• Journal of the Korean Society for Industrial and Applied Mathematics
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• v.17 no.4
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• pp.295-306
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• 2013

In this paper, we consider the adaptive multigrid method for solving the Black-Scholes equation to improve the efficiency of the option pricing. Adaptive meshing is generally regarded as an indispensable tool because of reduction of the computational costs. The Black-Scholes equation is discretized using a Crank-Nicolson scheme on block-structured adaptively refined rectangular meshes. And the resulting discrete equations are solved by a fast solver such as a multigrid method. Numerical simulations are performed to confirm the efficiency of the adaptive multigrid technique. In particular, through the comparison of computational results on adaptively refined mesh and uniform mesh, we show that adaptively refined mesh solver is superior to a standard method.

• Proceedings of the Korean Society For Composite Materials Conference
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• 2000.04a
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• pp.80-84
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• 2000

A mesh superposition technique is presented for an efficient analysis of structural behavior. Refined child mesh is superimposed over parent elements for the region of interest. It is a kind of adaptive mesh refinement, which allows locally refined mesh without introducing transition region or multipoint constraints. Proper boundary condition is necessary to avoid redundant rigid body motion and kinematic compatibility between neighbor elements. Delamination buckling analysis is conducted to demonstrate accuracy and efficiency of the present method.

• Proceedings of the Korean Society For Composite Materials Conference
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• 2000.04a
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• pp.89-92
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• 2000

In this paper, an effective procedure is presented for the local recovery of displacements and stresses in multilayered composite panels, which incorporate the local refinement using mesh superposition. The mesh superposition method is used to refine the global coarse mesh by superimposing refined mesh to the localized zone of interest without transition zones. The finite element model used is a solid element based on the Hellinger-Reissner variational principle. The a posteriori computation of the through-the-thickness distributions of displacements and stresses is achieved using a predictor-corrector procedure. The procedure utilizes the superconvergent stresses and nodal displacements of the finite element patch. The element patch is generated by locally superimposing a refined local mesh to the coarse global mesh.

• Wind and Structures
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• v.1 no.1
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• pp.111-125
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• 1998

This paper deals with the development of variable-node element and its application to the adaptive h-version mesh refinement-recovery for the incompressible viscous flow analysis. The element which has variable mid-side nodes can be used in generating the transition zone between the refined and unrefined element and efficiently used for the construction of a refined mesh without generating distorted elements. A modified Guassian quadrature is needed to evaluate the element matrices due to the discontinuity of derivatives of the shape functions used for the element. The penalty function method which can reduce the number of the independent variables is adopted for the purpose of computational efficiency and the selective reduced integration is carried out for the convection and pressure terms to preserve the stability of solution. For the economical analysis of transient problems in which the locations to be refined are changed in accordance with the dynamic distribution of velocity gradient, not only the mesh refinement but also the mesh recovery is needed. The numerical examples show that the optimal mesh for the finite element analysis of a wind around the structures can be obtained automatically by the proposed scheme.

• Proceedings of the Computational Structural Engineering Institute Conference
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• 1998.04a
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• pp.60-67
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• 1998

This paper deals with the development of a variable-node element and its application to the adaptive h-version mesh refinement-recovery for the incompressible viscous flow analysis. The element which has variable mid-side nodes can be used in generating the transition zone between the refined and unrefined elements and efficiently used for construction of a refined mesh without generating distorted elements. A modified Gaussian quadrature is needed to evaluate the element matrices due to the discontinuity of derivatives of the shape functions used for the element. The penalty function method which can reduce the number of independent variables is adopted for the purpose of computational efficiency and the selective reduced integration is carried out for the convection and pressure terms to preserve the stability of solution. For the economical analysis of transient problems, not only the mesh refinement but also the mesh recovery is needed. The numerical examples show that the optimal mesh for the finite element analysis of a wind around the structures can be obtained automatically by the proposed scheme.

• Transactions of Materials Processing
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• v.13 no.4
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• pp.334-341
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• 2004

In this study, a remeshing algorithm adapted to the mesh density map using the Delaunay mesh generation method is developed. In the finite element simulation of forging process, the numerical error increases as the process goes on because of discrete property of the finite elements and distortion of elements. Especially, in the region where stresses and strains are concentrated, the numerical error will be highly increased. However, it is not desirable to use a uniformly fine mesh in the whole domain. Therefore, it is necessary to reduce the analysis error by constructing locally refined mesh at the region where the error is concentrated such as at the die corner. In this paper, the point insertion algorithm is used and the mesh size is controlled by using a mesh density map constructed with a posteriori error estimation. An optimized smoothing technique is adopted to have smooth distribution of the mesh and improve the mesh element quality.

• Transactions of the Korean Society of Mechanical Engineers A
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• v.25 no.12
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• pp.1926-1932
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• 2001

In this second part of the paper, the automatic mesh generation and remeshing algorithm using bubble packing method is applied to the nonlinear problem. The remeshing/refinement procedure is necessary in the large deformation process especially because the mesh distortion deteriorates the convergence and accuracy. To perform the nonliear analysis, the transfer of state variables such as displacement and strain is added to the algorithm of Part 1. The equilibrium equation based on total Lagrangian formulation and elasto-viscoplastic model is used. For the numerical experiment, the upsetting process including the contact constraint condition is analyzed by two refinement criteria. And from the result, it is addressed that the present algorithm can generate the refined meshes easily at the largely deformed area with high error.

• The Transactions of the Korean Institute of Electrical Engineers
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• v.39 no.9
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• pp.921-927
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• 1990

This paper presents a mesh refinement scheme for 3-D adaptive finite element method. Firstly, the refinement of triangular meshes based on the bisection of triangles is discussed. And a new method to refine tetrahedral meshes employing the bisection method is presented. In two dimensional cases, it has been noted that all angles in the triangular meshes refined by the bisection method are greater than or equal to half the smallest angle in the original meshes. Through the examples where the newly proposed method is applied to three dimensional cases, it is shown that regarding the solid angles, the method gives nearly the same result as that in the two dimensional case. Accordingly, it can be concluded that the proposed method will be useful in the mesh refinements for 3-D adaptive finite element method.

• International Journal of Aeronautical and Space Sciences
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• v.8 no.1
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• pp.95-104
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• 2007

A new design approach of complex geometries such as wing/body configuration is arranged by using overset mesh techniques under large scale computing environment. For an in-depth study of the flow physics and highly accurate design, several special overlapped structured blocks such as collar grid, tip-cap grid, and etc. which are commonly used in refined drag prediction are adopted to consider the applicability of the present design tools to practical problems. Various pre- and post-processing techniques for overset flow analysis and sensitivity analysis are devised or implemented to resolve overset mesh techniques into the design optimization problem based on Gradient Based Optimization Method (GBOM). In the pre-processing, the convergence characteristics of the flow solver and sensitivity analysis are improved by overlap optimization method. Moreover, a new post-processing method, Spline-Boundary Intersecting Grid (S-BIG) scheme, is proposed by considering the ratio of cell area for more refined prediction of aerodynamic coefficients and efficient evaluation of their sensitivities under parallel computing environment. With respect to the sensitivity analysis, discrete adjoint formulations for overset boundary conditions are derived by a full hand-differentiation. A smooth geometric modification on the overlapped surface boundaries and evaluation of grid sensitivities can be performed by mapping from planform coordinate to the surface meshes with Hicks-Henne function. Careful design works for the drag minimization problems of a transonic wing and a wing/body configuration are performed by using the newly-developed and -applied overset mesh techniques. The results from design applications demonstrate the capability of the present design approach successfully.

• Proceedings of the KIEE Conference
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• 1993.07b
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• pp.935-937
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• 1993

This paper presents a three dimensional automatic mesh generation scheme for the boundary element method, and this scheme can be applicable to practical problems of complex shape. The geometry of the problem is expressed as an assemblage of linear Coon's surfaces, and each surface is made up of four edge curves which are defined in the form of a parametric function. Curves are automatically segmented according to their characteristics. With these segments of curves, interior points and triangular mesh elements are generated in the parametric plane using Lindholm's method, and then their projection on the real surface forms the initial mesh. The refinement of initial mesh is performed so that the discrete triangular planes are close to the real continuous surfaces. The bisection method is used for the refinement. Finally, interior points in the refined mesh are rearranged so as to make each element be close with an equilateral triangle. An attempt has been made to apply the proposed method to a DY(Deflection Yoke) model.