scholarly journals TWO-STAGE GRID METHOD OF SOLUTION OF BOUNDARY PROBLEMS OF STRUCTURAL MECHANICS WITH THE USE OF DISCRETE HAAR BASIS

2017 ◽  
Vol 13 (1) ◽  
pp. 69-85
Author(s):  
Marina L. Mozgaleva
Keyword(s):  
Three Dimensional ◽  
Initial Approximation ◽  
Grid Method ◽  
Coarse Grid ◽  
Discrete Problem ◽  
Two Stage ◽  
Boundary Problems ◽  
Fine Grid ◽  
Haar Basis ◽  
Dimensional Boundary

The distinctive paper is devoted to development of two-stage numerical method. At the first stage, the discrete problem is solved on a coarse grid, where the number of nodes in each direction is the same and is a pow-er of 2. Then the number of nodes in each direction is doubled and the resulting solution on a coarse grid using a discrete Haar basis is defined at the nodes of the fine grid as the initial approximation. At the second stage, we ob-tain a solution in the nodes of the fine grid using the most appropriate iterative method,. Test examples of the solu-tion of one-dimensional, two-dimensional and three-dimensional boundary problems are under consideration

scholarly journals ABOUT WAVELET-BASED MULTIGRID NUMERICAL METHOD OF STRUCTURAL ANALYSIS WITH THE USE OF DISCRETE HAAR BASIS

2018 ◽  
Vol 14 (3) ◽  
pp. 83-102
Author(s):  
Marina L. Mozgaleva ◽  
Pavel A. Akimov ◽  
Taymuraz B. Kaytukov
Keyword(s):  
Structural Analysis ◽  
Three Dimensional ◽  
Grid Method ◽  
Discrete Analogue ◽  
Algebraic Equations ◽  
Initial State ◽  
Haar Basis ◽  
Linear Algebraic Equations ◽  
Gauss Method ◽  
Original Grid

he distinctive paper is devoted to so-called multigrid (particularly two-grid) method of structural analysis based on discrete Haar basis (one-dimensional, two-dimensional and three-dimensional problems are under consideration). Approximations of the mesh functions in discrete Haar bases of zero and first levels are described (the mesh function is represented as the sum in which one term is its approximation of the first level, and the second term is so-called complement (up to the initial state) on the grid of the first level). Special projectors are constructed for the spaces of vector functions of the original grid to the space of their approximation on the first-level grid and its complement (the refinement component) to the initial state. Basic scheme of the two-grid method is presented. This method allows solution of boundary problems of structural mechanics with the use of matrix operators of significantly smaller dimension. It should be noted that discrete analogue of the initial operator equation is a system of linear algebraic equations which is constructed with the use of finite element method or finite difference method. Block Gauss method can be used for direct solution.

Sensitivity-Coefficient Based Inverse Heat Conduction Method for Identifying Hot Spots in Electronics Packages: A Comparison of Grid-Refinement Methods

2021 ◽  
Author(s):  
Patrick Krane ◽  
David Gonzalez Cuadrado ◽  
Francisco Lozano ◽  
Guillermo Paniagua ◽  
Amy Marconnet
Keyword(s):  
Heat Conduction ◽  
Hot Spots ◽  
Computation Time ◽  
Grid Method ◽  
Sensitivity Coefficient ◽  
Coarse Grid ◽  
Inverse Heat Conduction ◽  
Fine Grid ◽  
Temperature Maps ◽  
Electronics Packages

Abstract Estimating the distribution and magnitude of heat generation within electronics packages is pivotal for thermal packaging design and active thermal management systems. Inverse heat conduction methods can provide estimates using measured temperature profiles acquired using infrared imaging or discrete temperature sensors. However, if the heater locations are unknown, applying a fine grid of potential heater locations across the surface where heat generation is expected can result in prohibitively-large computation times. In contrast, using a more computationally-efficient coarse grid can reduce the accuracy of heat flux estimations. This paper evaluates two methods for reducing computation time using a sensitivity-coefficient method for solving the inverse heat conduction problem. One strategy uses a coarse grid that is refined near the hot spots, while the other uses a fine grid of potential heaters only near the hot spots. These grid-refinement methods are compared using both input temperature maps acquired from a "numerical experiment" (where the outputs of a 3D steady-state thermal model in FloTHERM are used for input temperatures) and temperature maps procured using infrared microscopy on a real electronics package. Compared to the coarse-grid method, the fine-grid method reduces computation time without significantly reducing accuracy, making it more convenient for designing and testing electronics packages. It also avoids the problem of "false hot spots" that occurs with the coarse-grid method. Overall, this approach provides a mechanism to predict hot spot locations during design and testing and a tool for active thermal management.

scholarly journals A Two-Grid Method for Finite Element Solutions of Nonlinear Parabolic Equations

2012 ◽  
Vol 2012 ◽  
pp. 1-11 ◽  
Author(s):  
Chuanjun Chen ◽  
Wei Liu
Keyword(s):  
Finite Element ◽  
Piecewise Linear ◽  
A Priori ◽  
Grid Method ◽  
Finite Element Approximation ◽  
Nonlinear Parabolic Equations ◽  
Coarse Grid ◽  
Element Approximation ◽  
Fine Grid ◽  
Nonlinear Parabolic

A two-grid method is presented and discussed for a finite element approximation to a nonlinear parabolic equation in two space dimensions. Piecewise linear trial functions are used. In this two-grid scheme, the full nonlinear problem is solved only on a coarse grid with grid sizeH. The nonlinearities are expanded about the coarse grid solution on a fine gird of sizeh, and the resulting linear system is solved on the fine grid. A priori error estimates are derived with theH1-normO(h+H2)which shows that the two-grid method achieves asymptotically optimal approximation as long as the mesh sizes satisfyh=O(H2). An example is also given to illustrate the theoretical results.

scholarly journals A Two-Grid Finite Element Method for a Second-Order Nonlinear Hyperbolic Equation

2014 ◽  
Vol 2014 ◽  
pp. 1-6
Author(s):  
Chuanjun Chen ◽  
Wei Liu ◽  
Xin Zhao
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Hyperbolic Equation ◽  
A Priori ◽  
Grid Method ◽  
Second Order ◽  
Coarse Grid ◽  
Nonlinear Hyperbolic Equation ◽  
Fine Grid ◽  
Element Method

We present a two-grid finite element scheme for the approximation of a second-order nonlinear hyperbolic equation in two space dimensions. In the two-grid scheme, the full nonlinear problem is solved only on a coarse grid of sizeH. The nonlinearities are expanded about the coarse grid solution on the fine gird of sizeh. The resulting linear system is solved on the fine grid. Some a priori error estimates are derived with theH1-normO(h+H2)for the two-grid finite element method. Compared with the standard finite element method, the two-grid method achieves asymptotically same order as long as the mesh sizes satisfyh=O(H2).

scholarly journals STABILITY ANALYSIS OF SEIDEL TYPE MULTICOMPONENT ITERATIVE METHOD

2002 ◽  
Vol 7 (1) ◽  
pp. 1-10
Author(s):  
V. N. Abrashin ◽  
R. Čiegis ◽  
V. Pakeniene ◽  
N. G. Zhadaeva
Keyword(s):  
Stability Analysis ◽  
Iterative Method ◽  
Iterative Methods ◽  
Convergence Rates ◽  
Splitting Method ◽  
Three Dimensional ◽  
Initial Approximation ◽  
Discrete Problem ◽  
Smooth Functions ◽  
The Stability

This paper deals with the stability analysis of multicomponent iterative methods for solving elliptic problems. They are based on a general splitting method, which decomposes a multidimensional parabolic problem into a system of one dimensional implicit problems. Error estimates in the L 2 norm are proved for the first method. For the stability analysis of Seidel type iterative method we use a spectral method. Two dimensional and three dimensional problems are investigated. Finally, we present results of numerical experiments. Our goal is to investigate the dependence of convergence rates of multicomponent iterative methods on the smoothness of the solution. Hence we solve a discrete problem, which approximates the 3D Poisson's problem. It is proved that the number of iterations depends weakly on the number of grid points if the exact solution and the initial approximation are smooth functions, both. The same problem is also solved by the Stability Correction iterative method. The obtained results indicate a similar behavior.

Applications of dynamic hybrid grid method for three-dimensional moving/deforming boundary problems

2012 ◽  
Vol 62 ◽  
pp. 45-63 ◽  
Author(s):  
Laiping Zhang ◽  
Xinghua Chang ◽  
Xupeng Duan ◽  
Zhong Zhao ◽  
Xin He
Keyword(s):  
Three Dimensional ◽  
Grid Method ◽  
Boundary Problems ◽  
Dynamic Hybrid ◽  
Hybrid Grid
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