Goal-oriented Mesh Adaptation Research Articles

Time accurate anisotropic goal-oriented mesh adaptation for unsteady flows

We present a new algorithm for combining an anisotropic goal-oriented error estimate with the mesh adaptation fixed point method for unsteady problems. The minimization of the error on a functional provides both the density and the anisotropy (stretching) of the optimal mesh. They are expressed in terms of state and adjoint. This method is used for specifying the mesh for a time sub-interval. A global fixed point iterates the re-evaluation of meshes and states over the whole time interval until convergence of the space–time mesh. Applications to unsteady blast-wave and acoustic-wave Euler flows are presented.

Goal-Oriented Mesh Adaptivity for Fluid-Structure Interaction with Application to Heart-Valve Settings

Goal-Oriented Mesh Adaptivity for Fluid-Structure Interaction with Application to Heart-Valve SettingsWe apply a fluid-structure interaction method to simulate prototypical dynamics of the aortic heart-valve. Our method of choice is based on a monolithic coupling scheme for fluid-structure interactions in which the fluid equations are rewritten in the ‘arbitrary Lagrangian Eulerian’ (ALE) framework. To prevent the backflow of structure waves because of their hyperbolic nature, a damped structure equation is solved on an artificial layer that is used to prolongate the computational domain. The increased computational cost in the presence of the artificial layer is resolved by using local mesh adaption. In particular, heuristic mesh refinement techniques are compared to rigorous goal-oriented mesh adaption with the dual weighted residual (DWR) method. A version of this method is developed for stationary settings. For the nonstationary test cases the indicators are obtained by a heuristic error estimator, which has a good performance for the measurement of wall stresses. The results for prototypical problems demonstrate that heart-valve dynamics can be treated with our proposed concepts and that the DWR method performs best with respect to a certain target functional.

Goal-Oriented h-Adaptivity for the Multigroup SPN Equations

We investigate application of goal-oriented mesh adaptivity to the SPN multigroup equations. This technique utilizes knowledge of the computational goal and combines it with mesh adaptivity to accurately and rapidly compute quantities of interest. Specifically, the local error is weighted by the importance of a given cell toward the computational goal, resulting in appropriate goal-oriented error estimates. Even though this approach requires the solution of an adjoint (dual) problem, driven by a specific source term for a given quantity of interest, the work reported here clearly shows the benefits of such a method.We demonstrate the level of accuracy this method can achieve using two-dimensional and three-dimensional numerical test cases for one-group and two-group models and compare results with more traditional mesh refinement and uniformly refined meshes. The test cases consider situations in which the radiative flux of a source is shielded and are designed to prototypically explore the range of conditions under which our methods improve on other refinement algorithms. In particular, they model strong contrasts in material properties, a situation ubiquitous in nuclear engineering.

Fully anisotropic goal-oriented mesh adaptation for 3D steady Euler equations

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 in three steps. First, a new a priori error estimate is proved in a formal framework adapted to goal-oriented mesh adaptation for output functionals. This estimate is based on a careful analysis of the contributions of the implicit error and of the interpolation error. 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. It involves the interpolation error of the Euler fluxes weighted by the gradient of the adjoint state associated with the observed functional. 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 continuous mesh is then derived analytically. Thus, it can be used as a metric tensor field to drive the mesh adaptation. From a numerical point of view, this method is completely automatic, intrinsically anisotropic, and does not depend on any a priori choice of variables to perform the adaptation. 3D examples of steady flows around supersonic and transsonic jets are presented to validate the current approach and to demonstrate its efficiency.

Goal-oriented mesh adaptation for flux-limited approximations to steady hyperbolic problems

The development of adaptive numerical schemes for steady transport equations is addressed. A goal-oriented error estimator is presented and used as a refinement criterion for conforming mesh adaptation. The error in the value of a linear target functional is measured in terms of weighted residuals that depend on the solutions to the primal and dual problems. The Galerkin orthogonality error is taken into account and found to be important whenever flux or slope limiters are activated to enforce monotonicity constraints. The localization of global errors is performed using a natural decomposition of the involved weights into nodal contributions. A nodal generation function is employed in a hierarchical mesh adaptation procedure which makes each refinement step readily reversible. The developed simulation tools are applied to a linear convection problem in two space dimensions.

Goal‐oriented mesh adaptation for FE‐vibration analyses of shell‐like structures

AbstractThe well–known semidiscrete approach in structural dynamics consists of two steps. First the discretization of the spatial domain using finite elements is performed. In the second step the numerical integration of the resulting system of ordinary differential equations is executed. The numerical schemes in both discretization steps contribute to the total discretization error. In this contribution the focus is on the spatial discretization error and on efficient strategies for error assessment and error reduction by goal–oriented mesh adaptation schemes. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)