Abstract
In this article, we present a 3D adaptive method for thermoelastic problems based on goal-oriented error estimation where the error is measured with respect to a pointwise quantity of interest. We developed a method for a posteriori error estimation and mesh adaptation based on dual weighted residual (DWR) method relying on the duality principles and consisting of an adjoint problem solution. Here, we consider the application of the derived estimator and mesh refinement to two-/three dimensional (2D/3D) thermo-mechanical multifield problems. In this study, the goal is considered to be given by singular pointwise functions, such as the point value or point value derivative at a specific point of interest (PoI). An adaptive algorithm has been adopted to refine the mesh to minimize the goal in the quantity of interest.
The mesh adaptivity procedure based on the DWR method is performed by adaptive local h-refinement/coarsening with allowed hanging nodes. According to the proposed DWR method, the error contribution of each element is evaluated. In the refinement process, the contribution of each element to the goal error is considered as the mesh refinement criterion.
In this study, we substantiate the accuracy and performance of this method by several numerical examples with available analytical solutions. Here, 2D and 3D problems under thermo-mechanical loadings are considered as benchmark problems. To show how accurately the derived estimator captures the exact error in the evaluation of the pointwise quantity of interest, in all examples, considering the analytical solutions, the goal error effectivity index as a standard measure of the quality of an estimator is calculated. Moreover, in order to demonstrate the efficiency of the proposed method and show the optimal behavior of the employed refinement method, the results of different conventional error estimators and refinement techniques (e.g., global uniform refinement, Kelly, and weighted Kelly techniques) are used for comparison.
•The DWR estimated goal error shows a monotonic decrease by increasing the total number of the DOFs, which indicates the stable convergence of the refinement process.•Comparing the error in the quantity of interest estimated by dual weighted residual (DWR) method with the exact goal error in examples with analytical solutions shows a very good accordance even for singular pointwise quantities of interest in 2D and 3D problems.•In the implemented method after few initial coarse meshes, goal error effectivity index lies in the acceptable range of 0.9–1.0 and by further refinements approaches to the ideal value of 1.•The comparison among the proposed method, global uniform, Kelly, and weighted Kelly (W-Kelly) refinement techniques shows that among the selected error estimation and refinement techniques, the employed DWR based mesh adaptation has the highest convergence rate and is the most optimum and efficient one.