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A Galerkin-based meshfree flat shell formulation is chosen to study natural frequency and eigenmode of thin-walled folded structures. Reproducing kernel is used as the interpolation function. Stabilized conforming nodal integration is employed for numerical integration of the weak form. Additionally, sub-domain stabilized conforming integration is adopted for the folded region to integrate the stiffness matrix accurately. The first order shear deformation theory is utilized considering in-plane deformation, out-of-plane deformation and drilling components. The singular kernel function is introduced to effectively handle the folded geometry. An advanced free vibration simulation can be achieved. Accuracy and effectiveness of the meshfree modeling are demonstrated through the numerical examples. [ABSTRACT FROM AUTHOR]

In this paper, a novel nodal integration technique for meshfree methods is introduced. This technique is based on the idea of the Cartesian transformation method, which prevents the presence of background cells during the numerical integration process. The Gauss–Lobatto quadrature is used instead of the conventional Gaussian quadrature to create the integration points so that the integration points coincide with the field nodes. This development gave rise to an innovative nodal integration method that eliminates the necessity for generating integration cells, thus advancing further toward a genuinely meshfree approach. The meshfree method chosen in this study is the radial point interpolation method due to the satisfaction of the Kronecker delta property. The curved shell is formulated by the first-order shear deformation theory (FSDT). The static and free vibration analyses of different geometry curved shells are conducted. Through several numerical examples, the accuracy and efficiency of the proposed technique are demonstrated and discussed. It is found that the current technique has better performance than the existing numerical integration techniques used in meshfree method. Furthermore, this new nodal integration technique also shows the ability to mitigate the shear-locking phenomenon when analyzing the problem of thin shells using the FSDT formulation. • A novel nodal integration based on the Cartesian transformation method is introduced. • The proposed nodal integration can mitigate the shear-locking phenomenon in thin shells. • A new discretization process is introduced. • High accuracy and computational efficiency are observed. [ABSTRACT FROM AUTHOR]

Numerical exploration of spatial fractional differential equations in three dimensions is not trivial due to their complexity, especially for complicated problem domains. In this work, the time-Caputo and space-Laplacian fractional diffusion equations in three dimensions are analyzed using a meshfree technique. In the temporal dimension, the classical L1 finite difference scheme is used to approach the Caputo fractional derivative. The spatial discretization is realized by a three dimensional reproducing kernel particle method (RKPM), which can eliminate the dependence of shape functions on certain meshes. Therefore RKPM is very suitable to approximate the field variable in complicated three dimensional domains compared with other mesh-dependent methods. For the purpose of increasing the computational efficiency, the stabilized conforming nodal integration (SCNI) and lumped mass matrix techniques are adopted in the Galerkin meshfree formulation. In the proposed method, the tedious derivatives computing for meshfree shape functions, numerical integration for Galerkin weak form and time-consuming inverse calculating for the large size mass matrix are all realized by more efficient approaches comparing with the conventional Galerkin RKPM. Several numerical examples in various domains with structured and unstructured discretization are studied to demonstrate the proposed methodology, and the results show very favorable performance of the proposed method regarding the accuracy and effectiveness. [ABSTRACT FROM AUTHOR]

In weak-form formulations of the local radial point interpolation method (LRPIM) for the solution of partial differential equations, space discretisation matrices have most often been obtained entirely through numerical integration. This work introduces a novel approach which derives closed-form expressions for obtaining the entries of the discretisation matrix for the solution of two-dimensional time-fractional diffusion problems. This analytical approach also yields a closed-form formula for the approximation of the Laplacian. These techniques are then applied for developing LRPIM-based numerical algorithms. Since the exact solutions usually have unbounded first-order time derivatives at time zero, a graded mesh is employed for a high-order in time approximation of the Caputo derivative. The analytical shape functions are used to develop a weak-form algorithm and a strong-form algorithm is developed using the analytical approximation of the Laplacian. We demonstrate that computed solutions obtained using the weak-form and strong-form algorithms have the accuracy levels consistent with the theoretical accuracy in space. An appropriate choice of the mesh grading parameter yields a high-order convergence rate in time. The unconditional stability and convergence of a LRPIM strong form algorithm on a uniform temporal mesh is established under the assumption that the exact solution has sufficient regularity. • New analytical formulas for two-dimensional LRPIM shape functions are derived. • Closed-form entries for LRPIM discretisation matrices are derived. • Graded meshes are employed for problems which have a layer at time zero. • Simplified L1-2 Caputo derivative formulas on graded meshes are derived. • Convergence of a strong-form algorithm on uniform meshes is established. [ABSTRACT FROM AUTHOR]

This study presents a novel numerical integration technique based on Artificial Neural Network (ANN) algorithms to overcome intrinsic limitations characterizing the Boundary Element Method (BEM). The proposed approach, taking advantage of some peculiar properties of the BEM equations, provides an effective alternative to traditional numerical techniques for evaluating the integrated kernels required to compute the displacements and stresses of a two-dimensional solid. Assuming isotropy and homogeneity, and modeling both the geometry and the mechanical parameters using quadratic shape functions, all the integrals in the classical BEM formulation can be expressed as the sum of two terms that are independent of the constitutive properties and solely dependent on four geometric parameters: three components of two distance vectors and a parameter representing the element's curvature. This interesting property of boundary integral equations in elasticity makes them particularly amenable to numerical evaluation using artificial neural networks. Results from numerical tests, which were conducted using increasingly complex integrals, demonstrate the high precision of the proposed approach as long as the integration and collocation points are sufficiently separated to avoid issues with singularity. [ABSTRACT FROM AUTHOR]

Particle shifting model plays a significant role in regularizing particle distribution in high-order accuracy Lagrangian particles methods. However, due to the incomplete neighbouring support of surface particles, the treatment of particle shifting models applied to free-surface particles still deserves attention from researchers. A new free-surface particle regularization scheme based on numerical integration (NIPS) for particle methods is proposed in this study. Specifically, high-resolution surrounding meshes are seeded outside the free surfaces to supplement the incomplete neighbouring support. Then, the contribution of the void region outside the free surface to the particle shifting vector is numerically integrated based on the surrounding meshes. The appropriate resolution and the screen scheme for constructing the surrounding meshes are optimized in this study. This scheme is easy to implement because the particle number density is only employed to screen the surrounding meshes. A typical high-order accuracy particle method, the LSMPS method is taken to evaluate the proposed scheme in this study. Numerical examples demonstrated that the proposed regularization method can keep uniform distribution of surface particles, thus the instability due to heterogeneous surface particles distribution can be effectively avoided. [ABSTRACT FROM AUTHOR]

The distributed-order fractional calculus (DOFC) is a generalization of the fractional calculus which its application can be found in viscoelasticity, transport processes and control theory. In the current paper, a system of time fractional distributed-order diffusion equation is investigated, numerically. In the first stage, the time derivative is approximated by a finite difference formulation. The integral terms are approximated by the numerical integration. Then, a semi-discrete scheme is constructed by this procedure. In the second stage, The stability and convergence of the time-discrete outline are analyzed by the energy method. In the third stage, the space derivative is discretized by the compact integrated radial basis function (CLIRBF) as a truly meshless method. Also, the numerical procedures are performed on regular and irregular computational domains. The numerical experiments verify the ability, efficiency and accuracy of the developed numerical formulation. [ABSTRACT FROM AUTHOR]

This paper discusses an adaptive framework based on the edge-based strain smoothing approach with polygonal meshes for largely deformable quasi-incompressible hyperelasticity. The proposed approach employs the quadtree decomposition for spatial discretization and the strain smoothing technique to compute the bilinear/linear form. The local refinement is based on the stress distribution and the element that has hanging nodes due to adaptive local refinement are treated as polygonal element within the strain smoothing framework. The accuracy and the robustness of the proposed framework are numerically studied with a few examples. When compared to uniform refinement, it is seen that the proposed framework yields comparable results with fewer degrees of freedom. [ABSTRACT FROM AUTHOR]

A novel and comprehensive method is proposed for calculating the dislocation Love numbers (DLNs), Green's functions (GFs), and the corresponding deformation in a transversely isotropic and layered elastic half-space. It is based on the newly introduced Fourier-Bessel series system of vector functions, along with the dual variable and position method. Two important features associated with this new system are: (1) it is much faster than the conventional cylindrical system of vector functions; (2) we can even pre-calculate the DLNs which are only possible in terms of this new system. This is due to the fact that the variables to be solved in the new system are functions of the simple discrete zero points of the Bessel functions, instead of the numerical integration of continuous Bessel functions between the neighboring zero points as in the conventional system. The introduced dual variable and position method is unconditionally stable as compared to the traditional propagator matrix method in dealing with layering. Exact asymptotic expressions of the DLNs for large wavenumber are further derived, which makes the Kummer's transformation applicable in accelerating the convergence of the corresponding GFs. For the reduced case of homogeneous and isotropic half-space, the present solutions amazingly reduce to the existing exact closed-form solutions. These new features are further seamlessly combined for calculating the deformation due to a finite dislocation (or a finite fault in geophysics) in the layered structure, which are demonstrated to be accurate and efficient. [ABSTRACT FROM AUTHOR]

A new accurate and efficient coupled method RKPM-DIEM is proposed. This is a stable and efficient meshfree nodally-integrated reproducing kernel particle method (RKPM) coupled with a dynamic infinite element method (DIEM) for solving half-space problems. The half-space domain is defined as the near field (bounded) and the far field (unbounded) analyzed by the RKPM and DIEM, respectively. Unlike the element-based methods, RKPM is constructed using only nodal data in the global Cartesian coordinates directly to avoid mesh issues such as mesh distortion and entanglement. Also, it provides flexible control of the local smoothness and order of basis, as well as easy construction for a higher-order gradient by changing the kernel function directly. DIEM is first used to show that this approach could solve not only dynamic but also static problems by setting the wave number and the decay coefficient properly. Furthermore, various meshfree integration methods, such as the Gaussian integration, the direct nodal integration, and the natural stabilized nodal integration, are tested to show accuracy and stability. Several benchmark problems are investigated to verify the effectiveness of the proposed method. It has been found that numerical results can achieve high accuracy and stability. • A coupled RKPM-DIEM model is proposed. • RKPM-DIEM is able to solve not only dynamic but also static heat and elastic problems. • The connection of RKPM and DIEM is considered. • Efficient numerical integration ensures stable and accurate results with varied discretizations. • The analytical solutions of Fourier and non-Fourier heat conduction has been developed. [ABSTRACT FROM AUTHOR]

A strong-form meshfree method is proposed for solving plane elastostatic equations of anisotropic functionally graded materials. Any general function may be the grading function and it is changing smoothly from location to location in the material. The proposed method is based on Pascal polynomial basis and multiple-scale technique and it is a genuinely meshfree method since no numerical integrations over domains and meshing processes are required for considered problems. Implementation of the proposed method is straightforward and the method gives very accurate results. Stability of the solutions are examined numerically in occurrence of random noise. Some certain test problems with known exact solutions are solved both on regular and irregular geometries. Acquired solutions by the suggested method are compared with the exact solutions as well as with solutions of some existing numerical techniques in literature, such as boundary element, meshless local Petrov–Galerkin and radial basis function based meshless methods, to show accuracy of the proposed method. [ABSTRACT FROM AUTHOR]

This paper describes a general approach to compute the boundary integral equations that appear when the boundary element method is applied for solving common engineering problems. The proposed procedure consists of a new quadrature rule to accurately evaluate singular and weakly singular integrals in the sense of the Cauchy Principal Value by an exclusively numerical procedure. This procedure is based on a system of equations that results from the finite part of known integrals, that include the shape functions used to approximate the field variables. The solution of this undetermined system of equations in the minimum norm sense provides the weights of the quadrature rule. A MATLAB script to compute the quadrature rule is included as supplementary material of this work. This approach is implemented in a boundary element method formulation based on the Bézier–Bernstein space as an approximation basis to represent both geometry and field variables for verification purposes. Specifically, heat transfer, elastostatic and elastodynamic problems are considered. • The power and versatility of the BEM is exhibited by this general approach. • BIE integrals evaluated in the sense of the CPV by a numerical procedure. • Numerical quadrature accounting for the element shape functions. • A MATLAB class to compute the quadrature rule is included. • The methodology is applied to solve four engineering problems. [ABSTRACT FROM AUTHOR]

In this paper, we present a hybrid numerical development to solve two-dimensional nonlinear structural problems. The proposed approach was developed by combining weak and strong formulations and using a High-Order Development, Continuation technique and Hybrid approximation (HODC-HYB). The hybrid approximation is based on meshless strong form method and Finite Element Method (FEM). This algorithm allows us to overcome several drawbacks such as the difficulties of implementing meshless strong form methods near the boundary of the structural domain, meshless methods can be unstable and less precise for problems with Neumann boundary conditions, but these methods can overcome the connectivity technique and numerical integration in a big part of the domain. Numerical tests are carried out to demonstrate the reliability and the performance of the proposed algorithm by setting up a comparative study with the solutions obtained by HODC-FEM and HODC-MESHLESS algorithms, which are based on the weak and strong forms, respectively. [ABSTRACT FROM AUTHOR]

The B-spline material point method (BSMPM) has proved to be a promising numerical method for modeling problems with large deformations. While the cell-crossing noise is alleviated by using high-order continuous B-spline basis functions, the BSMPM still suffers from reduced accuracy arising from quadrature errors when simulating large-deformation problems. In this work, a quadrature scheme for the information mapping and the internal force calculations in the BSMPM is developed to substitute for the widely adopted numerical integration at the material particles or the Gauss points within the particle domain. Representative numerical examples of elastic and elasto-plastic large-deformation problems demonstrate the highly enhanced accuracy and convergence of the BSMPM simulations for solid mechanics problems involving large deformations by the proposed quadrature scheme. Moreover, it is shown that, compared to the algorithm of applying numerical Gauss quadrature in the particle domain, the proposed integration scheme is a more favorable option for improving the capability and efficiency of the BSMPM in the analysis of large-deformation problems. [ABSTRACT FROM AUTHOR]

Although the radial point interpolation method (RPIM), which is a typical meshless numerical technique, usually behaves much better than the conventional FEM in addressing the numerical dispersion error issue for acoustic computation and more accurate solutions can generally be yielded with the identical node distributions, the related numerical error still can not be completely removed and further improvements are still required. In this work, we proposed a modified RPIM (M-RPIM) to enhance the abilities of the original RPIM in suppressing the numerical dispersion error. In this M-RPIM, a simple and straightforward scheme is employed to ensure that the integrands in performing the numerical integration are continuously differentiable, while in the original RPIM the quadrature cells usually do not align with the shape function supports and then results in the discontinuously differentiable numerical approximation in the quadrature cells, hence considerable numerical integration error can be generated. Since the discontinuously differentiable entities in the system stiffness matrix can be completely avoided in the present M-RPIM, it is found that the numerical dispersion can be markedly suppressed and more accurate numerical solutions can be yielded than the original RPIM in solving acoustic problems. It should be pointed out that the numerical treatments and conclusions in this work are also applicable for most of other meshfree approximations which are similar to the RPIM. [ABSTRACT FROM AUTHOR]

In this paper, we employ an improved Meshless Local Petrov–Galerkin (MLPG), namely Direct Meshless Local Petrov–Galerkin (DMLPG) method, for solving an inverse time-dependent source problem for two-dimensional fractal mobile/immobile solute transport equation on regular and irregular domains. In the weak form DMLPG method, numerical integrations apply over low-degree polynomial basis functions instead of complicated moving least squares (MLS) shape functions of MLPG method. Therefore, the computational costs often reduce in the DMLPG method. In space domain, we employ the DMLPG method and the time derivatives of problem are discretized based on finite difference schemes. Numerical results demonstrate efficiency and accuracy of the proposed algorithm for solving the inverse source problem. [ABSTRACT FROM AUTHOR]

Accurate numerical evaluation of integrals arising in the boundary element method is fundamental to achieving useful results via this solution technique. In this paper, a new technique is considered to evaluate the weak singular integrals that arise in the solution of three-dimensional Laplace's equation. This new application of the triple-reciprocity boundary element method is proposed for the calculation of singular integrals. A formulation of the boundary element method is utilized, and a method for the direct numerical integration of the two-dimensional surface using a two-dimensional interpolation method is proposed. In numerical integral calculation, the numerical integration of arbitrary shape is possible, and integration in the case of two-dimensional integration is approximately changed into a one-dimensional integration by using the Green's second identity. In the introduced line integral, there is no singularity. To evaluate the efficiency of this method, several numerical examples are given. [ABSTRACT FROM AUTHOR]

Meshfree methods have been used as alternatives to the Finite Element Method, due to their flexibility in building approximations without mesh alignment sensitivity. Another attractive feature is the capacity of obtaining approximate solutions of high regularity. On the other hand, the lack of the Kronecker-delta property, a more complex computation of the shape functions, and numerical integration issues represent drawbacks that can overburden the computational analysis. Aiming to conciliate the efficiency of the finite element analysis with the flexibility of meshfree methods, coupling techniques for both methods have been proposed. The coupling proposed here is based on the enrichment strategy of the Generalized Finite Element Method under the global–local approach. The global domain of the problem is represented by a coarse mesh of finite elements. A region of interest defines the local domain, discretized by a set of nodes of the Element Free Galerkin Method (EFG). This local discretization is responsible for providing a numerically obtained function used to enrich the approximate solution of the global problem. A two-dimensional numerical example is extensively evaluated to discuss the effectiveness of the approach and its behavior related to the quality of the boundary conditions of the local domain, penalty parameter, numerical integration and size of the EFG influence domain. • A novel global–local approach is presented, where the local models are discretized with the Element Free Galerkin Method (EFG). • This global–local approach is presented as a coupling technique for FEM and EFG, or any other meshless method. • The advantages of using a local model described by a meshless method are highlighted, as well as some issues related to this kind of approach. • Results sensitivity to a series of parameters is thoroughly studied in a linear plane-stress problem. [ABSTRACT FROM AUTHOR]

The Method of Fundamental Solutions (MFS) is an established technique for solving linear partial differential equations. In this paper it is used for a new purpose: the approximation of integrals over closed surfaces from a finite set of known points and values. The MFS is used to fit an implicit surface through the surface points, where the implicit equation is chosen such that a surface integral is provided by summing the weights of the fit. From the divergence theorem, these surface integrals can be related to specific integrals over the enclosed volume. As a demonstration, we calculate the surface area, volume, centroid and radius of gyration, for three solid geometries: a sphere, a torus, and an ellipsoid. Very quick convergence to analytical results is shown. Local surface properties, such as the components of curvature, can also be obtained accurately. The drawbacks and advantages of the method are discussed, and the potential to calculate properties of constant-density rigid bodies (e.g. the moment of inertia tensor) and averages of incompressible flow fields (e.g. average flow velocity and strain rate) is highlighted. [ABSTRACT FROM AUTHOR]

The meshless numerical manifold method (MNMM) inherits two covers of the numerical manifold method. A mathematical cover is composed of nodes' influence domains and a physical cover consists of physical patches, which are produced through cutting mathematical cover by physical boundaries. Because two covers are adopted, MNMM can naturally and uniquely solve both the continuous and discontinuous problems under Galerkin's variational framework. However, Galerkin's meshless method needs background integration grids to realize solving, which often does not match the nodes' influence domains, so the accuracy of numerical integration is reduced. Consider that MNMM allows even distribution of nodes and the physical cover contains the characteristics of the boundary and nodes' influence domains, the study presents a new numerical integration strategy to ensure that the background integration grids match the nodes' influence domains. The method can be applied to continuous and discontinuous problems, and is proved to be equivalent to the influence domain integration. At the same time, a reasonable arrangement of mathematical nodes is made to assure the background integration grid that it is accordant and straightforward. In this way, the number of physical patches of each integral point is the same, which improves the accuracy of interpolation calculation. The effectiveness of the proposed method is verified by numerical examples of both continuous and discontinuous problems. [ABSTRACT FROM AUTHOR]

Quadrilateral mesh is a commonly used mathematical cover for numerical manifold method (NMM). However, the NMM based on quadrilateral isoparametric mapping has some intrinsic shortcomings, which is revealed for the first time in this study. Therefore, a new NMM is established to overcome these drawbacks by adopting a quadrilateral area coordinate system. In the proposed NMM, the stiffness matrix can be analytically determined without resorting to cumbersome Jacobian inversion and numerical integration. Moreover, a cone-complementary-based contact model is formulated in the context of the new NMM framework, which enables accurate determination of frictional and cohesive contact forces in solving dynamic contact problems. Thus, artificial penalty and open-close iteration in the original NMM can be all avoided. Several benchmark examples are simulated to demonstrate the excellent performance of the presented NMM. [ABSTRACT FROM AUTHOR]

This paper describes a new efficient approach based on the concept of reduced basis for large deformation analysis. The domain problem is discretized using the meshfree particle radial point interpolation method (RPIM), which inherently possesses the Kronecker's delta property. Meshless numerical integration is evaluated by the Cartesian transformation method (CTM), which enhances the performance of the RPIM. In addition, we also introduce a new approach to further improve the capability of the current CTM in evaluation of numerical integration for problems with complex geometries, i.e., by incorporation of the non-uniform rational B-splines function (NURBS) into the CTM. The emphasis of the paper is on the nonlinear nature of the large deformation problems, which are often solved by an iterative scheme. Conventional Newton–Raphson technique usually requires high cost due to the fact that several load steps are usually performed, and multiple iterations are needed in each load step. This low computational efficiency can be overcome, as proposed in this work, by using the so-called combined approximation, which approximates the full-size solution by a set of reduced bases. In other words, reduction of the problem size can be obtained, leading to reduction of the computational time, while accuracy is almost preserved. • A novel meshfree RPIM approach for analysis of hyperelastic behaviors is presented. • The numerical integration scheme CTM is extended for complicated geometric shapes. • Computational process is accelerated by employing Combined Approximation (CA). • Elapsed time is efficiently saved, especially in fine discretization, while accuracy is maintained. • Desirable features of the present approach are illustrated via numerical experiments. [ABSTRACT FROM AUTHOR]

The second moment of inertia of a continuous axisymmetric object with an arbitrary shape made of a nonhomogeneous material is usually calculated by dividing it into small domains. However, it is a burdensome process to specify the density of the small domains. In this paper, a technique of easily calculating the second moment of inertia of an axisymmetric nonhomogeneous material using boundary integral equations is proposed. The calculations of the mass, primary moment, and center of mass of an arbitrarily shaped object made of a nonhomogeneous material are also shown. A formulization of the boundary element method is utilized, and a technique for the direct numerical integration of the axisymmetric domain using an axisymmetric interpolation method without the need to carry out domain division is proposed. Heterogeneous materials include laminated material composites, in which the density distribution is discontinuous. Axisymmetric interpolation using harmonic and biharmonic functions has a weak point that can be easily overcome using a scale method. [ABSTRACT FROM AUTHOR]

In the present paper, an efficient mesh-free approach is established to determine the stress intensity factors in the vicinity of the crack tip. This efficient mesh-free approach is based on the Weighted Least Squares method (WLS) combined with the stresses extrapolation method and with the visibility criterion to evaluate the Stress Intensity Factors (SIFs). This approach is applied on a strong formulation to avoid the technique of numerical integration and this permits us to compute the stresses with accuracy with the help of the collocation method. The accuracy and the robustness of the proposed approach are tested on several benchmarks in Linear Elastic Fracture Mechanics (LEFM). The obtained results have demonstrated that this approach is very accurate and stable even for complex problems and it can be an alternative way for the computation of SIFs. [ABSTRACT FROM AUTHOR]

• A GRK-SCM is introduced for solving the boundary value problems. • The method improves the accuracy and stability by the exact numerical integration. • Using GRK simplifies the computational complexity and promotes the efficiency. • GRK-SCM can outperform RK-CM, GRK-CM and RK-SCM. The meshfree strong form direct collocation method (DCM) can address the domain integration issues involved in the Galerkin meshfree methods and outperform them in the efficiency. However, the accuracy and stability of the DCM may not match the Galerkin meshfree methods in solving the complex problems. Moreover, strong form demands the high order derivatives calculation of the approximation function which increases its computational complexity and costs. Therefore, in the present study, we introduce a stabilized collocation method (SCM) associated with the gradient reproducing kernel (GRK) approximations. This method can improve the accuracy and stability of the DCM by proposing the exact numerical integration in the subdomains, and construct the GRK approximations instead of directly taking the reproducing kernel (RK) derivatives to simplify the computational complexity and promote the efficiency. The exact local integration of the SCM is much more efficient than the global integration of the Galerkin meshfree methods which makes the SCM keep the high efficiency as the DCM. Comparison analyses among the RK-CM, GRK-CM, RK-SCM and GRK-SCM are detailedly studied. Numerical simulations demonstrate that GRK-SCM can surpass the conventional RK-CM in accuracy, stability and efficiency, followed by the RK-SCM and then the GRK-CM, which provides an efficient and stable meshfree method for solving the boundary value problems. [ABSTRACT FROM AUTHOR]

Consider the Dirichlet problem for Laplace's/Poisson's equation in a bounded simply-connected domain S. The source function q ln | P Q * ¯ | is a fundamental solution (FS), and it can be found in many physical problems. The singularity occurs when the boundary value data affected by q ln | P Q * ¯ | as the source node Q * is located near the boundary Γ (= ∂ S). So far, there is no comprehensive study on this kind of singularity. In this paper, the solution singularity is explored and the reduced convergence rates are derived for the method of particular solutions (MPS) and the method of fundamental solutions (MFS). Classic domains, such as disks, ellipses and polygons, are discussed for analysis and computation. For this new kind of solution singularity, the convergence rates of the MFS and the MPS are very low. The errors caused by numerical integration are critical to the solution accuracy. A new analytic framework for the collocation Trefftz method (CTM) involving numerical integration is established in this paper; this is an advanced development of our previous study [19]. Since the numerical solutions are poor in accuracy, removal techniques are essential in applications. New removal techniques are proposed for a node Q * located near Γ. In this paper, an additional FS as, d 0 ln | P Q 0 ¯ | , is added to the original source nodes in the traditional MFS, and the point charge d 0 (= q) and the source node Q 0 are unknowns to be sought by nonlinear solvers (such as the secant method). When the source node Q * is located inside S but near Γ , both simple domains (such as disks, ellipses and squares) and complicated domains (such as amoeba-like domains) are studied. The validity of the new removal techniques is supported by numerical experiments. The removal techniques in this paper may also be applied to solve source identification problems. A comprehensive study has been completed in this paper for the solitary source function q ln | P Q * ¯ | as the source node Q * is located near Γ. [ABSTRACT FROM AUTHOR]

• A new meshfree boundary numerical model. • Mesh-free boundary analysis with a multi-objective optimization that automatically generates optimal discretization arrangements. • Highly efficient objective functions that control the performance of the multi-objective optimization. • Absolutely reliable and quite robust strategy of numerical modeling, with remarkably accurate results. The paper is concerned with the numerical solution of two-dimensional potential problems, through a mesh-free boundary model in a multi-objective optimization framework that automatically generates Pareto-optimal mesh-free discretization arrangements. This robust new strategy of analysis allows for simultaneously improving the solution accuracy, the conditioning of the numerical solver, the stability and efficiency of the mesh-free analysis. The boundary mesh-free model (BMFM) is built on the boundary integral equation of the Laplace potential, with a moving least squares (MLS) approximation of variables. The model considers independent MLS approximations in each boundary segment and performs integration with standard numerical quadrature. The main novelty of the paper is the automatic generation of Pareto-optimal nodal arrangements and corresponding compact supports of the mesh-free boundary model, by means of an evolutionary multi-objective optimization process, based on genetic algorithms, which uses reliable very efficient objective functions. A benchmark problem is presented to assess the accuracy and efficiency of the modeling strategy. The remarkably accurate results obtained, in perfect agreement with those of analytical solutions, make very reliable this robust new strategy of automatic mesh-free boundary analysis in a multi-objective optimization framework. [ABSTRACT FROM AUTHOR]

This article presents a model of the response of rigid circular plates supported by embedded elastic pile foundations under time-harmonic torsional excitation. The contact tractions at the pile-soil and plate-soil interfaces are resolved into circumferential components, which are then approximated by piece-wise constant distributions. The dynamic stress-displacement relation for the soil part, here modeled as a homogeneous, transversely isotropic half-space, is obtained by numerical integration of that medium's Green's functions. The pile is modeled as a series of interconnected one-dimensional circular shafts, and the plate is modeled by imposing rigid-body rotation conditions throughout the portion of the surface of the soil that it occupies. A tangential Winkler-type layer is incorporated between the plate and the soil, in order to represent potential imperfect bonding contact that may occur in engineering practice. This article presents an analysis of the influence of geometric and constitutive parameters of the system in its time-harmonic torsional response, as well as a study on the wave propagation from the piled plate system through the soil. [ABSTRACT FROM AUTHOR]

In this paper, we propose a robust framework based on the scaled boundary finite element method to model implicit interfaces in two-dimensional differential equations in nonhomegeneous media. The salient features of the proposed work are: (a) interfaces can be implicitly defined and need not conform to the background mesh; (b) Dirichlet boundary conditions can be imposed directly along the interface; (c) does not require special numerical integration technique to compute the bilinear and the linear forms and (d) can work with an efficient local mesh refinement using hierarchical background meshes. Numerical examples involving straight interface, circular interface and moving interface problems are solved to validate the proposed technique. Further, the presented technique is compared with conforming finite element method in terms of accuracy and convergence. From the numerical studies, it is seen that the proposed framework yields solutions whose error is O (h 2) in L 2 norm and O (h) in the H 1 semi-norm. Further the condition number increases with the mesh size similar to the FEM. [ABSTRACT FROM AUTHOR]

The concern of the paper is a local mesh free numerical method, implemented with automatic optimization of the discretization parameters, to solve problems of two-dimensional linear elasticity. For a nodal discretization, the mesh free local method uses a node-by-node process to generate the global system of equilibrium equations; in the local domain of each node, the respective equilibrium equations are generated with a reduced numerical integration and used as the formulation basis of the numerical method. Application of the local numerical method, to each node of a mesh free discretization, requires the use of two arbitrary parameters which specify the size of, respectively the compact support and the local domain of integration. In this paper, these parameters are automatically defined by means of a multi-objective optimization process, based on genetic algorithms, which is the novelty of the present numerical method. For the sake of efficiency a comparison between genetic algorithms and another robust evolutionary optimization method, the symbiotic organism search algorithm, is carried out. A benchmark problem was analyzed to assess the accuracy and efficiency of these techniques. The results presented in the paper are in perfect agreement with those of analytical solutions and therefore, make reliable and robust this mesh free numerical method, implemented with automatic parameter optimization. [ABSTRACT FROM AUTHOR]

A boundary element model of a piled raft foundation under horizontal and rocking excitations is presented. Excitations are time-harmonic and can be either horizontal loads and rocking moments applied to the raft, or a vertically-propagating seismic shear wave. The raft is modeled as a rigid circular plate bonded to the surface of the soil, and the pile is an elastic body embedded in the soil. The soil is a three-dimensional, transversely isotropic half-space. Contact tractions at the plate–soil and pile–soil interfaces are obtained by a numerical integration of Green's functions for the aforementioned half-space at discrete points. The equation of motion for the discretized piled raft is obtained upon establishing equilibrium and continuity conditions between foundation and soil throughout those interfaces. The paper investigates the effect of foundation and soil parameters in the response of the system to both external and seismic excitations. [ABSTRACT FROM AUTHOR]

In this paper, the singular integral equations are written for anisotropic plates with elastic anisotropic inclusions in a simple form based on simple dependencies between the Lekhnitskii complex potentials and stress and strain [5]. The numerical method for solving integral equations is developed by the mechanical quadrature formulas. Simplicity, precision of the application and stability of obtained numerical solution is illustrated in stress calculations with controlled accuracy in anisotropic plates: with inclusions of arbitrary shapes; with a system of inclusions, including periodic one; at tension at infinity and concentrated forces applied to inclusions. [ABSTRACT FROM AUTHOR]

The paper is concerned with a local mesh free numerical model (ILMF) developed to solve two-dimensional problems in linear elastic fracture mechanics. Based in the work theorem, the model formulation considers the approximation of the elastic field with moving least squares (MLS) and implements a reduced numerical integration, to improve the model accuracy. Linear elastic fracture mechanics applications of ILMF use the singularity subtraction technique (SST) which regularizes the elastic field, before the numerical solution, thus introducing the stress intensity factors (SIF) as additional primary unknowns of the problem. Hence, the numerical model performs a direct computation of the SIF and does not require a refined discretization to obtain accurate results which, therefore, is an efficient model strategy. Benchmark problems were solved, for an assessment of the accuracy and efficiency of these techniques. [ABSTRACT FROM AUTHOR]

Boundary element method formulations usually rely eventually on the calculation of weakly singular integrals, and hence robust and efficient algorithms for their evaluation are desirable. This paper proposes a numerical scheme based on a conformal polar transformation, four novel non-linear angular transformations, and a subdivision pattern which allows treating both triangular and quadrilateral elements in a common framework. It is shown that this scheme has small sensitivity to the location of the collocation point and the element aspect ratio and skewness because all sources of angular quasi–singularities have been removed by the non-linear transformations. The proposed methodology is compared against others where its robustness and efficacy is demonstrated. [ABSTRACT FROM AUTHOR]

Abstract The moment of inertia of a continuous object with an arbitrary shape made of a nonhomogeneous material is usually calculated by dividing it into small domains. However, it is a burdensome process to specify the density of the small domains. When the Monte Carlo method is used in the case of an arbitrary shape, the computation time increases. In this paper, a technique of easily calculating the moment of inertia of a 3D nonhomogeneous material using boundary integral equations is proposed. It is also shown how to calculate the mass, primary moment, and center of mass of an arbitrary object made of a nonhomogeneous material. A technique employed in the triple-reciprocity boundary element method is used to evaluate integral. In this paper, a formulization of the boundary element method is utilized, and a technique for the direct numerical integration of the three-dimensional domain using a three-dimensional interpolation method without carrying out domain division is proposed. To investigate the efficiency of this technique, several numerical examples are given. [ABSTRACT FROM AUTHOR]

Abstract In this paper, we generalize the recently developed Finite Integration Method (FIM) for the solutions of high-dimensional partial differential equations. Formulation of this Generalized Finite Integration Method (GFIM) can be derived due to the use of piecewise polynomials in the numerical integrations. The GFIM does not require the strict requirement for uniformly distributed nodal points in the original FIM. This robustness advantage extends the applicability of FIM to solve partial differential equations by using direct Kronecker product. Due to the unconditional stability of numerical integrations, the GFIM is effective and efficient to solve higher dimensional partial differential equations with stiffness. For numerical verification, we construct several 1D to 4D problems with different types of stiffness and make comparisons among existing numerical methods. [ABSTRACT FROM AUTHOR]

Abstract Consistent element-free Galerkin (CEFG) methods have improved the accuracy, efficiency and convergence of high-order EFG methods remarkably by developing integration schemes using less sampling points and corrected derivatives of nodal shape functions. However, the computation of these derivatives still takes considerable CPU time because it involves the evaluation of shape functions at relatively more points and the solution of equations. To reduce the evaluation of these functions substantially, an improved integration scheme is proposed for the second-order CEFG method. Moreover, these corrected derivatives are explicitly formulated in terms of the shape functions. In consequence, the computational efficiency of second-order CEFG method is further improved by this scheme. Furthermore, the high accuracy and convergence of the CEFG method is still maintained. Numerical results of elastic examples show that this improved CEFG method is evidently faster than the original and its efficiency approaches that of nesting sub-domains gradient smoothing (NSGS), which was developed recently for second-order meshfree Galerkin methods. An extension to small-strain elastoplasticity is also presented where the proposed method is demonstrated to perform much better than NSGS in elastoplastic computations. [ABSTRACT FROM AUTHOR]

In this paper a new scheme is investigated for solving partial differential equations via combination of radial basis functions (RBFs) and group preserving scheme (GPS), which takes advantage of two powerful methods. In this method, we use Kansas approach to approximate the spatial derivatives and then we apply GPS method to approximate first-order time derivative. An advantage of the developed method is that it can be applied to problems with non-regular geometrical domains. To show the efficiency of this method, some heat equations are solved in one, two and three dimension spaces. The two-dimensional version of heat equation on different geometries such as the rectangular, triangular and circular domains is solved. The three-dimensional case is solved on the cubical and spherical domains. To show the high accuracy of the method, a comparison study of the present method and method used in the paper of Dehghan [1] is given. [ABSTRACT FROM AUTHOR]

The singular edge-based smoothed finite element method (sES-FEM) using triangular (T3) mesh with a special layer of five-noded singular elements (sT5) connected to the singular point, was proposed to model fracture problems in solids. This paper aims to extend the previous studies on singular fields of any order from −0.5 to 0, by developing an analytical means for integration to obtain the smoothed strains. We provide a more efficient practical formulae to estimate the stress intensity factor(SIF) for singular fields of mentioned order. The sT5 element has an additional node at each of the two edges connected to the crack tip, and the displacements are enriched with necessary terms to simulate the singularity. A weakened weak (W2) formulation is used to avoid the differentiation to the assumed displacement functions. The stiffness matrix is computed by using the smoothed strains calculated analytically from the enriched shape functions. Furthermore, our analytical integration techniques reduces the dependency on the order of numerical integration during the computation of the smoothed strain matrix. Several examples have been presented to demonstrate the reliability of the proposed method, excellent agreement between numerical results and reference observations shows that sES-FEM is an efficient numerical tool for predicting the SIF for singular fields. [ABSTRACT FROM AUTHOR]

This paper introduces a weak meshless procedure combined with a multi-resolution numerical integration and its comparison with a strong local meshless formulation for approximating displacement and strain modeled in the form of Elliptic Boundary Value Problems (EBVPs) in one- and two-dimensional spaces. Assets and losses of both strong and weak meshless approaches are considered in detail. The meshless weak formulation considered in the current paper is the well-known Element Free Galerkin (EFG) method whereas the Local Radial Basis Functions Collocation Method (LRBFCM) is taken as a strong formulation. First aspect of the current work is implementation of the new numerical integration techniques introduced in Siraj-ul-Islam et al. (2010) and Aziz et al. (2011) [1,2] in the EFG method and its comparison with numerical integration based on standard Gaussian quadrature, adaptive integration and stabilized nodal integration techniques used in the context of EFG and other allied weak meshless formulations. Second aspect of the current work is analysis of comparative performance of the localized versions of strong and weak meshless formulations. Standard numerical tests are conducted to validate performance of both the approaches. [ABSTRACT FROM AUTHOR]

In this paper, the authors present a more efficient and robust implementation of conventional and hypersingular BIEs for potential problems in three dimensions under the framework of boundary face method (BFM). The focus is laid on the accurate evaluation of singular curved surface integrals, and three aspects related are considered simultaneously: (a) the near singularity caused by distorted element shape; (b) the near singularity derived from the angular direction; (c) the singularity in the radial direction. A conformal polar coordinate transformation (CPCT) is employed to eliminate the shape effect of distorted integration cells, which can retain the shape characteristic. Besides, an improved sigmoidal transformation is introduced to alleviate the near singularity in the angular direction. By combination of the two strategies with previous singularity subtraction method, an efficient numerical integration scheme has been obtained for various orders of singularities. Some numerical examples including parallelogram plate, sphere and hollow cylinder examples with coarse meshes are presented to demonstrate the accuracy and flexibility of the proposed method. [ABSTRACT FROM AUTHOR]

In this paper a stable and modified form of the Levin method based on Bessel radial basis functions is employed for numerical solution of highly oscillatory integrals. In the proposed technique, the multiquadric radial basis function (Levin, 1982 [1] ; Siraj-ul-Islam et al., 2013 [2] ) is replaced by Bessel radial basis functions (Fornberg et al., 2006 [3] ) and thin plate spline of order three. In this scheme the integration form is first transformed into differential form and then the numerical solution of the corresponding differential form is found. The accuracy and the algebraic stability in the form of well-conditioned coefficient matrices of the proposed methods are confirmed through numerical experiments. [ABSTRACT FROM AUTHOR]

In this paper we present a new numerical integration scheme for Element-Free Galerkin (EFG) methods used for solving elliptic problems. Integration points are distributed within the problem domain using an adaptive procedure, based on the characteristics of the shape functions. Existing numerical integration schemes for EFG methods do not offer any control over the integration accuracy. We devise a method of distributing the integration points which allows control over the integration accuracy for all elements of the stiffness matrix, while reducing the number of integration points required. The performance of the procedure is demonstrated on test problems in 1D and 2D. [ABSTRACT FROM AUTHOR]

Abstract: This paper proposed the new Isoparametric Reproducing Kernel Particle Method (IsoRKPM) for modeling nonlinear plate and shell deformation problems. The Reproducing Kernel shape functions are constructed on a two-dimensional parent domain. Following the concept of isoparametric mapping, the RK shape functions are directly used to approximate the plate geometry. A High Order Nodal Integration (HONI) is developed to integrate the Galerkin weak form of the Mindlin plate equilibrium equations. The proposed IsoRKPM with HONI is applied to solve several benchmark problems. Both modal and convergence analyses show that HONI provides more stable and accurate solutions than the nodal integration method. [Copyright &y& Elsevier]

Abstract: The accurate numerical evaluation of nearly singular boundary integrals is a major concerned issue in the implementation of the boundary element method (BEM). In this paper, the previous distance transformation method is extended into triangular elements both in polar and Cartesian coordinate systems. A new simple and efficient method using an approximate nearly singular point is proposed to deal with the case when the nearly singular point is located outside the element. In general, the results obtained using the polar coordinate system are superior to that in the Cartesian coordinate system when the nearly singular point is located inside the element. Besides, the accuracy of the results is influenced by the locations of the nearly singular point due to the special topology of triangular elements. However, when the nearly singular point is located outside the element, both the polar and Cartesian coordinate systems can get acceptable results. [Copyright &y& Elsevier]

Abstract: This work presents new variable transformations for accurate evaluation of the nearly singular integrals arising in the 3D boundary element method (BEM). The proposed method is an extension of the variable transformation method in Ref. [4] for 2D BEM to 3D BEM. In this paper, first a new system denoted as is introduced compared with the polar coordinate system. So the original transformations in Ref. [4] can be developed to 3D in or the polar coordinate system. Then, the new transformation is performed by four steps in case the source point coincides with the projection point or five steps otherwise. For each step, a new transformation is proposed based on the approximate distance function, so that all steps can finally be unified into a uniform formation. To perform integration on irregular elements, an adaptive integration scheme combined with the transformations is applied. Numerical examples compared with other methods are presented. The results demonstrate that our method is accurate and effective. [Copyright &y& Elsevier]

Abstract: The evaluation of volume integrals that arise in conjunction with a hypersingular boundary integral formulation is considered. In a recent work for the standard (singular) boundary integral equation, the volume term was decomposed into an easily computed boundary integral, plus a remainder volume integral with a modified source function. The key feature of this modified function is that it is everywhere zero on the boundary. In this work it is shown that the same basic approach is successful for the hypersingular equation, despite the stronger singularity in the domain integral. Specifically, the volume term can be directly evaluated without a body-fitted volume mesh, by means of a regular grid of cells that cover the domain. Cells that intersect the boundary are treated by continuously extending the integrand to be zero outside the domain. The method and error results for test problems are presented in terms of the three-dimensional Poisson problem, but the techniques are expected to be generally applicable. [Copyright &y& Elsevier]

Abstract: In this paper, we formulate and employ efficient and accurate methods for numerical evaluation of highly oscillatory integrals and integrals having stationary points. Two new approaches using radial basis function (RBF) and wavelets are discussed. The first approach is related to meshless method (MM) which is based on multiquadric (MQ) RBF, and is specially designed for integrands having oscillatory character. This approach stems from the Levin's method. In this procedure, the solution is obtained by solving the corresponding ODE or PDE instead of finding a numerical solution of the integration problem. In situations when the integrand has stationary points, MM fails to deliver. We opt for quadrature rules based on Haar wavelets and hybrid functions. The proposed methods are tested on a number of benchmark tests considered in available literature. The performance of the new methods is compared with the existing methods. Better accuracy of the proposed methods is reported for a variety of problems. [Copyright &y& Elsevier]

Abstract: Recently, sinh transformations have been proposed to evaluate nearly weakly singular integrals which arise in the boundary element method. These transformations have been applied to the evaluation of nearly weakly singular integrals arising in the solution of Laplace''s equation in both two and three dimensions and have been shown to evaluate the integrals more accurately than existing techniques. More recently, the sinh transformation was extended in an iterative fashion and shown to evaluate one dimensional nearly strongly singular integrals with a high degree of accuracy. Here the iterated sinh technique is extended to evaluate the two dimensional nearly singular integrals which arise as derivatives of the three dimensional boundary element kernel. The test integrals are evaluated for various basis functions and over flat elements as well as over curved elements forming part of a sphere. It is found that two iterations of the sinh transformation can give relative errors which are one or two orders of magnitude smaller than existing methods when evaluating two dimensional nearly strongly singular integrals, especially with the source point very close to the element of integration. For two dimensional nearly weakly singular integrals it is found that one iteration of the sinh transformation is sufficient. [Copyright &y& Elsevier]

Abstract: In this paper Haar wavelets (HWs), hybrid functions (HFs) and radial basis functions (RBFs) are used for the numerical solution of multi-dimensional mild, highly oscillatory and non-oscillatory integrals. Part of this study is extension of our earlier work to multi-dimensional oscillatory and non-oscillatory integrals. Second part of the study is focused on coupling Levin''s approach with meshless methods. In first part of the paper, application of the numerical algorithms based on HWs and HFs is extended to integrals having a varying oscillatory and non-oscillatory integrands defined on circular and rectangular domains. In second part of the paper, we propose a meshless method based on multiquadric (MQ) RBF for highly oscillatory multi-dimensional integrals. The first approach is directly related to numerical quadrature with wavelets basis. Like classical numerical quadrature, this approach does not need any intermediate numerical technique. The second approach based on meshless method of Levin''s type converts numerical integration problem to a partial differential equation (PDE) and subsequently finding numerical solution of the PDE by a meshless method. The computational algorithms thus derived are tested on a number of benchmark kernel functions having varying oscillatory character or integrands with critical points at the origin. The novel methods are compared with the existing methods as well. Accuracy of the methods is measured in terms of absolute and relative errors. The new methods are simple, more efficient and numerically stable. Theoretical and numerical convergence analysis of the HWs and HFs is also given. [Copyright &y& Elsevier]