scholarly journals Adaptive Local Refinement with Octree Load Balancing for the Parallel Solution of Three-Dimensional Conservation Laws

1997 ◽  
Vol 47 (2) ◽  
pp. 139-152 ◽  
Author(s):  
J.E. Flaherty ◽  
R.M. Loy ◽  
M.S. Shephard ◽  
B.K. Szymanski ◽  
J.D. Teresco ◽  
...  
Keyword(s):  
Load Balancing ◽  
Conservation Laws ◽  
Three Dimensional ◽  
Local Refinement ◽  
Parallel Solution ◽  
Adaptive Local Refinement

scholarly journals The Local and Parallel Finite Element Scheme for Electric Structure Eigenvalue Problems

2021 ◽  
Vol 2021 ◽  
pp. 1-11
Author(s):  
Fubiao Lin ◽  
Junying Cao ◽  
Zhixin Liu
Keyword(s):  
Finite Element ◽  
Eigenvalue Problems ◽  
Three Dimensional ◽  
Local Defect ◽  
Local Refinement ◽  
Element Discretization ◽  
One Dimensional ◽  
Correction Technique ◽  
Polar Coordinate ◽  
Multiscale Finite Element

In this paper, an efficient multiscale finite element method via local defect-correction technique is developed. This method is used to solve the Schrödinger eigenvalue problem with three-dimensional domain. First, this paper considers a three-dimensional bounded spherical region, which is the truncation of a three-dimensional unbounded region. Using polar coordinate transformation, we successfully transform the three-dimensional problem into a series of one-dimensional eigenvalue problems. These one-dimensional eigenvalue problems also bring singularity. Second, using local refinement technique, we establish a new multiscale finite element discretization method. The scheme can correct the defects repeatedly on the local refinement grid, which can solve the singularity problem efficiently. Finally, the error estimates of eigenvalues and eigenfunctions are also proved. Numerical examples show that our numerical method can significantly improve the accuracy of eigenvalues.

Conservation laws of linear elasticity in stress formulations

2005 ◽  
Vol 461 (2053) ◽  
pp. 99-116 ◽  
Author(s):  
Shaofan Li ◽  
Anurag Gupta ◽  
Xanthippi Markenscoff
Keyword(s):  
Conservation Laws ◽  
Linear Elasticity ◽  
Three Dimensional ◽  
Cauchy Stress ◽  
General Boundary Conditions ◽  
Cauchy Stress Tensor ◽  
Equilibrium Equations ◽  
Noether’S Theorem ◽  
Noether's Theorem ◽  
Three Dimensional Elasticity

In this paper, we present new conservation laws of linear elasticity which have been discovered. These newly discovered conservation laws are expressed solely in terms of the Cauchy stress tensor, and they are genuine, non–trivial conservation laws that are intrinsically different from the displacement conservation laws previously known. They represent the variational symmetry conditions of combined Beltrami–Michell compatibility equations and the equilibrium equations. To derive these conservation laws, Noether's theorem is extended to partial differential equations of a tensorial field with general boundary conditions. By applying the tensorial version of Noether's theorem to Pobedrja's stress formulation of three–dimensional elasticity, a class of new conservation laws in terms of stresses has been obtained.

scholarly journals Horizontal Momentum Diffusion in GCMs Using the Dynamic Smagorinsky Model

2013 ◽  
Vol 141 (3) ◽  
pp. 887-899 ◽  
Author(s):  
Urs Schaefer-Rolffs ◽  
Erich Becker
Keyword(s):  
Conservation Laws ◽  
Cross Sections ◽  
General Circulation ◽  
Three Dimensional ◽  
Circulation Model ◽  
Atmospheric Dynamics ◽  
Three Solutions ◽  
Smagorinsky Model ◽  
Test Filter ◽  
Tensor Norm

Abstract A dynamic version of Smagorinsky’s diffusion scheme is presented that is applicable for large-eddy simulations (LES) of the atmospheric dynamics. The approach is motivated (i) by the incompatibility of conventional hyperdiffusion schemes with the conservation laws, and (ii) because the conventional Smagorinsky model (which fulfills the conservation laws) does not maintain scale invariance, which is mandatory for a correct simulation of the macroturbulent kinetic energy spectrum. The authors derive a two-dimensional (horizontal) formulation of the dynamic Smagorinsky model (DSM) and present three solutions of the so-called Germano identity: the method of least squares, a solution without invariance of the Smagorinsky parameter, and a tensor-norm solution. The applicability of the tensor-norm approach is confirmed in simulations with the Kühlungsborn mechanistic general circulation model (KMCM). The standard spectral dynamical core of the model facilitates the implementation of the test filter procedure of the DSM. Various energy spectra simulated with the DSM and the conventional Smagorinsky scheme are presented. In particular, the results show that only the DSM allows for a reasonable spectrum at all scales. Latitude–height cross sections of zonal-mean fluid variables are given and show that the DSM preserves the main features of the atmospheric dynamics. The best ratio for the test-filter scale to the resolution scale is found to be 1.33, resulting in dynamically determined Smagorinsky parameters cS from 0.10 to 0.22 in the troposphere. This result is very similar to other values of cS found in previous three-dimensional applications of the DSM.

Sign in / Sign up

Export Citation Format

Share Document