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In recent years, one of the hottest topics in computational mechanics is the meshfree or meshless method. Increasing number of researchers are devoting themselves to the research of the meshfree methods, and a group of meshfree methods have been proposed and used to solve the ordinary differential equations (ODEs) or the partial differential equations (PDE). In the meantime, meshfree methods are being applied to a growing number of practical engineering problems. In this paper, a detailed discussion will be provided on the development of meshfree methods. First, categories of meshfree methods are introduced. Second, the methods for constructing meshfree shape functions are discussed, and the interpolation qualities of them are also studied using the surface fitting. Third, several typical meshfree methods are introduced and compared with each others in terms of their accuracy, convergence and effectivity. Finally, the major technical issues in meshfree methods are discussed, and the future development of meshfree methods is addressed.

Reducing the general problem of computing three-dimensional Green's function in a transversely isotropic plate to a finite summation of contributions from a series of planar problems can efficiently yield an accurate solution. Hence, solving the planar scattering problem, of the Pressure-Shear-Vertical (PSV) type or the Shear-Horizontal (SH) type, was performed by three different techniques: The boundary element method; the hybrid method; and the perfectly matched layer method. In the pursuit of these methods, the objective was to highlight their pros and cons in terms of accuracy and efficiency.

A boundary element method (BEM) is presented to study 3D wave scattering by cracks in a cylinder. Green's functions needed in the kernel of boundary integral equations in BEM are derived with the help of guided wave functions. Guided wave modes in the cylinder are obtained by a semi-analytical finite element (SAFE) method. Green's functions are constructed numerically by superposition of guided wave modes. In this method, the cylinder is discretized in the radial direction into several coaxial circular cylinders (sub-cylinders) and the radial dependence of the displacement in each sub-cylinder is approximated by quadratic interpolation polynomials. A numerical procedure is used here to accurately calculate the Cauchy's principal value (CPV) and weakly singular integrals. The multi-domain technique is employed here to model the crack surface. Numerical results are presented to show the effectiveness of the proposed solution.

Extreme elevation of temperature principally threatens tunnel linings and may cause fatal disaster; the recovery of it may take a long time and significant traffic troubles. System of equations is to be described and solution in terms of boundary element method (BEM) is suggested. Moreover, a technique of time-dependent eigenparameters enables one to apply parallel computations and converts the strongly nonlinear system to pseudo-linear one using the influence and polarization tensors. Consequently, instead of repeated solution of large systems of equations, the multiplication of pre-calculated influence matrices has to be carried out instead. In order to properly create the above-outlined procedure, internal cells are selected in the regions primarily connected by the change of temperature. Some examples follow the theory.

The boundary element method (BEM), along with the finite element and finite difference methods, is commonly used to carry out numerical simulations in a wide variety of subjects in science and engineering. The BEM, rooted in classical mathematics of integral equations, started becoming a useful computational tool around 50 years ago. Many researchers have worked on computational aspects of this method during this time.

This paper presents an overview of the BEM and related methods. It has three sections. The first, relatively short section, presents the governing equations for classical applications of the BEM in potential theory, linear elasticity and acoustics. The second describes specialized applications in bodies with thin features including micro-electro-mechanical systems (MEMS). The final section addresses current research. It has three subsections that present the boundary contour, boundary node and fast multipole methods (BCM, BNM and FMM), respectively. Several numerical examples are included in the second and third sections of this paper.

In this paper, a numerical model is developed based on coupled boundary element–finite element methods (BEM–FEM) to minimize liquid sloshing pressure in trapezoidal tank with different sidewall angles. Different geometric shapes such as rectangular, cylindrical, elliptical, spherical and circular conical have already been studied for ship storage tanks by other researchers. In this paper, a new arrangement, i.e., trapezoidal containers is suggested for liquid storage tanks. The tank shape is optimized based on sloshing pressures and forces for a range of frequencies and amplitudes of sway motion and tank configuration. Fluid is considered to be incompressible and inviscid. Therefore, Laplace equation and nonlinear free surface boundary conditions are used to model the sloshing phenomenon. The results validated using available data showed that a new arrangement of trapezoidal storage panels has a better efficiency against sloshing phenomenon than the conventional rectangular tanks.

A time-dependent boundary integral equation method named as pseudo-initial condition method is widely used to solve the transient heat conduction problems. Accurate evaluation of the domain integrals in the pseudo-initial condition formulation is of crucial importance for its successful implementation. As the time-dependent kernel in the domain integral is close to singular when small time step is used, a straightforward computation using Gaussian quadrature may produce large errors, and thus lead to instability of the analysis. To improve the computational accuracy of the domain integral, a coordinate transformation coupled with a domain cell subdivision technique is presented in this paper for 2D boundary element method. The coordinate transformation is denoted as (α, β) transformation, while the cell subdivision technique considers the position of the source point, the shape of the integration cell and the relations between the size of the cell and the time step. With the cell subdivision technique, more Gaussian points are shifted towards the source point, thus more accurate results can be obtained. Numerical examples have demonstrated the accuracy and efficiency of the proposed method.

A boundary condition (BC) related mixed element method is presented to address the corner problem in boundary element method (BEM) for 3D elastostatic problems. In this method, noncontinuous elements (NCEs) are only used at the displacement-prescribed corners/edges and continuous elements (CEs) in other places, which can decrease the degrees of freedom (DOFs) compared to the approach using NCEs at all corners/edges. Moreover, an automatic generation algorithm of BC related mixed linear triangular elements is implemented with the help of 3D modeling engine ACIS, and the boundary element analysis (BEA) is integrated into CAD systems. In order to solve large scale problems, the fast multipole BEM (FMBEM) with mixed elements is proposed and utilized in the BEA. The examples show that the node shift scheme adopting 1/4 is optimal and the BEM/FMBEM using mixed elements can produce more accurate results by only increasing a small number of DOFs.

The coefficient matrices of conventional boundary element method (CBEM) are dense and fully populated. Special techniques such as hierarchical matrices (H-matrices) format are required to extent its ability of handling large-scale problems. Adaptive cross approximation (ACA) algorithm is a widely adopted algorithm to obtain the H-matrices. However, the accuracy of the ACA boundary element method (ACABEM) cannot be adjusted by changing the tolerance ${𝜀}^{\prime }$ when it exceeds a certain value. In this paper, the degenerate kernel approximation idea for the low-rank matrices is developed to build a fast BEM for acoustic problems by exploring the multipole expansion of the kernel, which is referred as the multipole expansion H-matrices boundary element method (ME-H-BEM). The newly developed algorithm compresses the far-field submatrices into low rank submatrices with the expansion terms of Green’s function. The obtained H-matrices are applied in conjunction with the generalized minimal residual method (GMRES) to solve acoustic problems. Numerical examples are carefully set up to compare the accuracy, efficiency as well as memory consumption of the CBEM, ACABEM, fast multipole boundary element method (FMBEM) and ME-H-BEM. The results of a pulsating sphere indicate that the ME-H-BEM keeps both storage and operation logarithmic-linear complexity of the H-matrices format as the ACABEM does. Moreover, the ME-H-BEM can achieve better convergence and higher accuracy than the ACABEM. For the analyzed complicated large-scale model, the ME-H-BEM with appropriate number of expansion terms has an advantage in terms of efficiency as compared with the ACABEM. Compared with the FMBEM, the ME-H-BEM is easier to be implemented.

This paper outlines a new approach to identify a source term of a $2$D elliptic equation for anisotropic nonhomogenous media. The proposed methodology is based on the minimization of an objective function representing differences between the measured potential and those calculated by using the discontinuous dual reciprocity boundary element method, the measurements are required to render a unique solution and supposed to be pointwise in the problem domain. Since the additional data may be contaminated by measurement noises or the numerical computing errors, we adopt a regularizing Levenberg–Marquardt method to solve the nonlinear least-squares problem attained from the inverse source problem. The numerical performance of the proposed approach is studied at the end for both geometries: smooth and piecewise smooth one. The results show a very good agreement with the analytical solutions under exact and noisy data.

In this study, the influence of porosity on the elastic effective properties of polycrystalline materials is investigated using a 3D grain boundary micro mechanical model. The volume fraction of pores, their size and distribution can be varied to better simulate the response of real porous materials. The formulation is built on a boundary integral representation of the elastic problem for the grains, which are modeled as 3D linearly elastic orthotropic domains with arbitrary spatial orientation. The artificial polycrystalline morphology is represented using 3D Voronoi Tessellations. The formulation is expressed in terms of intergranular fields, namely displacements and tractions that play an important role in polycrystalline micromechanics. The continuity of the aggregate is enforced through suitable intergranular conditions. The effective material properties are obtained through material homogenization, computing the volume averages of micro-strains and stresses and taking the ensemble average over a certain number of microstructural samples. The obtained results show the capability of the model to assess the macroscopic effects of porosity.

A three-dimensional (3D) boundary element method for small strains crystal plasticity is described. The method, developed for polycrystalline aggregates, makes use of a set of boundary integral equations for modeling the individual grains, which are represented as anisotropic elasto-plastic domains. Crystal plasticity is modeled using an initial strains boundary integral approach. The integration of strongly singular volume integrals in the anisotropic elasto-plastic grain-boundary equations are discussed. Voronoi-tessellation micro-morphologies are discretized using nonstructured boundary and volume meshes. A grain-boundary incremental/iterative algorithm, with rate-dependent flow and hardening rules, is developed and discussed. The method has been assessed through several numerical simulations, which confirm robustness and accuracy.

Multiscale analyses considering the stretching problem in plates composed of metal matrix composites (MMC) have been performed using a coupled BEM/FEM model, where the boundary element method (BEM) and the finite element method (FEM) models, respectively, the macrocontinuum and the material microstructure, denoted as representative volume element (RVE). The RVE matrix zone behavior is governed by the von Mises elasto-plastic model while elastic inclusions have been incorporated to the matrix to improve the material mechanical properties. To simulate the microcracks evolution at the interface zone surrounding the inclusions, a modified cohesive fracture model has been adopted, where the interface zone is modeled by means of cohesive contact finite elements to capture the effects of phase debonding. Thus, this paper investigates how this phase debonding affects the microstructure mechanical behavior and consequently affects the macrostructure response in a multiscale analysis. For that, initially, only RVEs subjected to a generic strain are analyzed. Then, multiscale analyses of plates have been performed being each macro point represented by a RVE where the macro-strain must be imposed to solve its equilibrium problem and obtain the macroscopic constitutive response given by the homogenized values of stress and constitutive tensor fields over the RVE.

In this work, a unified scheme for computing the fundamental solutions of a three-dimensional homogeneous elliptic partial differential operator is presented. The scheme is based on the Rayleigh expansion and on the Fourier representation of a homogeneous function. The scheme has the advantage of expressing the fundamental solutions and their derivatives up to the desired order without any term-by-term differentiation. Moreover, the coefficients of the series need to be computed only once, thus making the presented scheme attractive for numerical implementation. The scheme is employed to compute the fundamental solution of isotropic elasticity showing that the spherical harmonics expansions provide the exact expressions. Then, the accuracy of the scheme is assessed by computing the fundamental solutions of a generally anisotropic magneto-electro-elastic material.

This paper presents a spectral boundary element formulation for analysis of structures subjected to dynamic loading. Two types of spectral elements based on Lobatto polynomials and Legendre polynomials are used. Two-dimensional analyses of elastic wave propagation in solids with and without cracks are carried out in the Laplace frequency domain with both conventional BEM and the spectral BEM. By imposing the requirement of the same level of accuracy, it was found that the use of spectral elements, compared with conventional quadratic elements, reduced the total number of nodes required for modeling high-frequency wave propagation. Benchmark examples included a simple one-dimensional bar for which analytical solution is available and a more complex crack problem where stress intensity factors were evaluated. Special crack tip elements are developed for the first time for the spectral elements to accurately model the crack tip fields. Although more integration points were used for the integrals associated with spectral elements than the conventional quadratic elements, shorter computation times were achieved through the application of the spectral BEM. This indicates that the spectral BEM is a more efficient method for the numerical modeling of structural health monitoring (SHM) processes, in which high-frequency waves are commonly used to detect damage, such as cracks, in structures.

This paper presents an advanced formulation of the time-domain two-dimensional (2D) boundary element method (BEM) for an elastic, homogeneous unsaturated soil subjected to dynamic loadings. Unlike the usual time-domain BEM, the present formulation applies a convolution quadrature which requires only the Laplace-domain instead of the time-domain fundamental solutions. The coupled equations governing the dynamic behavior of unsaturated soils ignoring contributions of the inertia effects of the fluids (water and air) are derived based on the poromechanics theory within the framework of a suction-based mathematical model. In this formulation, the solid skeleton displacements $u_i$, water pressure $p_w$ and air pressure $p_a$ are presumed to be independent variables. The fundamental solutions in Laplace transformed-domain for such a dynamic $u-p_w-p_a$ theory have been obtained previously by authors. Then, the BE formulation in time is derived after regularization by partial integrations and time and spatial discretizations. Thereafter, the BE formulation is implemented in a 2D boundary element code (PORO-BEM) for the numerical solution. To verify the accuracy of this implementation, the displacement response obtained by the boundary element formulation is verified by comparison with the elastodynamics problem.

A robust boundary element numerical scheme is presented to study crack-face frictional contact in cracked fiber reinforced composite materials. The dual boundary element method is considered for modeling fracture mechanics on these materials. The formulation is based on contact operators over the augmented Lagrangian to enforce contact constraints on the crack surface. Moreover, it considers a Halpin–Tsai macro model for fiber reinforced composite materials which makes it possible to take into account the influence of micromechanical aspects such as: the fibers’ orientation, the fiber’s aspect ratio or the fiber’s volume fraction, estimating the mechanical properties of these composite materials from the known values of the fiber and the matrix. After solving a crack face frictional contact benchmark problem, the capabilities of this methodology are illustrated by studying the influence of not only these micromechanical aspects but also crack face frictional contact conditions on a fractured carbon fiber-reinforced polymer under compression.

The Virtual Element Method (VEM) is a recent numerical technique capable of dealing with very general polygonal and polyhedral mesh elements, including irregular or non-convex ones. Because of this feature, the VEM ensures noticeable simplification in the data preparation stage of the analysis, especially for problems whose analysis domain features complex geometries, as in the case of computational micro-mechanics problems. The Boundary Element Method (BEM) is a well known, extensively used and effective numerical technique for the solution of several classes of problems in science and engineering. Due to its underlying formulation, the BEM allows reducing the dimensionality of the problem, resulting in substantial simplification of the pre-processing stage and in the reduction of the computational effort, without jeopardizing the solution accuracy. In this contribution, we explore the possibility of a coupling VEM and BEM for computational homogenization of heterogeneous materials with complex microstructures. The test morphologies consist of unit cells with irregularly shaped inclusions, representative e.g., of a fiber-reinforced polymer composite. The BEM is used to model the inclusions, while the VEM is used to model the surrounding matrix material. Benchmark finite element solutions are used to validate the analysis results.

Numerical tools which are able to predict and explain the initiation and propagation of damage at the microscopic level in heterogeneous materials are of high interest for the analysis and design of modern materials. In this contribution, we report the application of a recently developed numerical scheme based on the coupling between the Virtual Element Method (VEM) and the Boundary Element Method (BEM) within the framework of continuum damage mechanics (CDM) to analyze the progressive loss of material integrity in heterogeneous materials with complex microstructures. VEM is a novel numerical technique that, allowing the use of general polygonal mesh elements, assures conspicuous simplification in the data preparation stage of the analysis, notably for computational micro-mechanics problems, whose analysis domain often features elaborate geometries. BEM is a widely adopted and efficient numerical technique that, due to its underlying formulation, allows reducing the problem dimensionality, resulting in substantial simplification of the pre-processing stage and in the decrease of the computational effort without affecting the solution accuracy. The implemented technique has been applied to an artificial microstructure, consisting of the transverse section of a circular shaped stiff inclusion embedded in a softer matrix. BEM is used to model the inclusion that is supposed to behave within the linear elastic range, while VEM is used to model the surrounding matrix material, developing more complex nonlinear behaviors. Numerical results are reported and discussed to validate the proposed method.

Numerical prediction of composite damage behaviour at the microscopic level is still a challenging engineering issue for the analysis and design of modern materials. In this work, we document the application of a recently developed numerical technique based on the coupling between the virtual element method (VEM) and the boundary element method (BEM) within the framework of continuum damage mechanics (CDM) to model the in-plane damage evolution characteristics of composite materials. BEM is a widely adopted and efficient numerical technique that reduces the problem dimensionality due to its underlying formulation. It substantially simplifies the pre-processing stage and decreases the computational effort without affecting the solution’s accuracy. VEM is a recent generalization to general polygonal mesh elements of the finite element method that ensures noticeable simplification in the data preparation stage of the analysis, notably for computational micro-mechanics problems, whose analysis domain often features complex geometries. The numerical technique has been applied to artificial microstructures, representing the transverse section of composite material with stiffer circular-shaped inclusions embedded in a softer matrix. BEM is used to model the inclusions that are supposed to behave within the linear elastic range, while VEM is used to model the surrounding matrix material, developing nonlinear behaviours. Numerical results are reported and discussed to validate the proposed method.