A Finite Element Formulation for Unbounded Homogeneous Continua

1982 ◽  
Vol 49 (1) ◽  
pp. 136-140 ◽  
Author(s):  
G. Dasgupta
Keyword(s):  
Finite Element ◽  
Radiation Condition ◽  
Dynamic Responses ◽  
Element Formulation ◽  
Infinite Domain ◽  
Stiffness Matrices ◽  
Matrix Coefficients ◽  
The Matrix ◽  
Quadratic Matrix Equation ◽  
Quadratic Eigenvalue

Currently available finite element formulations to model dynamic responses of unbounded continua cannot accommodate the Sommerfeld radiation condition. Discrete numerical techniques explicitly rely on analytical expressions of frequency-dependent field variables to account for the energy loss associated with outgoing waves. In the proposed method, such a dependence is eliminated. For steady dynamic responses, the analog of the quadratic eigenvalue problem (occurring in continuum mechanics) is herein constructed in the form of a quadratic matrix equation. The unknown relates to the desired stiffness matrix pertaining to the infinite or semi-infinite domain. The matrix coefficients are obtained from the conventional mass and stiffness matrices of a suitably chosen, bounded finite element. A benchmark example is included in this paper to demonstrate the very high numerical accuracy and significant computational efficiency of the proposed cloning algorithm.

scholarly journals Reordering edges and elements in unstructured meshes to reduce execution time in Finite Element Computations

Nova Scientia ◽  
2018 ◽  
Vol 10 (20) ◽  
pp. 263-279
Author(s):  
Gerardo Mario Ortigoza Capetillo ◽  
Alberto Pedro Lorandi Medina ◽  
Alfonso Cuauhtemoc García Reynoso
Keyword(s):  
Finite Element ◽  
Sparse Matrix ◽  
Unstructured Meshes ◽  
Finite Element Formulation ◽  
Element Formulation ◽  
Vector Product ◽  
Stiffness Matrices ◽  
The Matrix ◽  
Compressed Sparse Row ◽  
Reduce Execution Time

Reverse Cuthill McKee (RCM) reordering can be applied to either edges or elements of unstructured meshes (triangular/tetrahedral) , in accordance to the respective finite element formulation,  to reduce the bandwidth of stiffness matrices . Grid generators are mainly designed for nodal based finite elements. Their output is a list of nodes (2d or 3d) and an array describing element connectivity, be it triangles or tetrahedra. However,  for edge-defined finite element formulations a numbering of the edges is required. Observations are reported for Triangle/Tetgen Delaunay grid generators and for the sparse structure of the assembled matrices in both edge- and element-defined formulations. The RCM is a renumbering algorithm traditionally applied to the nodal graph of the mesh. Thus, in order to apply this renumbering to either the edges or the elements of the respective finite element formulation,  graphs of the mesh were generated. Significant bandwidth reduction was obtained. This translates to reduction in the execution effort of the sparse-matrix-times-vector product. Compressed Sparse Row format was adopted and the matrix-times-vector product was implemented in an OpenMp parallel routine.

scholarly journals A Kollár and Pluzsik anisotropic composite beam theory for arbitrary multicelled cross sections

2016 ◽  
Vol 35 (23) ◽  
pp. 1696-1711 ◽  
Author(s):  
Danilo S Victorazzo ◽  
Andre De Jesus
Keyword(s):  
Finite Element ◽  
Cross Sections ◽  
Composite Beam ◽  
Linear Equations ◽  
Beam Theory ◽  
Element Formulation ◽  
Compliance Matrix ◽  
Asymmetric System ◽  
Anisotropic Composite ◽  
Stiffness Matrices

In this paper we extend Kollár and Pluzsik’s thin-walled anisotropic composite beam theory to include multiple cells with open branches and booms, and present a finite element formulation utilizing the stiffness matrix obtained from this theory. To recover the 4 × 4 compliance matrix of a beam containing N closed cells, we solve an asymmetric system of 2N + 4 linear equations four times with unitary section loads and extract influence coefficients from the calculated strains. Finally, we compare 4 × 4 stiffness matrices of a multicelled beam using this method against matrices obtained using the finite element method to demonstrate accuracy. Similarly to its originating theory, the effects of shear deformation and restrained warping are assumed negligible.

scholarly journals Analysis of Stationary Random Responses for Non-Parametric Probabilistic Systems

2010 ◽  
Vol 17 (3) ◽  
pp. 305-315 ◽  
Author(s):  
Y. Zhao ◽  
Y.H. Zhang ◽  
J.H. Lin ◽  
W.P. Howson ◽  
F.W. Williams
Keyword(s):  
Finite Element ◽  
Finite Element Model ◽  
Element Model ◽  
Physical Structure ◽  
Dynamic Responses ◽  
Response Analysis ◽  
Random Matrix Model ◽  
Stiffness Matrices ◽  
Element Technique ◽  
Non Parametric

The move from conceptual design, through fabrication to observation and measurement on the resulting physical structure is fraught with uncertainty. This, together with the necessary simplifications inherent when using the finite element technique, makes the development of a predictive model for the physical structure sufficiently approximate that the use of random structural models is often to be preferred. In this paper, the random uncertainties of the mass, damping and stiffness matrices in a finite element model are replaced by random matrices, and a highly efficient pseudo excitation method for the dynamic response analysis of non-parametric probability systems subjected to stationary random loads is developed. A numerical example shows that the dynamic responses calculated using a conventional (mean) finite element model may be quite different from those based on a random matrix model. For precise fabrication, the uncertainties of models cannot be ignored and the proposed method should be useful in the analysis of such problems.

scholarly journals A Numerical Method for Solving 3D Elasticity Equations with Sharp-Edged Interfaces

2013 ◽  
Vol 2013 ◽  
pp. 1-13 ◽  
Author(s):  
Liqun Wang ◽  
Songming Hou ◽  
Liwei Shi
Keyword(s):  
Finite Element ◽  
Piecewise Linear ◽  
Main Idea ◽  
Three Dimensions ◽  
Test Function ◽  
Linear System Of Equations ◽  
Elasticity Equations ◽  
Matrix Coefficients ◽  
The Matrix ◽  
3D Elasticity

Interface problems occur frequently when two or more materials meet. Solving elasticity equations with sharp-edged interfaces in three dimensions is a very complicated and challenging problem for most existing methods. There are several difficulties: the coupled elliptic system, the matrix coefficients, the sharp-edged interface, and three dimensions. An accurate and efficient method is desired. In this paper, an efficient nontraditional finite element method with nonbody-fitting grids is proposed to solve elasticity equations with sharp-edged interfaces in three dimensions. The main idea is to choose the test function basis to be the standard finite element basis independent of the interface and to choose the solution basis to be piecewise linear satisfying the jump conditions across the interface. The resulting linear system of equations is shown to be positive definite under certain assumptions. Numerical experiments show that this method is second order accurate in the L∞ norm for piecewise smooth solutions. More than 1.5th order accuracy is observed for solution with singularity (second derivative blows up).

scholarly journals Natural cubic element formulation and infinite domain modelling for potential flow problems

2003 ◽  
Vol 45 (1) ◽  
pp. 133-143
Author(s):  
G. A. Mohr ◽  
A. S. Power
Keyword(s):  
Finite Element ◽  
Potential Flow ◽  
Element Formulation ◽  
Infinite Domain ◽  
Simple Formulation ◽  
Infinite Domains ◽  
Flow Problems ◽  
Body Forces ◽  
Domain Modelling ◽  
Conditions At Infinity

AbstractA simple formulation of a 9 df cubic Hermitian finite element for potential flow problems is given, using the interpolation of the BCIZ element and after Argyris, defining natural velocities parallel to the element sides. Consistent loads for body forces are also derived and it is shown that these are necessary to obtain accurate results when body forces are significant. Example problems include those of infinite domains for which simpleconditions at infinityare used.

scholarly journals The stress analysis of a shear wall with matrix displacement method using rectangular finite element

2021 ◽  
Vol 4 (1) ◽  
pp. 18-27
Author(s):  
Mustafa Ergün ◽  
Şevket Ateş
Keyword(s):  
Finite Element ◽  
Stress Analysis ◽  
Plane Stress ◽  
Shear Wall ◽  
Element Formulation ◽  
The Finite Element Method ◽  
Displacement Method ◽  
Element Analysis ◽  
The Matrix ◽  
Characteristic Features

The aim in this study is to numerically present some characteristic features of the rectangular finite element using the matrix displacement method and to show the utility of this element in plane stress problems compared to the finite element method. The paper consisted of three parts. In the first part, all of the finite element formulation steps from choosing the convenient coordinate system to creating element stiffness matrix are presented respectively. In the second part of the study, a static finite element analysis of the shear wall is also made by ANSYS Mechanical APDL. In the final part, the results (displacements, strains and stresses) obtained from the previous parts are compared with each other by the help of tables and graphics. The results show that this method is effective and preferable for the stress analysis of shell structures. Further studies should be conducted in order to indicate the efficiency of the matrix displacement method for the solution of different types of plane stress problems using different finite elements.

Layer-wise finite-element analysis of a functionally graded hollow thick cylinder with a piezoelectric ring

2011 ◽  
Vol 225 (5) ◽  
pp. 1045-1060 ◽  
Author(s):  
M H Yas ◽  
M Shakeri ◽  
M Khanjani
Keyword(s):  
Finite Element ◽  
Potential Energy ◽  
Virtual Work ◽  
Electrical Potential ◽  
Element Model ◽  
Functionally Graded ◽  
Dynamic Responses ◽  
Element Formulation ◽  
Element Analysis ◽  
Thick Cylinder

In this work, a layer-wise finite-element formulation is developed for the analysis of a functionally graded material (FGM) hollow thick cylinder with one piezoactuator ring. The cylinder and ring is divided into many sublayers in the thickness direction and the full layer-wise shell theory is used to model a discretely stiffened FGM cylinder. In this model, the displacements are approximated linearly through each mathematical layer. This accounts for any discontinuities in the derivatives of the displacement at the interface of the ring and the cylindrical thick shell. This formulation is derived from the virtual work statement which includes the total structural potential energy and the electrical potential energy of the piezoelectric ring. Assembling stiffness and mass matrices, at each interface between two elements, stress and displacement continuity are forced, and then the finite-element model is solved. Static and dynamic responses of a functionally graded thick cylinder to electrical and mechanical loads with different exponent ‘ n’ of FGM are determined to show the significant influence of the material in homogeneity. The results obtained at a distance far from the ring are compared with the mechanical behaviour of an FGM cylindrical shell without a ring. Because of the Saint Venant effects, the piezoelectrically induced deformation of the shell is confined to a region close to the piezoelectric ring; thus agreements between these two results are observed.

A computerised design aid for drive shafts with some fatigue considerations

1986 ◽  
Vol 21 (2) ◽  
pp. 85-98 ◽  
Author(s):  
D Walton ◽  
S Prayoonrat
Keyword(s):  
Finite Element ◽  
Power Transmission ◽  
Cyclic Loads ◽  
Element Analysis ◽  
Bevel Gears ◽  
Stiffness Matrices ◽  
The Matrix ◽  
Machine Elements ◽  
Stress Concentration Factors ◽  
Design Factors

A method of designing power transmission drive shafts is described based on two interactive computer programs. The first program can model shafts of any configuration and under any system of loads and supports. The program is based on the finite element method. Reducing the size of the stiffness matrix by storing only the parameters in the matrix bandwidth and by overlaying the stiffness matrices, the problem of using finite element analysis on small memory microcomputers can be solved. A menu allows the designer to specify machine elements, such as spur or bevel gears or pulleys, to be positioned at any point along the shaft. The user can give information of the power transmitted, speed, and details of the physical dimensions of the machine elements which then enables the program to determine the force vectors and apply these to the shaft nodes. The second program determines the design factors of safety on the basis of both static and cyclic loads. For cyclic loads the factors of safety are based on infinite-life design, using Soderberg and Goodman criteria. The program contains extensive built-in data on endurance strengths, factors influencing the endurance limit, and stress concentration factors.

scholarly journals Dynamic Analysis of Thick Plates Including Deep Beams on Elastic Foundations Using Modified Vlasov Model

2013 ◽  
Vol 20 (1) ◽  
pp. 29-41 ◽  
Author(s):  
Korhan Ozgan
Keyword(s):  
Finite Element ◽  
Dynamic Analysis ◽  
Plate Theory ◽  
Beam Theory ◽  
Mindlin Plate ◽  
Element Formulation ◽  
Soil System ◽  
Stiffness Matrices ◽  
Foundation Model ◽  
Foundation Plate

Dynamic analysis of foundation plate-beam systems with transverse shear deformation is presented using modified Vlasov foundation model. Finite element formulation of the problem is derived by using an 8-node (PBQ8) finite element based on Mindlin plate theory for the plate and a 2-node Hughes element based on Timoshenko beam theory for the beam. Selective reduced integration technique is used to avoid shear locking problem for the evaluation of the stiffness matrices for both the elements. The effect of beam thickness, the aspect ratio of the plate and subsoil depth on the response of plate-beam-soil system is analyzed. Numerical examples show that the displacement, bending moments and shear forces are changed significantly by adding the beams.

Application of Composite Plate Theory and the Finite Element Method to the Dynamics and Stress Analysis of Spatial Flexible Mechanical Systems

1994 ◽  
Vol 116 (3) ◽  
pp. 952-960 ◽  
Author(s):  
J. M. Kremer ◽  
A. A. Shabana ◽  
G. E. O. Widera
Keyword(s):  
Finite Element ◽  
Rigid Body ◽  
Stress Analysis ◽  
Arbitrary Number ◽  
Plate Theory ◽  
Mass Matrix ◽  
Composite Plates ◽  
The Body ◽  
Element Formulation ◽  
Stiffness Matrices

This investigation concerns itself with the dynamic and stress analysis of thin, laminated composite plates consisting of layers of orthotropic laminae. It is assumed that the bonds between the laminae are infinitesimally thin and shear nondeformable. The finite element formulation presented is sufficiently general to accept an arbitrary number of layers and an arbitrary number of orthotropic material property sets. In the dynamic formulation presented, the laminae is assumed to undergo large arbitrary rigid body displacements and small elastic deformations. The nodal shape functions of the laminae are assumed to have rigid body modes that need to describe only large rigid body translations. Using the expressions for the kinetic and strain energies, the lamina mass and stiffness matrices are identified. The nonlinear mass matrix of the lamina is expressed in terms of a set of invariants that depend on the assumed displacement field. By summing the laminae kinetic and strain energies, the body mass and stiffness matrices are identified. It is shown that the body invariants can be expressed explicitly in terms of the invariants of its laminae. Numerical examples of a spatial RSSR mechanism are presented in order to demonstrate the use of the present formulation.

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