• Proceedings of the KSME Conference
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• 2001.06b
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• pp.590-595
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• 2001

To improve the modeling accuracy for the finite element method, this paper proposes a method to make a combined use of finite elements and exact dynamic elements. Exact interpolation functions for a Timoshenko beam element are derived and compared with interpolation functions of the finite element method (FEM). The exact interpolation functions are tested with the Laplace variable varied. The exact interpolation functions are used to gain more accurate mode shape functions for the finite element method. This paper also presents a combined use of finite elements and exact dynamic elements in design problems. A Timoshenko frame with tapered sections is tested to demonstrate the design procedure with the proposed method.

• Transactions of the Korean Society for Noise and Vibration Engineering
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• v.12 no.2
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• pp.141-149
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• 2002

Although the finite element method has become an indispensible tool for the dynamic analysis of structures, difficulty remains to quantify the errors associated with discretization. To improve the modeling accuracy, this paper proposes a method to make a combined use of finite elements and exact dynamic elements. Exact interpolation functions for the Timoshenko beam element are derived using the exact dynamic element modeling (EDEM) and compared with interpolation functions of the finite element method (FEM). The exact interpolation functions are tested with the Laplace variable varied. A combined use of finite element method and exact interpolation functions is presented to gain more accurate mode shape functions. This paper also presents a combined use of finite elements and exact dynamic elements in design/reanalysis problems. Timoshenko flames with tapered sections are tested to demonstrate the design procedure with the proposed method. The numerical study shows that the combined use of finite element model and exact dynamic element model is very useful.

• Structural Engineering and Mechanics
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• v.7 no.3
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• pp.259-276
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• 1999

An analytical solution for the shape functions of a beam segment supported on a generalized two-parameter elastic foundation is derived. The solution is general, and is not restricted to a particular range of magnitudes of the foundation parameters. The exact shape functions can be utilized to derive exact analytic expressions for the coefficients of the element stiffness matrix, work equivalent nodal forces for arbitrary transverse loads and coefficients of the consistent mass and geometrical stiffness matrices. As illustration, each distinct coefficient of the element stiffness matrix is compared with its conventional counterpart for a beam segment supported by no foundation at all for the entire range of foundation parameters.

• Proceedings of the Computational Structural Engineering Institute Conference
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• 2008.04a
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• pp.216-221
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• 2008

Shape design optimization for linear elasticity problem is performed using isogeometric analysis method. In many design optimization problems for real engineering models, initial raw data usually comes from CAD modeler. Then designer should convert this CAD data into finite element mesh data because conventional design optimization tools are generally based on finite element analysis. During this conversion there is some numerical error due to a geometry approximation, which causes accuracy problems in not only response analysis but also design sensitivity analysis. As a remedy of this phenomenon, the isogeometric analysis method is one of the promising approaches of shape design optimization. The main idea of isogeometric analysis is that the basis functions used in analysis is exactly same as ones which represent the geometry, and this geometrically exact model can be used shape sensitivity analysis and design optimization as well. In shape design sensitivity point of view, precise shape sensitivity is very essential for gradient-based optimization. In conventional finite element based optimization, higher order information such as normal vector and curvature term is inaccurate or even missing due to the use of linear interpolation functions. On the other hands, B-spline basis functions have sufficient continuity and their derivatives are smooth enough. Therefore normal vector and curvature terms can be exactly evaluated, which eventually yields precise optimal shapes. In this article, isogeometric analysis method is utilized for the shape design optimization. By virtue of B-spline basis function, an exact geometry can be handled without finite element meshes. Moreover, initial CAD data are used throughout the optimization process, including response analysis, shape sensitivity analysis, design parameterization and shape optimization, without subsequent communication with CAD description.

• Proceedings of the Computational Structural Engineering Institute Conference
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• 2008.04a
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• pp.391-396
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• 2008

A level set based topological shape optimization using extended B-spline basis functions is developed for steady state heat conduction problems. The only inside of complicated domain is identified by the level set functions and taken into account in computation. The solution of Hamilton-Jacobi equation leads to an optimal shape according to the normal velocity field determined from the sensitivity analysis, minimizing a thermal compliance while satisfying a volume constraint. To obtain exact shape sensitivity, the precise normal and curvature of geometry need to be determined using the level set and B-spline basis functions. The nucleation of holes is possible whenever and wherever necessary during the optimization using a topological derivative concept.

• Bulletin of the Korean Chemical Society
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• v.35 no.3
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• pp.805-810
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• 2014

The shape invariant potentials are proved to be exactly solvable, i.e. the wave functions and energies of a particle moving under the influence of the shape invariant potentials can be algebraically determined without any approximations. It is well known that the SWKB quantization is exact for all shape invariant potentials though the SWKB quantization itself is approximate. This mystery has not been mathematically resolved yet and may not be solved in a concrete fashion even in the future. Therefore, in the present work, to understand (not prove) the mystery an attempt of deriving exactly solvable potentials directly from the SWKB quantization has been made. And it turns out that all the derived potentials are shape invariant. It implicitly explains why the SWKB quantization is exact for all known shape invariant potentials. Though any new potential has not been found in this study, this brute-force derivation of potentials helps one understand the characteristics of shape invariant potentials.

• Structural Engineering and Mechanics
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• v.52 no.4
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• pp.787-813
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• 2014

Starting with Hamilton's variational principle, the governing field equations for the steady state response of thin-walled beams under harmonic forces are derived. The formulation captures shear deformation effects due to bending and warping, translational and rotary inertia effects and as well as torsional flexural coupling effects due to the cross section mono-symmetry. The equations of motion consist of four coupled differential equations in the unknown displacement field variables. A general closed form solution is then developed for the coupled system of equations. The solution is subsequently used to develop a family of shape functions which exactly satisfy the homogeneous form of the governing field equations. A super-convergent finite element is then formulated based on the exact shape functions. Key features of the element developed include its ability to (a) isolate the steady state response component of the response to make the solution amenable to fatigue design, (b) capture coupling effects arising as a result of section mono-symmetry, (c) eliminate spatial discretization arising in commonly used finite elements, (d) avoiding shear locking phenomena, and (e) eliminate the need for time discretization. The results based on the present solution are found to be in excellent agreement with those based on finite element solutions at a small fraction of the computational and modelling cost involved.

• Journal of Advanced Marine Engineering and Technology
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• v.14 no.4
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• pp.67-71
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• 1990

The traditional finite element method applied to dynamic problems employs shape functions which are based on a static displacement assumption. The more exact approach uses frequency-dependent shape functions and frequency-dependent mass and stiffness matrices. Such matrices are developed for a torsional vibration of shaft element. Numerical examples are presented for a cantilever beam.

• Journal of the Korean Mathematical Society
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• v.61 no.4
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• pp.761-778
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• 2024

A separable minimal surface is represented by the form of f(x) + g(y) + h(z) = 0, where f, g and h are real-valued functions of x, y and z, respectively. We provide exact equations for separable minimal surfaces with elliptic functions that are singly, doubly and triply periodic minimal surfaces and completely classify all them. In particular, parameters in the separable minimal surfaces change the shape of the surfaces, such as fundamental periods and its limit behavior, within the form f(x) + g(y) + h(z) = 0.

• Journal of the Computational Structural Engineering Institute of Korea
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• v.26 no.4
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• pp.255-262
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• 2013

In this paper, a shape design sensitivity analysis(DSA) method is presented for Mindlin plates using an isogeometric approach. The isogeometric method possesses desirable advantages; the representation of exact geometry and the higher order inter-element continuity, which lead to the fast convergence of solution as well as accurate sensitivity results. Unlike the finite element methods using linear shape functions, the isogeometric method considers the exact normal vector and curvature of the CAD geometry, taking advantages of higher order NURBS basis functions. A selective reduced integration(SRI) technique is incorporated to overcome the difficulty of 'shear locking' phenomenon. This simple technique is surprisingly helpful for the accuracy of the isogeometric shape sensitivity without complicated formulation. Through the numerical examples of plate bending problems, the accuracy of the proposed isogeometric analysis method is compared with that of finite element one. Also, the isogeometric shape sensitivity turns out to be very accurate when compared with finite difference sensitivity.