The solution of elastostatic and dynamic problems using the boundary element method based on spherical Hankel element framework

In this paper, new spherical Hankel shape functions are used to reformulate boundary element method for 2‐dimensional elastostatic and elastodynamic problems. To this end, the dual reciprocity boundary element method is reconsidered by using new spherical Hankel shape functions to approximate the state variables (displacements and tractions) of Navier's differential equation. Using enrichment of a class of radial basis functions (RBFs), called spherical Hankel RBFs hereafter, the interpolation functions of a Hankel boundary element framework has been derived. For this purpose, polynomial terms are added to the functional expansion that only uses spherical Hankel RBF in the approximation. In addition to polynomial function fields, the participation of spherical Bessel function fields has also increased robustness and efficiency in the interpolation. It is very interesting that there is no Runge phenomenon in equispaced Hankel macroelements, unlike equispaced classic Lagrange ones. Several numerical examples are provided to demonstrate the effectiveness, robustness and accuracy of the proposed Hankel shape functions and in comparison with the classic Lagrange ones, they show much more accurate and stable results.     

A note on the finite element method with singular basis functions

In this note, we make a few comments concerning the paper of Hughes and Akin (Int. J. Numer. Meth. Engng., 15 , 733–751 (1980)). Our primary goal is to demonstrate that the rate of convergence of numerical solutions of the finite element method with singular basis functions depends upon the location of additional collocation points associated with the singular elements. Copyright © 1999 John Wiley & Sons, Ltd.     

A new technique for numerical simulation of dendritic solidification using a meshfree interface finite element method

A new technique for sharp‐interface modeling of dendritic solidification is proposed using a meshfree interface finite element method such that the liquid–solid interface is represented implicitly and allowed to arbitrarily intersect the finite elements. At the interface‐embedded elements, meshfree interface points without connectivity are imposed directly at the zero level set while meshfree interpolants are constructed using radial basis functions. This ensures both the partition of unity and the Kronecker delta properties are satisfied allowing for precise and easy imposition of Dirichlet boundary conditions at the interface. The constructed meshfree interpolants are also used for solving a variational level set equation based on the Ginzburg–Landau energy functional minimization such that reinitialization is completely eliminated and fast marching algorithms for interfacial velocity extension are not necessary resulting in an efficient algorithm with excellent volume conservation. The meshfree interface finite element method is used for modeling dendritic solidification in a pure melt where it is found suitable in handling the complex interfacial dynamics often encountered in dendritic growth. Mathematical formulation and implementation followed by numerical results and analysis will be presented and discussed. Copyright © 2016 John Wiley & Sons, Ltd.     

Use of higher‐order shape functions in the scaled boundary finite element method

The scaled boundary finite element method is a novel semi‐analytical technique, whose versatility, accuracy and efficiency are not only equal to, but potentially better than the finite element method and the boundary element method for certain problems. This paper investigates the possibility of using higher‐order polynomial functions for the shape functions. Two techniques for generating the higher‐order shape functions are investigated. In the first, the spectral element approach is used with Lagrange interpolation functions. In the second, hierarchical polynomial shape functions are employed to add new degrees of freedom into the domain without changing the existing ones, as in the p‐version of the finite element method. To check the accuracy of the proposed procedures, a plane strain problem for which an exact solution is available is employed. A more complex example involving three scaled boundary subdomains is also addressed. The rates of convergence of these examples under p‐refinement are compared with the corresponding rates of convergence achieved when uniform h‐refinement is used, allowing direct comparison of the computational cost of the two approaches. The results show that it is advantageous to use higher‐order elements, and that higher rates of convergence can be obtained using p‐refinement instead of h‐refinement. Copyright © 2005 John Wiley & Sons, Ltd.     

Study of rectangular waveguides with grooves of irregular shape using the finite element method

Electromagnetic wave propagation through grooved waveguides is studied using the finite element method (FEM). The effect of grooves of irregular shape on TE₁₀, TE₂₀ mode frequencies and passband is studied. The variation in cutoff frequencies for TE₁₀, TE₂₀ mode and passband is observed.     

Iteratively-improved Robin boundary conditions for the finite element solution of scattering problems in unbounded domains

An iterative procedure is described for the finite-element solution of scalar scattering problems in unbounded domains. The scattering objects may have multiple connectivity, may be of different materials or with different boundary conditions. A fictitious boundary enclosing all the objects involved is introduced. An appropriate Robin (mixed) condition is initially guessed on this boundary and is iteratively improved making use of Green's formula. It will be seen that the best choice for the Robin boundary condition is an absorbing-like one. A theorem about the theoretical convergence of the procedure is demonstrated. An analytical study of the special case of a circular cylindrical scatterer is made. Comparisons are made with other methods. Some numerical examples are provided in order to illustrate and validate the procedure and to show its applicability whatever the frequency of the incident wave. Although particular emphasis is laid in the paper on electromagnetic problems, the procedure is fully applicable to other kinds of physical phenomena such as acoustic ones. © 1998 John Wiley & Sons, Ltd.     

Application of mixed variational formulations based on the finite element method to the solution of problems on natural vibrations of elastic bodies

We consider mixed variational formulations and the application of the mixed approximations of the finite element method to the solution of problems on natural vibrations of elastic bodies. To solve the generalized spectral problem, three forms of the mixed variational formulations are proposed. The correctness and stability of mixed variational formulations for displacements, strains and stresses are investigated. Matrix equations of the mixed method are given whose solution is performed using the modified algorithm of the steepest descent method. The results of calculations for natural frequencies of free vibrations of a straight and a circular beam are presented that are obtained in the solution of the problem in a two-dimensional formulation based on the classical and mixed finite-element method approaches. __________ Translated from Problemy Prochnosti, No. 2, pp. 121–140, March–April, 2008.     

Application of hybrid Trefftz finite element method to non‐linear problems of minimal surface

This investigation provides a hybrid Trefftz finite element approach for analysing minimal surface problems. The approach is based on combining Trefftz finite element formulation with radial basis functions (RBF) and the analogue equation method (AEM). In this method, use of the analogue equation approach avoids the difficulty of treating the non‐linear terms appearing in the soap bubble equation, making it possible to solve non‐linear problems with the Trefftz method. Global RBF is used to approximate the inhomogeneous term induced from non‐linear functions and other loading terms. Finally, some numerical experiments are implemented to verify the efficiency of this method. Copyright © 2006 John Wiley & Sons, Ltd.     

Testing complete and compact radial basis functions for solution of eigenvalue problems using the boundary element method with direct integration

This paper analyses the performance of the main radial basis functions in the formulation of the Boundary Element Method (DIBEM). This is an alternative for solving problems modeled by non-adjoint differential operators, since it transforms domain integrals in boundary integrals using radial basis functions. The solution of eigenvalue problem was chosen to performance evaluation. Natural frequencies are calculated numerically using several radial functions and their accuracy is evaluated by comparison with the available analytical solutions and with the Finite Element Method as well. The standard radial basis functions have presented similar performance to compact radial functions, being even slightly superior.     

Finite strain fracture analysis using the extended finite element method with new set of enrichment functions

Nonlinear fracture analysis of rubber‐like materials is computationally challenging due to a number of complicated numerical problems. The aim of this paper is to study finite strain fracture problems based on appropriate enrichment functions within the extended finite element method. Two‐dimensional static and quasi‐static crack propagation problems are solved to demonstrate the efficiency of the proposed method. Complex mixed‐mode problems under extreme large deformation regimes are solved to evaluate the performance of the proposed extended finite element analysis based on different tip enrichment functions. Finally, it is demonstrated that the logarithmic set of enrichment functions provides the most accurate and efficient solution for finite strain fracture analysis. Copyright © 2015 John Wiley & Sons, Ltd.     

The dual reciprocity method applied to free vibrations of 2D structures using compact supported radial basis functions

This paper presents a new boundary element application for free vibration analysis of 2D elastic structures. The dual reciprocity method is applied using four compact supported radial basis functions for approximating the domain inertia terms. The eigen-problem of displacement is then solved considering the traction contribution by means of static condensation. The formulation is also extended to consider additional internal nodes to improve accuracy. Three numerical problems are studied to demonstrate the validity and accuracy of the developed formulation. The results are compared to those obtained from analytical and other numerical solutions. A parametric study is set up to demonstrate the effect of the compact support radius on the final results and on the sparsity of system matrices.     

On the approximation in the smoothed finite element method (SFEM)

This letter aims at resolving the issues raised in the recent short communication (Int. J. Numer. Meth. Engng 2008; 76 (8):1285–1295. DOI: 10.1002/nme.2460 ) and answered by (Int. J. Numer. Meth. Engng 2009; DOI: 10.1002/nme.2587 ) by proposing a systematic approximation scheme based on non‐mapped shape functions, which both allows to fully exploit the unique advantages of the smoothed finite element method (SFEM) (Comput. Mech. 2007; 39 (6):859–877. DOI: 10.1007/s00466‐006‐0075‐4 ; Commun. Numer. Meth. Engng 2009; 25 (1):19–34. DOI: 10.1002/cnm.1098 ; Int. J. Numer. Meth. Engng 2007; 71 (8):902–930; Comput. Meth. Appl. Mech. Engng 2008; 198 (2):165–177. DOI: 10.1016/j.cma.2008.05.029 ; Comput. Meth. Appl. Mech. Engng 2007; submitted; Int. J. Numer. Meth. Engng 2008; 74 (2):175–208. DOI: 10.1002/nme.2146 ; Comput. Meth. Appl. Mech. Engng 2008; 197 (13–16):1184–1203. DOI: 10.1016/j.cma.2007.10.008 ) and resolve the existence, linearity and positivity deficiencies pointed out in (Int. J. Numer. Meth. Engng 2008; 76 (8):1285–1295). We show that Wachspress interpolants (A Rational Basis for Function Approximation. Academic Press, Inc.: New York, 1975) computed in the physical coordinate system are very well suited to the SFEM, especially when elements are heavily distorted (obtuse interior angles). The proposed approximation leads to results that are almost identical to those of the SFEM initially proposed in (Comput. Mech. 2007; 39 (6):859–877. DOI: 10.1007/s00466‐006‐0075‐4 ). These results suggest that the proposed approximation scheme forms a strong and rigorous basis for the construction of SFEMs. Copyright © 2009 John Wiley & Sons, Ltd.     

On the essence and the evaluation of the shape functions for the smoothed finite element method (SFEM)

This paper is written in response to the recently published paper (Int. J. Numer. Meth. Engng 2008; 76 :1285–1295) at IJNME entitled ‘On the smoothed finite element method’ (SFEM) by Zhang HH, Liu SJ, Li LX. In this paper we

• (1) repeat briefly the important essence of the original SFEM presented in (Comp. Mech. 2007; 39 : 859–877; Int. J. Numer. Meth. Engng 2007; 71 :902–930; Int. J. Numer. Meth. Engng 2008; 74 :175–208; Finite Elem. Anal. Des. 2007; 43 :847–860; J. Sound Vib. 2007; 301 :803–820), and
• (2) examine further issues in the evaluation of the shape functions used in the SFEM.

It will be shown that the ‘SFEM’ presented in paper (Int. J. Numer. Meth. Engng 2008; 76 :1285–1295) is not at all our original SFEM presented in (Comp. Mech. 2007; 39 :859–877; Int. J. Numer. Meth. Engng 2007; 71 :902–930; Int. J. Numer. Meth. Engng 2008; 74 :175–208; Finite Elem. Anal. Des. 2007; 43 :847–860; J. Sound Vib. 2007; 301 :803–820). Therefore, all these ‘Theorems’, ‘Corollaries’ and ‘Remarks’ presented in paper (Int. J. Numer. Meth. Engng 2008; 76 :1285–1295) have nothing to do with our original SFEM. The properties of the original SFEM stand as they were presented in our original papers (Comp. Mech. 2007; 39 :859–877; Int. J. Numer. Meth. Engng 2007; 71 :902–930; Int. J. Numer. Meth. Engng 2008; 74 :175–208; Finite Elem. Anal. Des. 2007; 43 :847–860; J. Sound Vib. 2007; 301 :803–820). Finally, we brief on our advancements made far beyond our original SFEM and our visions on future numerical methods. Copyright © 2009 John Wiley & Sons, Ltd.     

Construction of shape functions for the h- and p-versions of the FEM using tensorial product

This paper presents an uniform and unified approach to construct h- and p-shape functions for quadrilaterals, triangles, hexahedral and tetrahedral based on the tensorial product of one-dimensional Lagrange and Jacobi polynomials. The approach uses indices to denote the one-dimensional polynomials in each tensorization direction. The appropriate manipulation of the indices allows to obtain hierarchical or non-hierarchical and inter-element C⁰ continuous or non-continuous bases. For the one-dimensional elements, quadrilaterals, triangles and hexahedral, the optimal weights of the Jacobi polynomials are determined, the sparsity profiles of the local mass and stiffness matrices plotted and the condition numbers calculated. A brief discussion of the use of sum factorization and computational implementation is considered. Copyright © 2006 John Wiley & Sons, Ltd.     

Computational modeling of strong and weak discontinuities using the extended finite element method

In this article, the extended finite element method is employed to solve problems, including weak and strong discontinuities. To this end, a level set framework is used to represent the discontinuities location, and the Heaviside and Branch function are included in the standard finite element method. The case of two arbitrary curved cracks is solved numerically and stress intensity factor (SIF) values at the crack tips are calculated based on the evaluation of the crack tip opening displacement. Afterwards, J-integral methodology is adopted to evaluate the SIFs for isotropic and anisotropic bi-material interface crack problems. Numerical results are verified with those presented in the literature.     

A new coupling technique for the combination of wavelet‐Galerkin method with finite element method in solids and structures

In this paper, a coupling technique is developed for the combination of the wavelet‐Galerkin method (WGM) with the finite element method (FEM). In this coupled method, the WGM and FEM are respectively used in different sub‐domains. The WGM sub‐domain and the FEM sub‐domain are connected by a transition region that is described by B‐spline basis functions. The basis functions of WGM and FEM are modified in the transition region to ensure the basic polynomial reconstruction condition and the compatibility of displacements and their gradients. In addition, the elements of FEM and WGM are not necessary to conform to the transition region. This newly developed coupled method is applied to the stress analysis of 2D and 3D elasticity problems in order to investigate its performance and study parameters. Numerical results show that the present coupled method is accurate and stable. The new method has a promising potential for practical applications and can be easily extended for coupling of other numerical methods. Copyright © 2017 John Wiley & Sons, Ltd.     

A p-adaptive scaled boundary finite element method based on maximization of the error decrease rate

This study enhances the classical energy norm based adaptive procedure by introducing new refinement criteria, based on the projection-based interpolation technique and the steepest descent method, to drive mesh refinement for the scaled boundary finite element method. The technique is applied to p-adaptivity in this paper, but extension to h- and hp-adaptivity is straightforward. The reference solution, which is the solution of the fine mesh formed by uniformly refining the current mesh, is used to represent the unknown exact solution. In the new adaptive approach, a projection-based interpolation technique is developed for the 2D scaled boundary finite element method. New refinement criteria are proposed. The optimum mesh is assumed to be obtained by maximizing the decrease rate of the projection-based interpolation error appearing in the current solution. This refinement strategy can be interpreted as applying the minimisation steepest descent method. Numerical studies show the new approach out-performs the conventional approach.     

Application of the mixed approximation to the solution of two-dimensional problems of the theory of small elastoplastic strains using the finite element method

A triangular finite element that provides the stability and convergence of the mixed approximation is used for the solution of two-dimensional boundary-value problems of the theory of small elastoplastic strains. A system of resolving matrix equations of a mixed type is presented for the solution of which a three-layer iteration algorithm with a preconditioning matrix is used. Numerical results for the solution of model problems obtained by the classical and combined finite element methods are compared. __________ Translated from Problemy Prochnosti, No. 2, pp. 124–136, March–April, 2006.     

Application of the moment scheme of the finite element method to the solution of problems with initial stresses in the incremental elasticity theory

We discuss application of the finite element method to the solution of problems with initial stresses within the elasticity theory. Based on the incremental theory of deformable solids, the relationships of the finite element method are derived to calculate the stiffness matrix coefficients for a prestressed spatial element of the serendip family with quadratic approximation of displacements. The calculation of the stressed state of an eccentrically compressed beam and a round plate under conditions of longitudinal-transverse bending is carried out. Comparison of the numerical results with analytical solutions is presented. The variation in the compression and shear strains of a cylindrical damper is studied depending on the degree of deformation and the sequence of load application. __________ Translated from Problemy Prochnosti, No. 3, pp. 131–143, May–June, 2006.     

On computing upper and lower bounds on the outputs of linear elasticity problems approximated by the smoothed finite element method

Verification of the computation of local quantities of interest, e.g. the displacements at a point, the stresses in a local area and the stress intensity factors at crack tips, plays an important role in improving the structural design for safety. In this paper, the smoothed finite element method (SFEM) is used for finding upper and lower bounds on the local quantities of interest that are outputs of the displacement field for linear elasticity problems, based on bounds on strain energy in both the primal and dual problems. One important feature of SFEM is that it bounds the strain energy of the structure from above without needing the solutions of different subproblems that are based on elements or patches but only requires the direct finite element computation. Upper and lower bounds on two linear outputs and one quadratic output related with elasticity—the local reaction, the local displacement and the J‐integral—are computed by the proposed method in two different examples. Some issues with SFEM that remain to be resolved are also discussed. Copyright © 2010 John Wiley & Sons, Ltd.