Preconditioning Methods for the Penalty Function FEM Discretizations of 3D Nearly Incompressible Elasticity Problems

2016 ◽  
Vol 13 (05) ◽  
pp. 1650029
Author(s):  
Yingxiong Xiao ◽  
Zhenyou Li ◽  
Lei Zhou
Keyword(s):  
Penalty Function ◽  
Mixed Boundary ◽  
Fem Analysis ◽  
Three Dimensions ◽  
Preconditioned Conjugate Gradient ◽  
System Of Equations ◽  
Elasticity Problems ◽  
Overall Efficiency ◽  
Incompressible Elasticity ◽  
Nearly Incompressible Elasticity

Penalty function finite element method (FEM) is an efficient method for the solution of the nearly incompressible elasticity problems in three dimensions. In order to improve the overall efficiency, some faster solvers for the corresponding FEM system of equations must be presented. Since the resulting system of equations is symmetric positive definite and highly ill-conditioned, the preconditioned conjugate gradient (PCG) method is one of the most efficient methods for solving such FEM equations. In this paper, we have first presented some types of efficient PCG methods including [Formula: see text]-PCG and [Formula: see text]-PCG with two different block diagonal inverses as a preconditioner, respectively, and including the RS-GAMG-PCG based on the global matrix for the penalty function FEM equations of three-dimensional (3D) nearly incompressible elasticity problems with mixed boundary conditions. Furthermore, we apply these methods to the solution of the penalty function quadratic discretizations for the cantilever and the Cook’s membrane problems. The numerical results have shown that much more efficient PCG methods for 3D nearly incompressible problems can be obtained by using the known information that is easily available in most FEM applications, for instance, the type of the partial differential equations (PDEs) considered and the number of physical unknowns residing in each grid, and by combining the coarsening techniques used in the classic algebraic multigrid (AMG) method. This will greatly improve the overall efficiency of the penalty function FEM analysis.

Preconditioned Conjugate Gradient Methods for the Refined FEM Discretizations of Nearly Incompressible Elasticity Problems in Three Dimensions

2019 ◽  
Vol 17 (03) ◽  
pp. 1850136 ◽  
Author(s):  
Yingxiong Xiao ◽  
Zhenyou Li
Keyword(s):  
Conjugate Gradient ◽  
Discrete System ◽  
Gradient Methods ◽  
Three Dimensions ◽  
Preconditioned Conjugate Gradient ◽  
System Of Equations ◽  
Elasticity Problems ◽  
Incompressible Elasticity ◽  
Nearly Incompressible Elasticity ◽  
Element Method

Nearly incompressible problems in three dimensions are the important problems in practical engineering computation. The volume-locking phenomenon will appear when the commonly used finite elements such as linear elements are applied to the solution of these problems. There are many efficient approaches to overcome this locking phenomenon, one of which is the higher-order conforming finite element method. However, we often use the lower-order nonconforming elements as Wilson elements by considering the computational complexity for three-dimensional (3D) problems considered. In general, the convergence of Wilson elements will heavily rely on the quality of the meshes. It will greatly deteriorate or no longer converge when the mesh distortion is very large. In this paper, the refined element method based on Wilson element is first applied to solve nearly incompressible elasticity problems, and the influence of mesh quality on the refined element is tested numerically. Its validity is verified by some numerical examples. By using the internal condensation method, the refined element discrete system of equations is deduced into the one which is spectrally equivalent to an 8-node hexahedral element discrete system of equations. And then, a type of efficient algebraic multigrid (AMG) preconditioner is presented by combining both the coarsening techniques based on the distance matrix and the effective smoothing operators. The resulting preconditioned conjugate gradient (PCG) method is efficient for 3D nearly incompressible problems. The numerical results verify the efficiency and robustness of the proposed method.

Coupled FEM-BEM Approach for Axisymetrical Heat Transfer Problems

2008 ◽  
Vol 273-276 ◽  
pp. 740-745
Author(s):  
Gennady Mishuris ◽  
Michał Wróbel
Keyword(s):  
Heat Transfer ◽  
Half Space ◽  
Integral Transform ◽  
Transfer Problem ◽  
Mixed Boundary ◽  
Nonlocal Boundary ◽  
Fem Analysis ◽  
Nonlocal Boundary Conditions ◽  
Problem Solution ◽  
Infinite Domain

This work deals with a stationary axisymmetrical heat transfer problem in a combined domain. This domain consists of half-space joined with a bounded cylinder. An important feature of the problem is the possible flux singularity along the edge points of the transmission surface. Domain decomposition is used to separate the subdomains. The solution for an auxiliary mixed boundary value problem in the half space is found analytically by means of Hankel integral transform. This allows us to reduce the main problem in the infinite domain to another problem defined in the bounded subdomain. In turn, the new problem contains a nonlocal boundary conditions along the transmission surface. These conditions incorporate all basic information about the infinite sub-domain (material properties, internal sources etc.). The problem is solved then by means of the Finite Element Method. In fact it might be considered as a coupled FEM-BEM approach. We use standard MATLAB PDE toolbox for the FEM analysis. As it is not possible for this package to introduce directly a non-classical boundary condition, we construct an appropriate iterative procedure and show the fast convergence of the main problem solution. The possible solution singularity is taken into account and the corresponding intensity coefficient of the heat flux is computed with a high accuracy. Numerical examples dealing with heat transfer between closed reservoir (filled with some substance) and the infinite foundation are discussed.

A two-stage approximation strategy for piecewise smooth functions in two and three dimensions

2021 ◽  
Author(s):  
Sergio Amat ◽  
David Levin ◽  
Juan Ruiz-Álvarez
Keyword(s):  
Piecewise Smooth ◽  
Three Dimensions ◽  
System Of Equations ◽  
Global Approximation ◽  
Smooth Functions ◽  
Approximation Techniques ◽  
Second Stage ◽  
Principle Of Matching ◽  
The Given ◽  
Standard Approximation

Abstract Given values of a piecewise smooth function $f$ on a square grid within a domain $[0,1]^d$, $d=2,3$, we look for a piecewise adaptive approximation to $f$. Standard approximation techniques achieve reduced approximation orders near the boundary of the domain and near curves of jump singularities of the function or its derivatives. The insight used here is that the behavior near the boundaries, or near a singularity curve, is fully characterized and identified by the values of certain differences of the data across the boundary and across the singularity curve. We refer to these values as the signature of $f$. In this paper, we aim at using these values in order to define the approximation. That is, we look for an approximation whose signature is matched to the signature of $f$. Given function data on a grid, assuming the function is piecewise smooth, first, the singularity structure of the function is identified. For example, in the two-dimensional case, we find an approximation to the curves separating between smooth segments of $f$. Secondly, simultaneously, we find the approximations to the different segments of $f$. A system of equations derived from the principle of matching the signature of the approximation and the function with respect to the given grid defines a first stage approximation. A second stage improved approximation is constructed using a global approximation to the error obtained in the first stage approximation.

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