Nodal Distributions Research Articles

A meshless local RBF collocation method using integral operators for linear elasticity

This work presents a meshless numerical method for the solution of the linear elasticity equation, based on local collocation with radial basis functions and global reconstruction using an integral representational formula. Starting from a set of nodes distributed over the domain interior and boundaries, a small local stencil is formed around each interior node by connecting it to its neighbours. Over each local stencil a radial basis function (RBF) collocation is defined, which enforces the PDE governing and boundary operators. In this way the solution domain is covered by a series of small overlapping local RBF collocation systems. At each interior node the displacement field is approximated from the surrounding RBF collocation system and expressed in terms of its corresponding integral representational formula (Betti׳s formula) defined over a local contour. In this way the approach can be considered to be an improved local Boundary Element method (LBEM) where besides using a direct interpolation of the displacement field, the PDE and boundary conditions are also imposed within the local interpolation. The resulting sparse global assembly is then solved to obtain the displacement field at each interior node.Two benchmark problems with known analytical solutions are considered to test the performance of the method: a thin plate with a circular hole under traction, and a cylindrical annulus under internal and external pressure. Using these two numerical examples the accuracy of the proposed method is investigated, considering different configurations for the local integration contour, different quadrature schemes, the effect of basis function flatness, and spatial convergence rates. To allow a consistent analysis of these features regular nodal distributions are used for most investigations, however the capability of the method to operate on irregular datasets is also demonstrated.

Node shift method for stiffness-based optimization of single-layer reticulated shells

This paper presents a node shift method to find the optimal distribution of nodes in single-layer reticulated shells. The optimization process searches for the minimum strain energy configuration and this leads to reduced sensitivity in initial imperfections. Strain energy sensitivity numbers are derived for free shift and restricted shift where nodes can move freely in the 3D space or have to move within a predefined surface respectively. Numerical examples demonstrate the efficiency of the proposed approach. It was found that optimized structures achieve higher ultimate load and are less sensitive to imperfections than the initial structure. The configuration of the final structure is closely related to factors like the initial structural configuration, spatial conditions, etc. Based on different initial conditions, architects can be provided with diverse reasonable structures. Furthermore, by amending the defined shapes and nodal distributions, it is possible to improve the mechanical behavior of the structures.

Factors affecting accuracy of radial point interpolation meshfree method for 3-D solid mechanics

Recently, the radial point interpolation meshfree method has gained popularity owing to its advantages in large deformation and discontinuity problems, however, the accuracy of this method depends on many factors and their influences are not fully investigated yet. In this work, three main factors, i.e., the shape parameters, the influence domain size, and the nodal distribution, on the accuracy of the radial point interpolation method (RPIM) are systematically studied and conclusive results are obtained. First, the effect of shape parameters (R, q) of the multi-quadric basis function on the accuracy of RPIM is examined via global search. A new interpolation error index, closely related to the accuracy of RPIM, is proposed. The distribution of various error indexes on the R-q plane shows that shape parameters q ∈ [1.2, 1.8] and R ∈ [0, 1.5] can give good results for general 3-D analysis. This recommended range of shape parameters is examined by multiple benchmark examples in 3D solid mechanics. Second, through numerical experiments, an average of 30–40 nodes in the influence domain of a Gauss point is recommended for 3-D solid mechanics. Third, it is observed that the distribution of nodes has significant effect on the accuracy of RPIM although it has little effect on the accuracy of interpolation. Nodal distributions with better uniformity give better results. Furthermore, how the influence domain size and nodal distribution affect the selection of shape parameters and how the nodal distribution affects the choice of influence domain size are also discussed.

Solving an eigenvalue problem on a periodic domain using a radial basis function finite difference scheme

Local radial basis functions (RBFs) are becoming increasingly popular as an alternative to global RBFs, as the latter suffer from ill-conditioning. In this paper, a local meshless method based on RBFs in a finite-difference (FD) mode with better conditioned matrices has been developed for solving an eigenvalue problem with a periodic domain. Through numerical experiments, we examine the accuracy of the method as a result of variation in the number and layout of nodes in the domain and the effects of shape parameter, using various globally supported RBFs. The presented scheme has been validated on two different types of nodal arrangement, namely uniform and non-uniform node distributions. The results obtained from the method are found to be in good agreement with the benchmark analytical solutions. In addition, a higher-order RBF-FD scheme (which uses ideas from Hermite interpolation) is then proposed for solving the eigenvalue problem with a periodic domain. Tests show that both accuracy and convergence order can be improved dramatically by using higher-order RBF-FD formulae, which converge at a rate of O(h8.5) compared to the standard-order method which converges as O(h4.3) for uniformly distributed nodes with spacing h.

An implicit meshless scheme for the solution of transient non-linear Poisson-type equations

Abstract A meshfree point collocation method is used for the numerical simulation of both transient and steady state non-linear Poisson-type partial differential equations. Particular emphasis is placed on the application of the linearization method with special attention to the lagging of coefficients method and the Newton linearization method. The localized form of the Moving Least Squares (MLS) approximation is employed for the construction of the shape functions, in conjunction with the general framework of the point collocation method. Computations are performed for regular nodal distributions, stressing the positivity conditions that make the resulting system stable and convergent. The accuracy and the stability of the proposed scheme are demonstrated through representative and well-established benchmark problems.

A rotation-free beam element for beam and cable analyses

The rotation-free or RF element method represents a non-conventional finite element method in which the rotations are not used as dofs and the element interpolation domains are overlapping. Its obvious advantage is that the complication of finite rotations can be avoided. In this paper, the relatively unexplored RF plane beam element recently formulated by the authors in the course of developing a RF triangle for thin-shell analyses is revisited. Comparing with other RF plane beam elements, the present one is simple and physical yet its accuracy remains competitive. Using a corotational approach and the small strain assumption, its tangent bending stiffness matrix can be approximated by a constant matrix which does not require updating in geometric nonlinear analyses. The element is here extended to spatial cable analyses in which the torsional stiffness can often be neglected and the sectional properties are isotropic. Under the same nodal distributions, it is seen that the present element can tolerate much larger load increment and time step under static and dynamic analyses, respectively, than the two-node thin beam finite element.

Nodal superconducting-gap structure in ferropnictide superconductor BaFe2(As0.7P0.3)2

The Cooper pairs of conventional superconductors exhibit a nodeless s-wave symmetry, and most unconventional superconductors, including cuprates and heavy-fermion materials, exhibit nodal d-wave pairing. In contrast to both, angle-resolved photoemission spectroscopy measurements indicate that the iron-based superconductor BaFe2(As0.7P0.3)2 exhibits an unusual nodal s-wave pairing. The superconducting-gap distribution is a pivotal characteristic for a superconductor. Whereas the cuprates and conventional phonon-mediated superconductors are characterized by distinct d-wave and s-wave pairing symmetries with nodal and nodeless gap distributions respectively, the superconducting-gap distributions in iron-based superconductors are rather diversified. Whereas nodeless gap distributions have been directly observed in Ba1−xKxFe2As2, BaFe2−xCoxAs2, KxFe2−ySe2 and FeTe1−xSex (refs 1, 2, 3, 4), the signatures of nodal superconducting gaps have been reported in LaOFeP, LiFeP, KFe2As2, BaFe2(As1−xPx)2, BaFe2−xRuxAs2 and FeSe (refs 5, 6, 7, 8, 9, 10, 11, 12). We here report the angle-resolved photoemission spectroscopy measurements on the superconducting-gap structure of BaFe2(As0.7P0.3)2, and in particular the direct observation of a circular line node on the largest hole Fermi surface around the Z point at the Brillouin zone boundary. Our findings rule out a d-wave-pairing origin for the nodal gap, and establish the existence of nodes in iron pnictides under the s-wave pairing symmetry.

Anomalous sub-diffusion equations by the meshless collocation method

Recently, many new applications in engineering and science are governed by a series of fractional ordinary differential equation or fractional partial differential equations (FPDEs), in which the differential order is with a fractional order. The anomalous sub-diffusion equation (ASDE) is a typical FPDE. The current dominant numerical method for modelling ASDE is finite difference method, which is based on a pre-defined grid leading to inherited issues or shortcomings. Because of its distinguished advantages, the meshless method has good potential in simulation of ASDE. This paper aims to develop an implicit meshless collocation technique based on the moving least squares (MLS) approximation for numerical simulation of ASDE. The discrete system of equations is obtained by using the MLS meshless shape functions and the meshless collocation formulation. The stability and convergence of this meshless approach related to the time discretisation are investigated theoretically and numerically. The numerical examples with regular and irregular nodal distributions are used to validate and investigate accuracy and efficiency of the newly developed meshless formulation. It is concluded that the present meshless formulation is very effective for the modelling and simulation of ASDEs.

Mixed discrete least squares meshless method for planar elasticity problems using regular and irregular nodal distributions

A Mixed Discrete Least Square Meshless (MDLSM) method is proposed for the solution of planar elasticity problems. In this approach, the differential equations governing the planar elasticity problems are written in terms of the stresses and displacements which are approximated independently using the same shape functions. Since the resulting governing equations are of the first order, both the displacement and stress boundary conditions are of the Dirichlet-type which is easily incorporated via a penalty method. Because least squares based algorithm of MDLSM method, the proposed method does not need to be satisfied by the LBB condition. The performance of the proposed method is tested on a benchmark example from theory of elasticity namely the problem of infinite plate with a circular hole and the results are presented and compared with those of the analytical solution and the solutions obtained using the irreducible DLSM formulation. The results indicate that the proposed MDLSM method is more accurate than the DLSM method. The results show that the numerical solutions of the MDLSM method can be obtained with lower computational cost and with higher accuracy. Also its performance is marginally affected by the irregularity of the nodal distribution.

AN ADVANCED MESHLESS METHOD FOR TIME FRACTIONAL DIFFUSION EQUATION

Recently, because of the new developments in sustainable engineering and renewable energy, which are usually governed by a series of fractional partial differential equations (FPDEs), the numerical modeling and simulation for fractional calculus are attracting more and more attention from researchers. The current dominant numerical method for modeling FPDE is finite difference method (FDM), which is based on a pre-defined grid leading to inherited issues or shortcomings including difficulty in simulation of problems with the complex problem domain and in using irregularly distributed nodes. Because of its distinguished advantages, the meshless method has good potential in simulation of FPDEs. This paper aims to develop an implicit meshless collocation technique for FPDE. The discrete system of FPDEs is obtained by using the meshless shape functions and the meshless collocation formulation. The stability and convergence of this meshless approach are investigated theoretically and numerically. The numerical examples with regular and irregular nodal distributions are used to validate and investigate accuracy and efficiency of the newly developed meshless formulation. It is concluded that the present meshless formulation is very effective for the modeling and simulation of FPDEs.

Meshless point collocation for the numerical solution of Navier–Stokes flow equations inside an evaporating sessile droplet

The Navier–Stokes flow inside an evaporating sessile droplet is studied in the present paper, using sophisticated meshfree numerical methods for the computation of the flow field. This problem relates to numerous modern technological applications, and has attracted several analytical and numerical investigations that expanded our knowledge on the internal microflow during droplet evaporation. Two meshless point collocation methods are applied here to this problem and used for flow computations and for comparison with analytical and more traditional numerical solutions. Particular emphasis is placed on the implementation of the velocity-correction method within the meshless procedure, ensuring the continuity equation with increased precision. The Moving Least Squares (MLS) and the Radial Basis Function (RBF) approximations are employed for the construction of the shape functions, in conjunction with the general framework of the Point Collocation Method (MPC). An augmented linear system for imposing the coupled boundary conditions that apply at the liquid–gas interface, especially the zero shear-stress boundary condition at the interface, is presented. Computations are obtained for regular, Type-I embedded nodal distributions, stressing the positivity conditions that make the matrix of the system stable and convergent. Low Reynolds number (Stokes regime), and elevated Reynolds number (Navier–Stokes regime) conditions have been studied and the solutions are compared to those of analytical and traditional CFD methods. The meshless implementation has shown a relative ease of application, compared to traditional mesh-based methods, and high convergence rate and accuracy.

Many-Group Cross-Section Adjustment Techniques for Boiling Water Reactor Adaptive Simulation

Computational capability has been developed to adjust multigroup neutron cross sections, including self-shielding correction factors, to improve the fidelity of boiling water reactor (BWR) core modeling and simulation. The method involves propagating multigroup neutron cross-section uncertainties through various BWR computational models to evaluate uncertainties in key core attributes such as core keff, nodal power distributions, thermal margins, and in-core detector readings. Uncertainty-based inverse theory methods are then employed to adjust multigroup cross sections to minimize the disagreement between BWR core modeling predictions and observed (i.e., measured) plant data. For this paper, observed plant data are virtually simulated in the form of perturbed three-dimensional nodal power distributions with the perturbations sized to represent actual discrepancies between predictions and real plant data. The major focus of this work is to efficiently propagate multigroup neutron cross-section uncertainty through BWR lattice physics and core simulator calculations. The data adjustment equations are developed using a subspace approach that exploits the ill-conditioning of the multigroup cross-section covariance matrix to minimize computation and storage burden. Tikhonov regularization is also employed to improve the conditioning of the data adjustment equations. Expressions are also provided for posterior covariance matrices of both the multigroup cross-section and core attributes uncertainties.

Nodal Leaf Area Distribution in Soybean Plants Grown in High Yield Environments

At any given time, the leaf area index (LAI) of a soybean [Glycine max (L.) Merr.] crop consists of the summed contributions of each trifoliolate leaf present at each main stem node and on branches. No data are available on nodal LAI distributions in modern indeterminate (IN) or semi‐determinate (SD) cultivars grown in irrigated, early‐planted, high‐yield production systems. The impact of stem termination type and row spacing on that distribution was investigated in such environments at Lincoln, NE in 2003, 2004, and 2005. Trifoliolate LAI at each stem node followed a temporal pattern of rapid increase (after leaf initiation) to a peak before declining due to senescence‐driven leaf abscission, thus emulating, on a shorter time‐scale, the canopy LAI pattern. The post‐peak decline in nodal LAI was rapid in nodes initiated prebloom (i.e., nearly 100% abscission before seed‐fill), but was gradual in nodes initiated after pod initiation (i.e., little abscission before plant maturity). Nodal LAI peaked at the eighth node of the IN cultivar, but rapid leaf expansion at preflowering nodes of the SD cultivar led to a broad peak spanning the fifth to eighth node. Simulation of the Beer‐Lambert law of light attenuation in both canopies revealed that light penetration was deeper in the IN canopy than in the SD canopy. Although higher plant density suppressed branching (and thus branch leaf area) in the SD cultivar, this was not observed in the IN cultivar. These findings suggested that nodal LAI development can be used to mechanistically model canopy LAI.

Time-dependent fractional advection-diffusion equations by an implicit MLS meshless method

Recently, many new applications in engineering and science are governed by a series of fractional partial differential equations (FPDEs). Unlike the normal partial differential equations (PDEs), the differential order in a FPDE is with a fractional order, which will lead to new challenges for numerical simulation, because most existing numerical simulation techniques are developed for the PDE with an integer differential order. The current dominant numerical method for FPDEs is Finite Difference Method (FDM), which is usually difficult to handle a complex problem domain, and also hard to use irregular nodal distribution. This paper aims to develop an implicit meshless approach based on the moving least squares (MLS) approximation for numerical simulation of fractional advection-diffusion equations (FADE), which is a typical FPDE. The discrete system of equations is obtained by using the MLS meshless shape functions and the meshless strong-forms. The stability and convergence related to the time discretization of this approach are then discussed and theoretically proven. Several numerical examples with different problem domains and different nodal distributions are used to validate and investigate accuracy and efficiency of the newly developed meshless formulation. It is concluded that the present meshless formulation is very effective for the modeling and simulation of the FADE.

An implicit RBF meshless approach for time fractional diffusion equations

This paper aims to develop an implicit meshless approach based on the radial basis function (RBF) for numerical simulation of time fractional diffusion equations. The meshless RBF interpolation is firstly briefed. The discrete equations for two-dimensional time fractional diffusion equation (FDE) are obtained by using the meshless RBF shape functions and the strong-forms of the time FDE. The stability and convergence of this meshless approach are discussed and theoretically proven. Numerical examples with different problem domains and different nodal distributions are studied to validate and investigate accuracy and efficiency of the newly developed meshless approach. It has proven that the present meshless formulation is very effective for modeling and simulation of fractional differential equations.

Localized meshless point collocation method for time-dependent magnetohydrodynamics flow through pipes under a variety of wall conductivity conditions

In this article a numerical solution of the time dependent, coupled system equations of magnetohydrodynamics (MHD) flow is obtained, using the strong-form local meshless point collocation (LMPC) method. The approximation of the field variables is obtained with the moving least squares (MLS) approximation. Regular and irregular nodal distributions are used. Thus, a numerical solver is developed for the unsteady coupled MHD problems, using the collocation formulation, for regular and irregular cross sections, as are the rectangular, triangular and circular. Arbitrary wall conductivity conditions are applied when a uniform magnetic field is imposed at characteristic directions relative to the flow one. Velocity and induced magnetic field across the section have been evaluated at various time intervals for several Hartmann numbers (up to 105) and wall conductivities. The numerical results of the strong-form MPC method are compared with those obtained using two weak-form meshless methods, that is, the local boundary integral equation (LBIE) meshless method and the meshless local Petrov–Galerkin (MLPG) method, and with the analytical solutions, where they are available. Furthermore, the accuracy of the method is assessed in terms of the error norms L 2 and L ∞, the number of nodes in the domain of influence and the time step length depicting the convergence rate of the method. Run time results are also presented demonstrating the efficiency and the applicability of the method for real world problems.

Spectral element methods on unstructured meshes: which interpolation points?

In the field of spectral element approximations, the interpolation points can be chosen on the basis of different criteria, going from the minimization of the Lebesgue constant to the simplicity of the point generation procedure. In the present paper, we summarize some recent nodal distributions for a high order interpolation in the triangle. We then adopt these points as approximation points for the numerical solution of an elliptic partial differential equation on an unstructured simplicial mesh. The L 2-norm of the approximation error is then analyzed for a model problem.

Meshfree Point Collocation Schemes for 2D Steady State Incompressible Navier-Stokes Equations in Velocity-Vorticity Formulation for High Values of Reynolds Number

A meshfree point collocation method has been developed for the velocity- vorticity formulation of two-dimensional, steady state incompressible Navier-Stokes equations. Particular emphasis was placed on the application of the velocity-correc- tion method, ensuring the continuity equation. The Moving Least Squares (MLS) approximation is employed for the construction of the shape functions, in conjunc- tion with the general framework of the point collocation method. Computations are obtained for regular and irregular nodal distributions, stressing the positivity con- ditions that make the matrix of the system stable and convergent. The accuracy and the stability of the proposed scheme are demonstrated through two representative, well-known, and established benchmark problems. The numerical scheme was also applied to a case with irregular geometry for marginally high Reynolds numbers.

A pilot study of dual-isotope lymphoscintigraphy for breast sentinel node biopsy comparing intradermal and intraparenchymal injection

Aims Identification of sentinel lymph nodes (SLN) may depend on the tissue plane of tracer injection. To explore this, we developed a dual-isotope technique to compare the lymphatic drainage basins accessed by intradermal and parenchymal injections. Methods Fifteen breast cancer patients had simultaneous parenchymal and intradermal injections of 99mTc-labelled human immunoglobulin G (HIG) and 111In-HIG, respectively, 2–4 h before axillary lymph node clearance surgery. All 228 freshly dissected nodes were assayed by well counting and examined for metastatic disease by haematoxylin/eosin staining and immuno-histochemistry. Results Total nodal uptake following intradermal injection was 10 times more than after parenchymal injection. Tracer uptake within the first three draining nodes divided patients into three groups; four (group 1) had identical 1st, 2nd and 3rd echelon nodes, six (group 2) had identical 1st and 2nd echelon nodes and five (group 3) had different 1st echelon nodes. With respect to the first, second and third groups, there was close, moderate and poor correlation (Pearson), respectively, between individual nodal counts accumulated from the two injection sites. Of eight patients with nodal disease, the SLN identified by intradermal and parenchymal injections contained disease in seven and four patients, respectively. Conclusions Comparison of nodal tracer distributions from the two injection planes allows a functional model to be developed with two possible routes of drainage from the parenchymal plane, one joining the tract from the areolar plexus and the other passing independently to the axilla which builds upon Sappey's original anatomical model. This may explain the variable uptake, discordance and false negative SLN identification.

Electron-Attachment Resonances of Glycine Zwitterions from Quantum Scattering Calculations: Modeling Macrosolvation Effects

A computational study of the quantum dynamics for low-energy electrons scattered by the isolated zwitterionic species of the glycine molecule is carried out using a model interaction potential described in the main text. The macroscopic effects of water solvation on the target molecule in the electron scattering problem are described through a continuum polarizable model (CPCM) which modifies the target molecular structure. In such a way, realistic molecular orbitals depicting the glycine zwitterion in solution are used to model the electron-molecule interaction. The results of the calculations indicate the presence of five different transient negative ions (TNIs) formed at energies from the threshold and up to about 6 eV. Although no nuclear motion was explicitly considered in the ensuing decay processes, the analysis of the nodal structures and density distributions for the resonant excess electron wavefunctions over the molecular space suggests possible anionic fragmentations that produce (Gly-H)-, H-, -CO2-, and -NH3. The likely consequences of such releases into the medium are briefly discussed.