Error analysis for quadtree-type mesh coarsening algorithms adapted to pixelized heterogeneous microstructures

Abstract

Pixel-and voxel-based representations of microstructures obtained from tomographic imaging methods is an established standard in computational materials science. The corresponding highly resolved, uniform discretitization in numerical analysis is adequate to accurately describe the geometry of interfaces and defects in microstructures and, therefore, to capture the physical processes in these regions of interest. For the defect-free interior of phases and grains however, the high resolution is in view of only weakly varying field properties not necessary such that mesh-coarsening in these regions can improve efficiency without severe losses of accuracy in simulations. The present work proposes two different variants of adaptive, quadtree-based meshcoarsening algorithms applied to pixelized images that serves the purpose of a preprocessor for consecutive finite element analyses, here, in the context of numerical homogenization. Error analysis is carried out on the microscale by error estimation which itself is assessed by true error computation. A modified stress recovery scheme for a superconvergent error estimator is proposed which overcomes the deficits of the standard recovery scheme for nodal stress computation in cases of interfaces with stiffness jump. By virtue of error analysis the improved efficiency by the reduction of unknowns is put into relation to the increase of the discretization error. This quantitative analysis sets a rational basis for decisions on favorable meshes having the best trade-off between accuracy and efficiency as will be underpinned by various examples.

Decision Support for Removing Fractured Endodontic Instruments: A Patient-Specific Approach

“…Recent technologies such as micro CBCT [40] and the use of an intraoral scanner avoiding X-ray artifacts [41] could improve future simulations. In the present study, the mesh error was considered acceptable [27,28]. However, FEA results should also be carefully interpreted due to the technical impossibility to identify patientspecific parameters such as force intensity [42] or for ligament modeling [43].…”

“…Regarding stress distribution, high stresses around the instrument were found in the initial model, high stresses around the perforation were found in the enlargement models, and high stresses on the resected surface were found in apicoectomy models (Figure 3). The error indicator was considered as acceptable [27,28] for all models, indicating that this method provides valuable models for FEA. The apicoectomy models presented a lower fracture criterion than enlargement models and the model of the initial state of the patient.…”

From image data towards microstructure information – Accuracy analysis at the digital core of materials

“…FE‐HMM and FE 2 were developed independently, the latter in [21, 25, 31, 35, 39, 54]. The computation of the micro errors is carried out by projection of the approximate solution to the Gauss points of a reference mesh () with a consecutive numerical quadrature for approximating the integral element error as detailed in [22, 23]. The estimation of the discretization error is described in Section 5 with a focus on peculiarities at interfaces, which have undergone resolution coarsening as detailed in Section 4.…”

“…This type of microstructure‐informed adaptivity is frequently used for multiphase materials [33, 34, 37, 38, 47] for application in homogenization and fracture, in the context of the Scaled Boundary FEM [26, 48] and many more. But only very recently a rigorous accuracy analysis based on error estimation was presented in [23].…”

“…Suchlike preprocessing typically provides a favorable balance of accuracy and efficiency. The obtained, adaptively refined mesh can then be assessed by a posteriori error estimation for various loading conditions [23].…”