Diagonal PCG Method for FE Solution of 3D Biot's Consolidation Problem

2013 ◽  
Vol 838-841 ◽  
pp. 718-721
Author(s):  
Kun Yong Zhang ◽  
Gui Heng Xie
Keyword(s):  
Linear Systems ◽  
Large Scale ◽  
Solution Method ◽  
Preconditioned Conjugate Gradient ◽  
Finite Element Discretization ◽  
Element Discretization ◽  
Direct Solution Method ◽  
Indefinite Linear Systems ◽  
Pcg Method ◽  
Biot’S Consolidation

To solve large symmetric indefinite linear systems in finite element discretization of 3D Biot's consolidation equations.This paper adopted diagonal preconditioned conjugate gradient method to FE program. Several numerical examples show that the diagonal PCG method are significantly more efficient than direct solution method for large-scale symmetric indefinite linear systems.

scholarly journals Applications of Symmetric and Nonsymmetric MSSOR Preconditioners to Large-Scale Biot's Consolidation Problems with Nonassociated Plasticity

2012 ◽  
Vol 2012 ◽  
pp. 1-15 ◽  
Author(s):  
Xi Chen ◽  
Kok Kwang Phoon
Keyword(s):  
Linear Systems ◽  
Large Scale ◽  
Iterative Scheme ◽  
Linear Equations ◽  
Pile Group ◽  
Newton Iteration ◽  
Soil Consolidation ◽  
Considerable Potential ◽  
Stiffness Method ◽  
Biot’S Consolidation

Two solution schemes are proposed and compared for large 3D soil consolidation problems with nonassociated plasticity. One solution scheme results in the nonsymmetric linear equations due to the Newton iteration, while the other leads to the symmetric linear systems due to the symmetrized stiffness strategies. To solve the resulting linear systems, the QMR and SQMR solver are employed in conjunction with nonsymmetric and symmetric MSSOR preconditioner, respectively. A simple footing example and a pile-group example are used to assess the performance of the two solution schemes. Numerical results disclose that compared to the Newton iterative scheme, the symmetric stiffness schemes combined with adequate acceleration strategy may lead to a significant reduction in total computer runtime as well as in memory requirement, indicating that the accelerated symmetric stiffness method has considerable potential to be exploited to solve very large problems.

Limit Loads for Structural Elements Made of Heterogeneous Materials

2012 ◽  
Author(s):  
Min Chen ◽  
Abdelkader Hachemi ◽  
Dieter Weichert
Keyword(s):  
Material Properties ◽  
Large Scale ◽  
Numerical Approach ◽  
Interior Point Algorithm ◽  
Structural Elements ◽  
Finite Element Discretization ◽  
Element Discretization ◽  
Limit Loads ◽  
Heterogeneous Structures ◽  
Conforming Finite Element

A numerical method is presented for determining the limit loads of periodically heterogeneous structures subjected to variable loads. The Melan’s lower-bound shakedown theorem was applied to representative volume elements. Combined with the homogenization technique, the homogenized material properties were determined through transformation from the mesoscopic to macroscopic admissible loading domains. For the numerical applications, solid non-conforming finite element discretization and large-scale nonlinear optimization, based on an interior-point-algorithm were used. The methodology is illustrated by the application to pipes models. This way, the proposed method provides a direct numerical approach to evaluate the macroscopic strength of heterogeneous structures with periodic micro- or meso-structure as a useful tool for the design of structures.

scholarly journals Numerical Algorithms for Computing an Arbitrary Singular Value of a Tensor Sum

Axioms ◽  
2021 ◽  
Vol 10 (3) ◽  
pp. 211
Author(s):  
Asuka Ohashi ◽  
Tomohiro Sogabe
Keyword(s):  
Eigenvalue Problem ◽  
Large Scale ◽  
Numerical Algorithms ◽  
Direct Methods ◽  
Coefficient Matrix ◽  
Lanczos Method ◽  
Singular Value ◽  
Preconditioned Conjugate Gradient ◽  
Schur Decomposition ◽  
Pcg Method

We consider computing an arbitrary singular value of a tensor sum: T:=In⊗Im⊗A+In⊗B⊗Iℓ+C⊗Im⊗Iℓ∈Rℓmn×ℓmn, where A∈Rℓ×ℓ, B∈Rm×m, C∈Rn×n. We focus on the shift-and-invert Lanczos method, which solves a shift-and-invert eigenvalue problem of (TTT−σ˜2Iℓmn)−1, where σ˜ is set to a scalar value close to the desired singular value. The desired singular value is computed by the maximum eigenvalue of the eigenvalue problem. This shift-and-invert Lanczos method needs to solve large-scale linear systems with the coefficient matrix TTT−σ˜2Iℓmn. The preconditioned conjugate gradient (PCG) method is applied since the direct methods cannot be applied due to the nonzero structure of the coefficient matrix. However, it is difficult in terms of memory requirements to simply implement the shift-and-invert Lanczos and the PCG methods since the size of T grows rapidly by the sizes of A, B, and C. In this paper, we present the following two techniques: (1) efficient implementations of the shift-and-invert Lanczos method for the eigenvalue problem of TTT and the PCG method for TTT−σ˜2Iℓmn using three-dimensional arrays (third-order tensors) and the n-mode products, and (2) preconditioning matrices of the PCG method based on the eigenvalue and the Schur decomposition of T. Finally, we show the effectiveness of the proposed methods through numerical experiments.

scholarly journals Accelerating the solution of linear systems appearing in two-phase reservoir simulation by the use of POD-based deflation methods

2021 ◽  
Author(s):  
Gabriela Berenice Diaz Cortés ◽  
Cornelis Vuik ◽  
Jan-Dirk Jansen
Keyword(s):  
Linear Systems ◽  
Krylov Subspace ◽  
Computational Cost ◽  
Heterogeneous Porous Media ◽  
Preconditioned Conjugate Gradient ◽  
Two Phase ◽  
Transient Problems ◽  
Flow Through ◽  
Proper Orthogonal ◽  
Pcg Method

AbstractWe explore and develop a Proper Orthogonal Decomposition (POD)-based deflation method for the solution of ill-conditioned linear systems, appearing in simulations of two-phase flow through highly heterogeneous porous media. We accelerate the convergence of a Preconditioned Conjugate Gradient (PCG) method achieving speed-ups of factors up to five. The up-front extra computational cost of the proposed method depends on the number of deflation vectors. The POD-based deflation method is tested for a particular problem and linear solver; nevertheless, it can be applied to various transient problems, and combined with multiple solvers, e.g., Krylov subspace and multigrid methods.

Preconditioning for diffusion problem with small size high resolution inclusions

2018 ◽  
Vol 33 (4) ◽  
pp. 243-251
Author(s):  
Yuri A. Kuznetsov
Keyword(s):  
Solution Method ◽  
Diffusion Problem ◽  
Lanczos Method ◽  
Point System ◽  
High Contrast ◽  
Finite Element Discretization ◽  
Element Discretization ◽  
Preconditioning Technique ◽  
Saddle Point System ◽  
Diffusion Problems

Abstract In this paper, we propose and investigate a new preconditioning technique for diffusion problems with multiple small size high contrast inclusions. The inclusions partitioned into two groups. In the first group inclusions the value of diffusion coefficient can be very small, and in the inclusions of the second group it can be very large. The classical P1 finite element discretization is converted in the special algebraic saddle point system. The solution method combines elimination of the DOFs from the first group of inclusions with the Preconditioned Lanczos method with block diagonal preconditioner for the rest of the DOFs. Condition number estimates for the proposed preconditioner are given.

FINITE ELEMENT SURFACE FITTING FOR BRIDGE MANAGEMENT

2006 ◽  
Vol 05 (04) ◽  
pp. 671-681 ◽  
Author(s):  
XIAOJUN CHEN ◽  
TAKUMA KIMURA
Keyword(s):  
Finite Element ◽  
Large Scale ◽  
Piecewise Linear ◽  
Linear Equations ◽  
Surface Fitting ◽  
Quadratic Program ◽  
Bridge Management ◽  
Finite Element Discretization ◽  
Element Discretization ◽  
Very Large Databases

We propose a new finite element surface fitting method which can handle very large databases. This method uses finite element discretization to find an approximation of a smooth function which minimizes a sum of data residuals and second derivatives under some constraints on data. The finite element discretization derives a large scale constrained quadratic program, which can be reformulated as a system of piecewise linear equations. We develop a preconditioned Newton method to solve the system efficiently. We apply this method to form surfaces over Aomori Region in Japan by geographic databases, such that every bridge became associated with environmental information.

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