scaled boundary finite element method and its applications in structure analysis

Abstract: In this paper, an enhanced octree polyhedral scaled boundary finite element method (SBFEM) is proposed in which arbitrary convex polygon (pentagon, hexagon, heptagon, octagon etc.) can be directly served as boundary face elements. The presented method overcomes the existing SBFEM's limitation that boundary face is strictly restricted to be a quadrangle or triangle. The conforming shape functions are constructed using a polygon mean-value interpolation scheme for polyhedral face. A highly efficient octree mesh generation technology is introduced to accelerate the progress of pre-treatment, wherein the mesh information can be directly used in the enhanced SBFEM. The accuracy of the proposed method is first verified using a beam under shear and torsion load. Another three more complicated geometries including a nuclear power plant structure, as well as two sculptures named Terra-Cotta Warriors and Sioux Falls Church are presented to demonstrate the application and robustness of the proposed method. The new method possesses appealing versatility and offers a swift adaptive capacity in mesh generation, which can provide a powerful technique for the simulation of complex geometries, rapid-design analysis and multi-scale problems.

A meshless generalized finite difference method for inverse Cauchy problems associated with three-dimensional inhomogeneous Helmholtz-type equations

Abstract: The generalized finite difference method (GFDM) is a relatively new domain-type meshless method for the numerical solution of certain boundary value problems. The method involves a coupling between the Taylor series expansions and weighted moving least-squares method. The main idea here is to fully inherit the high-accuracy advantage of the former and the stability and meshless attributes of the latter. This paper makes the first attempt to apply the method for the numerical solution of inverse Cauchy problems associated with three-dimensional (3D) Helmholtz-type equations. Numerical results for three benchmark examples involving Helmholtz and modified Helmholtz equations in both smooth and piecewise smooth 3D geometries have been analyzed. The convergence, accuracy and stability of the method with respect to increasing the number of scatted nodes inside the whole domain and decreasing the amount of noise added into the input data, respectively, have been well-studied.