scholarly journals Gauss-Legendre Numerical Integrations over a Quadrilateral Element in Closed Form

1970 ◽  
Vol 46 (3) ◽  
pp. 399-405 ◽  
Author(s):  
MS Islam ◽  
G Saha ◽  
N Akter
Keyword(s):  
Closed Form ◽  
Coordinate Transformation ◽  
Stiffness Matrix ◽  
Quadrature Rule ◽  
Computational Time ◽  
Quadrilateral Element ◽  
Memory Space ◽  
The Matrix ◽  
Quadrilateral Finite Element ◽  
Legendre Quadrature

In this paper we investigate the stiffness matrix of a general quadrilateral element in closed form using n x n Gauss-Legendre quadrature rule. For this, we propose four types of nodal coordinate transformation. The terms of the matrix are divided into two groups, namely - diagonal and non-diagonal. Only one term (called leading) from each group is computed, and then the remaining fourteen terms are computed from these two leading terms exploited one of the proposed types of coordinate transformation. This leads us a great savings in computational time and memory space. In order to compute the matrix we use these transformations in two ways, and thus two algorithms are given to generate the matrix. Finally, numerical example is given to verify the effectiveness of the present formulation. Keywords: Gauss-Legendre quadrature; Numerical integration; Quadrilateral finite element; Stiffness matrix; Closed form. DOI: http://dx.doi.org/10.3329/bjsir.v46i3.9050 BJSIR 2011; 46(3): 399-405

A Simple and Efficient Large-Displacement Formulation for Beam Elements

1991 ◽  
Author(s):  
Xiaodong Tang
Keyword(s):  
Finite Element ◽  
Closed Form ◽  
Stiffness Matrix ◽  
Second Order ◽  
Small Displacement ◽  
Element Formulation ◽  
Beam Elements ◽  
The Matrix ◽  
Element Approach ◽  
Additive Relation

Abstract A simple and effective finite element formulation of beam-column stiffness matrix with the consideration of second-order displacement is presented. The derived second-order stiffness matrix is the simple addition of the small-displacement stiffness matrix and a few other component matrices. The matrix is derived closed form and no integration is necessary during the formation of the element stiffness matrix. The simple additive relation makes it easy to understand the composition of the second-order stiffness matrix and the meaning of each component matrix. The closed-form formulation eliminates the potential numerical error and considerably reduces the amount of computation thereby, increasing the computing speed. The standard finite element approach presented in this paper makes the beam-column compatible with other finite elements and can be incorporated into existing finite element programs with little modification. The analysis of a simply supported beam with axial restraints using the beam-column element shows a good correlation with the test results.

scholarly journals A Neural Network-Inspired Matrix Formulation of Chemical Kinetics for Acceleration on GPUs

Energies ◽  
2021 ◽  
Vol 14 (9) ◽  
pp. 2710
Author(s):  
Shivam Barwey ◽  
Venkat Raman
Keyword(s):  
Neural Network ◽  
Chemical Kinetics ◽  
Graphics Processing Units ◽  
Source Term ◽  
Turbulent Flames ◽  
Peak Performance ◽  
Computational Time ◽  
Detailed Chemical Kinetics ◽  
The Matrix ◽  
Artificial Neural

High-fidelity simulations of turbulent flames are computationally expensive when using detailed chemical kinetics. For practical fuels and flow configurations, chemical kinetics can account for the vast majority of the computational time due to the highly non-linear nature of multi-step chemistry mechanisms and the inherent stiffness of combustion chemistry. While reducing this cost has been a key focus area in combustion modeling, the recent growth in graphics processing units (GPUs) that offer very fast arithmetic processing, combined with the development of highly optimized libraries for artificial neural networks used in machine learning, provides a unique pathway for acceleration. The goal of this paper is to recast Arrhenius kinetics as a neural network using matrix-based formulations. Unlike ANNs that rely on data, this formulation does not require training and exactly represents the chemistry mechanism. More specifically, connections between the exact matrix equations for kinetics and traditional artificial neural network layers are used to enable the usage of GPU-optimized linear algebra libraries without the need for modeling. Regarding GPU performance, speedup and saturation behaviors are assessed for several chemical mechanisms of varying complexity. The performance analysis is based on trends for absolute compute times and throughput for the various arithmetic operations encountered during the source term computation. The goals are ultimately to provide insights into how the source term calculations scale with the reaction mechanism complexity, which types of reactions benefit the GPU formulations most, and how to exploit the matrix-based formulations to provide optimal speedup for large mechanisms by using sparsity properties. Overall, the GPU performance for the species source term evaluations reveals many informative trends with regards to the effect of cell number on device saturation and speedup. Most importantly, it is shown that the matrix-based method enables highly efficient GPU performance across the board, achieving near-peak performance in saturated regimes.

scholarly journals Explicit Dynamic Finite Element Method for Failure with Smooth Fracture Energy Dissipations

2013 ◽  
Vol 2013 ◽  
pp. 1-12 ◽  
Author(s):  
Jeong-Hoon Song ◽  
Thomas Menouillard ◽  
Alireza Tabarraei
Keyword(s):  
Finite Element ◽  
Fracture Energy ◽  
Computational Cost ◽  
Quadrature Rule ◽  
Dynamic Failure ◽  
Structural Dynamic ◽  
Integration Schemes ◽  
Hourglass Control ◽  
Quadrilateral Finite Element ◽  
Explicit Dynamic

A numerical method for dynamic failure analysis through the phantom node method is further developed. A distinct feature of this method is the use of the phantom nodes with a newly developed correction force scheme. Through this improved approach, fracture energy can be smoothly dissipated during dynamic failure processes without emanating noisy artifact stress waves. This method is implemented to the standard 4-node quadrilateral finite element; a single quadrature rule is employed with an hourglass control scheme in order to decrease computational cost and circumvent difficulties associated with the subdomain integration schemes for cracked elements. The effectiveness and robustness of this method are demonstrated with several numerical examples. In these examples, we showed the effectiveness of the described correction force scheme along with the applicability of this method to an interesting class of structural dynamic failure problems.

scholarly journals Collaborative Filtering Recommendation Using Nonnegative Matrix Factorization in GPU-Accelerated Spark Platform

2021 ◽  
Vol 2021 ◽  
pp. 1-15
Author(s):  
Bing Tang ◽  
Linyao Kang ◽  
Li Zhang ◽  
Feiyan Guo ◽  
Haiwu He
Keyword(s):  
Collaborative Filtering ◽  
Processing Speed ◽  
Matrix Factorization ◽  
Nonnegative Matrix Factorization ◽  
Nonnegative Matrix ◽  
Experimental Results ◽  
Computational Time ◽  
Data Sets ◽  
Heterogeneous Cluster ◽  
The Matrix

Nonnegative matrix factorization (NMF) has been introduced as an efficient way to reduce the complexity of data compression and its capability of extracting highly interpretable parts from data sets, and it has also been applied to various fields, such as recommendations, image analysis, and text clustering. However, as the size of the matrix increases, the processing speed of nonnegative matrix factorization is very slow. To solve this problem, this paper proposes a parallel algorithm based on GPU for NMF in Spark platform, which makes full use of the advantages of in-memory computation mode and GPU acceleration. The new GPU-accelerated NMF on Spark platform is evaluated in a 4-node Spark heterogeneous cluster using Google Compute Engine by configuring each node a NVIDIA K80 CUDA device, and experimental results indicate that it is competitive in terms of computational time against the existing solutions on a variety of matrix orders. Furthermore, a GPU-accelerated NMF-based parallel collaborative filtering (CF) algorithm is also proposed, utilizing the advantages of data dimensionality reduction and feature extraction of NMF, as well as the multicore parallel computing mode of CUDA. Using real MovieLens data sets, experimental results have shown that the parallelization of NMF-based collaborative filtering on Spark platform effectively outperforms traditional user-based and item-based CF with a higher processing speed and higher recommendation accuracy.

The Elastic Field of an Elliptic Inclusion With a Slipping Interface

1986 ◽  
Vol 53 (1) ◽  
pp. 103-107 ◽  
Author(s):  
E. Tsuchida ◽  
T. Mura ◽  
J. Dundurs
Keyword(s):  
Closed Form ◽  
Infinite Series ◽  
Elastic Field ◽  
Elliptic Inclusion ◽  
Numerical Examples ◽  
Elastic Fields ◽  
Bonded Interface ◽  
The Matrix ◽  
Displacement Potentials ◽  
Uniform Expansion

The paper analyzes the elastic fields caused by an elliptic inclusion which undergoes a uniform expansion. The interface between the inclusion and the matrix cannot sustain shear tractions and is free to slip. Papkovich–Neuber displacement potentials are used to solve the problem. In contrast to the perfectly bonded interface, the solution cannot be expressed in closed form and involves infinite series. The results are illustrated by numerical examples.

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