Search results for: least-squares polynomial approximation

Authors: Adrian W. Egger, Savvas P. Triantafyllou, Eleni N. Chatzi

Abstract:

The scaled boundary finite element method (SBFEM) is a semi-analytical numerical method, which introduces a scaling center in each element’s domain, thus transitioning from a Cartesian reference frame to one resembling polar coordinates. Consequently, an analytical solution is achieved in radial direction, implying that only the boundary need be discretized. The only limitation imposed on the resulting polygonal elements is that they remain star-convex. Further arbitrary p- or h-refinement may be applied locally in a mesh. The polygonal nature of SBFEM elements has been exploited in quadtree meshes to alleviate all issues conventionally associated with hanging nodes. Furthermore, since in 2D this results in only 16 possible cell configurations, these are precomputed in order to accelerate the forward analysis significantly. Any cells, which are clipped to accommodate the domain geometry, must be computed conventionally. However, since SBFEM permits polygonal elements, significantly coarser meshes at comparable accuracy levels are obtained when compared with conventional quadtree analysis, further increasing the computational efficiency of this scheme. The generalized stress intensity factors (gSIFs) are computed by exploiting the semi-analytical solution in radial direction. This is initiated by placing the scaling center of the element containing the crack at the crack tip. Taking an analytical limit of this element’s stress field as it approaches the crack tip, delivers an expression for the singular stress field. By applying the problem specific boundary conditions, the geometry correction factor is obtained, and the gSIFs are then evaluated based on their formal definition. Since the SBFEM solution is constructed as a power series, not unlike mode superposition in FEM, the two modes contributing to the singular response of the element can be easily identified in post-processing. Compared to the extended finite element method (XFEM) this approach is highly convenient, since neither enrichment terms nor a priori knowledge of the singularity is required. Computation of the gSIFs by SBFEM permits exceptional accuracy, however, when combined with hybrid quadtrees employing linear elements, this does not always hold. Nevertheless, it has been shown that crack propagation schemes are highly effective even given very coarse discretization since they only rely on the ratio of mode one to mode two gSIFs. The absolute values of the gSIFs may still be subject to large errors. Hence, we propose a post-processing scheme, which minimizes the error resulting from the approximation space of the cracked element, thus limiting the error in the gSIFs to the discretization error of the quadtree mesh. This is achieved by h- and/or p-refinement of the cracked element, which elevates the amount of modes present in the solution. The resulting numerical description of the element is highly accurate, with the main error source now stemming from its boundary displacement solution. Numerical examples show that this post-processing procedure can significantly improve the accuracy of the computed gSIFs with negligible computational cost even on coarse meshes resulting from hybrid quadtrees.

Keywords: linear elastic fracture mechanics, generalized stress intensity factors, scaled finite element method, hybrid quadtrees

Procedia PDF Downloads 146

Authors: Ece A. Irmak, Amdulla O. Mekhrabov, M. Vedat Akdeniz

Abstract:

There is a growing interest in the modeling and simulation of magnetic nanoalloys by various computational methods. Magnetic crystalline/amorphous nanoparticles (NP) are interesting materials from both the applied and fundamental points of view, as their properties differ from those of bulk materials and are essential for advanced applications such as high-performance permanent magnets, high-density magnetic recording media, drug carriers, sensors in biomedical technology, etc. As an important magnetic material, Fe-Ni based nanoalloys have promising applications in the chemical industry (catalysis, battery), aerospace and stealth industry (radar absorbing material, jet engine alloys), magnetic biomedical applications (drug delivery, magnetic resonance imaging, biosensor) and computer hardware industry (data storage). The physical and chemical properties of the nanoalloys depend not only on the particle or crystallite size but also on composition and atomic ordering. Therefore, computer modeling is an essential tool to predict structural, electronic, magnetic and optical behavior at atomistic levels and consequently reduce the time for designing and development of new materials with novel/enhanced properties. Although first-principles quantum mechanical methods provide the most accurate results, they require huge computational effort to solve the Schrodinger equation for only a few tens of atoms. On the other hand, molecular dynamics method with appropriate empirical or semi-empirical inter-atomic potentials can give accurate results for the static and dynamic properties of larger systems in a short span of time. In this study, structural evolutions, magnetic and electronic properties of Fe-Ni based nanoalloys have been studied by using molecular dynamics (MD) method in Large-scale Atomic/Molecular Massively Parallel Simulator (LAMMPS) and Density Functional Theory (DFT) in the Vienna Ab initio Simulation Package (VASP). The effects of particle size (in 2-10 nm particle size range) and temperature (300-1500 K) on stability and structural evolutions of amorphous and crystalline Fe-Ni bulk/nanoalloys have been investigated by combining molecular dynamic (MD) simulation method with Embedded Atom Model (EAM). EAM is applicable for the Fe-Ni based bimetallic systems because it considers both the pairwise interatomic interaction potentials and electron densities. Structural evolution of Fe-Ni bulk and nanoparticles (NPs) have been studied by calculation of radial distribution functions (RDF), interatomic distances, coordination number, core-to-surface concentration profiles as well as Voronoi analysis and surface energy dependences on temperature and particle size. Moreover, spin-polarized DFT calculations were performed by using a plane-wave basis set with generalized gradient approximation (GGA) exchange and correlation effects in the VASP-MedeA package to predict magnetic and electronic properties of the Fe-Ni based alloys in bulk and nanostructured phases. The result of theoretical modeling and simulations for the structural evolutions, magnetic and electronic properties of Fe-Ni based nanostructured alloys were compared with experimental and other theoretical results published in the literature.

Keywords: density functional theory, embedded atom model, Fe-Ni systems, molecular dynamics, nanoalloys

Procedia PDF Downloads 243

Authors: Artur Tyliszczak, Ewa Szymanek, Maciej Marek

Abstract:

Flow through granular materials is important to a vast array of industries, for instance in construction industry where granular layers are used for bulkheads and isolators, in chemical engineering and catalytic reactors where large surfaces of packed granular beds intensify chemical reactions, or in energy production systems, where granulates are promising materials for heat storage and heat transfer media. Despite the common usage of granulates and extensive research performed in this field, phenomena occurring between granular solid elements or between solids and fluid are still not fully understood. In the present work we analyze the heat exchange process between the flowing medium (gas, liquid) and solid material inside the granular layers. We consider them as a composite of isolated solid elements and inter-granular spaces in which a gas or liquid can flow. The structure of the layer is controlled by shapes of particular granular elements (e.g., spheres, cylinders, cubes, Raschig rings), its spatial distribution or effective characteristic dimension (total volume or surface area). We will analyze to what extent alteration of these parameters influences on flow characteristics (turbulent intensity, mixing efficiency, heat transfer) inside the layer and behind it. Analysis of flow inside granular layers is very complicated because the use of classical experimental techniques (LDA, PIV, fibber probes) inside the layers is practically impossible, whereas the use of probes (e.g. thermocouples, Pitot tubes) requires drilling of holes inside the solid material. Hence, measurements of the flow inside granular layers are usually performed using for instance advanced X-ray tomography. In this respect, theoretical or numerical analyses of flow inside granulates seem crucial. Application of discrete element methods in combination with the classical finite volume/finite difference approaches is problematic as a mesh generation process for complex granular material can be very arduous. A good alternative for simulation of flow in complex domains is an immersed boundary-volume penalization (IB-VP) in which the computational meshes have simple Cartesian structure and impact of solid objects on the fluid is mimicked by source terms added to the Navier-Stokes and energy equations. The present paper focuses on application of the IB-VP method combined with large eddy simulation (LES). The flow solver used in this work is a high-order code (SAILOR), which was used previously in various studies, including laminar/turbulent transition in free flows and also for flows in wavy channels, wavy pipes and over various shape obstacles. In these cases a formal order of approximation turned out to be in between 1 and 2, depending on the test case. The current research concentrates on analyses of the flows in dense granular layers with elements distributed in a deterministic regular manner and validation of the results obtained using LES-IB method and body-fitted approach. The comparisons are very promising and show very good agreement. It is found that the size, number of elements and their distribution have huge impact on the obtained results. Ordering of the granular elements (or lack of it) affects both the pressure drop and efficiency of the heat transfer as it significantly changes mixing process.

Keywords: granular layers, heat transfer, immersed boundary method, numerical simulations

Procedia PDF Downloads 136