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.     

Goal‐oriented adaptive refinement for phase field modeling with finite elements

Phase field modeling is very often performed with the finite‐difference method for equally spaced grids. Typically its solutions are highly non‐homogenous; and, therefore, non‐equally spaced grids with dense meshes at interfaces between different phases and coarse meshes in homogenous regions would be more advantageous with respect to both, efficiency and reliability of the numerical solutions. To this end, in the present work, an adaptive strategy with finite elements for phase field modeling is adopted, where the time step and the grid size are selected on the basis of goal‐oriented error estimation. In order to account for nonlinearity of the variational equations, we introduce a secant form for the dual problem, which for practical purposes is approximated by a tangent form. In a numerical example, we investigate transformation and retransformation for a two‐phase system in a square region subjected to thermal loading. Copyright © 2013 John Wiley & Sons, Ltd.     

A combined approach for goal‐oriented error estimation and adaptivity using operator‐customized finite element wavelets

We describe how wavelets constructed out of finite element interpolation functions provide a simple and convenient mechanism for both goal‐oriented error estimation and adaptivity in finite element analysis. This is done by posing an adaptive refinement problem as one of compactly representing a signal (the solution to the governing partial differential equation) in a multiresolution basis. To compress the solution in an efficient manner, we first approximately compute the details to be added to the solution on a coarse mesh in order to obtain the solution on a finer mesh (the estimation step) and then compute exactly the coefficients corresponding to only those basis functions contributing significantly to a functional of interest (the adaptation step). In this sense, therefore, the proposed approach is unified, since unlike many contemporary error estimation and adaptive refinement methods, the basis functions used for error estimation are the same as those used for adaptive refinement. We illustrate the application of the proposed technique for goal‐oriented error estimation and adaptivity for second and fourth‐order linear, elliptic PDEs and demonstrate its advantages over existing methods. Copyright © 2005 John Wiley & Sons, Ltd.     

Dispersion analysis and error estimation of Galerkin finite element methods for the Helmholtz equation

When applying numerical methods for the computation of stationary waves from the Helmholtz equation, one obtains ‘numerical waves’ that are dispersive also in non-dispersive media. The numerical wave displays a phase velocity that depends on the parameter k of the Helmholtz equation. In dispersion analysis, the phase difference between the exact and the numerical solutions is investigated. In this paper, the authors' recent result on the phase difference for one-dimensional problems is numerically evaluated and discussed in the context of other work directed to this topic. It is then shown that previous error estimates in H¹-norm are of nondispersive character but hold for medium or high wavenumber on extremely refined mesh only. On the other hand, recently proven error estimates for constant resolution contain a pollution term. With certain assumptions on the exact solution, this term is of the order of the phase difference. Thus a link is established between the results of dispersion analysis and the results of numerical analysis. Throughout the paper, the presentation and discussion of theoretical results is accompanied by numerical evaluation of several model problems. Special attention is given to the performance of the Galerkin method with a higher order of polynomial approximation p(h-p-version).     

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.     

A residual‐based finite element method for the Helmholtz equation

A new residual‐based finite element method for the scalar Helmholtz equation is developed. This method is obtained from the Galerkin approximation by appending terms that are proportional to residuals on element interiors and inter‐element boundaries. The inclusion of residuals on inter‐element boundaries distinguishes this method from the well‐known Galerkin least‐squares method and is crucial to the resulting accuracy of this method. In two dimensions and for regular bilinear quadrilateral finite elements, it is shown via a dispersion analysis that this method has minimal phase error. Numerical experiments are conducted to verify this claim as well as test and compare the performance of this method on unstructured meshes with other methods. It is found that even for unstructured meshes this method retains a high level of accuracy. Copyright © 2000 John Wiley & Sons, Ltd.     

A goal‐oriented hp‐adaptive discontinuous Galerkin approach for biharmonic problems

A goal‐oriented algorithm is developed and applied for hp‐adaptive approximations given by the discontinuous Galerkin finite element method for the biharmonic equation. The methodology is based on the dual problem associated with the target functional. We consider three error estimators and analyse their properties as basic tools for the design of the hp‐adaptive algorithm. To improve adaptation, the combination of two different error estimators is used, each one at its best efficiency, to guide the tasks of where and how to adapt the approximation spaces. The performance of the resulting hp‐adaptive schemes is illustrated by numerical experiments for two benchmark problems. Copyright © 2013 John Wiley & Sons, Ltd.     

Enhanced goal‐oriented error assessment and computational strategies in adaptive reduced basis solver for stochastic problems

This work focuses on providing accurate low‐cost approximations of stochastic finite elements simulations in the framework of linear elasticity. In a previous work, an adaptive strategy was introduced as an improved Monte‐Carlo method for multi‐dimensional large stochastic problems. We provide here a complete analysis of the method including a new enhanced goal‐oriented error estimator and estimates of CPU (computational processing unit) cost gain. Technical insights of these two topics are presented in details, and numerical examples show the interest of these new developments. Copyright © 2016 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.     

One‐dimensional dispersion analysis for the element‐free Galerkin method for the Helmholtz equation

The standard finite element method (FEM) is unreliable to compute approximate solutions of the Helmholtz equation for high wave numbers due to the dispersion, unless highly refined meshes are used, leading to unacceptable resolution times. The paper presents an application of the element‐free Galerkin method (EFG) and focuses on the dispersion analysis in one dimension. It shows that, if the basis contains the solution of the homogenized Helmholtz equation, it is possible to eliminate the dispersion in a very natural way while it is not the case for the finite element methods. For the general case, it also shows that it is possible to choose the parameters of the method in order to minimize the dispersion. Finally, theoretical developments are validated by numerical experiments showing that, for the same distribution of nodes, the element‐free Galerkin method solution is much more accurate than the finite element one. Copyright © 2000 John Wiley & Sons, Ltd.     

Forward-in-time goal-oriented adaptivity

In goal-oriented adaptive algorithms for partial differential equations, we adapt the finite element mesh to reduce the error of the solution in some quantity of interest. In time-dependent problems, this adaptive algorithm involves solving a dual problem that runs backward in time. This process is, in general, computationally expensive in terms of memory storage. In this work, we define a pseudo-dual problem that runs forward in time. We also describe a forward-in-time adaptive algorithm that works for some specific problems. Although it is not possible to define a general dual problem running forwards in time that provides information about future states, we provide numerical evidence via one-dimensional problems in space to illustrate the efficiency of our algorithm as well as its limitations. Finally, we propose a hybrid algorithm that employs the classical backward-in-time dual problem once and then performs the adaptive process forwards in time.     

Dual‐based adaptive FEM for inelastic problems with standard FE implementations

Goal‐oriented error estimation allows to refine meshes in space and time with respect to arbitrary quantities. The required dual problems that need to be solved usually require weak formulations and the Galerkin method in space and time to be established. Unfortunately, this does not obviously leads to structures of standard finite element implementations for solid mechanics. These are characterized by a combination of variables at nodes (e.g. displacements) and at integration points (e.g. internal variables) and are solved with a two‐level Newton method because of local uncoupled and global coupled equations. Therefore, we propose an approach to approximate the dual problem while maintaining these structures. The primal and the dual problems are derived from a multifield formulation. Discretization in time and space with appropriate shape functions and rearrangement yields the desired result. Details on practical implementation as well as applications to elasto‐plasticity are given. Numerical examples demonstrate the effectiveness of the procedure. Copyright © 2015 John Wiley & Sons, Ltd.     

Bounds for quantities of interest and adaptivity in the element‐free Galerkin method

A novel approach to implicit residual‐type error estimation in mesh‐free methods and an adaptive refinement strategy are presented. This allows computing upper and lower bounds of the error in energy norm with the ultimate goal of obtaining bounds for outputs of interest. The proposed approach precludes the main drawbacks of standard residual‐type estimators circumventing the need of flux‐equilibration and resulting in a simple implementation that avoids integrals on edges/sides of a domain decomposition (mesh). This is especially interesting for mesh‐free methods. The adaptive strategy proposed leads to a fast convergence of the bounds to the desired precision. Copyright © 2008 John Wiley & Sons, Ltd.     

A variational multiscale method with subscales on the element boundaries for the Helmholtz equation

In this paper, we apply the variational multiscale method with subgrid scales on the element boundaries to the problem of solving the Helmholtz equation with low‐order finite elements. The expression for the subscales is obtained by imposing the continuity of fluxes across the interelement boundaries. The stabilization parameter is determined by performing a dispersion analysis, yielding the optimal values for the different discretizations and finite element mesh configurations. The performance of the method is compared with that of the standard Galerkin method and the classical Galerkin least‐squares method with very satisfactory results. Some numerical examples illustrate the behavior of the method. Copyright © 2012 John Wiley & Sons, Ltd.     

A goal‐oriented hp‐adaptive finite element method with electromagnetic applications. Part I: electrostatics

We describe the development and application of a finite element (FE) self‐adaptive hp goal‐oriented algorithm for elliptic problems. The algorithm delivers (without any user interaction) a sequence of optimal hp‐grids. This sequence of grids minimizes the error of a prescribed quantity of interest with respect to the problem size. The refinement strategy is an extension of a fully automatic, energy‐norm based, hp‐adaptive algorithm. We illustrate the efficiency of the method with 2D numerical results. Among other problems, we apply the goal‐oriented hp‐adaptive strategy to simulate direct current (DC) resistivity logging instruments (including through casing resistivity tools) in a borehole environment and for the assessment of rock formation properties. Copyright © 2005 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.     

A two-level finite element method and its application to the Helmholtz equation

A two-level finite element method is introduced and its application to the Helmholtz equation is considered. The method retains the desirable features of the Galerkin method enriched with residual-free bubbles, while it is not limited to discretizations using elements with simple geometry. The method can be applied to other equations and to irregular-shaped domains. © 1998 John Wiley & Sons, Ltd.     

Influence functions and goal‐oriented error estimation for finite element analysis of shell structures

In this paper, we first present a consistent procedure to establish influence functions for the finite element analysis of shell structures, where the influence function can be for any linear quantity of engineering interest. We then design some goal‐oriented error measures that take into account the cancellation effect of errors over the domain to overcome the issue of over‐estimation. These error measures include the error due to the approximation in the geometry of the shell structure. In the calculation of the influence functions we also consider the asymptotic behaviour of shells as the thickness approaches zero. Although our procedures are general and can be applied to any shell formulation, we focus on MITC finite element shell discretizations. In our numerical results, influence functions are shown for some shell test problems, and the proposed goal‐oriented error estimation procedure shows good effectivity indices. Copyright © 2005 John Wiley & Sons, Ltd.     

Incomplete factorization‐based preconditionings for solving the Helmholtz equation

Preconditioning techniques based on incomplete factorization of matrices are investigated, to solve highly indefinite complex‐symmetric linear systems. A novel preconditioning is introduced. The real part of the matrix is made positive definite, or less indefinite, by adding properly defined perturbations to the diagonal entries, while the imaginary part is unaltered. The resulting preconditioning matrix, which is obtained by applying standard methods to the perturbed complex matrix, turns out to perform significantly better than classical incomplete factorization schemes. For realistic values of the GMRES restart parameter, spectacular reduction of iteration counts is observed. A theoretical spectral analysis is provided, in which the spectrum of the preconditioner applied to indefinite matrix is related to the spectrum of the same preconditioner applied to a Stieltjes matrix extracted from the indefinite matrix. Results of numerical experiments are reported, which display the efficiency of the new preconditioning. Copyright © 2001 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.