Shape Function Values Research Articles

Element Free Galerkin (EFG) Formulation in Solving Kinematic Wave Equation for Hydrological Modelling

Meshfree methods such as Element Free Galerkin Method (EFG) are the cutting edge of numerical methods as it does not require user to perform meshing which often is time consuming and tedious. The method being "meshfree" for the user can theoretically generate more accurate models which follow more complex boundary conditions such as natural rivers etc. The time saving and potential of higher accuracy in modelling makes EFG formulation an interesting research topic. EFG formulation is used to solve kinematic wave equations which is detailed in this paper. Galerkin weighted residual method using EFG’s shape function parameters is used to discretize the partial differential equation (PDE) of kinematic wave equations. To discretize the equations, a forward difference scheme is used for temporal discretization and Picard direct substitution method is employed to solve the nonlinear system at each time step. The formulations are validated by making a comparison between well-established numerical techniques against actual recorded datasets. After which it is found that EFG method agrees with conventional established techniques, thus validating the formulations as useful. To further investigate the usefulness of this formulation, an optimal performance study was also conducted to determine the most optimal shape function parameters. The optimal shape function values improved the performance to where it was found that established methods such as FDM were surpassed and on par with FEM. This shows that EFG is a good alternative for hydrological modelling.

Arbitrary polygon mesh for elastic and elastoplastic analysis of solids using smoothed finite element method

This work extends the smoothed finite elements method (S-FEM) to the use of arbitrarily complicated meshes, including nonconvex polygonal meshes for the first time. A novel formulation for arbitrarily polygonal elements based on the S-FEM is presented for analyses of elastic and elastoplastic problems. This enable the use of polygonal elements that are of conventional quadrilateral, convex polygonal, concave polygon elements and n-sides concave polygonal elements. This is achieved by making use of the exceptional unique features of S-FEM in terms of stability, accommodation to mesh distortion, and free from differentiation in strain field computation. The ear clipping technique is used to automatically generate smoothing domains without any dummy nodes. The stability in using such a complicated mesh is ensured by the gradient smoothing technique in S-FEM, and no additional stability control measure is required. In addition, domain integration is converted to boundary integral along sub-triangular domain, which requires only the shape function values at the field nodes. This avoiding coordinate mapping, and hence it is a Jacobian free formulation. The von Mises elastoplastic model is implemented in our S-FEM by decomposing the tangent stiffness matrix into three parts: elastic stiffness matrix, strain–displacement matrix and the derivative of the stress–strain matrix. The first two matrices are calculated only once at the initial stage, and only the derivative of the stress–strain matrix is updated in each iteration for capture the nonlinearity. The numerical results show that the proposed method can handle the general non-convex elements effectively with good accuracy and robustness. It offers an effective tool in dealing with polycrystalline metallic materials with crystal grains of arbitrary shapes.

An n-sided polygonal selective smoothed finite element method for nearly incompressible visco-hyperelastic soft materials

In this article, an n-sided polygonal smoothed finite element (nSFEM) is formulated for dynamic analyses of nonlinear problems of visco-hyperelastic materials undergoing large deformations. Two types of smoothing domains are constructed for mesh of n-sided polygonal elements. Using the gradient smoothing technique, the calculation of the strain-displacement matrix requires only the shape function values rather than the derivatives of shape function. The simple averaging point interpolation technique is used to calculate the values of the shape functions on the boundary of different smoothing domains. The Total Lagrangian formulation is used to deal with the nonlinear dynamic analysis of soft materials with finite strain. The polygonal FEM with Wachspress coordinate is also implemented for comparison. To overcome volumetric locking, selective smoothing domain scheme is used to calculate the smoothing nodal internal force. To analyze the time-dependent mechanical behavior of soft materials, a constitutive update algorithm for explicit time-integration is implemented based on the generalized Maxwell visco-hyperelastic model. A number of numerical examples are presented to demonstrate the high precision and super convergence, and robustness of the present nS-FEM for incompressible visco-hyperelastic problems.

An improved material point method using moving least square shape functions

In this study, moving least square (MLS) shape functions are employed to reduce the cell-crossing error occurred in conventional material point method (MPM) when material particles pass through grid cell boundaries. The level of smoothness of MLS shape functions for mapping information from material particles to a background grid can be controlled by the support size of MLS weight functions. A simple method is proposed to reduce the computational cost for evaluating MLS shape functions at material particles by interpolating pre-computed MLS shape function values at sampling points in a grid cell. Numerical results show that the present method is very effective to reduce the cell-crossing error in MPM computations.

Efficient isoparametric trimmed-hexahedral elements with explicit shape functions

In this study, we develop isoparametric trimmed-hexahedral (TH) elements with explicit shape functions for finite element analysis using trimmed meshes. TH elements are placed at the boundary of a trimmed mesh generated by cutting a regular hexahedral mesh with a geometric surface. Master TH elements are defined based on the marching cubes algorithm for an isoparametric mapping from the reference coordinates to the physical coordinates. Shape function values at the support points of master TH elements are precomputed by solving Laplace’s equation with appropriate boundary conditions. Explicit shape functions for master TH elements are constructed by interpolating the precomputed shape function values at the support points of master TH elements. The present explicit shape functions satisfy the required properties of FEM, and significantly reduce the computational cost compared with the previously developed polyhedral shape functions.

A Review on Analysis and Development of Mathematical Formulation Interpolants for a 4 Noded Polygon using Wachspress’ Interpolation Function

This paper presents a detailed view of a methodology of using Wachspress’ Interpolation Function for the analysis and development of Interpolants by means of a mathematical formulation for a 4 Noded Element i.e. A 4 Noded Quadrilateral Element which has been already addressed by Stephane P.A. Bordas, Sundararajan Natarajan in their research work from where the authors have took have inspiration and worked for advancement. Here a 4 Noded Quadrilateral is taken for analysis and is sub divided into 4 equal Quadrilaterals using Smoothed Finite Element Analysis (SFEM) and the respective coordinates of the discretized element will be same as the local coordinates of the parent element without and coordinate transformation. Once it is done then we form the C Matrix (Shape function Values Matrix) which remains constant. Here we consider the line that connects two nodes and using that we will use to generate the Wachspress’ Interpolants. Later on multiplication of the B Matrix (First Derivative Matrix) with the C Matrix, D Matrix (Material Matrix) and the Area Matrix we can find the Element Stiffness Matrix and later the results will be validated with that of an already existing generic example.

Steady-state characteristics of the coupled magneto-electro-thermo-elastic multi-physical system based on cell-based smoothed finite element method

A cell-based smoothed finite element method (CS-FEM) which incorporated the coupling among elastic, electric, magnetic and thermal properties was proposed to characterize the steady-state magneto-electro-elastic (MEE) structures in the thermal environment. Gradient smoothing was introduced into standard finite element method (FEM), to determine the accurate system stiffness. In CS-FEM, when the values of shape functions at the Gaussian integration point were directly utilized, the computation process was simplified; when no mapping procedure was involved, general shape elements could be employed, and mesh distortion and large deformation issues were dealt with more talentedly. CS-FEM could be carried out with customized subroutines in the business software. The accuracy and convergence of CS-FEM were proved through several numerical examples. Effects of clamped-free and clamped-clamped boundary conditions on the generalized displacements in the thermal environment were also determined. The simulation results can significantly contribute to enhancing the performance and applicability of MEE-based intelligence structures in the thermal environment.

An element-free smoothed radial point interpolation method (EFS-RPIM) for 2D and 3D solid mechanics problems

This paper presents a novel element-free smoothed radial point interpolation method (EFS-RPIM) for solving 2D and 3D solid mechanics problems. The idea of the present technique is that field nodes and smoothing cells (SCs) used for smoothing operations are created independently and without using a background grid, which saves tedious mesh generation efforts and makes the pre-process more flexible. In the formulation, we use the generalized smoothed Galerkin (GS-Galerkin) weak-form that requires only discrete values of shape functions that can be created using the RPIM. By varying the amount of nodes and SCs as well as their ratio, the accuracy can be improved and upper bound or lower bound solutions can be obtained by design. The SCs can be of regular or irregular polygons. In this work we tested triangular, quadrangle, n-sided polygon and tetrahedron as examples. Stability condition is examined and some criteria are found to avoid the presence of spurious zero-energy modes. This paper is the first time to create GS-Galerkin weak-form models without using a background mesh that tied with nodes, and hence the EFS-RPIM is a true meshfree approach. The proposed EFS-RPIM is so far the only technique that can offer both upper and lower bound solutions. Numerical results show that the EFS-RPIM gives accurate results and desirable convergence rate when comparing with the standard finite element method (FEM) and the cell-based smoothed FEM (CS-FEM).

Application of Smoothed Finite Element Method to Two-Dimensional Exterior Problems of Acoustic Radiation

In this work, the smoothed finite element method using four-node quadrilateral elements (SFEM-Q4) is employed to resolve underwater acoustic radiation problems. The SFEM-Q4 can be regarded as a combination of the standard finite element method (FEM) and the gradient smoothing technique (GST) from the meshfree methods. In the SFEM-Q4, only the values of shape functions (not the derivatives) at the quadrature points are needed and the traditional requirement of coordinate transformation procedure is not necessary to implement the numerical integration. Consequently, no additional degrees of freedom are required as compared with the original FEM. In addition, the original “overly-stiff” FEM model for acoustic problems (governed by the Helmholtz equation) is properly softened due to the gradient smoothing operations implemented over the smoothing domains and the present SFEM-Q4 possesses a relatively appropriate stiffness of the continuous system. Therefore, the well-known numerical dispersion error for Helmholtz equation is decreased significantly and very accurate numerical solutions can be obtained by using relatively coarse meshes. In order to truncate the unbounded domains and employ the domain-based numerical method to tackle the acoustic radiation in unbounded domains, the Dirichlet-to-Neumann (DtN) map is used to ensure that there are no spurious reflections from the far field. The numerical results from several numerical examples demonstrate that the present SFEM-Q4 is quite effective to handle acoustic radiation problems and can produce more accurate numerical results than the standard FEM.

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

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

A Detection Technique for Gas Concentration Based on the Spectral Line Shape Function

The methods that can rapidly and precisely measure concentrations of various gases have extensive applications in the fields such as air quality analysis, environmental pollution detection, and so on. The gas detection method based on the tunable laser absorption spectroscopy is considered a promising technique. For the infrared spectrum detection techniques, the line shape function of an absorption spectrum of a gas is an important parameter in qualitative and quantitative analysis of a gas. Specifically, how to obtain the line shape function of an absorption spectrum of a gas quickly and accurately is a key problem in the gas detection fields. In this paper we analyzed several existing line shape functions and proposed a method to calculate precisely the line shape function of a gas, and investigated the relation between the gas concentration and the peak value of a line shape function. Then we experimentally measured the absorption spectra of an acetylene gas in the wavelength range of 1,515-1,545 nm with a tunable laser source and a built-in spectrometer. With Lambert-Beer law we calculated the peak values of the line shape function of the gas at the given frequencies, and obtained a fitting curve for the line shape function in the whole waveband by using a computer program. Comparing the measured results with the calculated results of the Voigt function, we found that there was a deviation-between the experimental results and the calculated results. And we found that the measured concentration of the acetylene gas by using the fitting curve of the line shape function was more accurate and compatible with the actual situation. Hence, the empirical formula for the line shape function obtained from the experimental results would be more suitable for the concentration measurement of a gas. As the fitting curve for the line shape function of the acetylene gas has been deduced from the experiment, the corresponding peak values of the spectral lines can be immediately calculated out from the curve and used for the measurements of different concentrations of acetylene gases. Therefore, the calculation for the line shape function values is greatly simplified. The obtained data of the line shape function of the acetylene gas can be used for remote sensing of the gas, and the proposed method can also be applied for the measurements of line shape functions of other gases.

A three-dimensional ES-FEM for fracture mechanics problems in elastic solids

The edge-based smoothed finite element method (ES-FEM) using triangular mesh was recently proposed to model the fracture problems in 2D solids. This paper contains the following ingredients: (1) the ES-FEM is extended to three-dimensional (3D) ES-FEM using tetrahedral elements to compute the stress intensity factors and simulate crack propagation in 3D elastic solids; (2) to model the singular fields of arbitrary order near the crack front, a layer of specially designed seven-noded crack-tip element is constructed; (3) the displacement is then enriched with ease to reproduce the necessary order of stress singularity; and (4) the enrichment is done without losing the essential properties of partition-of-unity and the linear function reproduction. Because the singular ES-FEM uses the strain smoothing technique and it is a typical weakened weak (W2) formulation, the system stiffness matrix is computed employing only the shape function values on the surface of the smoothing domains created based on the edges of elements. No derivatives of the shape functions are needed, and thus no mapping and integration for the W2 form is required. Several numerical examples are presented to validate the effectiveness of the proposed method.

A smoothed radial point interpolation method for application in porodynamics

A meshfree numerical method for the dynamic analysis of porous media is presented. The u, p form of Biot's theory is adopted to mathematically model the dynamic interaction of the solid and the fluid phase within the porous media. The obtained partial differential equations (PDEs) are discretized by the generalized smoothed Galerkin weak form, which is established based on smoothed strains and fluxes. Therefore, edge-based and cell-based smoothing domains are used and a T3-scheme is employed for the selection of support nodes. The shape functions are generated by the radial point interpolation method. The focus of this work lies on the spatial integration of the mass/compressibility and coupling terms of the discrete PDE system. A new algorithm is introduced, which reuses the shape function values that are needed for the construction of the stiffness/permeability matrix to keep the computational effort at a minimum. Numerical problems are analyzed in order to test the algorithm regarding accuracy and efficiency.

A Node-Based Smoothed eXtended Finite Element Method (NS-XFEM) for Fracture Analysis

This paper aims to incorporate the node-based smoothed finite element method (NS-FEM) into the extended finite element method (XFEM) to form a novel numerical method (NS-XFEM) for analyzing fracture problems of 2D elasticity. NS-FEM uses the strain smoothing technique over the smoothing domains associated with nodes to compute the system stiffness matrix, which leads to the line integrations using directly the shape function values along the boundaries of the smoothing domains. As a result, we avoid integration of the stress singularity at the crack tip. It is not necessary to divide elements cut by cracks when we replace interior integration by boundary integration, simplifying integration of the discontinuous approximation. The key advantage of the NS-XFEM is that it provides more accurate solutions compared to the XFEM-T3 element. We will show for two numerical examples that the NS-XFEM significantly improves the results in the energy norm and the stress intensity factors. For the examples studied, we obtain super-convergent results

A novel singular node‐based smoothed finite element method (NS‐FEM) for upper bound solutions of fracture problems

AbstractIt is well known that the lower bound to exact solutions in linear fracture problems can be easily obtained by the displacement compatible finite element method (FEM) together with the singular crack tip elements. It is, however, much more difficult to obtain the upper bound solutions for these problems. This paper aims to formulate a novel singular node‐based smoothed finite element method (NS‐FEM) to obtain the upper bound solutions for fracture problems. In the present singular NS‐FEM, the calculation of the system stiffness matrix is performed using the strain smoothing technique over the smoothing domains (SDs) associated with nodes, which leads to the line integrations using only the shape function values along the boundaries of the SDs. A five‐node singular crack tip element is used within the framework of NS‐FEM to construct singular shape functions via direct point interpolation with proper order of fractional basis. The mix‐mode stress intensity factors are evaluated using the domain forms of the interaction integrals. The upper bound solutions of the present singular NS‐FEM are demonstrated via benchmark examples for a wide range of material combinations and boundary conditions. Copyright © 2010 John Wiley & Sons, Ltd.

New concepts for moving least squares: An interpolating non-singular weighting function and weighted nodal least squares

In this paper two new concepts for the classical moving least squares (MLS) approach are presented. The first one is an interpolating weighting function, which leads to MLS shape functions fulfilling the interpolation condition exactly. This enables a direct application of essential boundary conditions in the element-free Galerkin method without additional numerical effort. In contrast to existing approaches using singular weighting functions, this new weighting type leads to regular values of the weights and coefficients matrices in the whole domain even at the support points. The second enhancement is an approach, where the computation of the polynomial coefficient matrices is performed only at the nodes. At the interpolation point then a simple operation leads to the final shape function values. The basis polynomial of each node can be chosen independently which enables the simple realization of a p-adaptive scheme.

An improved interface element with variable nodes for non-matching finite element meshes

Extensive improvements of the interface element method (IEM) are proposed for the efficient treatment of non-matching finite element meshes. Our approach enables us to establish the master element via the moving least-square (MLS) approximation, and so to remove the cumbersome process of constructing interface elements. The values of shape functions and their derivatives are therefore mapped from the master element, as in the conventional finite element method. For the assurance of convergence and compatibility condition, a patch test is demonstrated. Through several examples of 2D linear elasticity, the convergence rate is compared between the present interface element and the previous version.

Active control of higher-order acoustic modes in ducts

The active control of higher-order acoustic modes within a semi-infinite hard-walled rectangular duct is examined analytically for harmonic constant volume velocity sound sources. The effect of source location, size, strength, and relative phase upon the total mean acoustic power flow is investigated for both single and dual control source configurations, at a number of excitation frequencies. For the analysis, the primary source is mounted in the plane of the duct cross section, which may be a uniform surface of arbitrary finite impedance. It was found that the reduction in total acoustic power is dependent upon the relative location of the primary and control sources, both laterally across the wall of the duct and axially along the duct. The reduction in total acoustic power was found to be a maximum at axial source separation distances corresponding to multiples of half-wavelengths of the propagating modes, and at relative lateral locations such that the primary and control source mode shape function values were equivalent.

AN ALGORITHM FOR GENERATION OF SHAPE FUNCTIONS IN SERENDIPITY ELEMENTS

An algorithm to calculate shape function values at specific points is presented. It applies to three‐dimensional serendipity elements with variable node numbers per side and, as a particular case, to plane and truss elements. The procedure is shown for the two‐dimensional case using the natural orthogonal reference system of the element and is then generalized to the three‐dimensional case. The source code of the described algorithms written in Fortran 77 is included.

Sensitivity analysis and shape optimization of axisymmetric structures

AbstractAn efficient method is developed for sensitivity analysis in shape optimization of axisymmetric structures. The technique of isoparametric mapping is used to generate the finite element mesh from a small set of master elements and master nodes. Co‐ordinates of selected master nodes are used as design variables. Shape function values of master elements at derived finite element nodes obtained during the isoparametric mapping process are utilized to calculate the gradients of weight and response of the structures with respect to the design variables. Analytic formulations of the gradients are developed for sensitivity analysis of axisymmetric structures. An optimization procedure using a sequential linear programming method is applied to effectively utilize the calculated gradients. Numerical examples of optimum design of disks subject to thermo‐mechanical loadings are presented.