scholarly journals Fast Quadrature Techniques for Retarded Potentials Based on TT/QTT Tensor Approximation

2001 ◽  
Vol 11 (3) ◽  
pp. 342-362 ◽  
Author(s):  
Boris Khoromskij ◽  
Stefan Sauter ◽  
Alexander Veit
Keyword(s):  
Substantial Reduction ◽  
Computational Cost ◽  
Three Dimensional ◽  
Unbounded Domains ◽  
Boundary Integral ◽  
Basis Functions ◽  
Quadrature Rules ◽  
Galerkin Approach ◽  
Tensor Approximation ◽  
Element Matrix

AbstractWe consider the Galerkin approach for the numerical solution of retarded boundary integral formulations of the three dimensional wave equation in unbounded domains. Recently smooth and compactly supported basis functions in time were introduced which allow the use of standard quadrature rules in order to compute the entries of the boundary element matrix. In this paper, we use TT and QTT tensor approximations to increase the efficiency of these quadrature rules. Various numerical experiments show the substantial reduction of the computational cost that is needed to obtain accurate approximations for the arising integrals.

Reduced Material Model of Composite Laminates for 3D Finite Element Analysis

2014 ◽  
Author(s):  
Goldy Kumar ◽  
Vadim Shapiro
Keyword(s):  
Finite Element ◽  
Structural Analysis ◽  
Composite Laminates ◽  
Substantial Reduction ◽  
Computational Cost ◽  
Material Model ◽  
Basis Functions ◽  
Constitutive Relationship ◽  
Element Analysis ◽  
3D Fea

Laminate composites are widely used in automotive, aerospace, medical, and increasingly in consumer industries, due to their reduced weight, superior structural properties and cost-effectiveness. However, structural analysis of complex laminate structures remains challenging. 2D finite element methods based on plate and shell theories may be accurate and efficient, but they generally do not apply to the whole structure, and require identification and preprocessing (dimensional reduction) of the regions where the underlying assumptions hold. Differences in and limitations of theories for thin/thick plates and shells further complicate modeling and simulation of composites. Fully automated structural analysis using 3D elements with sufficiently high order basis functions is possible in principle, but is rarely practiced due to the significant increase in computational integration cost in the presence of a large number of laminate plies. We propose to replace the actual layup of the laminate structure by a simplified material model, allowing for a substantial reduction of the computational cost of 3D FEA. The reduced model, under the usual assumptions made in lamination theory, has the same constitutive relationship as the corresponding 2D plate model of the original laminate, but requires only a small fraction of computational integration costs in 3D FEA. We describe implementation of 3D FEA using the reduced material model in a meshfree system using second order B-spline basis functions. Finally, we demonstrate its validity by showing agreement between computed and known results for standard problems.

A Parallel Domain Decomposition BEM Algorithm for Three-Dimensional Exponentially Graded Elasticity

2008 ◽  
Vol 75 (5) ◽  
Author(s):  
J. E. Ortiz ◽  
W. A. Shelton ◽  
V. Mantič ◽  
R. Criado ◽  
L. J. Gray ◽  
...  
Keyword(s):  
Analytical Solution ◽  
Domain Decomposition ◽  
Functionally Graded Material ◽  
Computational Cost ◽  
Three Dimensional ◽  
Boundary Integral ◽  
Functionally Graded ◽  
Graded Material ◽  
High Computational Cost ◽  
Exponentially Graded

A parallel domain decomposition boundary integral algorithm for three-dimensional exponentially graded elasticity has been developed. As this subdomain algorithm allows the grading direction to vary in the structure, geometries arising from practical functionally graded material applications can be handled. Moreover, the boundary integral algorithm scales well with the number of processors, also helping to alleviate the high computational cost of evaluating the Green’s functions. For axisymmetric plane strain states in a radially graded material, the numerical results for cylindrical geometries are in excellent agreement with the analytical solution deduced herein.

scholarly journals Stress Analysis of Three-Dimensional Media Containing Localized Zone by FEM-SGBEM Coupling

2011 ◽  
Vol 2011 ◽  
pp. 1-27
Author(s):  
Jaroon Rungamornrat ◽  
Sakravee Sripirom
Keyword(s):  
Stress Analysis ◽  
Boundary Element ◽  
Integral Equations ◽  
Boundary Integral Equations ◽  
Numerical Technique ◽  
Computational Cost ◽  
Three Dimensional ◽  
Boundary Integral ◽  
Boundary Element Methods ◽  
Element Method

This paper presents an efficient numerical technique for stress analysis of three-dimensional infinite media containing cracks and localized complex regions. To enhance the computational efficiency of the boundary element methods generally found inefficient to treat nonlinearities and non-homogeneous data present within a domain and the finite element method (FEM) potentially demanding substantial computational cost in the modeling of an unbounded medium containing cracks, a coupling procedure exploiting positive features of both the FEM and a symmetric Galerkin boundary element method (SGBEM) is proposed. The former is utilized to model a finite, small part of the domain containing a complex region whereas the latter is employed to treat the remaining unbounded part possibly containing cracks. Use of boundary integral equations to form the key governing equation for the unbounded region offers essential benefits including the reduction of the spatial dimension and the corresponding discretization effort without the domain truncation. In addition, all involved boundary integral equations contain only weakly singular kernels thus allowing continuous interpolation functions to be utilized in the approximation and also easing the numerical integration. Nonlinearities and other complex behaviors within the localized regions are efficiently modeled by utilizing vast features of the FEM. A selected set of results is then reported to demonstrate the accuracy and capability of the technique.

scholarly journals Alternating directions parallel hybrid memory iGRM direct solver for non-stationary simulations

2020 ◽  
Vol 21 (4) ◽  
Author(s):  
Maciej Woźniak ◽  
Anna Janina Bukowska
Keyword(s):  
Isogeometric Analysis ◽  
Parallel Machines ◽  
Sparse Matrix ◽  
Computational Cost ◽  
Three Dimensional ◽  
Modern Method ◽  
Basis Functions ◽  
Computational Domain ◽  
Hybrid Memory ◽  
Hybrid Parallelization

The three-dimensional isogeometric analysis (IGA-FEM) is a modern method for simulation. The idea is to utilize B-splines or NURBS basis functions for both computational domain descriptions and the engineering computations. Refined isogeometric analysis (rIGA) employs a mixture of patches of elements with B-spline basis functions, and $C^0$ separators between them. It enables a reduction of the computational cost of direct solvers. Both IGA and rIGA come with challenging sparse matrix structure, that is expensive to generate. In this paper, we show a hybrid parallelization method to reduce the computational cost of the integration phase using hybrid-memory parallel machines. The two-level parallelization includes the partitioning of the computational mesh into sub-domains on the first level (MPI), and loop parallelization on the second level (OpenMP). We show that hybrid parallelization of the integration reduces the contribution of this phase significantly. Thus, alternative algorithms for fast isogeometric integration are not necessary.

scholarly journals Three-Dimensional Elastodynamic Analysis Employing Partially Discontinuous Boundary Elements

Algorithms ◽  
2021 ◽  
Vol 14 (5) ◽  
pp. 129
Author(s):  
Yuan Li ◽  
Ni Zhang ◽  
Yuejiao Gong ◽  
Wentao Mao ◽  
Shiguang Zhang
Keyword(s):  
Side Effect ◽  
Domain Boundary ◽  
Three Dimensional ◽  
Boundary Integral ◽  
Integration Scheme ◽  
Computational Accuracy ◽  
Time Step ◽  
Corner Points ◽  
Discontinuous Elements ◽  
The Time Domain

Compared with continuous elements, discontinuous elements advance in processing the discontinuity of physical variables at corner points and discretized models with complex boundaries. However, the computational accuracy of discontinuous elements is sensitive to the positions of element nodes. To reduce the side effect of the node position on the results, this paper proposes employing partially discontinuous elements to compute the time-domain boundary integral equation of 3D elastodynamics. Using the partially discontinuous element, the nodes located at the corner points will be shrunk into the element, whereas the nodes at the non-corner points remain unchanged. As such, a discrete model that is continuous on surfaces and discontinuous between adjacent surfaces can be generated. First, we present a numerical integration scheme of the partially discontinuous element. For the singular integral, an improved element subdivision method is proposed to reduce the side effect of the time step on the integral accuracy. Then, the effectiveness of the proposed method is verified by two numerical examples. Meanwhile, we study the influence of the positions of the nodes on the stability and accuracy of the computation results by cases. Finally, the recommended value range of the inward shrink ratio of the element nodes is provided.

scholarly journals The ICON Single-Column Mode

Atmosphere ◽  
2021 ◽  
Vol 12 (7) ◽  
pp. 906
Author(s):  
Ivan Bašták Ďurán ◽  
Martin Köhler ◽  
Astrid Eichhorn-Müller ◽  
Vera Maurer ◽  
Juerg Schmidli ◽  
...  
Keyword(s):  
Data Storage ◽  
Model Development ◽  
Computational Cost ◽  
Three Dimensional ◽  
Shallow Convection ◽  
Modeling Framework ◽  
Additional Tool ◽  
Eddy Simulation ◽  
Single Column ◽  
Stratocumulus Clouds

The single-column mode (SCM) of the ICON (ICOsahedral Nonhydrostatic) modeling framework is presented. The primary purpose of the ICON SCM is to use it as a tool for research, model evaluation and development. Thanks to the simplified geometry of the ICON SCM, various aspects of the ICON model, in particular the model physics, can be studied in a well-controlled environment. Additionally, the ICON SCM has a reduced computational cost and a low data storage demand. The ICON SCM can be utilized for idealized cases—several well-established cases are already included—or for semi-realistic cases based on analyses or model forecasts. As the case setup is defined by a single NetCDF file, new cases can be prepared easily by the modification of this file. We demonstrate the usage of the ICON SCM for different idealized cases such as shallow convection, stratocumulus clouds, and radiative transfer. Additionally, the ICON SCM is tested for a semi-realistic case together with an equivalent three-dimensional setup and the large eddy simulation mode of ICON. Such consistent comparisons across the hierarchy of ICON configurations are very helpful for model development. The ICON SCM will be implemented into the operational ICON model and will serve as an additional tool for advancing the development of the ICON model.

scholarly journals Retinal vessel segmentation with constrained-based nonnegative matrix factorization and 3D modified attention U-Net

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Yang Yu ◽  
Hongqing Zhu
Keyword(s):  
Matrix Factorization ◽  
Nonnegative Matrix Factorization ◽  
Computational Cost ◽  
Three Dimensional ◽  
Nonnegative Matrix ◽  
Retinal Vessel ◽  
Classification Problem ◽  
Image Resolution ◽  
Retinal Vessels ◽  
Segmentation Task

AbstractDue to the complex morphology and characteristic of retinal vessels, it remains challenging for most of the existing algorithms to accurately detect them. This paper proposes a supervised retinal vessels extraction scheme using constrained-based nonnegative matrix factorization (NMF) and three dimensional (3D) modified attention U-Net architecture. The proposed method detects the retinal vessels by three major steps. First, we perform Gaussian filter and gamma correction on the green channel of retinal images to suppress background noise and adjust the contrast of images. Then, the study develops a new within-class and between-class constrained NMF algorithm to extract neighborhood feature information of every pixel and reduce feature data dimension. By using these constraints, the method can effectively gather similar features within-class and discriminate features between-class to improve feature description ability for each pixel. Next, this study formulates segmentation task as a classification problem and solves it with a more contributing 3D modified attention U-Net as a two-label classifier for reducing computational cost. This proposed network contains an upsampling to raise image resolution before encoding and revert image to its original size with a downsampling after three max-pooling layers. Besides, the attention gate (AG) set in these layers contributes to more accurate segmentation by maintaining details while suppressing noises. Finally, the experimental results on three publicly available datasets DRIVE, STARE, and HRF demonstrate better performance than most existing methods.

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