Begell House Inc.
International Journal for Multiscale Computational Engineering
JMC
1543-1649
14
1
2016
SPARSE GENERALIZED MULTISCALE FINITE ELEMENT METHODS AND THEIR APPLICATIONS
1-23
Eric T.
Chung
Department of Mathematics, The Chinese University of Hong Kong, Shatin, New Territories,
Hong Kong SAR, China
Yalchin
Efendiev
Department of Mathematics and Institute for Scientific Computation (ISC),
Texas A&M University, College Station, TX 77840, USA; Multiscale Model Reduction Laboratory, North-Eastern Federal University,
Yakutsk, Russia, 677980
Wing Tat
Leung
Department of Mathematics and Institute for Scientific Computation (ISC), Texas A&M University, College Station, Texas 77843-3368, USA
Guanglian
Li
Department of Mathematics & Institute for Scientific Computation (ISC), Texas A&M University, College Station, Texas, USA; Institute for Numerical Simulation, The University of Bonn, Wegelerstrasse 6,53115 Bonn,
Germany
In a number of previous papers, local (coarse grid) multiscale model reduction techniques are developed using a Generalized Multiscale Finite Element Method. In these approaches, multiscale basis functions are constructed using local snapshot spaces, where a snapshot space is a large space that represents the solution behavior in a coarse block. In a number of applications (e.g., those discussed in the paper), one may have a sparsity in the snapshot space for an appropriate choice of a snapshot space. More precisely, the solution may only involve a portion of the snapshot space. In this case, one can use sparsity techniques to identify multiscale basis functions. In this paper, we consider two such sparse local multiscale model reduction approaches. In the first approach (which is used for parameter-dependent multiscale PDEs), we use local minimization techniques, such as sparse POD, to identify multiscale basis functions, which are sparse in the snapshot space. These minimization techniques use l1 minimization to find local multiscale basis functions, which are further used for finding the solution. In the second approach (which is used for the Helmholtz equation), we directly apply l1 minimization techniques to solve the underlying PDEs. This approach is more expensive as it involves a large snapshot space; however, in this example, we cannot identify a local minimization principle, such as local generalized SVD. All our numerical results assume the sparsity and we discuss this assumption for the snapshot spaces. Moreover, we discuss the computational savings provided by our approach. The sparse solution allows a fast evaluation of stiffness matrices and downscaling the solution to the fine grid since the reduced dimensional solution representation is sparse in terms of local snapshot vectors. Numerical results are presented, which show the convergence of the proposed method and the sparsity of the solution.
A MULTISCALE APPROACH FOR THERMO-MECHANICAL SIMULATIONS OF LOADING COURSES IN CAST IRON BRAKE DISCS
25-43
Stefan
Schmid
Institute of Materials and Processes, Karlsruhe University of Applied Science, Moltkestrasse 30, D-76133 Karlsruhe, Germany
Daniel
Schneider
IAM-CMS, Karlsruhe Institute of Technology, Kaiserstrasse 12, D-76131 Karlsruhe, Germany
Christoph
Herrmann
Institute of Materials and Processes, Karlsruhe University of Applied Science, Moltkestrasse 30, D-76133 Karlsruhe, Germany
Michael
Selzer
Institute of Materials and Processes, Karlsruhe University of Applied Science, Moltkestrasse 30, D-76133 Karlsruhe, Germany; IAM-CMS, Karlsruhe Institute of Technology, Kaiserstrasse 12, D-76131 Karlsruhe, Germany
Britta
Nestler
Institute of Materials and Processes, Karlsruhe University of Applied Science, Moltkestrasse 30, D-76133 Karlsruhe, Germany; IAM-CMS, Karlsruhe Institute of Technology, Kaiserstrasse 12, D-76131 Karlsruhe, Germany
This article presents a multiscale approach for the simulation of coupled heat and stress evolution induced by different loading courses in gray cast iron brake discs. The concept integrates the microstructural properties as homogenized material laws into the macroscopic computations. Extensive experimental testing is required to establish a complete set of material parameters needed to conduct thermo-mechanical simulations on a macroscopic length scale. In addition, the microstructure can vary within the disc due to differences in wall thicknesses and cooling rates. In order to reduce the experimental effort and to estimate the influence of microstructure characteristics on macroscopic heat and stress distributions, simulations on the mesoscopic scale resolving the heterogeneous microstructure with graphite flakes in a pearlite matrix are conducted. The workflow to derive the elasto-plastic properties according to its microstructure is demonstrated for a typical cast iron material. Geometrical parameters of the graphite phase distributions and shape factors composed from micrographic analysis are used to generate representative volume elements (RVE) and to define the metallographic constituents. The information serves as input parameters to algorithmically construct a 3D cast iron microstructure. The elastic and elasto-plastic material models of the constituents are briefly elucidated. In order to simulate the different material behavior in tension and compression, a crack opening and crack closure mechanism is included. The potential of complementing and substituting experimental testing is shown by a quantitative comparison of the simulation results with experimental data at ambient temperature. Both virtual tension and compression tests are executed as well as a tension-compression cycle and the determination of the yield surface of the material. The presented approach provides a first step into a versatile range of applications and illustrates a broad potential for future challenges of multiscale modeling in the field of thermo-mechanical failure analysis.
BUCKLING ANALYSIS OF CURVED NANOTUBE STRUCTURES BASED ON THE NONLOCAL SHELL THEORY
45-54
Hamidreza
Yazdani Sarvestani
Concordia Centre for Composites (CONCOM), Department of Mechanical and Industrial Engineering, Concordia University, Montreal, Canada H3G 1M8
In reality, nanotubes are not straight. In the present work, the buckling behavior of curved single-walled nanotubes (SWNTs), double-walled nanotubes (DWNTs) and multi-walled nanotubes (MWNTs) under axial compression is studied. The buckling analysis for the curved nanotube structures is performed by applying a nonlocal shell theory based on the constitutive relations of Eringen. The governing equations of curved SWNTs, DWNTs and MWNTs are developed. Solutions are obtained using Fourier series expansion. The effects of the curved nanotube length, bend angle, diameter and nonlocal parameter on the buckling loads are investigated. The numerical results indicate that the nonlocal parameter is important for the buckling analysis of curved nanotube structures.
SIMULATION OF DYNAMIC STRAIN AGING PROCESS AT THE MICROSCOPIC SCALE BY MONTE CARLO DYNAMIC MODEL
55-63
Yansheng
He
CAS Key Laboratory of Mechanical Behavior and Design of Material, University of Science and Technology of China, Hefei 230027, China
Shihua
Fu
CAS Key Laboratory of Mechanical Behavior and Design of Material, University of Science and Technology of China, Hefei 230027, China
Qingchuan
Zhang
CAS Key Laboratory of Mechanical Behavior and Design of Materials, University of Science and Technology of China, Hefei 230027, China
A two-dimensional Monte Carlo dynamic model has been developed to simulate the interaction between mobile dislocations and solute atoms. When there was a single dislocation with a constant stress rate, the dislocation was pinned under low stress rate conditions, moved continuously under high stress rate conditions, and moved intermittently at an intermediate stress rate, where the step-shaped curve of dislocation displacement showed dynamic strain aging. Under multi-dislocation unstressed conditions, the solute atoms gathered below positive dislocations and above negative dislocations. Under the multi-dislocation conditions with constant stress, the dislocation displacement became longer with increasing stress. Under multi-dislocation conditions with constant stress rate, the collective pinning and unpinning results showed a step-shaped curve of total dislocation displacement. The simulation results present a microscopic view of the pinning and unpinning in dynamic strain aging.
SIZE-DEPENDENT POSTBUCKLING OF ANNULAR NANOPLATES WITH DIFFERENT BOUNDARY CONDITIONS SUBJECTED TO THE AXISYMMETRIC RADIAL LOADING INCORPORATING SURFACE STRESS EFFECTS
65-80
R.
Ansari
Department of Mechanical Engineering, University of Guilan, P.O. Box 3756, Rasht, Iran
V.
Mohammadi
Department of Mechanical Engineering, University of Guilan, P.O. Box 3756, Rasht, Iran
M. Faghih
Shojaei
Department of Mechanical Engineering, University of Guilan, P.O. Box 3756, Rasht, Iran
R.
Gholami
Department of Mechanical Engineering, Lahijan Branch, Islamic Azad University, P.O. Box 1616, Lahijan, Iran
This paper addresses the problem of size-dependent axisymmetric postbuckling behavior of annular shear deformable nanoplates by taking into consideration surface effects. A size-dependent continuum plate model is developed based on the Gurtin−Murdoch elasticity theory, the first-order shear deformation theory, and the von Karman geometrically nonlinear relations. It is assumed that the annular nanoplate is subjected to compressive axisymmetric radial loads. By using the Gurtin−Murdoch theory, the influences of surface stress and residual surface stress are incorporated into the formulation. Afterward, according to the virtual work principle, the size-dependent geometrically nonlinear governing equations and associated boundary conditions of first-order shear deformable nanoplates are obtained. The obtained set of nonlinear equations is discretized and solved via the generalized differential quadrature method and pseudo-arc-length continuation method. Then, the postbuckling behavior of nanoplates made of silicon and aluminum with different boundary conditions is carefully studied. The results obtained from classical and non-classical theories are compared for the first three postbuckling modes. In addition, the effects of the surface elastic modulus, residual surface stress, thickness, and radius ratio on the postbuckling response of annular nanoplates are examined.
COMPARISON OF MULTIRESOLUTION CONTINUUM THEORY AND NONLOCAL DAMAGE MODEL FOR USE IN SIMULATION OF MANUFACTURING PROCESSES
81-94
Olufunminiyi
Abiri
Lulea University of Technology, 97187 Lulea, Sweden
Hao
Qin
Lulea University of Technology, 97187 Lulea, Sweden
Lars-Erik
Lindgren
Lulea University of Technology, 97187 Lulea, Sweden
Modelling and simulation of manufacturing processes may require the capability to account for localization behavior, often associated with damage/fracture. It may be unwanted localization indicating a failure in the process or, as in the case of machining and cutting, a wanted phenomenon to be controlled. The latter requires a higher accuracy regarding the modelling of the underlying physics, as well as the robustness of the simulation procedure. Two different approaches for achieving mesh-independent solutions are compared in this paper. They are the multiresolution continuum theory (MRCT) and nonlocal damage model. The MRCT theory is a general multilength-scale finite element formulation, while the nonlocal damage model is a specialized method using a weighted averaging of softening internal variables over a spatial neighborhood of the material point. Both approaches result in a converged finite element solution of the localization problem upon mesh refinement. This study compares the accuracy and robustness of their numerical schemes in implicit finite element codes for the plane strain shear deformation test case. Final remarks concerning ease of implementation of the methods in commercial finite element packages are also given.