A posteriori error estimation techniques in practical finite element analysis

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

Advances in Adaptive Computational Methods in Mechanics

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

This paper studies the coupling between anisotropic mesh adaptation and goal-oriented error estimate. The former is very well suited to the control of the interpolation error. It is generally interpreted as a local geometric error estimate. On the contrary, the latter is preferred when studying approximation errors for PDEs. It generally involves non local error contributions. Consequently, a full and strong coupling between both is hard to achieve due to this apparent incompatibility. This paper shows how to achieve this coupling. This is done in three steps. First, a new ap riorierror estimate is proved in a formal framework adapted to goal-oriented mesh adaptation for output functionals. Second, the error estimate is applied to the set of steady compressible Euler equations which are solved by a stabilized Galerkin finite element discretization. A goal-oriented error estimation is derived. Third, rewritten in the continuous mesh framework, the previous estimate is minimized on the set of continuous meshes thanks to a calculus of variations. The optimal mesh is then derive. 3D examples of steady flows around supersonic and transsonic aircrafts are presented to validate the current approach and to demonstrate its efficiency.

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.

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.

Over the last decades the finite element method, which was introduced by engineers in the 1960s, has become the perhaps most important numerical method for partial differential equations, particularly for equations of elliptic and parabolic types. This method is based on the variational form of the boundary value problem and approximates the exact solution by a piecewise polynomial function. It is more easily adapted to the geometry of the underlying domain than the finite difference method, and for symmetric positive definite elliptic problems it reduces to a finite linear system with a symmetric positive definite matrix. We first introduce this method in Sect. 5.1 for the case of a two-point boundary value problem and show a number of error estimates. In Sect. 5.2 we then formulate the method for a two-dimensional model problem. Here the piecewise polynomial approximations are defined on triangulations of the spatial domain, and in the following Sect. 5.3 we study such approximation in more detail. In Sect. 5.4 we show basic error estimates for the finite element method for the model problem, using piecewise linear approximating functions. All error bounds derived up to this point contain a norm of the unknown exact solution and are therefore often referred to as a priori error estimates. In Sect. 5.5 we show a so-called a posteriori error estimate in which the error bound is expressed in terms of the data of the problem and the computed solution. In Sect. 5.6 we analyze the effect of numerical integration, which is often used when the finite element equations are assembled in a computer program. In Sect. 5.7 we briefly describe a so-called mixed finite element method.KeywordsFinite Element MethodElliptic EquationQuadrature FormulaInterpolation ErrorPosteriori Error EstimateThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

We introduce a goal-oriented strategy for multiscale computations performed using the Multiscale Finite Element Method (MsFEM). In a previous work, we have shown how to use, in the MsFEM framework, the concept of Constitutive Relation Error (CRE) to obtain a guaranteed and fully computable a posteriori error estimate in the energy norm (as well as error indicators on various error sources). Here, the CRE concept is coupled with the solution of an adjoint problem to control the error and drive an adaptive procedure with respect to a given output of interest. Furthermore, a local and non-intrusive enrichment technique is proposed to enhance the accuracy of error bounds. The overall strategy, which is fully automatic and robust, enables to reach an appropriate trade-off between certified reliability and computational cost in the MsFEM context. The performances of the proposed method are investigated on several illustrative numerical test cases. In particular, the error estimation is observed to be very accurate, yielding a very efficient adaptive procedure.

Many rate constant measurements, including some “direct” measurements, involve fitting a complex reaction mechanism to experimental data. Two techniques for estimating the error in such measurements were compared. In the first technique, local first‐order elementary sensitivities were used to rapidly estimate the sensitivity of the fitted rate constants to the remaining mechanism parameters. Our group and others have used this technique for error estimation and experimental design. However, the nonlinearity and strong coupling found in reaction mechanisms make verification against globally valid results desirable. Here, the local results were compared with analogous importance‐sampled Monte Carlo calculations in which the parameter values were distributed according to their uncertainties. Two of our published rate measurements were examined. The local uncertainty estimates were compared with Monte Carlo confidence intervals. The local sensitivity coefficients were compared with coefficients from first and second‐degree polynomial regressions over the whole parameter space. The first‐order uncertainty estimates were found to be sufficiently accurate for experimental design, but were subject to error in the presence of higher order sensitivities. In addition, global uncertainty estimates were found to narrow when the quality of the fit was used to weight the randomly distributed points. For final results, the global technique was found to provide efficient, accurate values without the assumptions inherent in the local analysis. The rigorous error estimates derived in this way were used to address literature criticism of one of the studies discussed here. Given its efficiency and the variety of problems it can detect, the global technique could also be used to check local results during the experimental design phase. The global routine, coded using SENKIN, can easily be extended to different types of data, and therefore can serve as a valuable tool for assessing error in rate constants determined using complex mechanisms. © 2001 John Wiley & Sons, Inc. Int J Chem Kinet 33: 784–802, 2001

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

Data mining and probabilistic models for error estimate analysis of finite element method

Error estimates for finite element solutions of elliptic boundary value problems

New a posteriori error estimator and adaptive mesh refinement strategy for 2-D crack problems

A posteriori error estimation for the method of lumped masses applied to second-order hyperbolic problems

It is often necessary in optical analysis software to trace millions of rays in order to determine flux transfer, color chromaticity, and the distributions of intensity, irradiance, or radiance onto one or more targets. The key question of "have I traced enough rays?" cannot be answered unless the statistical error in the final simulated result is below some measurement criteria; for instance, if you are to determine irradiance uniformity to less than 1%, having a 2% statistical error across the target will wash out what you are trying to analyze. Some optical analysis software packages do not provide error estimation methods, while others use error estimation algorithms having assumptions that are not valid for all cases. This paper describes how subdividing and recombining raytraces provides a robust method for estimating error. We will show that this error estimation technique can be used with most optical analysis packages and we will compare it with algorithms employed currently. Example systems will be analyzed and presented.