Application of Groebner Basis Methodology to Nonlinear Analysis of an Underwater Cable

2011 ◽  
Author(s):  
Y. Jane Liu ◽  
George R. Buchanan
Keyword(s):  
Galerkin Method ◽  
Nonlinear Analysis ◽  
Analytical Solutions ◽  
Large Deflection ◽  
Geometrically Nonlinear ◽  
Groebner Basis ◽  
Governing Equations ◽  
Geometrically Nonlinear Analysis ◽  
Highly Nonlinear ◽  
The Galerkin Method

The governing equations for a large deflection cable analysis have a highly nonlinear and coupled nature. As a result, analytical solutions are unavailable or limited to a few simplified cases. The present study is to apply the Groebner Basis methodology combined with the Galerkin method to a geometrically nonlinear analysis of an underwater cable as an example.

Application of Groebner Basis Methodology to Nonlinear Static Cable Analysis

2013 ◽  
Vol 135 (4) ◽  
Author(s):  
Y. Jane Liu ◽  
George R. Buchanan ◽  
John Peddieson
Keyword(s):  
Nonlinear Analysis ◽  
Analytical Solutions ◽  
Geometrically Nonlinear ◽  
Large Deflections ◽  
Groebner Basis ◽  
Governing Equations ◽  
Geometrically Nonlinear Analysis ◽  
Exact Analytical Solutions ◽  
Highly Nonlinear ◽  
Nonlinear Static

The governing equations for large deflections of cables have a highly nonlinear and coupled nature, which precludes exact analytical solutions except in a few simplified cases. The present study demonstrates the utility of Groebner Basis methodology in facilitating the construction of approximate analytical and semianalytical Galerkin solutions in the geometrically nonlinear analysis of cable statics.

Extension of the unsymmetric 8-node hexahedral solid element US-ATFH8 to geometrically nonlinear analysis

2021 ◽  
Vol ahead-of-print (ahead-of-print) ◽  
Author(s):  
Zhi Li ◽  
Song Cen ◽  
Chenfeng Li
Keyword(s):  
Nonlinear Analysis ◽  
Linear Elasticity ◽  
Analytical Solutions ◽  
Geometrically Nonlinear ◽  
Governing Equations ◽  
Geometrically Nonlinear Analysis ◽  
Content Type ◽  
Solid Element ◽  
Key Innovation ◽  
Updated Lagrangian

Purpose The purpose of this paper is to extend a recent unsymmetric 8-node, 24-DOF hexahedral solid element US-ATFH8 with high distortion tolerance, which uses the analytical solutions of linear elasticity governing equations as the trial functions (analytical trial function) to geometrically nonlinear analysis. Design/methodology/approach Based on the assumption that these analytical trial functions can still properly work in each increment step during the nonlinear analysis, the present work concentrates on the construction of incremental nonlinear formulations of the unsymmetric element US-ATFH8 through two different ways: the general updated Lagrangian (UL) approach and the incremental co-rotational (CR) approach. The key innovation is how to update the stresses containing the linear analytical trial functions. Findings Several numerical examples for 3D structures show that both resulting nonlinear elements, US-ATFH8-UL and US-ATFH8-CR, perform very well, no matter whether regular or distorted coarse mesh is used, and exhibit much better performances than those conventional symmetric nonlinear solid elements. Originality/value The success of the extension of element US-ATFH8 to geometrically nonlinear analysis again shows the merits of the unsymmetric finite element method with analytical trial functions, although these functions are the analytical solutions of linear elasticity governing equations.

A GALERKIN MESHFREE METHOD WITH STABILIZED CONFORMING NODAL INTEGRATION FOR GEOMETRICALLY NONLINEAR ANALYSIS OF SHEAR DEFORMABLE PLATES

2011 ◽  
Vol 08 (04) ◽  
pp. 685-703 ◽  
Author(s):  
DONGDONG WANG ◽  
YUE SUN
Keyword(s):  
Nonlinear Analysis ◽  
Large Deflection ◽  
Lagrangian Description ◽  
Geometrically Nonlinear ◽  
Geometrically Nonlinear Analysis ◽  
Nodal Integration ◽  
Galerkin Meshfree ◽  
Shear Deformable ◽  
Shear Deformable Plates ◽  
Stabilized Conforming Nodal Integration

A Galerkin meshfree approach formulated within the framework of stabilized conforming nodal integration (SCNI) is presented for geometrically nonlinear analysis of large deflection shear deformable plates. This method is based upon a Lagrangian curvature smoothing in which the smoothed curvature is constructed within a nodal representative domain on the initial configuration. It is shown that the Lagrangian smoothed nodal gradients of the meshfree shape function is capable of exactly representing arbitrary constant curvature fields in the discrete sense of nodal integration. The consistent linearization is performed on the weak form of large deflection plate in the context of the total Lagrangian description. Subsequently, the discrete incremental equations are obtained by the method of SCNI in which to relieve the locking as well as ensure the stability of the present scheme, the bending contribution is evaluated using the smoothed nodal gradients, while the membrane and shear contributions are computed with the direct nodal gradients. The effectiveness of the present method is thoroughly demonstrated through several numerical examples.

scholarly journals Vibration of rotating circular cylindrical shells with distributed springs

2021 ◽  
Vol 37 ◽  
pp. 346-358
Author(s):  
Fuchun Yang ◽  
Xiaofeng Jiang ◽  
Fuxin Du
Keyword(s):  
Boundary Conditions ◽  
Galerkin Method ◽  
Shell Theory ◽  
Rotation Speed ◽  
Cylindrical Shells ◽  
Free Vibrations ◽  
Governing Equations ◽  
Natural Response ◽  
Circular Cylindrical Shells ◽  
The Galerkin Method

Abstract Free vibrations of rotating cylindrical shells with distributed springs were studied. Based on the Flügge shell theory, the governing equations of rotating cylindrical shells with distributed springs were derived under typical boundary conditions. Multicomponent modal functions were used to satisfy the distributed springs around the circumference. The natural responses were analyzed using the Galerkin method. The effects of parameters, rotation speed, stiffness, and ratios of thickness/radius and length/radius, on natural response were also examined.

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