A composite triangular equilibrium element for modelling thin plate behaviour is investigated when shear deformation, according to Reissner's theory, is included in addition to…
Abstract
A composite triangular equilibrium element for modelling thin plate behaviour is investigated when shear deformation, according to Reissner's theory, is included in addition to bending deformation. The element flexibility matrix is formed as the sum of two component matrices containing the contributions from piecewise linear moment fields and piecewise constant shear fields separately. Properties of the shear component matrix, including its rank, are determined, and the influence of load basis on the pattern of this matrix is studied. The way these properties affect the condition of a flexibility matrix is investigated when the element size/thickness ratio varies, and particularly when this ratio tends towards zero. Load bases are suggested which could avoid certain problems of ill‐conditioning. Condition parameters are evaluated for an equilateral element. Finally, the application of kinematic boundary conditions to the equilibrium is considered.
This paper offers a simple valid alternative approach to the formulation of finite element models. Using two‐dimensional steady state heat conduction as a representative problem…
Abstract
This paper offers a simple valid alternative approach to the formulation of finite element models. Using two‐dimensional steady state heat conduction as a representative problem, the paper formalizes and extends a direct approach, normally restricted to constant strain triangular (CST) elements, to the construction of general finite element models. The approach is illustrated with the conventional 4, 8, and 9 noded quadrilateral elements. The sufficient conditions for convergence are explained in straightforward physical terms. The direct approach is based on the dual roles of shape functions, and does not require the use of variational methods or functional analysis. The formulation is shown to be equivalent to a weak form of Galerkin which makes explicit use of three forms of residual. This approach is intended to enable engineering students to gain a sound understanding of basic finite element methods with a minimum of mathematics.
Presents a review on implementing finite element methods on supercomputers, workstations and PCs and gives main trends in hardware and software developments. An appendix included…
Abstract
Presents a review on implementing finite element methods on supercomputers, workstations and PCs and gives main trends in hardware and software developments. An appendix included at the end of the paper presents a bibliography on the subjects retrospectively to 1985 and approximately 1,100 references are listed.
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Li Wang, Mengwu Guo and Hongzhi Zhong
– The purpose of this paper is to acquire strict upper and lower bounds on quantities of slender beams on Winkler foundation in finite element analysis.
Abstract
Purpose
The purpose of this paper is to acquire strict upper and lower bounds on quantities of slender beams on Winkler foundation in finite element analysis.
Design/methodology/approach
It leans on the dual analysis wherein the constitutive relation error (CRE) is used to perform goal-oriented error estimation. Due to the coupling of the displacement field and the stress field in the equilibrium equations of the beam, the prolongation condition for the stress field which is the key ingredient of CRE estimation is not directly applicable. To circumvent this difficulty, an approximate problem and the solution thereof are introduced, enabling the CRE estimation to proceed. It is shown that the strict bounding property for CRE estimation is preserved and strict bounds of quantities of the beam are obtainable thereafter.
Findings
Numerical examples are presented to validate the strict upper and lower bounds for quantities of beams on elastic foundation by dual analysis.
Research limitations/implications
This paper deals with one-dimensional (1D) beams on Winkler foundation. Nevertheless, the present work can be naturally extended to analysis of shells and 2D and 3D reaction-diffusion problems for future research.
Originality/value
CRE estimation is extended to analysis of beams on elastic foundation by a decoupling strategy; strict upper bounds of global energy norm error for beams on elastic foundation are obtained; strict bounds of quantities for beams on elastic foundation are also obtained; unified representation and corresponding dual analysis of various quantities of the beam are presented; rigorous derivation of admissible stresses for beams is given.
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Yan Shang, Song Cen and Wengen Ouyan
The purpose of this paper is to propose a new finite element method (FEM) solving strategy for efficient analysis of the challenging edge effect problem in plate structures. Its…
Abstract
Purpose
The purpose of this paper is to propose a new finite element method (FEM) solving strategy for efficient analysis of the challenging edge effect problem in plate structures. Its main ideas are to develop special-purpose plate element models to effectively simulate the behaviors in the plate’s edge zones near free/SS1 edges.
Design/methodology/approach
These new elements are developed based on the hybrid-Trefftz element method. During their construction procedures, the analytical solutions of the edge effect problem, which are in exponential forms, are used to enhance the interior displacement fields. Besides, the Lagrangian multipliers are introduced into the modified hybrid-Trefftz functional for considering the stress resultant constraints at free/SS1 edges. Thus, these elements theoretically possess the abilities to correctly capture the very steep gradients of the resultant distributions in the boundary layers.
Findings
These new specialized hybrid-Trefftz plate elements can very efficiently solve the edge effect problem with high accuracy, even when distorted meshes are used. Moreover, because these elements’ construction procedures contain only boundary integrals, the computation expense for accurately integrating the exponential trial functions can be considerably saved.
Originality/value
This work presents an alternative novel idea for using the FEM to more effectively handle the local stress problems by incorporating the use of the analytical trial functions.
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Maryam Daei and S. Hamid Mirmohammadi
The efficiency of the finite element analysis via force method depends on the overall flexibility matrix of the structure, while this matrix is directly affected from null bases…
Abstract
Purpose
The efficiency of the finite element analysis via force method depends on the overall flexibility matrix of the structure, while this matrix is directly affected from null bases vectors. As the null bases for an indeterminate structure are not unique, for an optimal analysis, the selected null bases should be sparse and banded corresponding to sparse, banded and well-conditioned flexibility matrix. This paper aims to present an efficient method for the formation of optimal flexibility matrix of finite element models comprising tetrahedron elements via mathematical optimization technique.
Design/methodology/approach
For this purpose, a linear mixed integer programming model is presented for finding sparse solution of underdetermined linear system, which is correspond to sparse null vector. The charged system search algorithm is improved and used to find the best generator for formation of null bases.
Findings
The efficiency of the present method is illustrated through some examples. The proposed method leads to highly sparse, banded and accurate null basis matrices. It makes an efficient force method feasible for the analysis of finite element model comprising tetrahedron elements.
Originality/value
The force method, in which the member forces are used as unknowns, can be appealing to engineers. The main problem in the application of the force method is the formation of a self-stress matrix corresponding to a sparse flexibility matrix. In this paper, the highly sparse, banded and accurate null basis matrices gains by using mathematical optimization technique.
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Farzad Shafiei Dizaji and Mehrdad Shafiei Dizaji
The purpose is to reduce round-off errors in numerical simulations. In the numerical simulation, different kinds of errors may be created during analysis. Round-off error is one…
Abstract
Purpose
The purpose is to reduce round-off errors in numerical simulations. In the numerical simulation, different kinds of errors may be created during analysis. Round-off error is one of the sources of errors. In numerical analysis, sometimes handling numerical errors is challenging. However, by applying appropriate algorithms, these errors are manageable and can be reduced. In this study, five novel topological algorithms were proposed in setting up a structural flexibility matrix, and five different examples were used in applying the proposed algorithms. In doing so round-off errors were reduced remarkably.
Design/methodology/approach
Five new algorithms were proposed in order to optimize the conditioning of structural matrices. Along with decreasing the size and duration of analyses, minimizing analytical errors is a critical factor in the optimal computer analysis of skeletal structures. Appropriate matrices with a greater number of zeros (sparse), a well structure and a well condition are advantageous for this objective. As a result, a problem of optimization with various goals will be addressed. This study seeks to minimize analytical errors such as rounding errors in skeletal structural flexibility matrixes via the use of more consistent and appropriate mathematical methods. These errors become more pronounced in particular designs with ill-suited flexibility matrixes; structures with varying stiffness are a frequent example of this. Due to the usage of weak elements, the flexibility matrix has a large number of non-diagonal terms, resulting in analytical errors. In numerical analysis, the ill-condition of a matrix may be resolved by moving or substituting rows; this study examined the definition and execution of these modifications prior to creating the flexibility matrix. Simple topological and algebraic features have been mostly utilized in this study to find fundamental cycle bases with particular characteristics. In conclusion, appropriately conditioned flexibility matrices are obtained, and analytical errors are reduced accordingly.
Findings
(1) Five new algorithms were proposed in order to optimize the conditioning of structural flexibility matrices. (2) A JAVA programming language was written for all five algorithms and a friendly GUI software tool is developed to visualize sub-optimal cycle bases. (3) Topological and algebraic features of the structures were utilized in this study.
Research limitations/implications
This is a multi-objective optimization problem which means that sparsity and well conditioning of a matrix cannot be optimized simultaneously. In conclusion, well-conditioned flexibility matrices are obtained, and analytical errors are reduced accordingly.
Practical implications
Engineers always finding mathematical modeling of real-world problems and make them as simple as possible. In doing so, lots of errors will be created and these errors could cause the mathematical models useless. Applying decent algorithms could make the mathematical model as precise as possible.
Social implications
Errors in numerical simulations should reduce due to the fact that they are toxic for real-world applications and problems.
Originality/value
This is an original research. This paper proposes five novel topological mathematical algorithms in order to optimize the structural flexibility matrix.
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Gives a bibliographical review of the error estimates and adaptive finite element methods from the theoretical as well as the application point of view. The bibliography at the…
Abstract
Gives a bibliographical review of the error estimates and adaptive finite element methods from the theoretical as well as the application point of view. The bibliography at the end contains 2,177 references to papers, conference proceedings and theses/dissertations dealing with the subjects that were published in 1990‐2000.
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This paper seeks to present an efficient algorithm for the formation of null basis for finite element model discretized as rectangular bending elements. The bases obtained by this…
Abstract
Purpose
This paper seeks to present an efficient algorithm for the formation of null basis for finite element model discretized as rectangular bending elements. The bases obtained by this algorithm correspond to highly sparse and narrowly banded flexibility matrices and such bases can be considered as an efficient tool for optimal analysis of structures.
Design/methodology/approach
In the present method, two graphs are associated with finite element mesh consisting of an “interface graph” and an “associate digraph”. The underlying subgraphs of the self‐equilibrating systems (SESs) (null vectors) are obtained by graph theoretical approaches forming a null basis. Application of unit loads (moments) at the end of the generator of each subgraph results in the corresponding null vector.
Findings
In the present hybrid method, graph theory is used for the formation of null vectors as far as it is possible and then algebraic method is utilized to find the complementary part of the null basis.
Originality/value
This hybrid approach makes the use of pure force method in the finite element analysis feasible. Here, a simplified version of the algorithm is also presented where the SESs for weighted graphs are obtained using an analytical approach. Thus, the formation of null bases is achieved using the least amount of algebraic operations, resulting in substantial saving in computational time and storage.
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A bibliographical review of the finite element methods (FEMs) applied for the linear and nonlinear, static and dynamic analyses of basic structural elements from the theoretical…
Abstract
A bibliographical review of the finite element methods (FEMs) applied for the linear and nonlinear, static and dynamic analyses of basic structural elements from the theoretical as well as practical points of view is given. The bibliography at the end of the paper contains 1,726 references to papers, conference proceedings and theses/dissertations dealing with the analysis of beams, columns, rods, bars, cables, discs, blades, shafts, membranes, plates and shells that were published in 1996‐1999.