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

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

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

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.

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.

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

A posteriori error estimation and adaptive mesh refinement in chemoplastic analyses of shotcrete tunnel linings

Mesh quality strongly affects the solution accuracy in electromagnetic finite-element analysis. Hence, the realization of adequate mesh generation becomes a very important task. Several adaptive meshing methods for automatic adjustments of the mesh density in accordance with the shape and complexity of the analyzed problem have been proposed. However, the most of them are not enough robust, some are quite laborious and could not be universally used for adaptive meshing of complex analysis models. In this paper, a new adaptive mesh refinement method based on magnetic field conservation at the border between finite elements is proposed. The proposed error estimation method provides easy mesh refinements, and generates smaller element within regions with large curvature of the magnetic flux lines. The proposed adaptive mesh refinement method based on non-conforming edge finite elements, which could avoid generation of flat or ill-shaped elements, was applied to a simple magnetostatic permanent magnet model. To confirm the validity and accuracy, the obtained results were compared with those obtained by means of the Zienkiewicz-Zhu (ZZ) error estimator. The results show that the computational error using the proposed method was reduced down to 1% compared with that of the ZZ method, which yields error of 8.6%, for the same model.

Towards adaptive kinetic-fluid simulations of weakly ionized plasmas

<p>We present an accurate and fast finite element solver for global electromagnetic induction forward modeling problems in spherical Earth. We solve for the electric field equation using the first-order Nedelec elements. The magnetic field is then obtained by computing the curl of the electric field. The computational domain composed of the air space and the conductive Earth is discretized by disjoint unstructured tetrahedral elements. To improve the accuracy with an optimal number of unknowns, we propose a simple two-step goal-oriented adaptive mesh refinement (AMR) strategy. In the first step, an h-type AMR procedure is used to obtain an optimal finite element mesh. The mesh refinement is accomplished by bisection to generate a set of hierarchal tetrahedral meshes. The AMR procedure is driven by a goal-oriented error estimator, which is based on face jumps of normal components of current density. In the second step, we adopt the high-order finite elements at the last iteration to update the accuracy of final numerical solutions. This simple two-step adaptive strategy takes advantage of both h-type AMR and high-order basis functions, and in the meanwhile, it is also computationally economical. To improve efficiency, the solver is parallelized with an MPI-based domain decomposition technique. The sophisticated auxiliary space preconditioned linear solver is adopted to efficiently solve the linear system of equations. This new solver is verified on both semi-analytic and realistic 3-D Earth models. It can be used as a core to derive the inversion of global electromagnetic induction data.</p><p> </p>

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.

SummaryThis article introduces a novel error estimator for the proper generalized decomposition (PGD) approximation of parametrized equations. The estimator is intrinsically random: it builds on concentration inequalities of Gaussian maps and an adjoint problem with random right‐hand side, which we approximate using the PGD. The effectivity of this randomized error estimator can be arbitrarily close to unity with high probability, allowing the estimation of the error with respect to any user‐defined norm as well as the error in some quantity of interest. The performance of the error estimator is demonstrated and compared with some existing error estimators for the PGD for a parametrized time‐harmonic elastodynamics problem and the parametrized equations of linear elasticity with a high‐dimensional parameter space.

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.

Mesh adaptation is essential for accurate computational fluid dynamics (CFD) simulations, especially when computational resources are limited.In simulating the advection-diffusion equation using the finite volume method (FVM), mesh adaptation is often necessary to accurately capture sharp gradients and complex flow features.Although the metric-based mesh adaptation used in this study is highly effective with true errors, these are typically unavailable, requiring the use of error estimators.However, adjoint-based error estimators, though effective, are computationally expensive as they require solving the larger adjoint system.To address this, an inexpensive error estimator based on the gradient of the numerical solution and the embedded method is proposed.Specifically, the central difference and upwind difference schemes are utilized for error estimation within this framework.Since errors are predominantly influenced by sharp gradients in the scalar boundary layer, a target functional based on the product of the embedded error and the solution gradient to compute the error estimator is proposed.Consequently, these proposed error estimators generate meshes closely resembling those produced using true error values, effectively resolving the scalar boundary layer.As a result, the L 2 and L norms of the error typically decrease with successive adaptation cycles.Furthermore, the L 1 , L 2 , and L norms of error per element also reduce with successive adaptation cycles.In conclusion, the proposed error estimator facilitates mesh adaptation without needing true errors or computationally expensive adjoint-based error estimators, thus ensuring accurate numerical solutions, especially in scenarios with sharp boundary layers.

The intensity variation in a scanning electron microscope is a complex function of sample topography and composition. Measurement accuracy is improved when an explicit accounting for the relationship between signal and measurand is made. Because the determinants of the signal are many, the theoretical understanding usually takes the form of a simulator. For samples with nonconducting regions that charge, one phase of the simulation is finite element analysis to compute the electric field. The size of the finite element mesh, and consequently computation time, can be reduced through the use of adaptive mesh refinement. We present a new a posteriori local error estimator and adaptive mesh refinement algorithm for the scanning electron microscope simulation. This error estimate is designed to minimize the error in the electron trajectories, rather than the energy norm of the error that traditional error estimators minimize. Using a test problem with a known exact solution, we show that the adaptive mesh can achieve the same error in electron trajectories as a carefully designed hand-graded mesh while using 3.5 times fewer vertices and 2.25 times less computation time.