A priori error estimator of the generalized‐α method for structural dynamics

An a priori error estimator for the generalized‐α time‐integration method is developed to solve structural dynamic problems efficiently. Since the proposed error estimator is computed with only information in the previous and current time‐steps, the time‐step size can be adaptively selected without a feedback process, which is required in most conventional a posteriori error estimators. This paper shows that the automatic time‐stepping algorithm using the a priori estimator performs more efficient time integration, when compared to algorithms using an a posteriori estimator. In particular, the proposed error estimator can be usefully applied to large‐scale structural dynamic problems, because it is helpful to save computation time. To verify efficiency of the algorithm, several examples are numerically investigated. Copyright © 2003 John Wiley & Sons, Ltd.     

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.     

PetFMM—A dynamically load‐balancing parallel fast multipole library

Fast algorithms for the computation of N‐body problems can be broadly classified into mesh‐based interpolation methods, and hierarchical or multiresolution methods. To this latter class belongs the well‐known fast multipole method (FMM ), which offers ??(N) complexity. The FMM is a complex algorithm, and the programming difficulty associated with it has arguably diminished its impact, being a barrier for adoption. This paper presents an extensible parallel library for N‐body interactions utilizing the FMM algorithm. A prominent feature of this library is that it is designed to be extensible, with a view to unifying efforts involving many algorithms based on the same principles as the FMM and enabling easy development of scientific application codes. The paper also details an exhaustive model for the computation of tree‐based N‐body algorithms in parallel, including both work estimates and communications estimates. With this model, we are able to implement a method to provide automatic, a priori load balancing of the parallel execution, achieving optimal distribution of the computational work among processors and minimal inter‐processor communications. Using a client application that performs the calculation of velocity induced by N vortex particles in two dimensions, ample verification and testing of the library was performed. Strong scaling results are presented with 10 million particles on up to 256 processors, including both speedup and parallel efficiency. The largest problem size that has been run with the P etFMM library at this point was 64 million particles in 64 processors. The library is currently able to achieve over 85% parallel efficiency for 64 processes. The performance study, computational model, and application demonstrations presented in this paper are limited to 2D. However, the software architecture was designed to make an extension of this work to 3D straightforward, as the framework is templated over the dimension. The software library is open source under the PETS c license, even less restrictive than the BSD license; this guarantees the maximum impact to the scientific community and encourages peer‐based collaboration for the extensions and applications. Copyright © 2010 John Wiley & Sons, Ltd.     

Indirect T‐Trefftz and F‐Trefftz methods for solving boundary value problem of Poisson equation

Abstract

The Poisson equation can be solved by first finding a particular solution and then solving the resulting Laplace equation. In this paper, a computational procedure based on the Trefftz method is developed to solve the Poisson equation for two‐dimensional domains. The radial basis function approach is used to find an approximate particular solution for the Poisson equation. Then, two kinds of Trefftz methods, the T‐Trefftz method and F‐Trefftz method, are adopted to solve the resulting Laplace equation. In order to deal with the possible ill‐posed behaviors existing in the Trefftz methods, the truncated singular value decomposition method and L‐curve concept are both employed. The Poisson equation of the type, ?² u = f(x, u), in which x is the position and u is the dependent variable, is solved by the iterative procedure. Numerical examples are provided to show the validity of the proposed numerical methods and some interesting phenomena are carefully discussed while solving the Helmholtz equation as a Poisson equation. It is concluded that the F‐Trefftz method can deal with a multiply connected domain with genus p(p > 1) while the T‐Trefftz method can only deal with a multiply connected domain with genus 1 if the domain partition technique is not adopted.     

A Fourier–Karhunen–Loève discretization scheme for stationary random material properties in SFEM

In order to overcome the computational difficulties in Karhunen–Loève (K–L) expansions of stationary random material properties in stochastic finite element method (SFEM) analysis, a Fourier–Karhunen–Loève (F–K–L) discretization scheme is developed in this paper, by following the harmonic essence of stationary random material properties and solving a series of specific technical challenges encountered in its development. Three numerical examples are employed to investigate the overall performance of the new discretization scheme and to demonstrate its use in practical SFEM simulations. The proposed F–K–L discretization scheme exhibits a number of advantages over the widely used K–L expansion scheme based on FE meshes, including better computational efficiency in terms of memory and CPU time, convenient a priori error‐control mechanism, better approximation accuracy of random material properties, explicit methods for predicting the associated eigenvalue decay speed and geometrical compatibility for random medium bodies of different shapes. Copyright © 2007 John Wiley & Sons, Ltd.     

A mesh adaptation procedure for periodic domains

In this paper, an original technique is developed in order to build adaptive meshes on periodic domains. The new approach has the important property that it is code‐reused. The procedure is used against three different algorithms, namely, MAdLib ( Int. J. Numer. Meth. Engng 2000; in press), mmg (Proc. 17th Int. Meshing Roundtable, 2008) and the couple Yams (Rapport Technique RT‐0252, 2001) /Ghs3d (Proc. 8th Int. Meshing Roundtable, 1999). None of the latter algorithms needs to be adapted before it is applied to periodic domains. Some examples of adaptation are presented based on analytical, isotropic and anisotropic mesh‐size fields. Periodicity in translation and rotation both are considered. Finally, the mesh adaptation strategy is used in order to reduce the computational cost of a prediction of strain heterogeneity throughout a periodic polycrystalline aggregate deforming by dislocation slip. Copyright © 2011 John Wiley & Sons, Ltd.     

Landweber iteration regularization method for identifying unknown source on a columnar symmetric domain

In this paper, an inverse source problem for the Helium Production–Diffusion Equation on a columnar symmetric domain is investigated. Based on an a priori assumption, the optimal error bound analysis and a conditional stability result are given. This problem is ill-posed and Landweber iteration regularization method is used to deal with this problem. Convergence estimates are presented under the priori and the posteriori regularization choice rules. For the a priori and the a posteriori regularization parameters choice rules, the convergence error estimates are all order optimal. Numerical examples are given to show that the regularization method is effective and stable for dealing with this ill-posed problem.     

Comparison of a phase‐field model and of a thick level set model for brittle and quasi‐brittle fracture

This paper provides a comparison between one particular phase‐field damage model and a thick level set (TLS) damage model for the simulation of brittle and quasi‐brittle fractures. The TLS model is recasted in a variational framework, which allows comparison with the phase‐field model. Using this framework, both the equilibrium equations and the damage evolution laws are guided by the initial choice of the potential energy. The potentials of the phase‐field model and of the TLS model are quite different. TLS potential enforces a priori a bound on damage gradient whereas the phase‐field potential does not. The TLS damage model is defined such that the damage profile fits to the one of the phase‐field model for a beam of infinite length. The model parameters are calibrated to obtain the same surface fracture energy. Numerical results are provided for unidimensional and bidimensional tests for both models. Qualitatively, similar results are observed, although TLS model is observed to be less sensible to boundary conditions. Copyright © 2015 John Wiley & Sons, Ltd.     

Assessment of explicit and semi‐explicit classes of model‐based algorithms for direct integration in structural dynamics

The ‘model‐based’ algorithms available in the literature are primarily developed for the direct integration of the equations of motion for hybrid simulation in earthquake engineering, an experimental method where the system response is simulated by dividing it into a physical and an analytical domain. The term ‘model‐based’ indicates that the algorithmic parameters are functions of the complete model of the system to enable unconditional stability to be achieved within the framework of an explicit formulation. These two features make the model‐based algorithms also potential candidates for computations in structural dynamics. Based on the algorithmic difference equations, these algorithms can be classified as either explicit or semi‐explicit, where the former refers to the algorithms with explicit difference equations for both displacement and velocity, while the latter for displacement only. The algorithms pertaining to each class are reviewed, and a new family of second‐order unconditionally stable parametrically dissipative semi‐explicit algorithms is presented. Numerical characteristics of these two classes of algorithms are assessed under linear and nonlinear structural behavior. Representative numerical examples are presented to complement the analytical findings. The analysis and numerical examples demonstrate the advantages and limitations of these two classes of model‐based algorithms for applications in structural dynamics. Copyright © 2015 John Wiley & Sons, Ltd.     

Boundary element method and internal electrodes in electrical impedance tomography

Traditionally in electrical impedance tomography an approximation for the internal resistivity distribution is computed based on the knowledge of the injected currents and measured voltages on the surface of the object. However, in certain applications it is also possible to use internal current sources and voltage measurements. In this paper we propose a boundary element‐based method which utilizes data also from internal electrodes in the image reconstruction. The proposed approach assumes that the internal geometry is known a priori and only the conductivities of the predetermined regions are estimated. Two‐dimensional simulations with four additional sources/measurement locations show clear improvement in the reconstructed images when the results are compared to those based only on boundary data. Copyright © 2001 John Wiley & Sons, Ltd.     

Analysis of three‐dimensional crack initiation and propagation using the extended finite element method

An Erratum has been published for this article in International Journal for Numerical Methods in Engineering 2005, 63(8): 1228. We present a new formulation and a numerical procedure for the quasi‐static analysis of three‐dimensional crack propagation in brittle and quasi‐brittle solids. The extended finite element method (XFEM) is combined with linear tetrahedral elements. A viscosity‐regularized continuum damage constitutive model is used and coupled with the XFEM formulation resulting in a regularized ‘crack‐band’ version of XFEM. The evolving discontinuity surface is discretized through a C⁰ surface formed by the union of the triangles and quadrilaterals that separate each cracked element in two. The element's properties allow a closed form integration and a particularly efficient implementation allowing large‐scale 3D problems to be studied. Several examples of crack propagation are shown, illustrating the good results that can be achieved. Copyright © 2005 John Wiley & Sons, Ltd.     

Incremental harmonic balance method with multiple time variables for dynamical systems with cubic non‐linearities

The incremental harmonic balance method with multiple time variables is developed for analysis of almost periodic oscillations in multi‐degree‐of‐freedom dynamical systems with cubic non‐linearities, subjected to the external multi‐tone excitation. The method is formulated to treat non‐autonomous as well as autonomous dynamical systems. The almost periodic oscillations, which coexist with periodic oscillations in a rotating system model with cubic restoring force and an electromagnetic eddy‐current damper are analysed. The closed form solutions based on generalized Fourier series containing two incommensurate frequencies are obtained in the case of small non‐dimensional stiffness ratio. Almost periodic oscillations of a rotating system model in dependence on variable parameters are also analysed, where solutions are computed through an augmentation process including a greater number of harmonics and combination frequencies involved. Copyright © 2003 John Wiley & Sons, Ltd.     

Generalized finite element method using proper orthogonal decomposition

A methodology is presented for generating enrichment functions in generalized finite element methods (GFEM) using experimental and/or simulated data. The approach is based on the proper orthogonal decomposition (POD) technique, which is used to generate low‐order representations of data that contain general information about the solution of partial differential equations. One of the main challenges in such enriched finite element methods is knowing how to choose, a priori, enrichment functions that capture the nature of the solution of the governing equations. POD produces low‐order subspaces, that are optimal in some norm, for approximating a given data set. For most problems, since the solution error in Galerkin methods is bounded by the error in the best approximation, it is expected that the optimal approximation properties of POD can be exploited to construct efficient enrichment functions. We demonstrate the potential of this approach through three numerical examples. Best‐approximation studies are conducted that reveal the advantages of using POD modes as enrichment functions in GFEM over a conventional POD basis. Copyright © 2009 John Wiley & Sons, Ltd.     

On the adaptive finite element analysis of the Kohn–Sham equations: methods,algorithms, and implementation

In this paper, details of an implementation of a numerical code for computing the Kohn–Sham equations are presented and discussed. A fully self‐consistent method of solving the quantum many‐body problem within the context of density functional theory using a real‐space method based on finite element discretisation of realspace is considered. Various numerical issues are explored such as (i) initial mesh motion aimed at co‐aligning ions and vertices; (ii) a priori and a posteriori optimization of the mesh based on Kelly's error estimate; (iii) the influence of the quadrature rule and variation of the polynomial degree of interpolation in the finite element discretisation on the resulting total energy. Additionally, (iv) explicit, implicit and Gaussian approaches to treat the ionic potential are compared. A quadrupole expansion is employed to provide boundary conditions for the Poisson problem. To exemplify the soundness of our method, accurate computations are performed for hydrogen, helium, lithium, carbon, oxygen, neon, the hydrogen molecule ion and the carbon‐monoxide molecule. Our methods, algorithms and implementation are shown to be stable with respect to convergence of the total energy in a parallel computational environment. Copyright © 2015 John Wiley & Sons, Ltd.     

On the scattering of light by a periodic structure in the presence of randomness II. On the detection of weak periodicities

Abstract

Light scattering by a rough grating is examined in the context of detecting a hidden periodic part. We make use of the fact that differentiated signals enhance the structures contained in the signal so that these derivatives help to obtain a priori ideas about the structure of the surface from which details about the surface profile can be worked out. The suggested numerical methods to extract the statistical properties of the surface and of the periodic grating involve simple algebraic operations and are similar to ‘matched filtering’ in signal detection problems. Numerical illustrations are provided to elucidate the method and to explain its usefulness. The suggested method, although simple, is seen to be an improvement vis-à-vis existing notions about detectability of periodic structures hidden behind a disorder.     

Homotopy approach for random eigenvalue problem

A novel approach, referred to as the homotopy stochastic finite element method, is proposed to solve the eigenvalue problem of a structure associated with some amount of uncertainty based on the homotopy analysis method. For this approach, an infinite multivariate series of the involved random variables is proposed to express the random eigenvalue or even a random eigenvector. The coefficients of the multivariate series are determined using the homotopy analysis method. The convergence domain of the derived series is greatly expanded compared with the Taylor series due to the use of an approach function of the parameter h. Therefore, the proposed method is not limited to random parameters with small fluctuation. However, in practice, only single‐variable and double‐variable approximations are employed to simplify the calculation. The numerical examples show that with a suitable choice of the auxiliary parameter h, the suggested approximations can produce very accurate results and require reduced or similar computational efforts compared with the existing methods.     

Variational principles in dissipative electro‐magneto‐mechanics: A framework for the macro‐modeling of functional materials

This paper presents a general framework for the macroscopic, continuum‐based formulation and numerical implementation of dissipative functional materials with electro‐magneto‐mechanical couplings based on incremental variational principles. We focus on quasi‐static problems, where mechanical inertia effects and time‐dependent electro‐magnetic couplings are a priori neglected and a time‐dependence enters the formulation only through a possible rate‐dependent dissipative material response. The underlying variational structure of non‐reversible coupled processes is related to a canonical constitutive modeling approach, often addressed to so‐called standard dissipative materials. It is shown to have enormous consequences with respect to all aspects of the continuum‐based modeling in macroscopic electro‐magneto‐mechanics. At first, the local constitutive modeling of the coupled dissipative response, i.e. stress, electric and magnetic fields versus strain, electric displacement and magnetic induction, is shown to be variational based, governed by incremental minimization and saddle‐point principles. Next, the implications on the formulation of boundary‐value problems are addressed, which appear in energy‐based formulations as minimization principles and in enthalpy‐based formulations in the form of saddle‐point principles. Furthermore, the material stability of dissipative electro‐magneto‐mechanics on the macroscopic level is defined based on the convexity/concavity of incremental potentials. We provide a comprehensive outline of alternative variational structures and discuss details of their computational implementation, such as formulation of constitutive update algorithms and finite element solvers. From the viewpoint of constitutive modeling, including the understanding of the stability in coupled electro‐magneto‐mechanics, an energy‐based formulation is shown to be the canonical setting. From the viewpoint of the computational convenience, an enthalpy‐based formulation is the most convenient setting. A numerical investigation of a multiferroic composite demonstrates perspectives of the proposed framework with regard to the future design of new functional materials. Copyright © 2011 John Wiley & Sons, Ltd.     

A comprehensive catastrophe theory for non‐linear buckling of simple systems exhibiting fold and cusp catastrophes

Non‐linear static buckling of simple systems associated with typical discrete critical points is comprehensively presented using elementary Catastrophe Theory. Attention is focused on the Fold and Cusp Catastrophe, all local properties of which are assessed in detail. Hence, in dealing with stability problems of potential systems there is no need to seek any of these properties since all of these are known a priori. Then, one has only to classify, after reduction, the total potential energy of a system into one of the universal unfoldings of the above types of catastrophe. Two illustrative numerical examples show the methodology of the proposed technique. Copyright © 2002 John Wiley & Sons, Ltd.     

Novel partitioned time integration methods for DAE systems based on L‐stable linearly implicit algorithms

Real‐time applications based on the principle of Dynamic Substructuring require integration methods that can deal with constraints without exceeding an a priori fixed number of steps. For these applications, first we introduce novel partitioned algorithms able to solve DAEs arising from transient structural dynamics. In particular, the spatial domain is partitioned into a set of disconnected subdomains and continuity conditions of acceleration at the interface are modeled using a dual Schur formulation. Interface equations along with subdomain equations lead to a system of DAEs for which both staggered and parallel procedures are developed. Moreover under the framework of projection methods, also a parallel partitioned method is conceived. The proposed partitioned algorithms enable a Rosenbrock‐based linearly implicit LSRT2 method, to be strongly coupled with different time steps in each subdomain. Thus, user‐defined algorithmic damping and subcycling strategies are allowed. Secondly, the paper presents the convergence analysis of the novel schemes for linear single‐Degree‐of‐Freedom (DoF) systems. The algorithms are generally A‐stable and preserve the accuracy order as the original monolithic method. Successively, these results are validated via simulations on single‐ and three‐DoFs systems. Finally, the insight gained from previous analyses is confirmed by means of numerical experiments on a coupled spring–pendulum system. Copyright © 2011 John Wiley & Sons, Ltd.     

Space–time meshfree collocation method: Methodology and application to initial‐boundary value problems

A novel space–time meshfree collocation method (STMCM) for solving systems of non‐linear ordinary and partial differential equations by a consistent discretization in both space and time is proposed as an alternative to established mesh‐based methods. The STMCM belongs to the class of truly meshfree methods, i.e. the methods that do not have any underlying mesh, but work on a set of nodes only without any a priori node‐to‐node connectivity. Instead, the neighbouring information is established on‐the‐fly. The STMCM is constructed using the Interpolating Moving Least‐squares technique, which allows a simplified implementation of boundary conditions due to fulfillment of the Kronecker delta property by the kernel functions, which is not the case for the major part of other meshfree methods. The method is validated by several examples ranging from interpolation problems to the solution of PDEs, whereas the STMCM solutions are compared with either analytical or reference ones. Copyright © 2009 John Wiley & Sons, Ltd.