Schur complement spectral bounds for large hybrid FETI-DP clusters and huge three-dimensional scalar problems

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Zdeněk Dostál ◽  
Tomáš Brzobohatý ◽  
Oldřich Vlach
Keyword(s):  
Condition Number ◽  
Three Dimensional ◽  
Schur Complement ◽  
Decomposition Methods ◽  
Stiffness Matrices ◽  
Spectral Bounds ◽  
Schur Complements ◽  
Linear Problems ◽  
Primal Method ◽  
Large Clusters

Abstract Bounds on the spectrum of Schur complements of subdomain stiffness matrices with respect to the interior variables are key ingredients of the convergence analysis of FETI (finite element tearing and interconnecting) based domain decomposition methods. Here we give bounds on the regular condition number of Schur complements of “floating” clusters arising from the discretization of 3D Laplacian on a cube decomposed into cube subdomains. The results show that the condition number of the cluster defined on a fixed domain decomposed into m × m × m cube subdomains connected by face and optionally edge averages increases proportionally to m. The estimates support scalability of unpreconditioned H-FETI-DP (hybrid FETI dual-primal) method. Though the research is most important for the solution of variational inequalities, the results of numerical experiments indicate that unpreconditioned H-FETI-DP with large clusters can be useful also for the solution of huge linear problems.

Three-Dimensional Parametric Finite-Volume Homogenization of Periodic Materials with Multi-Scale Structural Applications

2018 ◽  
Vol 10 (04) ◽  
pp. 1850045 ◽  
Author(s):  
Qiang Chen ◽  
Guannan Wang ◽  
Xuefeng Chen
Keyword(s):  
Finite Volume ◽  
Three Dimensional ◽  
Functionally Graded ◽  
Attractive Alternative ◽  
Matrix Assembly ◽  
Functionally Graded Composite ◽  
Multi Scale ◽  
Hexahedral Elements ◽  
Stiffness Matrices ◽  
Structural Applications

In order to satisfy the increasing computational demands of micromechanics, the Finite-Volume Direct Averaging Micromechanics (FVDAM) theory is developed in three-dimensional (3D) domain to simulate the multiphase heterogeneous materials whose microstructures are distributed periodically in the space. Parametric mapping, which endorses arbitrarily shaped and oriented hexahedral elements in the microstructure discretization, is employed in the unit cell solution. Unlike the finite-element (FE) technique, the expressions for local stiffness matrices are derived explicitly, enabling efficient global stiffness matrix assembly using an easily implementable algorithm. To demonstrate the accuracy and efficiency of the proposed theory, the homogenized moduli and localized stress distributions produced by the FE analyses are given for comparisons, where excellent agreement is always obtained for the 3D microstructures with different geometrical and material properties. Finally, a multi-scale stress analysis of functionally graded composite cylinders is conducted. This extension further increases the FVDAM’s range of applicability and opens new opportunities for pursuing other areas, providing an attractive alternative to the FE-based approaches that may be compared.

scholarly journals Multi-Aspect Incremental Tensor Decomposition Based on Distributed In-Memory Big Data Systems

2020 ◽  
Vol 5 (2) ◽  
pp. 13-32
Author(s):  
Hye-Kyung Yang ◽  
Hwan-Seung Yong
Keyword(s):  
Execution Time ◽  
Three Dimensional ◽  
Large Data ◽  
Tensor Decomposition ◽  
Decomposition Methods ◽  
Apache Spark ◽  
Decomposition Algorithms ◽  
Data Systems ◽  
Big Data Systems ◽  
Spark Framework

AbstractPurposeWe propose InParTen2, a multi-aspect parallel factor analysis three-dimensional tensor decomposition algorithm based on the Apache Spark framework. The proposed method reduces re-decomposition cost and can handle large tensors.Design/methodology/approachConsidering that tensor addition increases the size of a given tensor along all axes, the proposed method decomposes incoming tensors using existing decomposition results without generating sub-tensors. Additionally, InParTen2 avoids the calculation of Khari–Rao products and minimizes shuffling by using the Apache Spark platform.FindingsThe performance of InParTen2 is evaluated by comparing its execution time and accuracy with those of existing distributed tensor decomposition methods on various datasets. The results confirm that InParTen2 can process large tensors and reduce the re-calculation cost of tensor decomposition. Consequently, the proposed method is faster than existing tensor decomposition algorithms and can significantly reduce re-decomposition cost.Research limitationsThere are several Hadoop-based distributed tensor decomposition algorithms as well as MATLAB-based decomposition methods. However, the former require longer iteration time, and therefore their execution time cannot be compared with that of Spark-based algorithms, whereas the latter run on a single machine, thus limiting their ability to handle large data.Practical implicationsThe proposed algorithm can reduce re-decomposition cost when tensors are added to a given tensor by decomposing them based on existing decomposition results without re-decomposing the entire tensor.Originality/valueThe proposed method can handle large tensors and is fast within the limited-memory framework of Apache Spark. Moreover, InParTen2 can handle static as well as incremental tensor decomposition.

Fast Domain Decomposition Type Solver for Stiffness Matrices of Reference p-Elements

2013 ◽  
Vol 13 (2) ◽  
pp. 161-183 ◽  
Author(s):  
Vadim Korneev
Keyword(s):  
Finite Difference ◽  
Domain Decomposition ◽  
Elliptic Equations ◽  
Harmonic Functions ◽  
Schur Complement ◽  
P Elements ◽  
Stiffness Matrices ◽  
The Family ◽  
Discrete Harmonic Functions

Abstract. A key component of domain decomposition solvers for hp discretizations of elliptic equations is the solver for internal stiffness matrices of p-elements. We consider an algorithm which belongs to the family of secondary domain decomposition solvers, based on the finite-difference like preconditioning of p-elements, and was outlined by the author earlier. We remove the uncertainty in the choice of the coarse (decomposition) grid solver and suggest the new interface Schur complement preconditioner. The latter essentially uses the boundary norm for discrete harmonic functions induced by orthotropic discretizations on slim rectangles, which was derived recently. We prove that the algorithm has linear arithmetical complexity.

An efficient coupled pressure–velocity solver for three-dimensional injection molding simulation using Schur complement preconditioned FGMRES

2019 ◽  
Vol 36 (4) ◽  
pp. 1101-1120
Author(s):  
Xiang Liu ◽  
Fei Guo ◽  
Yun Zhang ◽  
Junjie Liang ◽  
Dequn Li ◽  
...  
Keyword(s):  
Injection Molding ◽  
Three Dimensional ◽  
Simple Algorithm ◽  
Schur Complement ◽  
Gradient Term ◽  
Content Type ◽  
Coupled Problem ◽  
Parallel Performance ◽  
Injection Molding Simulation ◽  
Fully Coupled

Purpose The purpose of this paper is to develop a coupled approach to solve the pressure–velocity-coupled problem efficiently in the three-dimensional injection molding simulation. Design/methodology/approach A fully coupled pressure–velocity algorithm is developed to solve the coupled problem, by treating the pressure gradient term implicitly. And, the Schur complement preconditioned FGMRES is applied to decompose the resulting coupled pressure–velocity equation into pressure and velocity subsystems. Then, BoomerAMG is adopted to solve the pressure subsystem, and block Jacobi preconditioned FGMRES is applied to the velocity subsystem. Findings According to the several experiments, the fully coupled pressure–velocity algorithm was demonstrated to have faster convergence than the traditional SIMPLE algorithm, and the calculating time was reduced by up to 70 per cent. And, the Schur complement preconditioned FGMRES worked more efficiently than block Gauss–Seidel preconditioned FGMRES, block-selective AMG and AMG with block ILU(0) smoother and could take at least 47.4 per cent less time. The proposed solver had good scalability for different-size problems, including various cases with different numbers of elements. It also kept good speedup and efficiency in parallel performance. Originality/value A coupled solver has been proposed to effectively solve the coupled problem in the three-dimensional injection molding simulation, which is more robust and efficient than existing methods.

scholarly journals Three-Dimensional Innervation Zone Imaging from Multi-Channel Surface EMG Recordings

2015 ◽  
Vol 25 (06) ◽  
pp. 1550024 ◽  
Author(s):  
Yang Liu ◽  
Yong Ning ◽  
Sheng Li ◽  
Ping Zhou ◽  
William Z. Rymer ◽  
...  
Keyword(s):  
High Performance ◽  
Unmet Need ◽  
Three Dimensional ◽  
Motor Units ◽  
Decomposition Methods ◽  
Surface Emg ◽  
Source Imaging ◽  
Dynamic Propagation ◽  
Channel Surface ◽  
Density Surface

There is an unmet need to accurately identify the locations of innervation zones (IZs) of spastic muscles, so as to guide botulinum toxin (BTX) injections for the best clinical outcome. A novel 3D IZ imaging (3DIZI) approach was developed by combining the bioelectrical source imaging and surface electromyogram (EMG) decomposition methods to image the 3D distribution of IZs in the target muscles. Surface IZ locations of motor units (MUs), identified from the bipolar map of their MU action potentials (MUAPs) were employed as a prior knowledge in the 3DIZI approach to improve its imaging accuracy. The performance of the 3DIZI approach was first optimized and evaluated via a series of designed computer simulations, and then validated with the intramuscular EMG data, together with simultaneously recorded 128-channel surface EMG data from the biceps of two subjects. Both simulation and experimental validation results demonstrate the high performance of the 3DIZI approach in accurately reconstructing the distributions of IZs and the dynamic propagation of internal muscle activities in the biceps from high-density surface EMG recordings.

Dexterity Analysis of 6-dof 3-urps Parallel Mechanism

2012 ◽  
Vol 591-593 ◽  
pp. 754-757
Author(s):  
Hong Li Yun ◽  
Xiao Na Song
Keyword(s):  
Condition Number ◽  
Parallel Mechanism ◽  
Three Dimensional ◽  
Inverse Solution ◽  
Kinematics Analysis ◽  
Characteristic Analysis ◽  
Kinematic Design ◽  
Ps I

The kinematics analysis is carried out on a kind of 3-urps parallel mechanism (pm) with 6-dof. The positional inverse solution and Jacobean matrixes which is used to analyze the dexterity of mechanism are derived. Condition number is used as evaluating indicator of dexterity in this article, and its’ three-dimensional graphs are drawn when this mechanism in different attitudes. It is indicated that dexterity of moving is excelled than the rotary movement’s for this pm. The optimum dexterity can be obtained in the center of mechanism’s workroom, which can be applied to further characteristic analysis and kinematic design of mechanisms.

Dynamic Finite Element Analysis of a Highly Parallel Robotic Surface

2011 ◽  
Author(s):  
Chris Salisbury
Keyword(s):  
Finite Element ◽  
Degrees Of Freedom ◽  
Angular Displacement ◽  
Three Dimensional ◽  
Element Analysis ◽  
Revolute Joints ◽  
Stiffness Matrices ◽  
Joint Forces ◽  
Ricatti Equation ◽  
Optimal Dynamic

A novel three-dimensional robotic surface is devised using triangular modules connected by revolute joints that mimic the constraints of a spherical joint at each triangle intersection. The finite element method (FEM) is applied to the dynamic loading of this device using three dimensional (6 degrees of freedom) beam elements to not only calculate the cartesian displacement and force, but also the angular displacement and torque at each joint. In this way, the traditional methods of finding joint forces and torques are completely bypassed. An effiecient algorithm is developed to linearly combine local mass and stiffness matrices into a full structural stiffness matrix for the easy application of loads. An analysis of optimal dynamic joint forces is carried out in Simulink® with the use of an algebraic Ricatti equation.

Sign in / Sign up

Export Citation Format

Share Document