A meshless singular boundary method for three‐dimensional elasticity problems

This study documents the first attempt to extend the singular boundary method, a novel meshless boundary collocation method, for the solution of 3D elasticity problems. The singular boundary method involves a coupling between the regularized BEM and the method of fundamental solutions. The main idea here is to fully inherit the dimensionality and stability advantages of the former and the meshless and integration‐free attributes of the later. This makes it particularly attractive for problems in complex geometries and three dimensions. Four benchmark 3D problems in linear elasticity are well studied to demonstrate the feasibility and accuracy of the proposed method. The advantages, disadvantages, and potential applications of the proposed method, as compared with the FEM, BEM, and method of fundamental solutions, are also examined and discussed. Copyright © 2015 John Wiley & Sons, Ltd.     

A hybrid element formulation for the three‐dimensional penalty finite element analysis of incompressible fluids

This work presents a hybrid element formulation for the three‐dimensional penalty finite element analysis of incompressible Newtonian fluids. The formulation is based on a mixed variational statement in which velocity and stresses are treated as independent field variables. The main advantage of this formulation is that it bypasses the use of ad hoc techniques such as selective reduced integration that are commonly used in penalty‐based finite element formulations, and directly yields high accuracy for the velocity and stress fields without the need to carry out smoothing. In addition, since the stress degrees of freedom are condensed out at an element level, the cost of solving for the global degrees of freedom is the same as in a standard penalty finite element method, although the gain in accuracy for both the velocity and stress (including the pressure) fields is quite significant. Copyright © 2007 John Wiley & Sons, Ltd.     

Mapped finite element methods: High‐order approximations of problems on domains with cracks and corners

Linear elasticity problems posed on cracked domains, or domains with re‐entrant corners, yield singular solutions that deteriorate the optimality of convergence of finite element methods. In this work, we propose an optimally convergent finite element method for this class of problems. The method is based on approximating a much smoother function obtained by locally reparameterizing the solution around the singularities. This reparameterized solution can be approximated using standard finite element procedures yielding optimal convergence rates for any order of interpolating polynomials, without additional degrees of freedom or special shape functions. Hence, the method provides optimally convergent solutions for the same computational complexity of standard finite element methods. Furthermore, the sparsity and the conditioning of the resulting system is preserved. The method handles body forces and crack‐face tractions, as well as multiple crack tips and re‐entrant corners. The advantages of the method are showcased for four different problems: a straight crack with loaded faces, a circular arc crack, an L‐shaped domain undergoing anti‐plane deformation, and lastly a crack along a bimaterial interface. Optimality in convergence is observed for all the examples. A proof of optimal convergence is accomplished mainly by proving the regularity of the reparameterized solution. Copyright © 2016 John Wiley & Sons, Ltd.     

Computational electro‐elasticity and magneto‐elasticity for quasi‐incompressible media immersed in free space

In this work, a mixed variational formulation to simulate quasi‐incompressible electro‐active or magneto‐active polymers immersed in the surrounding free space is presented. A novel domain decomposition is used to disconnect the primary coupled problem and the arbitrary free‐space mesh update problem. Exploiting this decomposition, we describe a block‐iterative approach to solving the linearised multiphysics problem, and a physically and geometrically based, three‐parameter method to update the free space mesh. Several application‐driven example problems are implemented to demonstrate the robustness of the mixed formulation for both electro‐elastic and magneto‐elastic problems involving both finite deformations and quasi‐incompressible media. Copyright © 2016 John Wiley & Sons, Ltd.     

Accuracy of Galerkin finite elements for groundwater flow simulations in two and three‐dimensional triangulations

In standard finite element simulations of groundwater flow the correspondence between hydraulic head gradients and groundwater fluxes is represented by the stiffness matrix. In two‐dimensional problems the use of linear triangular elements on Delaunay triangulations guarantees a stiffness matrix of type M. This implies that the local numerical fluxes are physically consistent with Darcy's law. This condition is fundamental to avoid the occurrence of local maxima or minima, and is of crucial importance when the calculated flow field is used in contaminant transport simulations or pathline evaluation. In three spatial dimensions, the linear Galerkin approach on tetrahedra does not lead to M‐matrices even on Delaunay meshes. By interpretation of the Galerkin approach as a subdomain collocation scheme, we develop a new approach (OSC, orthogonal subdomain collocation) that is shown to produce M‐matrices in three‐dimensional Delaunay triangulations. In case of heterogeneous and anisotropic coefficients, extra mesh properties required for M‐stiffness matrices will also be discussed. Copyright © 2001 John Wiley & Sons, Ltd.     

Complex three‐dimensional microstructural permeability prediction of porous fibrous media with and without compaction

Much of the work in permeability prediction has so far been done with respect to the in‐plane properties. Guided by Darcy's law, the in‐plane permeabilities (i.e. Kxx and Kyy) have been well characterized by researchers both experimentally and, to some extent, analytically and numerically. Work on transverse or through thickness permeability, however, has been sparse. Owing to the fact that the limited length scale in the through thickness direction of most fibre preforms makes transverse permeability a difficult value to measure experimentally, the objective of the present development is to study the feasibility of applying the methodology proposed by the authors in a previous work for in‐plane permeability prediction to the estimation of transverse, as well as the in‐plane, permeability of a typical 3D woven fibre preform. The additional objective of this work is to present a preliminary study on the effects of fibre mat compression on the fibre preform permeability. Copyright © 2004 John Wiley & Sons, Ltd.     

A spline strip kernel particle method and its application to two‐dimensional elasticity problems

In this paper we present a novel spline strip kernel particle method (SSKPM) that has been developed for solving a class of two‐dimensional (2D) elasticity problems. This new approach combines the concepts of the mesh‐free methods and the spline strip method. For the interpolation of the assumed displacement field, we employed the kernel particle shape functions in the transverse direction, and the B₃‐spline function in the longitudinal direction. The formulation is validated on several beam and semi‐infinite plate problems. The numerical results of these test problems are then compared with the existing solutions obtained by the exact or numerical methods. From this study we conclude that the SSKPM is a potential alternative to the classical finite strip method (FSM). Copyright © 2003 John Wiley & Sons, Ltd.     

An implicit–explicit solution method for electro‐osmotic flow through three‐dimensional micro‐channels

This paper presents a comprehensive finite‐element modelling approach to electro‐osmotic flows on unstructured meshes. The non‐linear equation governing the electric potential is solved using an iterative algorithm. The employed algorithm is based on a preconditioned GMRES scheme. The linear Laplace equation governing the external electric potential is solved using a standard pre‐conditioned conjugate gradient solver. The coupled fluid dynamics equations are solved using a fractional step‐based, fully explicit, artificial compressibility scheme. This combination of an implicit approach to the electric potential equations and an explicit discretization to the Navier–Stokes equations is one of the best ways of solving the coupled equations in a memory‐efficient manner. The local time‐stepping approach used in the solution of the fluid flow equations accelerates the solution to a steady state faster than by using a global time‐stepping approach. The fully explicit form and the fractional stages of the fluid dynamics equations make the system memory efficient and free of pressure instability. In addition to these advantages, the proposed method is suitable for use on both structured and unstructured meshes with a highly non‐uniform distribution of element sizes. The accuracy of the proposed procedure is demonstrated by solving a basic micro‐channel flow problem and comparing the results against an analytical solution. The comparisons show excellent agreement between the numerical and analytical data. In addition to the benchmark solution, we have also presented results for flow through a fully three‐dimensional rectangular channel to further demonstrate the application of the presented method. Copyright © 2007 John Wiley & Sons, Ltd.     

Stress intensity factors for surface cracks in single‐edge notch bend specimen by a three‐dimensional weight function method

Stress intensity factors for half‐elliptical surface cracks at a semi‐circular notch in a recently developed single‐edge notch bend specimen are determined for a wide range of geometrical parameters using a three‐dimensional weight function method. Two load cases of pin loading and uniform remote tension are considered. The results are in good agreement with abaqus/franc3d finite element analysis. It is found that the Ziegler–Newman engineering similitude approach (programmed into the Fatigue Crack Growth Structural Analysis life‐prediction code) produces good results for a wide range in a/c ratios. Expressions by multi‐variable curve fitting to the weight function results are presented for easy engineering applications.     

On the a priori and a posteriori error analysis of a two‐fold saddle‐point approach for nonlinear incompressible elasticity

In this paper, we reconsider the a priori and a posteriori error analysis of a new mixed finite element method for nonlinear incompressible elasticity with mixed boundary conditions. The approach, being based only on the fact that the resulting variational formulation becomes a two‐fold saddle‐point operator equation, simplifies the analysis and improves the results provided recently in a previous work. Thus, a well‐known generalization of the classical Babu?ka–Brezzi theory is applied to show the well‐posedness of the continuous and discrete formulations, and to derive the corresponding a priori error estimate. In particular, enriched PEERS subspaces are required for the solvability and stability of the associated Galerkin scheme. In addition, we use the Ritz projection operator to obtain a new reliable and quasi‐efficient a posteriori error estimate. Finally, several numerical results illustrating the good performance of the associated adaptive algorithm are presented. Copyright © 2006 John Wiley & Sons, Ltd.     

Comparative accuracy of DQ and HDQ methods for three‐dimensional vibration analysis of rectangular plates

An accuracy study between the Differential Quadrature (DQ) and Harmonic Differential Quadrature (HDQ) methods for three‐dimensional elasticity solutions of free vibration of rectangular plates is carried out. The solution capability of the DQ and HDQ methods is first studied. Then the numerical performance of both the methods is compared. It is found that the DQ method displays more superior convergence characteristics over the HDQ method for the lower modes of vibration. However, the HDQ method is advantageous over the DQ method for computing higher modes of vibration. It is also discovered that the DQ and HDQ methods produce better convergent solutions than the Finite Element Method (FEM) when a similar number of discrete points/nodes are used. Copyright © 1999 John Wiley & Sons, Ltd.     

Theory and numerics of three‐dimensional beams with elastoplastic material behaviour

A theory of space curved beams with arbitrary cross‐sections and an associated finite element formulation is presented. Within the present beam theory the reference point, the centroid, the centre of shear and the loading point are arbitrary points of the cross‐section. The beam strains are based on a kinematic assumption where torsion‐warping deformation is included. Each node of the derived finite element possesses seven degrees of freedom. The update of the rotational parameters at the finite element nodes is achieved in an additive way. Applying the isoparametric concept the kinematic quantities are approximated using Lagrangian interpolation functions. Since the reference curve lies arbitrarily with respect to the centroid the developed element can be used to discretize eccentric stiffener of shells. Due to the implemented constitutive equations for elastoplastic material behaviour the element can be used to evaluate the load‐carrying capacity of beam structures. Copyright © 2000 John Wiley & Sons, Ltd.     

Optimal layout design of three‐dimensional geometrically non‐linear structures using the element connectivity parameterization method

The topology design optimization of ‘three‐dimensional geometrically‐non‐linear’ continuum structures is still difficult not only because of the size of the problem but also because of the unstable continuum finite elements that arise during the optimization. To overcome these difficulties, the element connectivity parameterization (ECP) method with two implementation formulations is proposed. In ECP, structural layouts are represented by inter‐element connectivity, which is controlled by the stiffness of element‐connecting zero‐length links. Depending on the link location, ECP may be classified as an external ECP (E‐ECP) or an internal ECP (I‐ECP). In this paper, I‐ECP is newly developed to substantially enhance computational efficiency. The main idea in I‐ECP is to reduce system matrix size by eliminating some internal degrees of freedom associated with the links at voxel level. As for ECP implementation with commercial software, E‐ECP, developed earlier for two‐dimensional problems, is easier to use even for three‐dimensional problems because it requires only numerical analysis results for design sensitivity calculation. The characteristics of the I‐ECP and E‐ECP methods are compared, and these methods are validated with numerical examples. Copyright © 2006 John Wiley & Sons, Ltd.     

Semi‐analytic solution for multiple interacting three‐dimensional inhomogeneous inclusions of arbitrary shape in an infinite space

This paper develops a semi‐analytic solution for multiple arbitrarily shaped three‐dimensional inhomogeneous inclusions embedded in an infinite isotropic matrix under external load. All interactions between the inhomogeneous inclusions are taken into account in this solution. The inhomogeneous inclusions are discretized into small cuboidal elements, each of which is treated as a cuboidal inclusion with initial eigenstrain plus unknown equivalent eigenstrain according to the Equivalent Inclusion Method. All the unknown equivalent eigenstrains are determined by solving a set of simultaneous constitutive equations established for each equivalent cuboidal inclusion. The final solution is obtained by summing up the closed‐form solutions for each individual equivalent cuboidal inclusion in an infinite space. The solution evaluation is performed by application of the fast Fourier transform algorithm, which greatly increases the computational efficiency. Finally, the solution is validated by taking Eshelby's analytic solution of an ellipsoidal inhomogeneous inclusion as a benchmark and by the finite element analysis. A few sample results are also given to demonstrate the generality of the solution. The solution may have potentially significant applications in solving a wide range of inhomogeneity‐related problems. Copyright © 2011 John Wiley & Sons, Ltd.     

On a high‐order Newton linearization method for solving the incompressible Navier–Stokes equations

The present study aims to accelerate the non‐linear convergence to incompressible Navier–Stokes solution by developing a high‐order Newton linearization method in non‐staggered grids. For the sake of accuracy, the linearized convection–diffusion–reaction finite‐difference equation is solved line‐by‐line using the nodally exact one‐dimensional scheme. The matrix size is reduced and, at the same time, the CPU time is considerably saved owing to the reduction of stencil points. This Newton linearization method is computationally efficient and is demonstrated to outperform the classical Newton method through computational exercises. Copyright © 2005 John Wiley & Sons, Ltd.     

Extension of the method of auxiliary mapping for three‐dimensional elliptic boundary value problems

In this paper, we extend the method of auxiliary mapping (MAM), introduced by Babuška and Oh, to three dimensions so that the extended MAM (3‐D MAM) can effectively handle three‐dimensional elliptic problems containing the singularities caused by the non‐smooth domains. There are three type of singularities caused by non‐smoothness of domains in R ³: the vertex, the edge, and the vertex–edge combined singularities. To deal with the singularities of these types, we present three auxiliary mappings and formulas for the transformed bilinear forms and the transformed linear functionals by these auxiliary mappings. Then we present 3‐D MAM and constructions of the blending‐type elemental mappings for elements containing singularities. Numerical experiments that show the effectiveness of 3‐D MAM are provided. Copyright © 2001 John Wiley & Sons, Ltd.     

Three‐dimensional theory of elasticity for free vibration analysis of composite laminates via layerwise differential quadrature modelling

In this paper, the free vibration analysis of simply‐supported and clamped composite laminates, especially thick laminates, is carried out. The three‐dimensional theory of elasticity is integrated into a layerwise model via differential quadrature discretization. All physical governing equations are satisfied, including the additional constraints of the characteristics of continuity and discontinuity of interfacial transverse and in‐plane strains and stresses along the interfaces of composite laminates. Effects of plate aspect and thickness ratios on the free vibration of these laminates are examined in detail. This study demonstrates the applicability, accuracy, and stability of the present methodology, for vibration analyses of composite structures of thick laminated constitution. Copyright © 2003 John Wiley & Sons, Ltd.     

Efficient computation of order and mode of corner singularities in 3D‐elasticity

A general numerical procedure is presented for the efficient computation of corner singularities, which appear in the case of non‐smooth domains in three‐dimensional linear elasticity. For obtaining the order and mode of singularity, a neighbourhood of the singular point is considered with only local boundary conditions. The weak formulation of the problem is approximated by a Galerkin–Petrov finite element method. A quadratic eigenvalue problem ( P +λ Q +λ² R ) u = 0 is obtained, with explicitly analytically defined matrices P , Q , R . Moreover, the three matrices are found to have optimal structure, so that P , R are symmetric and Q is skew symmetric, which can serve as an advantage in the following solution process. On this foundation a powerful iterative solution technique based on the Arnoldi method is submitted. For not too large systems this technique needs only one direct factorization of the banded matrix P for finding all eigenvalues in the interval ?e(λ)∈(?0.5,1.0) (no eigenpairs can be ‘lost’) as well as the corresponding eigenvectors, which is a great improvement in comparison with the normally used determinant method. For large systems a variant of the algorithm with an incomplete factorization of P is implemented to avoid the appearance of too much fill‐in. To illustrate the effectiveness of the present method several new numerical results are presented. In general, they show the dependence of the singular exponent on different geometrical parameters and the material properties. Copyright © 2001 John Wiley & Sons, Ltd.     

Numerical method for the simulation of the Brownian dynamics of rod‐like microstructures with three‐dimensional nonlinear beam elements

In several fast‐growing research areas such as material science, macromolecular chemistry, bioengineering and biophysics, the dynamics of rod‐like microstructures such as polymers, rod‐like viruses, flagella of bacteria or thin glass fibers plays a crucial role. These microstructures are typically embedded into some viscous fluid and kept by stochastic thermal forces in an incessant motion, the so‐called Brownian dynamics. In this article, we introduce a novel approach wherein three‐dimensional finite beam elements can be employed for the computer simulation of the Brownian dynamics of rod‐like microstructures. This new approach is superior to state‐of‐the‐art methods by its transparent theoretical foundation and its remarkable efficiency. By means of numerical examples, we demonstrate the applicability of this approach to problems of biophysics and macromolecular chemistry. Copyright © 2012 John Wiley & Sons, Ltd.     

Numerical simulation of the motion of a two‐dimensional glacier

The motion of a two‐dimensional glacier is considered. At each time step, given the shape of the glacier, ice is modelled as an incompressible non‐Newtonian fluid and a non‐linear elliptic problem has to be solved to obtain the horizontal velocity field. Then, the upper surface of the glacier is updated by solving a transport equation. Finite element techniques are used to compute the velocity field whereas the transport equation is solved using a Lax–Wendroff scheme. Numerical results are compared to experiments on Gries glacier (Wallis, Switzerland) between 1961 and 1991. Then, a predition for 2021 is proposed. Copyright 2004 John Wiley & Sons, Ltd.