• Structural Engineering and Mechanics
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• v.42 no.1
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• pp.39-53
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• 2012

This paper presents an alternative way to derive the exact element stiffness matrix for a beam on Winkler foundation and the fixed-end force vector due to a linearly distributed load. The element flexibility matrix is derived first and forms the core of the exact element stiffness matrix. The governing differential compatibility of the problem is derived using the virtual force principle and solved to obtain the exact moment interpolation functions. The matrix virtual force equation is employed to obtain the exact element flexibility matrix using the exact moment interpolation functions. The so-called "natural" element stiffness matrix is obtained by inverting the exact element flexibility matrix. Two numerical examples are used to verify the accuracy and the efficiency of the natural beam element on Winkler foundation.

• The Transactions of the Korean Institute of Electrical Engineers C
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• v.49 no.4
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• pp.249-257
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• 2000

In this paper, we propose the use of stochastic finite element method, that is popularly employed in mechanical structure analysis, for more practical designing purpose of RF device. The proposed method is formulated based on the vector finite element method cooperated by pertubation analysis. The method utilizes sensitivity analysis algorithm with covariance matrix of the random variables that represent for uncertain physical quantities such as length or various electrical constants to compute the probabilities of the measure of performance of the structure. For this computation one need to know the variance and covariance of the random variables that might be determined by practical experiences. The presenting algorithm has been verified by analyzing several device with different be determined by practical experiences. The presenting algorithm has been verified by analysis several device with different measure of performanes. For the convenience of formulation, two dimensional analysis has been performed to apply it into waveguide with dielectric slab. In the problem the dielectric constant of the dielectric slab is considered as random variable. Another example is matched waveguide and cavity problem. In the problem, the dimension of them are assumed to be as random variables and the expectations and variances of quality factor have been computed.

• Transactions of the Korean Society of Mechanical Engineers A
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• v.25 no.11
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• pp.1730-1736
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• 2001

A finite element analysis for a rotating cantilever beam is presented in this study. Based on a dynamic modeling method using the stretch deformation instead of the conventional axial deformation, three linear partial differential equations are (derived from Hamilton's principle. Two of the linear differential equations show the coupling effect between stretch and chordwise deformations. The other equation is an uncoupled one for the flapwise deformation. From these partial differential equations and the associated boundary conditions, two weak forms are derived: one is for the chordwise motion and the other is fur the flptwise motion. The weak farms are spatially discretized with newly defined two-node beam elements. With the discretized equations or the matrix-vector equations, the behaviors of the natural frequencies are investigated for the variation of the rotating speed.

• Transactions of Materials Processing
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• v.26 no.5
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• pp.286-292
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• 2017

Magnesium alloy shows strong anisotropy and asymmetric behavior in tension and compression curve, especially at room temperature. These characteristics limit the application of finite element method (FEM) which is based on conventional continuum mechanics. To accurately predict the material behavior of magnesium alloy at microstructural level, a methodology of fully coupled multiscale simulation is presented and a crystal plasticity model as a constitutive equation in the simulation of metal forming process is introduced in this study. The existing constitutive equation for rigid plastic FEM is modified to accommodate deviatoric stress component and its derivatives with respect to strain rate components. Viscoplastic self-consistent (VPSC) polycrystal model was selected as a constitutive model because it was regarded as the most robust model compared to Taylor model or Sachs model. Stiffness matrix and load vector were derived based on the new approach and implemented into $DEFORM^{TM}-3D$ via a user subroutine handling stiffness matrix at an elemental level. The application to extrusion and rolling process of pure magnesium is presented in this study to assess the validity of the proposed multiscale process.

• Structural Engineering and Mechanics
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• v.85 no.2
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• pp.147-161
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• 2023

Based on the ideas of variational differential quadrature (VDQ) and finite element method (FEM), a numerical approach named as VDQFEM is applied herein to study the large deformations of plate-type structures under static loading with arbitrary shape hole made of functionally graded graphene platelet-reinforced composite (FG-GPLRC) in the context of higher-order shear deformation theory (HSDT). The material properties of composite are approximated based upon the modified Halpin-Tsai model and rule of mixture. Furthermore, various FG distribution patterns are considered along the thickness direction of plate for GPLs. Using novel vector/matrix relations, the governing equations are derived through a variational approach. The matricized formulation can be efficiently employed in the coding process of numerical methods. In VDQFEM, the space domain of structure is first transformed into a number of finite elements. Then, the VDQ discretization technique is implemented within each element. As the last step, the assemblage procedure is performed to derive the set of governing equations which is solved via the pseudo arc-length continuation algorithm. Also, since HSDT is used herein, the mixed formulation approach is proposed to accommodate the continuity of first-order derivatives on the common boundaries of elements. Rectangular and circular plates under various boundary conditions with circular/rectangular/elliptical cutout are selected to generate the numerical results. In the numerical examples, the effects of geometrical properties and reinforcement with GPL on the nonlinear maximum deflection-transverse load amplitude curve are studied.

• Proceedings of the Computational Structural Engineering Institute Conference
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• 2007.04a
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• pp.577-582
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• 2007

A variational formulation for plane elasticity problems is derived based on an isogeometric approach. The isogeometric analysis is an emerging methodology such that the basis functions in analysis domain arc generated directly from NURBS (Non-Uniform Rational B-Splines) geometry. Thus. the solution space can be represented in terms of the same functions to represent the geometry. The coefficients of basis functions or the control variables play the role of degrees-of-freedom. Furthermore, due to h-. p-, and k-refinement schemes, the high order geometric features can be described exactly and easily without tedious re-meshing process. The isogeometric sensitivity analysis method enables us to analyze arbitrarily shaped structures without re-meshing. Also, it provides a precise construction method of finite element model to exactly represent geometry using B-spline base functions in CAD geometric modeling. To obtain precise shape sensitivity, the normal and curvature of boundary should be taken into account in the shape sensitivity expressions. However, in conventional finite element methods, the normal information is inaccurate and the curvature is generally missing due to the use of linear interpolation functions. A continuum-based adjoint sensitivity analysis method using the isogeometric approach is derived for the plane elasticity problems. The conventional shape optimization using the finite element method has some difficulties in the parameterization of boundary. In isogeometric analysis, however, the geometric properties arc already embedded in the B-spline shape functions and control points. The perturbation of control points in isogeometric analysis automatically results in shape changes. Using the conventional finite clement method, the inter-element continuity of the design space is not guaranteed so that the normal vector and curvature arc not accurate enough. On tile other hand, in isogeometric analysis, these values arc continuous over the whole design space so that accurate shape sensitivity can be obtained. Through numerical examples, the developed isogeometric sensitivity analysis method is verified to show excellent agreement with finite difference sensitivity.

• Proceedings of the Korean Society for Noise and Vibration Engineering Conference
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• 2006.05a
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• pp.660-665
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• 2006

This paper presents a finite element method to analyze the free and forced vibration of a hard disk drive (HDD) considering the flexibility of a spinning disk-spindle with fluid dynamic bearings (FDBs), an actuator with pivot bearings, an air bearing between head-disk interface and the base with complicated geometry. Finite element equation of each component is consistently derived with the satisfaction of the geometric compatibility of the internal boundary between each component. The spinning disk, hub and FDBs are modeled by annular sector elements, beam elements and stiffness and damping elements, respectively. The actuator am, E-block, suspension and base plate are modeled by tetrahedral elements. The pivot bearing in the actuator and the air bearing between head-disk interfaces are modeled by the stiffness element with five degrees of freedom and the axial stiffness, respectively. A global matrix equation obtained by assembling the finite element equations of each substructure is transformed to a state-space matrix-vector equation, and both damped natural frequencies and modal damping ratios are calculated by solving the associated eigenvalue problem with the restarted Arnoldi iteration method. Modal and shock testing are performed to show that the proposed method well predicts the vibration characteristics of a HDD.

• Structural Engineering and Mechanics
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• v.81 no.4
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• pp.411-428
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• 2022

This paper presents the Campbell diagram analysis of the rotordynamic system using the full order model (FOM) and the reduced order model (ROM) techniques to determine the critical speeds, identify the stability and reduce the computational time. Due to the spin-speed-dependent matrices (e.g., centrifugal stiffening matrix), several model order reduction (MOR) techniques may be considered, such as the modal superposition (MS) method and the Krylov subspace-based MOR techniques (e.g., Ritz vector (RV), quasi-static Ritz vector (QSRV), multifrequency quasi-static Ritz vector (MQSRV), multifrequency/ multi-spin-speed quasi-static Ritz vector (MMQSRV) and the combined Ritz vector & modal superposition (RV+MS) methods). The proposed MMQSRV method in this study is extended from the MQSRV method by incorporating the rotational-speed-dependent stiffness matrices into the Krylov subspace during the MOR process. Thus, the objective of this note is to respond to the question of whether to use the MS method or the Krylov subspace-based MOR technique in establishing the Campbell diagram of the shaft-disc-blade assembly systems in three-dimensional (3D) finite element analysis (FEA). The Campbell diagrams produced by the FOM and various MOR methods are presented and discussed thoroughly by computing the norm of relative errors (ER). It is found that the RV and the MS methods are dominant at low and high rotating speeds, respectively. More precisely, as the spinning velocity becomes large, the calculated ER produced by the RV method is significantly increased; in contrast, the ER produced by the MS method is smaller and more consistent. From a computational point of view, the MORs have substantially reduced the time computing considerably compared to the FOM. Additionally, the verification of the 3D FE rotordynamic model is also provided and found to be in close agreement with the existing solutions.

• KIEE International Transaction on Electrical Machinery and Energy Conversion Systems
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• v.5B no.3
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• pp.248-252
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• 2005

This paper deals with a new design method for a small-size rotating machine with high power. In order to achieve high performance, secondary excitation by Nd-Fe-B magnets and the grain oriented electrical steel sheets were selected and a new design using dual rotors is proposed. The outline of the high-performance rotating machine will be presented and the results of the finite element analysis by using this method combined with the E&SS modeling will be shown in the paper.

• Communications of the Korean Mathematical Society
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• v.20 no.3
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• pp.563-578
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• 2005

In [Z. Cai and B. C. Shin, SIAM J. Numer. Anal. 40 (2002), 307-318], we developed the discrete first-order system least squares method for the second-order elliptic boundary value problem by directly approximating $H(div){\cap}H(curl)-type$ space based on the Helmholtz decomposition. Under general assumptions, error estimates were established in the $L^2\;and\;H^1$ norms for the vector and scalar variables, respectively. Such error estimates are optimal with respect to the required regularity of the solution. In this paper, we study solution methods for solving the system of linear equations arising from the discretization of variational formulation which possesses discrete biharmonic term and focus on numerical results including the performances of multigrid preconditioners and the finite element accuracy.