scholarly journals Numerical evaluation of Coulomb integrals for 1, 2 and 3-electron distance operators, RC1-Nrd1-M, RC1-Nr12-M and R12-Nr13-M with real (N, M) and the Descartes product of 3 dimension common density functional numerical integration scheme

2019 ◽  
Author(s):  
Sandor Kristyan
Keyword(s):  
Numerical Integration ◽  
Density Functional ◽  
Numerical Evaluation ◽  
Integration Scheme ◽  
Numerical Integration Scheme ◽  
Coulomb Integrals

scholarly journals Numerical Evaluation of Coulomb Integrals For 1, 2 and 3-electron Distance Operators, RC1 -nRD1 -m , RC1 -n r12 -m and r12 -n r13 -m with Real (N, M) and The Descartes Product of 3 Dimension Common Density Functional Numerical Integration Scheme

2018 ◽  
Author(s):  
Sandor Kristyan
Keyword(s):  
Quantum Chemistry ◽  
Numerical Integration ◽  
Analytical Solutions ◽  
Density Functional ◽  
Integration Scheme ◽  
Functional Theory ◽  
Computational Quantum Chemistry ◽  
Numerical Integration Scheme ◽  
Analytical Integration ◽  
Coulomb Integrals

Analytical solutions to integrals are far more useful than numeric, however, the former is not available in many cases. We evaluate integrals indicated in the title numerically that are necessary in some approaches in quantum chemistry. In the title, where R stands for nucleus-electron and r for electron-electron distances, the n, m= 0 case is trivial, the (n, m)= (1,0) or (0,1) cases are well known, a fundamental milestone in the integration and widely used in computational quantum chemistry, as well as analytical integration is possible if Gaussian functions are used. For the rest of the cases the analytical solutions are restricted, but worked out for some, e.g. for n, m= 0,1,2 with Gaussians. In this work we generalize the Becke-Lebedev-Voronoi 3 dimensions numerical integration scheme (commonly used in density functional theory) to 6 and 9 dimensions via Descartes product to evaluate integrals indicated in the title, and test it. This numerical recipe (up to Gaussian integrands with seed exp(-|<b>r</b><sub>1</sub>|<sup>2</sup>), as well as positive and negative real n and m values) is useful for manipulation with higher moments of inter-electronic distances, for example, in correlation calculations; more, our numerical scheme works for Slaterian type functions with seed exp(-|<b>r</b><sub>1</sub>|) as well.

scholarly journals Numerical Evaluation of Coulomb Integrals For 1, 2 and 3-electron Distance Operators, RC1 -nRD1 -m , RC1 -n r12 -m and r12 -n r13 -m with Real (N, M) and The Descartes Product of 3 Dimension Common Density Functional Numerical Integration Scheme

2018 ◽  
Author(s):  
Sandor Kristyan
Keyword(s):  
Quantum Chemistry ◽  
Numerical Integration ◽  
Analytical Solutions ◽  
Density Functional ◽  
Integration Scheme ◽  
Functional Theory ◽  
Computational Quantum Chemistry ◽  
Numerical Integration Scheme ◽  
Analytical Integration ◽  
Coulomb Integrals

Analytical solutions to integrals are far more useful than numeric, however, the former is not available in many cases. We evaluate integrals indicated in the title numerically that are necessary in some approaches in quantum chemistry. In the title, where R stands for nucleus-electron and r for electron-electron distances, the n, m= 0 case is trivial, the (n, m)= (1,0) or (0,1) cases are well known, a fundamental milestone in the integration and widely used in computational quantum chemistry, as well as analytical integration is possible if Gaussian functions are used. For the rest of the cases the analytical solutions are restricted, but worked out for some, e.g. for n, m= 0,1,2 with Gaussians. In this work we generalize the Becke-Lebedev-Voronoi 3 dimensions numerical integration scheme (commonly used in density functional theory) to 6 and 9 dimensions via Descartes product to evaluate integrals indicated in the title, and test it. This numerical recipe (up to Gaussian integrands with seed exp(-|<b>r</b><sub>1</sub>|<sup>2</sup>), as well as positive and negative real n and m values) is useful for manipulation with higher moments of inter-electronic distances, for example, in correlation calculations; more, our numerical scheme works for Slaterian type functions with seed exp(-|<b>r</b><sub>1</sub>|) as well.

Dynamic Analysis of Shipping Cask Subjected to Sequential Bolt Preload and Impact Loads

2009 ◽  
Author(s):  
Tsu-te Wu
Keyword(s):  
Finite Element ◽  
Dynamic Analysis ◽  
Numerical Integration ◽  
Dynamic Responses ◽  
Integration Scheme ◽  
Impact Loads ◽  
Element Mesh ◽  
Bolt Preload ◽  
Numerical Integration Scheme ◽  
Structural Responses

This paper presents an improved methodology for evaluating the dynamic responses of shipping casks subjected to the sequential HAC impact loads. The methodology utilizes the import technique of the finite-element mesh and the analytical results form one dynamic analysis using explicit numerical integration scheme into another dynamic analysis also using explicit numerical integration scheme. The new methodology presented herein has several advantages over conventional methods. An example problem is analyzed to illustrate the application of the present methodology in evaluating the structural responses of a shipping cask to the sequential HAC loading.

Inelastic, Nonlinear Analysis of Stiffened Shells of Revolution by Numerical Integration

1974 ◽  
Vol 96 (2) ◽  
pp. 121-130 ◽  
Author(s):  
H. S. Levine ◽  
V. Svalbonas
Keyword(s):  
Numerical Integration ◽  
Cross Sections ◽  
Thermal Loading ◽  
Large Deflection ◽  
Kinematic Hardening ◽  
Integration Scheme ◽  
Shells Of Revolution ◽  
Stiffened Shells ◽  
Numerical Integration Scheme ◽  
Deflection Analysis

This paper describes the latest addition to the STARS system of computer programs, STARS-2P, for the plastic, large deflection analysis of axisymmetrically loaded shells of revolution. The STARS system uses a numerical integration scheme to solve the governing differential equations. Several unique features for shell of revolution programs that are included in the STARS-2P program are described. These include orthotropic nonlinear kinematic hardening theory, a variety of shell wall cross sections and discrete ring stiffeners, cyclic and nonproportional mechanical and thermal loading capability, the coupled axisymmetric large deflection elasto-plastic torsion problem, an extensive restart option, arbitrary branching capability, and the provision for the inelastic treatment of smeared stiffeners, isogrid, and waffle wall constructions. To affirm the validity of the results, comparisons with available theoretical and experimental data are presented.

A Modified Gaussian Integration Scheme in Element Free Method

2002 ◽  
Vol 18 (1) ◽  
pp. 17-27
Author(s):  
Jopan Sheng ◽  
Chung-Yue Wang ◽  
Kuo-Jui Shen
Keyword(s):  
Numerical Integration ◽  
Solid Mechanics ◽  
Gaussian Quadrature ◽  
Quadrature Rule ◽  
Integration Scheme ◽  
Quadrature Point ◽  
Physical Domain ◽  
Gaussian Integration ◽  
Numerical Integration Scheme ◽  
Element Free

ABSTRACTIn this paper, a modified numerical integration scheme is presented that improves the accuracy of the numerical integration of the Galerkin weak form, within the integration cells of the analyzed domain in the element-free methods. A geometrical interpretation of the Gaussian quadrature rule is introduced to map the effective weighting territory of each quadrature point in an integration cell. Then, the conventional quadrature rule is extended to cover the overlapping area between the weighting territory of each quadrature point and the physical domain. This modified numerical integration scheme can lessen the errors due to misalignment between the integration cell and the boundary or interface of the physical domain. Some numerical examples illustrate that this newly proposed integration scheme for element-free methods does effectively improve the accuracy when solving solid mechanics problems.

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