A non-intrusive approach of goal-oriented error estimation for evolution problems solved by the finite element method

In this article, we set up a non-intrusive procedure that yields for strict and highquality error bounds of quantities of interest in linear viscoelasticity problems solved by means of the Finite Element Method. The non-intrusive feature is achieved by introducing, via a partition of unity, enrichment functions in the solution of the adjoint problem (handbook techniques). The resulting goal-oriented error estimation method is thus easy to implement in a FE code and enables to consider trully pointwise quantities of interest.

A goal-oriented a-posteriori error estimator is developed for transient coupled thermo-hydro-mechanical (THM) parametric problems solved with a reduced basis approximation. The estimator assesses the error in some specific Quantity of Interest (QoI). The goal-oriented error estimate is derived based on explicitly-computed weak residual of the primal problem and implicitly-computed adjoint solution associated with the QoI. The time-dependence of the coupled THM system poses an additional complexity as the auxiliary adjoint problem evolves backwards in time. The error estimator guides a greedy adaptive procedure that constructs progressively an optimal reduced basis by smartly selecting snapshot points over a given parametric training sample. The reduced basis obtained is used to drastically reduce the coupled system spatial degrees of freedom by several orders of magnitude. The computational gain obtained from the developed methodology is demonstrated through applications in 2D and 3D parametrized problems simulating the evolution of coupled THM processes in rock masses.

Goal-oriented error estimation and adaptivity for the finite element method

A stochastic computational method based on goal-oriented error estimation for heterogeneous geological materials

Goal-oriented mesh adaptivity for inverse problems in linear micromorphic elasticity

In many engineering applications, the global energy of the problem may be a quantity of limited relevance. This motivated the design of goal-oriented adaptive finite element methods (see e.g. [1,4-6]). They intend to approximate a particular Quantity of Interest (QoI) using an error estimator based on the solution of the adjoint problem to guide the refinements. In this framework, Darrigrand et al. [3] presented an alternative dual operator for the representations of the error in the QoI. In the context of hp-adaptivity, Darrigrand et al. [2] proposed an algorithm for symmetric and positive definite (SPD) problems based on performing global refinements followed by optimal unrefinements. This algorithm marks the basis functions with the lowest contributions to the energy of the solution and remove them. The resulting automatic hp-adaptive strategy employs a multi-level hierarchical data-structure proposed by Zander et al. [7]. In this presentation, we extended Darrigrand et al. [2] algorithm to non-SPD problems in the framework of goal-oriented adaptivity. To do so, we evaluate the contribution to the energy in the alternative dual operator proposed by Darrigrand et al. [3], thus, considering both the direct and the adjoint problem. As a result, we obtain an hp-adaptive algorithm for SPD and non-SPD problems in the context of goal-oriented adaptivity. We test and analyze our algorithm on two-dimensional (2D) Laplace, Helmholtz, and convection-dominated problems. We also describe the main features and limitations of the proposed method. In particular, our algorithm is robust and simple to implement; therefore, it can be used for industrial applications.

Generalized or extended finite element method (G/XFEM) models the crack by enriching functions of partition of unity type with discontinuous functions that represent well the physical behavior of the problem. However, this enrichment functions are not available for all problem types. Thus, one can use numerically-built (global-local) enrichment functions to have a better approximate procedure. This paper investigates the effects of micro-defects/inhomogeneities on a main crack behavior by modeling the micro-defects/inhomogeneities in the local problem using a two-scale G/XFEM. The global-local enrichment functions are influenced by the micro-defects/inhomogeneities from the local problem and thus change the approximate solution of the global problem with the main crack. This approach is presented in detail by solving three different linear elastic fracture mechanics problems for different cases: two plane stress and a Reissner–Mindlin plate problems. The numerical results obtained with the two-scale G/XFEM are compared with the reference solutions from the analytical, numerical solution using standard G/XFEM method and ABAQUS as well, and from the literature.

Goal-oriented error estimation in the analysis of fluid flows with structural interactions

The Orthonormalized Generalized Finite Element Method–OGFEM: Efficient and stable reduction of approximation errors through multiple orthonormalized enriched basis functions

We deal with a posteriori error control of discontinuous Galerkin approximations for linear boundary value problems. The computational error is estimated in the framework of the Dual Weighted Residual method (DWR) for goal-oriented error estimation which requires to solve an additional (adjoint) problem. We focus on the control of the algebraic errors arising from iterative solutions of algebraic systems corresponding to both the primal and adjoint problems. Moreover, we present two different reconstruction techniques allowing an efficient evaluation of the error estimators. Finally, we propose a complex algorithm which controls discretization and algebraic errors and drives the adaptation of the mesh in the close to optimal manner with respect to the given quantity of interest.

Optimization problems constrained by complex dynamics can lead to computationally challenging problems especially when high accuracy and efficiency are required. We present an approach to adaptively control numerical errors in optimization problems approximated using the finite element method. The discrete adjoint equation serves as a key tool to efficiently compute both parameter sensitivities and goal-oriented error estimates at the same discretized levels. By using a recovery method for the error estimators, we avoid expensive higher order adjoint calculations. We nest the adaptivity of the mesh within the optimization algorithm, which is responsible for converging both the state and optimization algorithms and thereby allowing the reuse of state, parameters, and reduced Hessian in subsequent optimization iterations. Our approach is demonstrated on a parameter estimation problem for contamination transport in a contact tank reactor. Significant efficiency and accuracy improvements are realized in comparison to uniform grid refinement strategies and black-box optimization methods. A flexible and maintainable software interface was developed to provide access between the underlying linear algebra of a production simulator and advanced numerical algorithms such as optimization and error estimation.

When designing permanent magnet synchronous machines, the voltage waveforms and mechanical forces are important in many applications. Many analysts use the finite-element (FE) method to determine fields in the air gap of machines. These fields are then used to calculate vector forces, voltages, or other quantities of interest. Although the FE method has been used for decades, it requires significant computational resources, especially when transient analyses are required. This limits its application to a tool for analysis rather than a large design space, iterative optimization. To address this, an enhanced field reconstruction technique (EFRT) has been developed. Using the EFRT, the fields in the air gap of a machine are computed using a minimum number of FE evaluations. Using the results, a set of basis functions are established that enable the calculation of fields under arbitrary excitation and rotor position. The EFRT greatly reduces the computational effort required to compute the vector fields inside machines and compares favorably with the FE method and hardware results.

In this paper, nonconforming finite element methods (FEMs) are proposed for the constrained optimal control problems (OCPs) governed by the nonsmooth elliptic equations, in which the popular $$EQ_1^{rot}$$ element is employed to approximate the state and adjoint state, and the piecewise constant element is used to approximate the control. Firstly, the convergence and superconvergence properties for the nonsmooth elliptic equation are obtained by introducing an auxiliary problem. Secondly, the goal-oriented error estimates are obtained for the objective function through establishing the negative norm error estimate. Lastly, the methods are extended to some other well-known nonconforming elements.

In this paper we review the basic concepts to obtain a posteriori error estimates for the finite element solution of an elliptic linear model problem. We give the basic ideas to establish global error estimates for the energy norm as well as goal-oriented error estimates. While we show how these error estimation techniques are employed for our simple model problem, the emphasis of the paper is on assessing whether these procedures are ready for use in practical linear finite element analysis. We conclude that the actually practical error estimation techniques do not provide mathematically proven bounds on the error and need to be used with care. The more accurate estimation procedures also do not provide proven bounds that, in general, can be computed efficiently. We also briefly comment upon the state of error estimations in nonlinear and transient analyses and when mixed methods are used.

The workshop Nonstandard Finite Element Methods , organized by Susanne C. Brenner (Baton Rouge), Carsten Carstensen (Berlin) and Peter Monk (Newark) was held August 10 – 16, 2008. This meeting was well attended with over 40 participants with broad geographic representation. Although Courant is often credited with the discovery of finite elements [2, 3], the first practical use of such methods in engineering seems to date back to the work of Argyris and Clough et. al. (apparently Clough coined the name “finite elements”) in the 1950s [1, 4]. By the 1970s the finite element method was firmly established in engineering practice and the basic theory of conforming elements for elliptic problems was well understood [2]. Engineers and mathematicians have since expanded the use of finite elements to a wide variety of new applications and have improved the theoretical underpinnings of the method. In addition, efficient computational algorithms have been developed, such as the multigrid method, that allow for the application of finite elements to sophisticated three dimensional problems. Within current research, we can distinguish classical or standard finite element methods, and newer nonstandard finite elements The former include conforming elements applied to “standard” Galerkin discretizations of problems. Usually accuracy is obtained by mesh refinement. As the range of applications has increased, and to alleviate perceived shortcomings in standard conforming elements, numerical analysts and engineers have sought more general finite element methods based, for example, on mixed formulations, and non-conforming or even discontinuous Galerkin schemes. These methods may be superior to standard finite elements in particular applications because of enhancements to stability, robustness or conservation properties. We refer to them as non-standard finite element methods. This Oberwolfach workshop was devoted to non-standard finite element methods and their analyses. Inevitability such a snapshot of the subject is biased, but we have tried to include non-standard methods in the broadest sense including mixed, discontinuous Galerkin, generalized, partition of unity and mortar FEM methods. All these schemes have in common that stability and convergence is not obvious and requires mathematical analysis. This is even more true for developing fast solvers, a posteriori error estimation and adaptive mesh design. Through the week there were 29 presentations on various non-standard topics. These provided an overview of current research directions, new developments and open problems.