Guaranteed error bounds on pointwise quantities of interest for transient viscodynamics problems

A new method is developed to obtain guaranteed error bounds on pointwise quantities of interest for linear transient viscodynamics problems. The calculation of strict error bounds is based on the concept of “constitutive relation error” (CRE) and the solution of an adjoint problem. The central and original point of this work is the treatment of the singularity in space and time introduced by the loading of the adjoint problem. Hence, the adjoint solution is decomposed into two parts: (i) an analytical part determined from Green’s functions; (ii) a residual part approximated with classical numerical tools (finite element method, Newmark integration scheme). The capabilities and the limits of the proposed approach are analyzed on a 2D example.     

Modal‐based goal‐oriented error assessment for timeline‐dependent quantities in transient dynamics

This article presents a new approach to assess the error in specific quantities of interest in the framework of linear elastodynamics. In particular, a new type of quantities of interest (referred as timeline‐dependent quantities) is proposed. These quantities are scalar time‐dependent outputs of the transient solution, which are better suited to time‐dependent problems than the standard scalar ones, frozen in time. The proposed methodology furnishes error estimates for both the standard scalar and the new timeline‐dependent quantities of interest. The key ingredient is the modal‐based approximation of the associated adjoint problems, which allows efficiently computing and storing the adjoint solution. The approximated adjoint solution is readily post‐processed to produce an enhanced solution, requiring only one spatial post‐process for each vibration mode and using the time‐harmonic hypothesis to recover the time dependence. Thus, the proposed goal‐oriented error estimate consists in injecting this enhanced adjoint solution into the residual of the direct problem. The resulting estimate is very well suited for transient dynamic simulations because the enhanced adjoint solution is computed before starting the forward time integration of the direct problem. Thus, the cost of the error estimate at each time step is very low. Copyright © 2013 John Wiley & Sons, Ltd.     

Error bounds on block Gauss–Seidel solutions of coupled multiphysics problems

Mathematical models in many fields often consist of coupled sub‐models, each of which describes a different physical process. For many applications, the quantity of interest from these models may be written as a linear functional of the solution to the governing equations. Mature numerical solution techniques for the individual sub‐models often exist. Rather than derive a numerical solution technique for the full coupled model, it is therefore natural to investigate whether these techniques may be used by coupling in a block Gauss–Seidel fashion. In this study, we derive two a posteriori bounds for such linear functionals. These bounds may be used on each Gauss–Seidel iteration to estimate the error in the linear functional computed using the single physics solvers, without actually solving the full, coupled problem. We demonstrate the use of the bound first by using a model problem from linear algebra, and then a linear ordinary differential equation example. We then investigate the effectiveness of the bound using a non‐linear coupled fluid‐temperature problem. One of the bounds derived is very sharp for most linear functionals considered, allowing us to predict very accurately when to terminate our block Gauss–Seidel iteration. Copyright © 2011 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.     

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.     

Pointwise error estimation and adaptivity for the finite element method using fundamental solutions

In this paper, we present a goal-oriented a posteriori error estimation technique for the pointwise error of finite element approximations using fundamental solutions. The approach is based on an integral representation of the pointwise quantity of interest using the corresponding Green's function, which is decomposed into an unknown regular part and a fundamental solution. Since only the regular part must be approximated with finite elements, very accurate results are obtained. The approach also allows the derivation of error bounds for the pointwise quantity, which are expressed in terms of the primal problem and the regular part problem. The presented technique is applied to linear elastic test problems in two-dimensions, but it can be applied to any linear problem for which fundamental solutions exist.     

Numerical solution of second‐order,two‐point boundary value problems using continuous genetic algorithms

Second‐order, two‐point boundary‐value problems are encountered in many engineering applications including the study of beam deflections, heat flow, and various dynamic systems. Two classical numerical techniques are widely used in the engineering community for the solution of such problems; the shooting method and finite difference method. These methods are suited for linear problems. However, when solving the non‐linear problems, these methods require some major modifications that include the use of some root‐finding technique. Furthermore, they require the use of other basic numerical techniques in order to obtain the solution. In this paper, the author introduces a novel method based on continuous genetic algorithms for numerically approximating a solution to this problem. The new method has the following characteristics; first, it does not require any modification while switching from the linear to the non‐linear case; as a result, it is of versatile nature. Second, this approach does not resort to more advanced mathematical tools and is thus easily accepted in the engineering application field. Third, the proposed methodology has an implicit parallel nature which points to its implementation on parallel machines. However, being a variant of the finite difference scheme with truncation error of the order O(h²), the method provides solutions with moderate accuracy. Numerical examples presented in the paper illustrate the applicability and generality of the proposed method. Copyright © 2004 John Wiley & Sons, Ltd.     

Goal‐oriented adaptivity using unconventional error representations for the multidimensional Helmholtz equation

In goal‐oriented adaptivity, the error in the quantity of interest is represented using the error functions of the direct and adjoint problems. This error representation is subsequently bounded above by element‐wise error indicators that are used to drive optimal refinements. In this work, we propose to replace, in the error representation, the adjoint problem by an alternative operator. The main advantage of the proposed approach is that, when judiciously selecting such alternative operator, the corresponding upper bound of the error representation becomes sharper, leading to a more efficient goal‐oriented adaptivity. While the method can be applied to a variety of problems, we focus here on two‐ and three‐dimensional (2‐D and 3‐D) Helmholtz problems. We show via extensive numerical experimentation that the upper bounds provided by the alternative error representations are sharper than the classical ones and lead to a more robust p‐adaptive process. We also provide guidelines for finding operators delivering sharp error representation upper bounds. We further extend the results to a convection‐dominated diffusion problem as well as to problems with discontinuous material coefficients. Finally, we consider a sonic logging‐while‐drilling problem to illustrate the applicability of the proposed method.     

Element‐wise a posteriori estimates based on hierarchical bases for non‐linear parabolic problems

We show that the issue of a posteriori estimate the errors in the numerical simulation of non‐linear parabolic equations can be reduced to a posteriori estimate the errors in the approximation of an elliptic problem with the right‐hand side depending on known data of the problem and the computed numerical solution. A procedure to obtain local error estimates for the p version of the finite element method by solving small discrete elliptic problems with right‐hand side the residual of the p‐FEM solution is introduced. The boundary conditions are inherited by those of the space of hierarchical bases to which the error estimator belongs. We prove that the error in the numerical solution can be reduced by adding the estimators that behave as a locally defined correction to the computed approximation. When the error being estimated is that of a elliptic problem constant free local lower bounds are obtained. The local error estimation procedure is applied to non‐linear parabolic differential equations in several space dimensions. Some numerical experiments for both the elliptic and the non‐linear parabolic cases are provided. Copyright © 2005 John Wiley & Sons, Ltd.     

Structural design optimization on thermally induced vibration

The numerical method of design optimization for structural thermally induced vibration is originally studied in this paper and implemented in the software JIFEX. The direct and adjoint methods of sensitivity analysis for thermal‐induced vibration coupled with both linear and non‐linear transient heat conduction is firstly proposed. Based on the finite element method, the linear structural dynamics is treated simultaneously with linear and non‐linear transient heat conduction. In the heat conduction, the non‐linear factors include the radiation and temperature‐dependent materials. The sensitivity analysis of transient linear and non‐linear heat conduction is performed with the precise time integration method; and then, the sensitivity analysis of structural transient responses is performed by the Newmark method. Both the direct method and the adjoint method are employed to derive the sensitivity equations of thermal vibration. In the adjoint method, two adjoint vectors of structure and of heat conduction are used to derive the adjoint equations. The coupling effect of heat conduction on thermal vibration in the sensitivity analysis is particularly investigated. With the coupling sensitivity analysis, the optimization model is constructed and solved by the sequential linear programming or sequential quadratic programming algorithm. Numerical examples are given to validate the proposed methods and to demonstrate the importance of the coupled design optimization. Copyright © 2003 John Wiley & Sons, Ltd.     

Guaranteed error bounds in homogenisation: an optimum stochastic approach to preserve the numerical separation of scales

This paper proposes a new methodology to guarantee the accuracy of the homogenisation schemes that are traditionally employed to approximate the solution of PDEs with random, fast evolving diffusion coefficients. More precisely, in the context of linear elliptic diffusion problems in randomly packed particulate composites, we develop an approach to strictly bound the error in the expectation and second moment of quantities of interest, without ever solving the fine‐scale, intractable stochastic problem. The most attractive feature of our approach is that the error bounds are computed without any integration of the fine‐scale features. Our computations are purely macroscopic, deterministic and remain tractable even for small scale ratios. The second contribution of the paper is an alternative derivation of modelling error bounds through the Prager–Synge hypercircle theorem. We show that this approach allows us to fully characterise and optimally tighten the interval in which predicted quantities of interest are guaranteed to lie. We interpret our optimum result as an extension of Reuss–Voigt approaches, which are classically used to estimate the homogenised diffusion coefficients of composites, to the estimation of macroscopic engineering quantities of interest. Finally, we make use of these derivations to obtain an efficient procedure for multiscale model verification and adaptation. Copyright © 2016 John Wiley & Sons, Ltd.     

Error bounds for the reliability index in finite element reliability analysis

This work presents an extension of the goal‐oriented error estimation techniques to the reliability analysis of a linear elastic structure. We use a first‐order reliability method in conjunction with a finite element analysis (FEA) to compute the failure probability of the structure. In such a situation the output of interest that is computed from the FEA is the reliability index β. The accuracy of this output, and thus of the reliability analysis, depends, in particular, on the accuracy of the FEA. In this paper, upper and lower bounds of the reliability index are proposed, as well as simple bounds of the failure probability. An application to linear fracture mechanics is presented. Copyright © 2011 John Wiley & Sons, Ltd.     

Output error estimation strategies for discontinuous Galerkin discretizations of unsteady convection‐dominated flows

We study practical strategies for estimating numerical errors in scalar outputs calculated from unsteady simulations of convection‐dominated flows, including those governed by the compressible Navier–Stokes equations. The discretization is a discontinuous Galerkin finite element method in space and time on static spatial meshes. Time‐integral quantities are considered for scalar outputs and these are shown to superconverge with temporal refinement. Output error estimates are calculated using the adjoint‐weighted residual method, where the unsteady adjoint solution is obtained using a discrete approach with an iterative solver. We investigate the accuracy versus computational cost trade‐off for various approximations of the fine‐space adjoint and find that exact adjoint solutions are accurate but expensive. To reduce the cost, we propose a local temporal reconstruction that takes advantage of superconvergence properties at Radau points, and a spatial reconstruction based on nearest‐neighbor elements. This inexact adjoint yields output error estimates at a computational cost of less than 2.5 times that of the forward problem for the cases tested. The calculated error estimates account for numerical error arising from both the spatial and temporal discretizations, and we present a method for identifying the percentage contributions of each discretization to the output error. Copyright © 2011 John Wiley & Sons, Ltd.     

A constitutive relation error estimator based on traction‐free recovery of the equilibrated stress

A new methodology for recovering equilibrated stress fields is presented, which is based on traction‐free subdomains' computations. It allows a rather simple implementation in a standard finite element code compared with the standard technique for recovering equilibrated tractions. These equilibrated stresses are used to compute a constitutive relation error estimator for a finite element model in 2D linear elasticity. A lower bound and an upper bound for the discretization error are derived from the error in the constitutive relation. These bounds in the discretization error are used to build lower and upper bounds for local quantities of interest. Copyright © 2008 John Wiley & Sons, Ltd.     

Computable exact bounds for linear outputs from stabilized solutions of the advection–diffusion–reaction equation

The paper introduces a methodology to compute strict upper and lower bounds for linear‐functional outputs of the exact solutions of the advection–diffusion–reaction equation. The bounds are computed using implicit a posteriori error estimators from stabilized finite element approximations of the exact solution. The new methodology extends the a posteriori error estimates yielding bounds for the standard Galerkin formulation to be able to obtain bounds for stabilized formulations. This methodology is combined with both hybrid‐flux and flux‐free techniques for error assessment. The application to stabilized formulations provides sharper estimates than when applied to Galerkin methods. The best results are found in combination with the flux‐free technique. Copyright © 2012 John Wiley & Sons, Ltd.     

Efficient non‐linear model reduction via a least‐squares Petrov–Galerkin projection and compressive tensor approximations

A Petrov–Galerkin projection method is proposed for reducing the dimension of a discrete non‐linear static or dynamic computational model in view of enabling its processing in real time. The right reduced‐order basis is chosen to be invariant and is constructed using the Proper Orthogonal Decomposition method. The left reduced‐order basis is selected to minimize the two‐norm of the residual arising at each Newton iteration. Thus, this basis is iteration‐dependent, enables capturing of non‐linearities, and leads to the globally convergent Gauss–Newton method. To avoid the significant computational cost of assembling the reduced‐order operators, the residual and action of the Jacobian on the right reduced‐order basis are each approximated by the product of an invariant, large‐scale matrix, and an iteration‐dependent, smaller one. The invariant matrix is computed using a data compression procedure that meets proposed consistency requirements. The iteration‐dependent matrix is computed to enable the least‐squares reconstruction of some entries of the approximated quantities. The results obtained for the solution of a turbulent flow problem and several non‐linear structural dynamics problems highlight the merit of the proposed consistency requirements. They also demonstrate the potential of this method to significantly reduce the computational cost associated with high‐dimensional non‐linear models while retaining their accuracy. Copyright © 2010 John Wiley & Sons, Ltd.     

New bounding techniques for goal‐oriented error estimation applied to linear problems

The paper deals with the accuracy of guaranteed error bounds on outputs of interest computed from approximate methods such as the finite element method. A considerable improvement is introduced for linear problems, thanks to new bounding techniques based on Saint‐Venant's principle. The main breakthrough of these optimized bounding techniques is the use of properties of homothetic domains that enables to cleverly derive guaranteed and accurate bounding of contributions to the global error estimate over a local region of the domain. Performances of these techniques are illustrated through several numerical experiments. Copyright © 2012 John Wiley & Sons, Ltd.     

Multi‐time‐step and two‐scale domain decomposition method for non‐linear structural dynamics

In this paper we propose a method to improve the means of taking into account the specific time‐scale and space‐scale characteristics in time‐dependent non‐linear problems. This method enables the use of arbitrary time steps in each subdomain: these can be coupled by prescribing continuous velocities at the interfaces, which are modelled using a dual Schur formulation. For certain subdomains, in space, we adopt a two‐scale resolution technique inspired by the multigrid methods in order to obtain the part of the solution related to small variation lengths on a refined scale and the part corresponding to large variation lengths on a coarse scale. For non‐linear problems, we propose an algorithm with a single iteration level to deal with both the non‐linear equilibrium and the two space scales thanks to a two‐grid method in which the relaxation steps are performed using a non‐linear, preconditioned conjugate gradient algorithm. Finally, we present an example which demonstrates the feasibility of the method. Copyright © 2003 John Wiley & Sons, Ltd.     

Goal‐oriented adaptivity for linear elastic micromorphic continua based on primal and adjoint consistency analysis

Microscopic considerations are drawing increasing attention for modern simulation techniques. Micromorphic continuum theories, considering micro degrees of freedom, are usually adopted for simulation of localization effects like shear bands. The increased number of degrees of freedom clearly motivates an application of adaptive methods. In this work, the adaptive FEM is tailored for micromorphic elasticity. The proposed adaptive procedure is driven by a goal‐oriented a posteriori error estimator based on duality techniques. For efficient computation of the dual solution, a patch‐based recovery technique is proposed and compared to a reference approach. In order to theoretically ensure optimal convergence order of the proposed adaptive procedure, adjoint consistency of the FE‐discretized solution for the linear elastic micromorphic continua is shown. For illustration, numerical examples are provided. Copyright © 2017 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.