Guaranteed energy error bounds for the Poisson equation using a flux‐free approach: Solving the local problems in subdomains

A method to compute guaranteed upper bounds for the energy norm of the exact error in the finite element solution of the Poisson equation is presented. The bounds are guaranteed for any finite element mesh however coarse it may be, not just in the asymptotic regime. The bounds are constructed by employing a subdomain‐based a posteriori error estimate which yields self‐equilibrated residual loads in stars (patches of elements). The proposed approach is an alternative to standard equilibrated residual methods providing sharper bounds. The use of a flux‐free error estimator improves the effectivities of the upper bounds for the energy while retaining the certainty of the bounds. Copyright © 2009 John Wiley & Sons, Ltd.     

Estimation of the dispersion error in the numerical wave number of standard and stabilized finite element approximations of the Helmholtz equation

An estimator for the error in the wave number is presented in the context of finite element approximations of the Helmholtz equation. The proposed estimate is an extension of the ideas introduced in Steffens and D'?ez (Comput. Methods Appl. Mech. Engng 2009; 198 :1389–1400). In the previous work, the error assessment technique was developed for standard Galerkin approximations. Here, the methodology is extended to deal also with stabilized approximations of the Helmholtz equation. Thus, the accuracy of the stabilized solutions is analyzed, including also their sensitivity to the stabilization parameters depending on the mesh topology. The procedure builds up an inexpensive approximation of the exact solution, using post‐processing techniques standard in error estimation analysis, from which the estimate of the error in the wave number is computed using a simple closed expression. The recovery technique used in Steffens and Díez (Comput. Methods Appl. Mech. Engng 2009; 198 :1389–1400) is based in a polynomial least‐squares fitting. Here a new recovery strategy is introduced, using exponential (in a complex setup, trigonometric) local approximations respecting the nature of the solution of the wave problem. Copyright © 2011 John Wiley & Sons, Ltd.     

Strict lower bounds with separation of sources of error in non‐overlapping domain decomposition methods

This article deals with the computation of guaranteed lower bounds of the error in the framework of finite element and domain decomposition methods. In addition to a fully parallel computation, the proposed lower bounds separate the algebraic error (due to the use of a domain decomposition iterative solver) from the discretization error (due to the finite element), which enables the steering of the iterative solver by the discretization error. These lower bounds are also used to improve the goal‐oriented error estimation in a substructured context. Assessments on 2D static linear mechanic problems illustrate the relevance of the separation of sources of error and the lower bounds' independence from the substructuring. We also steer the iterative solver by an objective of precision on a quantity of interest. This strategy consists in a sequence of solvings and takes advantage of adaptive remeshing and recycling of search directions. Copyright © 2016 John Wiley & Sons, Ltd.     

Variational multiscale methods to embed the macromechanical continuum formulation with fine‐scale strain gradient theories

A variational basis is presented to link fine‐scale theories of material behaviour with the classical, macromechanical continuum theory. The approach is based on the weak form of the linear momentum balance equations, and a separation of the weighting function and displacement fields into coarse and fine‐scale components. Coarse and fine‐scale weak forms are defined. The latter is used to introduce a strain gradient theory that operates at finer scales of deformation. Attention is focused upon applications requiring the enhanced physical accuracy of the fine‐scale strain gradient theory, without the computational cost of discretization that spans the range from coarse to fine scales. A variationally consistent method is developed to embed the fine‐scale strain gradient theory in the macromechanical formulation. The embedding is achieved by eliminating the fine‐scale displacement field from the problem. Two examples demonstrate the numerical efficiency of the method, while retaining physical and mathematical properties of the fine‐scale strain gradient theory. Copyright © 2003 John Wiley & Sons, Ltd.     

Guaranteed computable bounds on quantities of interest in finite element computations

We develop and compare a number of alternative approaches to obtain guaranteed and fully computable bounds on the error in quantities of interest of arbitrary order finite element approximations in the context of a linear second‐order elliptic problem. In each case, the bounds are fully computable and do not involve any unknown multiplicative factors. Guaranteed computable bounds are also obtained for the case when the Dirichlet boundary conditions are non‐homogeneous. This is achieved by taking account of the error incurred by the approximation of the Dirichlet data in the functional used to approximate the quantity of interest itself, which is found to generally give better results. Numerical examples are presented to show that the resulting estimators provide tight bounds with the effectivity index tending to unity from above. Copyright © 2012 John Wiley & Sons, Ltd.     

Error estimation and model adaptation for a stochastic‐deterministic coupling method based on the Arlequin framework

The paper deals with the issue of accuracy for multiscale methods applied to solve stochastic problems. It more precisely focuses on the control of a coupling, performed using the Arlequin framework, between a deterministic continuum model and a stochastic continuum one. By using residual‐type estimates and adjoint‐based techniques, a strategy for goal‐oriented error estimation is presented for this coupling and contributions of various error sources (modeling, space discretization, and Monte Carlo approximation) are assessed. Furthermore, an adaptive strategy is proposed to enhance the quality of outputs of interest obtained by the coupled stochastic‐deterministic model. Performance of the proposed approach is illustrated on 1D and 2D numerical experiments. Copyright © 2013 John Wiley & Sons, Ltd.     

Certification of projection‐based reduced order modelling in computational homogenisation by the constitutive relation error

In this paper, we propose upper and lower error bounding techniques for reduced order modelling applied to the computational homogenisation of random composites. The upper bound relies on the construction of a reduced model for the stress field. Upon ensuring that the reduced stress satisfies the equilibrium in the finite element sense, the desired bounding property is obtained. The lower bound is obtained by defining a hierarchical enriched reduced model for the displacement. We show that the sharpness of both error estimates can be seamlessly controlled by adapting the parameters of the corresponding reduced order model. Copyright © 2013 John Wiley & Sons, Ltd.     

A new stochastic residual error based homotopy approach for stability analysis of structures with large fluctuation of random parameters

The large fluctuation of uncertain parameters introduces a great challenge in the stability analysis of structures. To address this problem, a novel stochastic residual error based homotopy method is proposed in this article. This new method used the concept of homotopy to reconstruct a new governing equation for stochastic elastic buckling analysis, and the closed-form solutions of the isolated buckling eigenvalues and eigenvectors are obtained by the stochastic homotopy analysis method. On this basis, a pth order origin moment of the stochastic residual error with respect to the elastic buckling equation is defined. Then, the optimal form of the homotopy series can be determined automatically by minimizing the pth order origin moment, which overcomes the disadvantage of highly relying on sample values of the existing homotopy stochastic finite element method. Moreover, the proposed method is developed to deal with the stochastic closely spaced buckling eigenvalue problem. Three mathematical examples and three buckling eigenvalue examples, including a variable cross-section column, a 7-story frame, and a Kiewitt single-layer latticed spherical shell, are performed to illustrate the accuracy and effectiveness of the proposed method by comparing with the existing methods when dealing with large fluctuation of random parameters.     

Analysis of geometrical error in the stereolithography process using a stochastic approach

The geometrical error in the stereolithography process is analysed using a stochastic approach. This approach is based on a unified methodology, developed by the authors, for studying the mechanical error in different rapid prototyping processes. The tolerances and clearances have been assumed to be random variables. The coordinates of a point on the resin surface, traced by the laser beam, are expressed as a function of random variables. In a numerical example, the geometrical error has been found for a grid of points traced by the laser beam. The three-sigma error bands are plotted when tracing example curves. This is the band in which the laser beams of 99.73% of machines, produced on a mass scale, lie on the work surface for the given tolerances and clearances. Stringent values of tolerances and clearances reduce the error at the tool tip, but the cost of manufacturing and assembling the machines may become prohibitive.     

Strict and effective bounds in goal‐oriented error estimation applied to fracture mechanics problems solved with XFEM

In this work, we analyze a method that leads to strict and high‐quality local error bounds in the context of fracture mechanics. We investigate in particular the capability of this method to evaluate the discretization error for quantities of interest computed using the extended finite element method (XFEM). The goal‐oriented error estimation method we are focusing on uses the concept of constitutive relation error along with classical extraction techniques. The main innovation in this paper resides in the methodology employed to construct admissible fields in the XFEM framework, which involves enrichments with singular and level set basis functions. We show that this construction can be performed through a generalization of the classical procedure used for the standard finite element method. Thus, the resulting goal‐oriented error estimation method leads to relevant and very accurate information on quantities of interest that are specific to fracture mechanics, such as mixed‐mode stress intensity factors. The technical aspects and the effectiveness of the method are illustrated through two‐dimensional numerical examples. Copyright © 2009 John Wiley & Sons, Ltd.     

Equilibrated patch recovery error estimates: simple and accurate upper bounds of the error

This paper introduces a new recovery‐type error estimator ensuring local equilibrium and yielding a guaranteed upper bound of the error. The upper bound property requires the recovered solution to be both statically equilibrated and continuous. The equilibrium is obtained locally (patch‐by‐patch) and the continuity is enforced by a postprocessing based on the partition of the unity concept. This postprocess is expected to preserve the features of the locally equilibrated stress field. Nevertheless, the postprocess phase modifies the equilibrium, which is no longer exactly fulfilled. A new methodology is introduced that yields upper bound estimates by taking into account this lack of equilibrium. This requires computing the ??₂ norm of the error or relating it with the energy norm. The guaranteed upper bounds are obtained by using a pessimistic bound of the error ??₂ norm, derived from an eigenvalue problem. Nevertheless, these bounds are not sharp. An additional strategy based on a more accurate assessment of the error ??₂ norm is introduced, providing sharp estimates, which are practical upper bounds as it is demonstrated in the numerical tests. Copyright © 2006 John Wiley & Sons, Ltd.     

On the optimum support size in meshfree methods: A variational adaptivity approach with maximum‐entropy approximants

We present a method for the automatic adaption of the support size of meshfree basis functions in the context of the numerical approximation of boundary value problems stemming from a minimum principle. The method is based on a variational approach, and the central idea is that the variational principle selects both the discretized physical fields and the discretization parameters, here those defining the support size of each basis function. We consider local maximum‐entropy approximation schemes, which exhibit smooth basis functions with respect to both space and the discretization parameters (the node location and the locality parameters). We illustrate by the Poisson, linear and non‐linear elasticity problems the effectivity of the method, which produces very accurate solutions with very coarse discretizations and finds unexpected patterns of the support size of the shape functions. Copyright © 2009 John Wiley & Sons, Ltd.     

On error estimator and p‐adaptivity in the generalized finite element method

This paper addresses the issue of a p‐adaptive version of the generalized finite element method (GFEM). The technique adopted here is the equilibrated element residual method, but presented under the GFEM approach, i.e., by taking into account the typical nodal enrichment scheme of the method. Such scheme consists of multiplying the partition of unity functions by a set of enrichment functions. These functions, in the case of the element residual method are monomials, and can be used to build the polynomial space, one degree higher than the one of the solution, in which the error functions is approximated. Global and local measures are defined and used as error estimator and indicators, respectively. The error indicators, calculated on the element patches that surrounds each node, are used to control a refinement procedure. Numerical examples in plane elasticity are presented, outlining in particular the effectivity index of the error estimator proposed. Finally, the ‐adaptive procedure is described and its good performance is illustrated by the last numerical example. Copyright © 2004 John Wiley & Sons, Ltd.     

Analysis of mechanical error in a fused deposition process using a stochastic approach

A stochastic model has been developed for studying the mechanical error in different rapid prototyping (RP) processes. Tolerances and clearances, which cause mechanical error, have been assumed to be random variables. The coordinates of a point on the work surface traced by the laser beam or the tip of the extruder head is expressed as a function of the random variables involved in the process. Using a unified approach for the RP processes, the mechanical error in the fused deposition process is analysed. In a numerical example, the mechanical error has been found for a grid of points traced by the nozzle tip. The three-sigma bands of the error in tracing example curves are plotted. This is the band in which the nozzle tips of 99.73% of machines, produced on a mass scale, lie for the given tolerances and clearances. Stringent values of tolerances and clearances reduce the error at the nozzle tip, but the cost of manufacturing and assembling the machines may become prohibitive.     

Enhancing piecewise-defined surrogate response surfaces with adjoints on sets of unstructured samples to solve stochastic inverse problems

Many approaches for solving stochastic inverse problems suffer from both stochastic and deterministic sources of error. The finite number of samples used to construct a solution is a common source of stochastic error. When computational models are expensive to evaluate, surrogate response surfaces are often employed to increase the number of samples available for approximating the solution. This leads to a reduction in finite sampling errors while the deterministic error in the evaluation of each sample is potentially increased. The pointwise accuracy of sampling the surrogate is primarily impacted by two sources of deterministic error: the local order of accuracy in the surrogate and the numerical error from the numerical solution of the model. In this work, we use adjoints to simultaneously give a posteriori error and derivative estimates in order to construct low-order, piecewise-defined surrogates on sets of unstructured samples. Several examples demonstrate the computational gains of this approach in obtaining accurate estimates of probabilities for events in the design space of model input parameters. This lays the groundwork for future studies on goal-oriented adaptive refinement of such surrogates.     

Filter approach to the stochastic analysis of MDOF wind-excited structures

In this paper, an approach useful for stochastic analysis of the Gaussian and non-Gaussian behavior of the response of multi-degree-of-freedom (MDOF) wind-excited structures is presented. This approach is based on a particular model of the multivariate stochastic wind field based upon a particular diagonalization of the power spectral density (PSD) matrix of the fluctuating part of wind velocity. This diagonalization is performed in the space of eigenvectors and eigenvalues that are called here wind-eigenvalues and wind-eigenvectors, respectively. From the examination of these quantities it can be recognized that the wind-eigenvectors change slowly with frequency while the first wind-eigenvalue dominates all the others in the low-frequency range. It is shown that the wind field can be modeled in a satisfactory way by taking the first wind-eigenvector as constant and by retaining only the first eigenvalue in the calculations. The described model is then used for stochastic analysis in the time domain of MDOF wind-excited structures. This is accomplished by modeling each element of the diagonalized wind-PSD matrix as the velocity PSD function of a set of second-order digital filters with viscous damping driven by white noise of selected intensity. This approach makes it easy to obtain in closed form the statistical moments of every order of the structural response, taking full advantage of the Itô calculus. Moreover, in the proposed approach, it is possible to reduce the computational effort by appropriately selecting the number of wind modes retained in the calculation.     

Mitigating the integration error in numerical simulations of Newtonian systems

We introduce a method for mitigating the numerical integration errors of linear, second‐order initial value problems. We propose a methodology for constructing an optimal state‐space representation that gives minimum numerical truncation error, and in this sense, is the optimal state‐space representation for modelling given phase‐space dynamics. To that end, we utilize a simple transformation of the state‐space equations into their variational form. This process introduces an inherent freedom, similar to the gauge freedom in electromagnetism. We then utilize the gauge function to reduce the numerical integration error. We show that by choosing an appropriate gauge function the numerical integration error dramatically decreases and one can achieve much better accuracy compared to the standard state variables for a given time‐step. Moreover, we derive general expressions yielding the optimal gauge functions given a Newtonian one degree‐of‐freedom ODE. For the n degrees‐of‐freedom case we describe MATLAB^® code capable of finding the optimal gauge functions and integrating the given system using the gauge‐optimized integration algorithm. In all of our illustrating examples, the gauge‐optimized integration outperforms the integration using standard state variables by a few orders of magnitude. Copyright © 2006 John Wiley & Sons, Ltd.     

A posteriori estimation and adaptive control of the pollution error in the h-version of the finite element method

In References 1 and 2 we showed that the error in the finite-element solution has two parts, the local error and the pollution error, and we studied the effect of the pollution error on the quality of the local error-indicators and the quality of the derivatives recovered by local post-processing. Here we show that it is possible to construct a posteriori estimates of the pollution error in any patch of elements by employing the local error-indicators over the mesh outside the patch. We also give an algorithm for the adaptive control of the pollution error in any patch of elements of interest.     

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.