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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]

In this study, an optimal L -stable time-efficient hybrid block method with a relative measure of stability is developed for solving stiff systems in ordinary differential equations. The derivation resorts to interpolation and collocation techniques over a single step with two intermediate points, resulting in an efficient one-step method. The optimization of the two off-grid points is achieved by means of the local truncation error (LTE) of the main formula. The theoretical analysis shows that the method is consistent, zero-stable, seventh-order convergent for the main formula, and L -stable. The highly stiff systems solved with the proposed and other algorithms (even of higher-order than the proposed one) proved the efficiency of the former in the context of several types of errors, precision factors, and computational time. • Optimized block method is proposed having main formula of seventh-order convergence. • Both fixed and adaptive stepsize approaches are used for the proposed family. • The proposed method is found to have L -stability and A -acceptability. • The nonlinear, and stiff models are numerically solved with the proposed and some existing methods. • Several differential systems from scientific fields are solved by the proposed method. [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]

We propose a simple meshfree method for two-dimensional and three-dimensional second order elliptic boundary value problems in heterogeneous media based on equilibrated basis functions. The domain is discretized by a regular nodal grid, over which the degrees of freedom are defined. The boundary is also discretized by simply introducing some boundary points over it, independent from the domain nodes, granting the method the ability of application for arbitrarily shaped domains without the drawback of irregularity in the nodal grid. In heterogeneous media, the governing Partial Differential Equation (PDE) has non-constant coefficients for the partial derivatives, preventing Trefftz-based techniques to be applicable. The proposed method satisfies the PDE independent from the boundary conditions, as in Trefftz approaches. Meanwhile, by applying the weak form of the PDE instead of the strong form through weighted residual integration, the inconvenience of variable coefficients shall be resolved, since the bases will not need to analytically satisfy the PDE. The weighting in the weak form integrals is such that the boundary integrals vanish. The boundary conditions are thus collocated over the defined boundary points. Domain integrals break into algebraic combination of one-dimensional pre-evaluated integrals, thus omitting the numerical integration from the solution process. Each node corresponds to a local sub-domain called cloud. The overlap between adjacent clouds ensures integrity of the solution and its derivatives throughout the domain, an advantage with respect to C 0 formulations. [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]

This work is devoted to the construction and analysis of a new nonlinear technique that allows obtaining accurate numerical integrations of any order using data that contains discontinuities, and when the integrand is only known at grid points. The novelty of the technique consists in the inclusion of correction terms with a closed expression that depend on the size of the jumps of the function and its derivatives at the discontinuities, that are supposed to be known. The addition of these terms allows recovering the accuracy of classical numerical integration formulas close to the discontinuities, as these correction terms account for the error that the classical integration formulas commit up to their accuracy at smooth zones. Thus, the correction terms can be added during the integration or as post-processing, which is useful if the main calculation of the integral has been already done using classical formulas. We include several numerical experiments that confirm the theoretical conclusions reached in this article. [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]

Recently, a time-splitting Fourier pseudo-spectral (TSFP) scheme for solving numerically the Klein–Gordon–Dirac equation (KGDE) has been proposed (Yi et al., 2019). However, that paper only gives numerical experiments and lacks rigorous convergence analysis and error estimates for the scheme. In addition, the time symmetry of the scheme has not been proved. This is not satisfactory from the perspective of geometric numerical integration. In this paper, we proposed a new TSFP scheme for the KGDE with periodic boundary conditions by reformulating the Klein–Gordon part into a relativistic nonlinear Schrödinger equation. The new scheme is time symmetric, fully explicit and conserves the discrete mass exactly. We make a rigorously convergence analysis and establish error estimates by comparing semi-discretization and full-discretization using the mathematical induction. The convergence rate of the scheme is proved to be second-order in time and spectral-order in space, respectively, in a generic norm under the specific regularity conditions. The numerical experiments support our theoretical analysis. The conclusion is also applicable to high-dimensional problems under sufficient regular conditions. Our scheme can also serve as a reference for solving some other coupled equations or systems such as Klein–Gordon–Schrödinger equation. [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]

In this work, we propose a new n + 1 integration scheme over arbitrary polygonal elements based on centroid approximation and Richardson extrapolation scheme. For the purpose of numerical integration, the polygonal element is divided into quadrilateral subcells by connecting the centroid of the polygon with the mid-point of the edges. The bilinear form is then computed in a two-stage approximation: as a first approximation, the bilinear form is computed at the centroid of the given polygonal element and in the second approximation, it is computed at the center of the quadrilateral cells. Both steps can be computed independently and thus parallelization is possible. When compared to commonly used approach, numerical integration based on sub-triangulation, the proposed scheme requires less computational time and fewer integration points. The accuracy, convergence properties and the efficiency are demonstrated with a few standard benchmark problems in two dimensional linear elasto-statics. From the systematic numerical study, it can be inferred that the proposed numerical scheme converges with an optimal rate in both L 2 norm and H 1 semi-norm at a fraction of computational time when compared to existing approaches, without compromising the accuracy. • First time, We introduce n + 1 integration scheme over arbitrary polygonal elements. • The stiffness matrix is computed in a two stage approximation and extrapolated with weighted functions. • Both the steps can be computed independently and thus parallelization is possible and requires less computational time. • The accuracy, convergence properties and the efficacy are demonstrated with a few standard benchmark problems. • It converges with optimal rate in both L 2 norm and H 1 semi-norm at a fraction of computational time. [ABSTRACT FROM AUTHOR]

This paper presents a two-level nesting smoothed extended mesh-free method (S-XMM-N) for the static and dynamic fracture mechanics analysis in orthotropic materials. To capture the jump of displacement fields across the crack surface, as well as singularities of stress fields in the vicinity of the crack tip, both Heaviside function and asymptotic solutions from the orthotropic fracture modeling are incorporated into the radial point interpolation shape functions associated with the partition of unity approach. To improve the accuracy and accelerate the computation of the extended mesh-free method, the two-level nesting smoothing integration scheme based on the first and second-level triangular smoothing sub-domains is developed to perform the stiffness integration. Compared to the extended mesh-free method using the conventional Gaussian quadrature, the presented S-XMM-N uses even less integration sampling points. At the same time, the complex and time-consuming derivative calculation of extended shape functions is completely avoided. This leads to the dramatically improvement of efficiency. Moreover, the presented method gives very exactly numerical solutions by optimally combining the contributions from the two-level nesting smoothing sub-domains based on the Richardson extrapolation method. We use the interaction integral method in conjunction with the asymptotic near crack tip fields of orthotropic materials to extract the static and dynamic stress intensity factors. The numerical solutions obtained from the S-XMM-N are further compared with reference solutions from the literature to verify the accuracy of the proposed method. • S-XMM-N formulation for fracture mechanics analysis in 2D orthotropic materials. • Numerical integration of two-level nesting smoothing sub-domains is presented. • Static and dynamic stress intensity factors are analyzed. • Effect of node distribution, time step and geometric parameters on the DSIFs. • The S-XMM-N is accurate, efficient and stable. [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]

Synchronization is one of the most striking instances of collective behavior, occurring in many natural phenomena. For example, in some ant species, ants are inactive within the nest most of the time, but their bursts of activity are highly synchronized and involve the entire nest population. Here we revisit a simulation model that generates this synchronized rhythmic activity through autocatalytic behavior, i.e., active ants can activate inactive ants, followed by a period of rest. We derive a set of delay differential equations that provide an accurate description of the simulations for large ant colonies. Analysis of the fixed-point solutions, complemented by numerical integration of the equations, indicates the existence of stable limit-cycle solutions when the rest period is greater than a threshold and the event of spontaneous activation of inactive ants is very unlikely, so that most of the arousal of ants is done by active ants. Furthermore, we argue that the persistent oscillations observed in the simulations for colonies of finite size are due to resonant amplification of demographic noise. • Bursts of activity within the nest of some ant species are highly synchronized and involve the entire nest population. • The short-term activity cycles can be explained by stimulation by nest mates after a refractory rest period. • In the limit of infinitely large colonies, the dynamics of ant activity can be described by a set of delay differential equations. • Stable limit cycle solutions occur when the rest period is greater than a threshold and the event of spontaneous activation of inactive ants is very rare. • The synchronized rhythmic activity observed in the simulations for small colony sizes is a finite-size effect. [ABSTRACT FROM AUTHOR]

A time-dependent periodic chain of coupled nonlinear oscillators including local and global potentials is studied. The continuous form of system equations is projected on an arbitrary mode. Detection of different dynamics are highlighted leading to clarification of equilibrium and singular points of the reduced system. The analytical studies permit prediction/design of periodic and quasi-periodic regimes of the system validated by results obtained through direction numerical integrations. • A time-dependent nonlinear periodic chain is investigated. • Fast and slow dynamics of the system around one of its arbitrary modes is studied. • Periodic and non-periodic regimes are predicted. • This study provides design tools for development of new time-dependent systems. [ABSTRACT FROM AUTHOR]

A recent study conducted by Merck & Company in Rahway, New Jersey, proposes a new procedure called forest search for identifying subgroups with large treatment effects in survival analysis applications. The procedure focuses on subgroups where treatment may be harmful and aims to identify substantial benefits by reversing the role of treatment. The researchers evaluated the procedure's performance in simulations and compared it to other methods, finding favorable results. The study includes real data applications from clinical trials in oncology and HIV. The research has been peer-reviewed and published in Statistics in Medicine. [Extracted from the article]

We compare two different numerical methods to integrate in time spatially delocalized initial densities using the Schrödinger-Poisson equation system as the evolution law. The basic equation is a nonlinear Schrödinger equation with an auto-gravitating potential created by the wave function density itself. The latter is determined as a solution of Poisson's equation modelling, e.g., non-relativistic gravity. For reasons of complexity, we treat a one-dimensional version of the problem whose numerical integration is still challenging because of the extreme long-range forces (being constant in the asymptotic limit). Both of our methods, a Strang splitting scheme and a basis function approach using B-splines, are compared in numerical convergence and effectivity. Overall, our Strang-splitting evolution compares favourably with the B-spline method. In particular, by using an adaptive time-stepper rather large one-dimensional boxes can be treated. These results give hope for extensions to two spatial dimensions for not too small boxes and large evolution times necessary for describing, for instance, dark matter formation over cosmologically relevant scales. [ABSTRACT FROM AUTHOR]

In recent papers, a simple harmonic oscillator with additive noise has been studied by several researchers, and it has been shown that its mean total energy increases linearly as time goes to infinity. In contrast to them, we consider an underdamped harmonic oscillator with additive noise. Our analysis reveals that the mean total energy of the stochastic underdamped harmonic oscillator remains bounded and it asymptotically tends to a certain value. In addition, we give a relation between the mean kinetic energy and the growth rate of the mean total energy. Whereas all stochastic θ -methods preserve this relation as they are of weak second local order, we show that only the stochastic trapezoidal method can attain the asymptotic values of the mean total energy and its derivative given by the exact solution. Numerical experiments are carried out to confirm these results. [ABSTRACT FROM AUTHOR]

• Stochastic configuration networks have superior network performance. • An SCN-based numerical integral method is present. • The proposed method is extended to structural system reliability analysis. • The method shows high computational accuracy and efficiency. Complex multi-dimensional integrals are widely used in various engineering problems. This paper proposes a novel numerical integration method based on stochastic configuration networks (SCNs), which is constructed by learning the integrand function. A corresponding primitive function based on a simple functional expression of the trained SCN can be analytically derived, and a general functional relation between the integrand and the primitive function is established based on SCN parameters. By repeatedly applying the derived functional relations, we can successfully evaluate many complex multi-dimensional integrals. The SCN-based numerical integral method provides a powerful tool for solving complex multi-dimensional integrals. Effectiveness of the proposed method in terms of both computational accuracy and stability is demonstrated through numerical experiments. [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]

We present a practical algorithm to approximate the exponential of skew-Hermitian matrices up to round-off error based on an efficient computation of Chebyshev polynomials of matrices and the corresponding error analysis. It is based on Chebyshev polynomials of degrees 2, 4, 8, 12 and 18 which are computed with only 1, 2, 3, 4 and 5 matrix–matrix products, respectively. For problems of the form exp (− i A) , with A a real and symmetric matrix, an improved version is presented that computes the sine and cosine of A with a reduced computational cost. The theoretical analysis, supported by numerical experiments, indicates that the new methods are more efficient than schemes based on rational Padé approximants and Taylor polynomials for all tolerances and time interval lengths. The new procedure is particularly recommended to be used in conjunction with exponential integrators for the numerical time integration of the Schrödinger equation. [ABSTRACT FROM AUTHOR]

• Nearly dynamically spherical rigid body motion with viscoelastic part is studied. • We study such dynamical case of rigid body motion about its center of mass. • Viscoelasticity is due moving mass linked by spring-damper to point on main axes. • Numerical integration of asymptotic equations is conducted for the body motion. • Solutions are obtained over infinite time range with asymptotically small error. The paper develops an approximate solution by means of an averaging method to the system of Euler's equations with additional perturbation terms for a nearly dynamically spherical rigid body containing a viscoelastic element. The averaging method is used. The asymptotic approach permits to obtain some qualitative results and to describe evolution of angular motion using simplified averaged equations and numerical solution. The main objective of this paper is to extend the previous results for the problem of motion about a center of mass of a rigid body under the influence of small internal torque (cavity filled with a fluid of high viscosity) or external torques (resistive medium, constant body-fixed torque). This paper can be considered as mainstreaming of previous works. The advantage of this paper is in receiving the original asymptotic and numerical calculations, as well as solutions that describe the evolution of motion a rigid body with a moving mass over an infinite time interval with an asymptotically small error. The paper presents a contribution in several areas, partially in the problems of spacecraft and satellite motion, and the activities of crew members about the vehicles. The importance of the results is in the progress of moving mass control, and in the motion of spinning projectiles. [ABSTRACT FROM AUTHOR]

Fast and accurate numerical integration always represented a bottleneck in high-performance computational physics, especially in large and multiscale industrial simulations involving Finite (FEM) and Boundary Element Methods (BEM). The computational demand escalates significantly in problems modelled by irregular or endpoint singular behaviours which can be approximated with generalised polynomials of real degree. This is due to both the practical limitations of finite-arithmetic computations and the inefficient samples distribution of traditional Gaussian quadrature rules. We developed a non-iterative mathematical software implementing an innovative numerical quadrature which largely enhances the precision of Gauss-Legendre formulae (G-L) for integrands modelled as generalised polynomial with the optimal amount of nodes and weights capable of guaranteeing the required numerical precision. This methodology avoids to resort to more computationally expensive techniques such as adaptive or composite quadrature rules. From a theoretical point of view, the numerical method underlying this work was preliminary presented in [1] by constructing the monomial transformation itself and providing all the necessary conditions to ensure the numerical stability and exactness of the quadrature up to machine precision. The novel contribution of this work concerns the optimal implementation of said method, the extension of its applicability at run-time with different type of inputs, the provision of additional insights on its functionalities and its straightforward implementation, in particular FEM applications or other mathematical software either as an external tool or embedded suite. The open-source, cross-platform C++ library Monomial Transformation Quadrature Rule (MTQR) has been designed to be highly portable, fast and easy to integrate in larger codebases. Numerical examples in multiple physical applications showcase the improved efficiency and accuracy when compared to traditional schemes. Program title: MTQR CPC Library link to program files: https://doi.org/10.17632/276f78wzsw.1 Developer's repository link: https://github.com/MTQR/MTQR Licensing provisions: GNU General Public License 3 Programming language: C++ (C++17 standard) Supplementary material: User manual (for installation and execution) Nature of problem: Accuracy and time of execution of the current implementations of high-precision numerical integration routines for singular and irregular integrands modelled by generalised polynomials are restricted by: limitations of the floating-point (f.p.) finite-arithmetic of the machine; inability of classical Gaussian quadrature rules to efficiently capture irregular behaviours or end-point singularities using an optimal number of samples; relying on significantly expensive techniques as adaptive or composite quadrature rules that severely increase the number of steps necessary to converge to the desired accuracy threshold. However by precisely manipulating the G-L samples using an ad-hoc monomial transformation we achieve a one-shot, non-iterative, machine-precise quadrature rule with straightforward scalability in higher dimensions. The advantages brought by a non-adaptive technique are greatly emphasised whenever the problem on hand is characterised by a numerous set of integrand functions that behave similarly to sets of generalised polynomials. Solution method: The underlying algorithm is an application of a monomial transformation of (real) order to the n min ∈ N quadrature samples (nodes and weights) of the G-L rule. The order of the transformation depends on the infimum (λ min) and supremum (λ max) of the Müntz sequence of real degrees of the integrands modelled by generalised polynomials. With an a-priori full analysis of the set of integrands to be integrated, the design and the action of such transformation ensures that the bounding monomials are integrated with machine-epsilon precision in double floating-point (f.p.) format [2] without resorting to less efficient schemes as adaptive or composite quadrature rules, singularity subtraction and cancellation methods with limited and uncontrolled precision. The effects of the action of the monomial transformation onto the original G-L samples are clearly visible by the clustering of the nodes around the endpoint singularity of the integrand function (see Fig. 1.2 of the user manual shipped with the source code and located in the MTQR/doc sub-directory as indicated in the root tree in Fig. 2.1 of said manual.) Additional comments including restrictions and unusual features: To strictly secure the integration with finite-arithmetic precision at the order of the machine epsilon in double f.p. representation, higher-than-double data types are necessary to build the quadrature rule; furthermore to compute the optimal number of quadrature samples, the sole real root of a 7th degree polynomial [1, Equation (62)] has to be computed. For these reasons the proposed software depends on both the Boost's [3] Multiprecision header-only library and on the GNU Scientific Library, GSL [4]. Any source code using MTQR must be linked to those libraries which can be easily installed and configured for the system specific compiler in both Windows and Linux. Detailed information and assistance for successfully compiling and building applications with this library can be found in the aforementioned user manual. • Quadrature of functions with endpoint singularities. • Machine precision. • Mathematical software: C++ library. • Generalised (Müntz) polynomials. [ABSTRACT FROM AUTHOR]

In this paper the dynamics of a fractional order system modeling the interaction between dark matter and dark energy is analytically and numerically studied. It is shown for the first time that systems modeling the interaction between dark matter and dark energy, chaotic hidden attractors can be present. The chaotic attractor coexists with two asymptotically stable equilibria. Equilibria of the linearized system exhibit a center-like behavior. The numerical integration is done by means of the Adams–Bashforth–Moulton scheme and the finite Lyapunov exponents are numerically determined with a dedicated Matlab code. The 3D representation of the chaotic hidden attractor reveals the fact it is not connected with the equilibria, being "hidden" somewhere in the considered phase space. • FO system modeling interaction between dark matter and dark energy. • Hidden chaotic attractor coexists with two stable equilibria. • Locally Lyapunov exponents as two-parameter surfaces. [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]

We study the efficient construction of good polynomial lattice rules, which are special instances of quasi-Monte Carlo (QMC) methods. The integration rules obtained are of particular interest for the approximation of multivariate integrals in weighted Walsh spaces. In particular, we study a construction algorithm which assembles the components of the generating vector, which is in this case a vector of polynomials over a finite field, of the polynomial lattice rule in a component-wise fashion. We show that the constructed QMC rules achieve the almost optimal error convergence order in the function spaces under consideration and prove that the obtained error bounds can, under certain conditions on the involved weights, be made independent of the dimension. We also demonstrate that our alternative component-by-component construction, which is independent of the underlying smoothness of the function space, can be implemented relatively easily in a fast manner. Numerical experiments confirm our theoretical findings. [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]

We propose a safe approximation to joint chance-constrained programming, where the constraint functions are additively dependent on a normally-distributed random vector. The approximation is analytical, meaning that it requires neither numerical integrations nor sampling-based probability approximations. Under mild assumptions, the approximation is a standard nonlinear program. We compare this new safe approximation to another analytical safe approximation for joint chance-constrained programming based on Boole's inequality and to a scenario approach through two numerical examples representing the constrained control of linear Gauss–Markov models. It is shown that our proposed safe approximation has a lower degree of conservativeness compared to these two alternative approaches. [ABSTRACT FROM AUTHOR]

In the response analysis of uncertain structural models with limited information, probability-boxes can be effectively employed to address the aleatory and epistemic uncertainty together. This paper presents a copula-based uncertainty propagation method which can accurately perform uncertainty propagation analysis with correlated parametric probability-boxes. Firstly, the parameter estimation and Akaike information criterion analysis are utilized to select an optimal copula based on available samples, by which the joint cumulative distribution function is constructed for the correlated input variables. Then, using the obtained joint cumulative distribution function, the correlated parametric probability-boxes are transformed into independent normal variables, and an efficient method based on sparse grid numerical integration is proposed to calculate the bounds on statistical moments of a response function. Finally, numerical examples and an engineering application are provided to verify the effectiveness of the presented method. [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]

The trapezoidal formula is known to achieve exponential convergence when calculating infinite integrals of bilateral rapidly decreasing functions. Even when considering unilateral rapidly decreasing functions, the trapezoidal formula can be made to converge exponentially by applying an appropriate conformal map to the integrand, as proposed by Stenger. This study modifies the conformal map to achieve a better convergence rate. Furthermore, aiming for verified numerical integration, we specify a rigorous error bound for the modified quadrature formula. Numerical examples comparing the modified to the existing formula are considered. [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]

Discrete particle simulations have become the standard in science and industrial applications exploring the properties of particulate systems. Most of such simulations rely on the concept of interacting spherical particles to describe the properties of particulates, although, the correct representation of the nonspherical particle shape is crucial for a number of applications. In this work we describe the implementation of clumps, i.e. assemblies of rigidly connected spherical particles, which can approximate given nonspherical shapes, within the MercuryDPM particle dynamics code. MercuryDPM contact detection algorithm is particularly efficient for polydisperse particle systems, which is essential for multilevel clumps approximating complex surfaces. We employ the existing open-source CLUMP library to generate clump particles. We detail the pre-processing tools providing necessary initial data, as well as the necessary adjustments of the algorithms of contact detection, collision/migration and numerical time integration. The capabilities of our implementation are illustrated for a variety of examples. [ABSTRACT FROM AUTHOR]

In this work, an accelerated novel meshless coupled algorithm is first presented for the numerical study of high-dimensional space-fractional nonlinear Schrödinger or Gross-Pitaevskii equations (SFNLS/GPEs). The proposed meshless method is motivated by the novel coupled concepts: (a) a time splitting-step is adopted to decompose the NLSE; (b) the spatial Riesz fractional derivative under the Riemann-Liouville operator is discretized by the combination of numerical integration and meshless weighted-least-square schemes; (c) an accurate absorbing boundary condition is employed to treat the unbounded domain. Meanwhile, a parallel algorithm based on the above approximated schemes is designed to reduce the computing cost on a CUDA-program-based GPU card. Subsequently, the validation and numerical convergence rate of the proposed method for SFNLSE are discussed and illustrated by solving two 2D/3D examples. Furthermore, the proposed parallel meshless algorithm is extended to predict the nonlinear dynamic behaviors dominated by the space-fractional Gross-Pitaevskii equations (SFGPEs), and the influence of nonlocal effect on the nonlinear phenomena is also discussed. The presented numerical results illustrate that the proposed algorithm for multi-dimensional SFNLSEs is reliability and high-efficient. [ABSTRACT FROM AUTHOR]