• Structural Engineering and Mechanics
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• 제42권4호
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• pp.471-488
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• 2012

An efficient one dimensional finite element model has been presented for the dynamic analysis of composite laminated beams, using the efficient layerwise zigzag theory. To meet the convergence requirements for the weak integral formulation, cubic Hermite interpolation is used for the transverse displacement ($w_0$), and linear interpolation is used for the axial displacement ($u_0$) and shear rotation (${\psi}_0$). Each node of an element has four degrees of freedom. The expressions of variationally consistent inertia, stiffness matrices and the load vector are derived in closed form using exact integration. The formulation is validated by comparing the results with the 2D-FE results for composite symmetric and sandwich beams with various end conditions. The employed finite element model is free of shear locking. The present zigzag finite element results for natural frequencies, mode shapes of cantilever and clamped-clamped beams are obtained with a one-dimensional finite element codes developed in MATLAB. These 1D-FE results for cantilever and clamped beams are compared with the 2D-FE results obtained using ABAQUS to show the accuracy of the developed MATLAB code, for zigzag theory for these boundary conditions. This comparison establishes the accuracy of zigzag finite element analysis for dynamic response under given boundary conditions.

• Journal of the Computational Structural Engineering Institute of Korea
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• 제20권2호
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• pp.113-125
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• 2007

Improved numerical method to obtain the exact stiffness matrices is newly proposed to perform the spatially coupled elastic and stability analyses of non-symmetric and open/closed thin-walled beam on elastic foundation. This method overcomes drawbacks of the previous method to evaluate the exact stiffness matrix for the spatially coupled stability analysis of thin-walled beam-column This numerical technique is accomplished via a generalized eigenproblem associated with 14 displacement parameters by transforming equilibrium equations to a set of first order simultaneous ordinary differential equations. Next polynomial expressions as trial solutions are assumed for displacement parameters corresponding to zero eigenvalues and the eigenmodes containing undetermined parameters equal to the number of zero eigenvalues are determined by invoking the identity condition. And then the exact displacement functions are constructed by combining eigensolutions and polynomial solutions corresponding to non-zero and zero eigenvalues, respectively. Consequently an exact stiffness matrix is evaluated by applying the member force-deformation relationships to these displacement functions. In order to illustrate the accuracy and the practical usefulness of this study, the numerical solutions are compared with results obtained from the thin-walled beam and shell elements.

• Journal of the Korean Vacuum Society
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• 제21권3호
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• pp.171-177
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• 2012

The computer program (EPR-NMR program version 6.2) employed here sets up the spin Hamiltonian matrices and determines their eigenvalues using exact diagonalization. We study the electron spin resonance for $Mn^{2+}$ in ferroelectric $LiNbO_3$ single crystals. The self-energy is obtained using the projection operator method developed by Argyres and Sigel. The self-energy is calculated to be axially symmetric about the by the spin Hamiltonian. The line-widths decreased as the temperature increased; we assume that the hyperfine structure transition is a more dominant scattering than the other transitions. We conclude that the calculation process presented in this study is useful for quantum optical transitions.

• Transactions of the Korean Society of Mechanical Engineers A
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• 제21권5호
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• pp.707-718
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• 1997

This paper presents an efficient numerical method for the computation of eigenpair derivatives for a real symmetric eigenvalue problem with distinct and multiple eigenvalues. The method has a very simple algorithm and gives an exact solution. Furthermore, it saves computer sotrage and CPU time. The algorithm preserves not only the symmetricity but also the band width of the matrices, allowing efficient computer storage and solution techniques. Results from the proposed method for calculating the eigenpair derivatives are compared with those from Rudisill and Chu's method and Nelson's method which is known efficient one in the case of distinct natural frequencies. As an example to demonstrate the efficiency of the proposed method in the case of distinct eigenvalues, a cantilever plate is considered. The design parameter of the cantilever plate is its thickness. For the eigenvalue problem with multiple natural frequencies, the adjacent eigenvectors are used in the algebraic equation as side conditions, lying adjacent to the multiplicity of multiple natural frequency distinct eigenvalues, which appear when design parameter varies. A cantilever beam is used to demonstrate the efficiency of the proposed method in the case of multiple natural frequencies. Results form the proposed method for calculating the eigenpair derivatives are compared with those from Dailey's method(an amendation of Ojalvo's work) which finds the exact eigenvector derivatives. The design parameter of the cantilever beam is its height. Data is presented showing the amount of CPU time used to compute the first ten eigenpair derivatives by each method. It is important to note that the numerical stability of the proposed method is proved.

• Structural Engineering and Mechanics
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• 제58권1호
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• pp.59-91
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• 2016

Laminated composite shells are commonly used in various engineering applications including aerospace and marine structures. In this paper, using semi-analytical finite strip method, the buckling behavior of laminated composite deep as well as thick shells of revolution under follower forces which remain normal to the shell is investigated. The stiffness caused by pressure is calculated for the follower forces subjected to external fibers in thick shells. The shell is divided into several closed strips with alignment of their nodal lines in the circumferential direction. The governing equations are derived based on first-order shear deformation theory which accounts for through thickness-shear flexibility. Displacements and rotations in the middle surface of shell are approximated by combining polynomial functions in the meridional direction as well as truncated Fourier series with an appropriate number of harmonic terms in the circumferential direction. The load stiffness matrix which accounts for variation of loads direction will be derived for each strip of the shell. Assembling of these matrices results in global load stiffness matrix which may be un-symmetric. Upon forming linear elastic stiffness matrix called constitutive stiffness matrix, geometric stiffness matrix and load stiffness matrix, the required elements for the second step analysis which is an eigenvalue problem are provided. In this study, different parameter effects are investigated including shell geometry, material properties, and different boundary conditions. Afterwards, the outcomes are compared with other researches. By considering the results of this article, it can be concluded that the deformation-dependent pressure assumption can entail to decrease the calculated buckling load in shells. This characteristic is studied for different examples.

• KSCE Journal of Civil and Environmental Engineering Research
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• 제8권2호
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• pp.13-24
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• 1988

A geometrically nonlinear analysis procedure for plane frame structures in order to study the static and dynamic post-buckling behavior of these structures subjected to circulatory forces is presented. The elastic and geometric stiffness matrices, the mass matrix and load correction stiffness matrix are derived from the extended virtual work principle, where the tangent stiffness matrix becomes non-symmetric due to the effects of non-conservative circulatory forces. The dynamic analysis of plane frame structures subjected to circulatory forces in pre- and post-buckling ranges is carried out by integrating the equations of motion directly by the numerically stable Newmark method. Numerical results are presented in order to demonstrate the vality and accuracy of the proposed procedure.

• Proceedings of the Computational Structural Engineering Institute Conference
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• 한국전산구조공학회 1995년도 가을 학술발표회 논문집
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• pp.225-233
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• 1995

This paper presents an efficient numerical method for computation of eigenpair derivatives for the real symmetric eigenvalue problem with distinct and multiple eigenvalues. The method has very simple algorithm and gives an exact solution. Furthermore, it saves computer storage and CPU time. The algorithm preserves the symmetry and band of the matrices, allowing efficient computer storage and solution techniques. Thus, the algorithm of the proposed method will be inserted easily in the commercial FEM codes. Results of the proposed method for calculating the eigenpair derivatives are compared with those of Rudisill and Chu's method and Nelson's method which is efficient one in the case of distinct natural frequencies. As an example to demonstrate the efficiency of the proposed method in the case of distinct eigenvalues, a cantilever plate is considered. The design parameter of the cantilever plate is its thickness. For the eigenvalue problem with multiple natural frequencies, the adjacent eigenvectors are used in the algebraic equation as side conditions, they lie adjacent to the m (multiplicity of multiple natural frequency) distinct eigenvalues, which appear when design parameter varies. As an example to demonstrate the efficiency of the proposed method in the case of multiple natural frequencies, a cantilever beam is considered. Results of the proposed method fDr calculating the eigenpair derivatives are compared with those of Bailey's method (an amendation of Ojalvo's work) which finds the exact eigenvector derivatives. The design parameter of the cantilever beam is its height. Data is persented showing the amount of CPU time used to compute the first ten eigenpair derivatives by each method. It is important to note that the numerical stability of the proposed method is proved.

• Computational Structural Engineering
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• 제10권4호
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• pp.205-216
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• 1997

In order to trace the lateral post-buckling behaviors of thin-wafled space frames, a geometrically nonlinear finite element formulation is presented by applying incremental equilibrium equations based on the updated Lagrangian formulation and introducing Vlasov's assumption. The improved displacement field for symmetric thin-walled cross sections is introduced based on inclusion of second order terms of finite rotations, and the potential energy corresponding to the semitangential rotations and moments is consistently derived. For finite element analysis, tangent stiffness matrices of the thinwalled space frame element with 7 degrees of freedom including the restrained warping for each node are derived by using the Hermition polynomials as shape functions. A co-rotational formulation in order to evaluate the unbalanced loads is presented by separating the rigid body rotations and pure deformations from incremental displacements and evaluating the updated direction cosines of the frame element due to rigid body rotations and incremental member forces from pure deformations. Finite element solutions for the spatial buckling and post-buckling analysis of thin-walled space frames are presented and compared with available solutions and other researcher's results.

• Computational Structural Engineering
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• 제10권1호
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• pp.201-211
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• 1997

A clearly consistent finite element formulation for geometrically non-linear analysis of space frames is presented by applying incremental equilibrium equations based on the updated Lagrangian formulation and introducing Vlasov's assumption. The improved displacement field for symmetric cross sections is introduced based on inclusion of second order terms of finite rotations, and the potential energy corresponding to the semitangential rotations and moments is consistently derived. For finite element analysis, elastic and geometric stiffness matrices of the space frame element are derived by using the Hermitian polynomials as shape functions. A co-rotational formulation in order to evaluate the unbalanced loads is presented by separating the rigid body rotations and pure deformations from incremental displacements and evaluating the updated direction cosines of the frame element due to rigid body rotations and incremental member forces from pure deformaions. Finite element solutions for the spatial buckling and post-buckling analysis of space frames are compared with available solutions and other researcher's results.

• Journal of the Korea Society of Computer and Information
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• 제26권1호
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• pp.1-9
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• 2021

OpenGL compute shader is a shader stage that operate differently from other shader stage and it can be used for the calculating purpose of any data in parallel. This paper proposes a GPU-based parallel algorithm for computing sparse linear systems through conjugate gradient using an iterative method, which perform calculation on OpenGL compute shader. Basically, this sparse linear solver is used to solve large linear systems such as symmetric positive definite matrix. Four well-known matrix formats (Dense, COO, ELL and CSR) have been used for matrix storage. The performance comparison from our experimental tests using eight sparse matrices shows that GPU-based linear solving system much faster than CPU-based linear solving system with the best average computing time 0.64ms in GPU-based and 15.37ms in CPU-based.