scholarly journals Variational Multiscale Stabilization and the Exponential Decay of Fine-Scale Correctors

2016 ◽  
pp. 343-369 ◽  
Author(s):  
Daniel Peterseim
Keyword(s):  
Exponential Decay ◽  
Fine Scale ◽  
Variational Multiscale

scholarly journals A variational multiscale immersed meshfree method for heterogeneous materials

2021 ◽  
Vol 67 (4) ◽  
pp. 1059-1097
Author(s):  
Tsung-Hui Huang ◽  
Jiun-Shyan Chen ◽  
Michael R. Tupek ◽  
Frank N. Beckwith ◽  
Jacob J. Koester ◽  
...  
Keyword(s):  
Reproducing Kernel ◽  
Arbitrary Order ◽  
Heterogeneous Materials ◽  
Meshfree Method ◽  
Fine Scale ◽  
Coarse Scale ◽  
Variational Equations ◽  
Variational Multiscale ◽  
Discrete Equations ◽  
Interfacial Compatibility

AbstractWe introduce an immersed meshfree formulation for modeling heterogeneous materials with flexible non-body-fitted discretizations, approximations, and quadrature rules. The interfacial compatibility condition is imposed by a volumetric constraint, which avoids a tedious contour integral for complex material geometry. The proposed immersed approach is formulated under a variational multiscale based formulation, termed the variational multiscale immersed method (VMIM). Under this framework, the solution approximation on either the foreground or the background can be decoupled into coarse-scale and fine-scale in the variational equations, where the fine-scale approximation represents a correction to the residual of the coarse-scale equations. The resulting fine-scale solution leads to a residual-based stabilization in the VMIM discrete equations. The employment of reproducing kernel (RK) approximation for the coarse- and fine-scale variables allows arbitrary order of continuity in the approximation, which is particularly advantageous for modeling heterogeneous materials. The effectiveness of VMIM is demonstrated with several numerical examples, showing accuracy, stability, and discretization efficiency of the proposed method.

High Resolution Stereography of Complementary Freeze-Fracture Replicas Reveals the Presence of Depressions Opposite Membrane Associated Particles of Split Membranes

1971 ◽  
Vol 29 ◽  
pp. 442-443
Author(s):  
Russell L. Steere
Keyword(s):  
High Resolution ◽  
Cardiac Muscle ◽  
Electron Micrographs ◽  
Chromic Acid ◽  
Freeze Fracture ◽  
Distilled Water ◽  
Fine Scale ◽  
Acid Cleaning ◽  
Cleaning Solution

Complementary replicas have revealed the fact that the two common faces observed in electron micrographs of freeze-fracture and freeze-etch specimens are complementary to each other and are thus the new faces of a split membrane rather than the original inner and outer surfaces (1, 2 and personal observations). The big question raised by published electron micrographs is why do we not see depressions in the complementary face opposite membrane-associated particles? Reports have appeared indicating that some depressions do appear but complementarity on such a fine scale has yet to be shown.Dog cardiac muscle was perfused with glutaraldehyde, washed in distilled water, then transferred to 30% glycerol (material furnished by Dr. Joaquim Sommer, Duke Univ., and VA Hospital, Durham, N.C.). Small strips were freeze-fractured in a Denton Vacuum DFE-2 Freeze-Etch Unit with complementary replica tooling. Replicas were cleaned in chromic acid cleaning solution, then washed in 4 changes of distilled water and mounted on opposite sides of the center wire of a Formvar-coated grid.

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