Simulation of Nonlinear Magnetic Systems by the Finite Element Method Using BLR-Factorization

2021 ◽  
Vol 64 (4-5) ◽  
pp. 14-19
Author(s):  
Artem Khoroshev ◽  
Keyword(s):  
Finite Element ◽  
Electromagnetic Field ◽  
Finite Element Modeling ◽  
Low Rank ◽  
Low Rank Approximation ◽  
Magnetic Systems ◽  
Numerical Problem ◽  
Element Modeling ◽  
The Matrix ◽  
Rank Approximation

The possibility of practical application of BLR-factorization (low-rank approximation of the matrix of un-knowns of a system of linear equations) for finite element modeling of the electromagnetic field topology of nonlinear magnetic systems is considered. A method for estimating the accuracy of the computed solution of the SLAE and the nature of the influence of the given accuracy of the low-rank approximation of the matrix of un-knowns on the upper limit of the relative forward error of the computed solution of the SLAE are shown. Using a model problem as an example, the dependence of the accuracy of calculating the integral characteristics of an electromechanical apparatus on the tolerance of the low-rank approximation of the matrix of unknowns is shown, as well as its effect on the convergence of the process of solving a nonlinear numerical problem. A quantitative assessment of the reduction in the computational complexity of the process of solving a numerical problem and the required amount of computer memory for solving the SLAE is carried out. The applicability of BLR-factorization for finite element modeling of the topology of the electromagnetic field without the use of numerical methods of the Krylov subspace is estimated.

Modeling of Cure-Induced Residual Stresses in 3D Woven Composites of Different Reinforcement Architectures

2013 ◽  
Vol 577-578 ◽  
pp. 253-256 ◽  
Author(s):  
Igor Tsukrov ◽  
Borys Drach ◽  
Harun Bayraktar ◽  
Jon Goering
Keyword(s):  
Residual Stress ◽  
Finite Element ◽  
Residual Stresses ◽  
Finite Element Modeling ◽  
Room Temperature ◽  
Woven Composites ◽  
Element Modeling ◽  
The Matrix ◽  
3D Woven Composites ◽  
Resin Curing

This paper presents finite element modeling effort to predict possible microcracking of the matrix in 3D woven composites during curing. Three different reinforcement architectures are considered: a ply-to-ply weave, a one-by-one and a two-by-two orthogonal through-thickness reinforcement. To realistically reproduce the as-woven geometry of the fabric, the data from the Digital Fabric Mechanics Analyzer software is used as input for finite element modeling. The curing processed is modeled in a simplified way as a uniform drop in temperature from the resin curing to room temperature. The simulations show that the amount of residual stress is strongly influenced by the presence of through-thickness reinforcement.

scholarly journals A parallel implementation of the matrix cross approximation method

2015 ◽  
pp. 369-375
Author(s):  
Д.А. Желтков ◽  
Е.Е. Тыртышников
Keyword(s):  
Approximation Method ◽  
Parallel Implementation ◽  
Computational Cost ◽  
Low Rank ◽  
Fast Method ◽  
Low Rank Approximation ◽  
Rank Matrix ◽  
The Matrix ◽  
Rank Approximation ◽  
Low Rank Matrix

Матричный крестовый метод является быстрым методом аппроксимации матриц матрицами малого ранга, его сложность составляет $O((m+n)r^2)$ операций. Важной особенностью является то, что если матрица задана не как хранящийся в памяти массив, а как функция от двух целочисленных аргументов, то можно найти еe малоранговое приближение, вычислив лишь $O((m+n)r)$ значений этой функции. Однако в случае сверхбольших размеров матрицы или крайней затратности вычисления еe элементов аппроксимация может занимать существенное время. Ускорить метод для подобных случаев можно с помощью параллельных алгоритмов. В настоящей статье предложен эффективный параллельный алгоритм для случая одинаковой сложности вычисления любого элемента матрицы. The matrix cross approximation method is a fast method based on low-rank matrix approximations with complexity $O((m+n)r^2)$ arithmetic operations. Its main feature consists in the following: if a matrix is not given as an array but is given as a function of two integer arguments, then this method allows one to compute the low-rank approximation of the given matrix by evaluating only $O((m+n)r)$ values of this function. However, if the matrix is extremely large or the evaluation of its elements is computationally expensive, then such an approximation becomes timeconsuming. For such cases, the performance of the method can be improved via parallelization. In this paper we propose an efficient parallel algorithm for the case of an equal computational cost for the evaluation of each matrix element.

scholarly journals An unconventional robust integrator for dynamical low-rank approximation

2021 ◽  
Author(s):  
Gianluca Ceruti ◽  
Christian Lubich
Keyword(s):  
Differential Equation ◽  
Time Integration ◽  
Singular Values ◽  
Time Dependent ◽  
Low Rank ◽  
Low Rank Approximation ◽  
Numerical Integrator ◽  
The Matrix ◽  
Rank Approximation ◽  
Matrix Differential

AbstractWe propose and analyse a numerical integrator that computes a low-rank approximation to large time-dependent matrices that are either given explicitly via their increments or are the unknown solution to a matrix differential equation. Furthermore, the integrator is extended to the approximation of time-dependent tensors by Tucker tensors of fixed multilinear rank. The proposed low-rank integrator is different from the known projector-splitting integrator for dynamical low-rank approximation, but it retains the important robustness to small singular values that has so far been known only for the projector-splitting integrator. The new integrator also offers some potential advantages over the projector-splitting integrator: It avoids the backward time integration substep of the projector-splitting integrator, which is a potentially unstable substep for dissipative problems. It offers more parallelism, and it preserves symmetry or anti-symmetry of the matrix or tensor when the differential equation does. Numerical experiments illustrate the behaviour of the proposed integrator.

scholarly journals GNS: Forge High Anonymity Graph by Nonlinear Scaling Spectrum

2021 ◽  
Vol 2021 ◽  
pp. 1-11
Author(s):  
Yong Zeng ◽  
Yixin Li ◽  
Zhongyuan Jiang ◽  
Jianfeng Ma
Keyword(s):  
Data Availability ◽  
Low Rank ◽  
Data Sets ◽  
Low Rank Approximation ◽  
High Data ◽  
Spectral Graph ◽  
The Matrix ◽  
Rank Approximation ◽  
Distance Vector ◽  
Better Than

It is crucial to generate random graphs with specific structural properties from real graphs, which could anonymize graphs or generate targeted graph data sets. The state-of-the-art method called spectral graph forge (SGF) was proposed at INFOCOM 2018. This method uses a low-rank approximation of the matrix by throwing away some spectrums, which provides privacy protection after distributing graphs while ensuring data availability to a certain extent. As shown in SGF, it needs to discard at least 20% spectrum to defend against deanonymous attacks. However, the data availability will be significantly decreased after more spectrum discarding. Thus, is there a way to generate a graph that guarantees maximum spectrum and anonymity at the same time? To solve this problem, this paper proposes graph nonlinear scaling (GNS). We firmly prove that GNS can preserve all eigenvectors meanwhile providing high anonymity for the forged graph. Precisely, the GNS scales the eigenvalues of the original spectrum and constructs the forged graph with scaled eigenvalues and original eigenvectors. This approach maximizes the preservation of spectrum information to guarantee data availability. Meanwhile, it provides high robustness towards deanonymous attacks. The experimental results show that when SGF discards only 10% of the spectrum, the forged graph has high data availability. At this time, if the distance vector deanonymity algorithm is used to attack the forged graph, almost 100% of the nodes can be identified, while when achieving the same availability, only about 20% of the nodes in the forged graph obtained from GNS can be identified. Moreover, our method is better than SGF in capturing the real graph’s structure in terms of modularity, the number of partitions, and average clustering.

Exact First Order Finite Element Modeling for Eddy Current NDE

1991 ◽  
Vol 3 (1) ◽  
pp. 235-253 ◽  
Author(s):  
L. D. Philipp ◽  
Q. H. Nguyen ◽  
D. D. Derkacht ◽  
D. J. Lynch ◽  
A. Mahmood
Keyword(s):  
Finite Element ◽  
Finite Element Modeling ◽  
Eddy Current ◽  
Element Modeling ◽  
First Order ◽  
Eddy Current Nde
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