• Journal of the Korean Mathematical Society
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• v.34 no.1
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• pp.87-118
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• 1997

This paper is concerned with a mixed Lagrangian formulation of the wiscous, stationary, incompressible Navier-Stokes equations $$ (1.1) -\nu\Delta u + (u \cdot \nabla)u + \nabla_p = f in \Omega $$ and $$ (1.2) \nubla \cdot u = 0 in \Omega $$ along with inhomogeneous Dirichlet boundary conditions on a portion of the boundary $$ (1.3) u = ^{0 on \Gamma_0 _{g on \Gamma_g, $$ where $\Omega$ is a bounded open domain in $R^d, d = 2 or 3$, or with a boundary $\Gamma = \partial\Omega$, which is composed of two disjoint parts $\Gamma_0$ and $\Gamma_g$.

• Journal of the Korean Society for Industrial and Applied Mathematics
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• v.6 no.1
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• pp.1-15
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• 2002

Advection-dominated transport problems possess difficulties in the design of numerical methods for solving them. Because of the hyperbolic nature of advective transport, many characteristic numerical methods have been developed such as the classical characteristic method, the Eulerian-Lagrangian method, the transport diffusion method, the modified method of characteristics, the operator splitting method, the Eulerian-Lagrangian localized adjoint method, the characteristic mixed method, and the Eulerian-Lagrangian mixed discontinuous method. In this paper relationships among these characteristic methods are examined. In particular, we show that these sometimes diverse methods can be given a unified formulation. This paper focuses on characteristic finite element methods. Similar examination can be presented for characteristic finite difference methods.

• Journal of the Computational Structural Engineering Institute of Korea
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• v.25 no.6
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• pp.469-476
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• 2012

In this paper, a finite element domain decomposition method using local and mixed Lagrange multipliers for a large scal structural analysis is presented. The proposed algorithms use local and mixed Lagrange multipliers to improve computational efficiency. In the original FETI method, classical Lagrange multiplier technique was used. In the dual-primal FETI method, the interface nodes are used at the corner nodes of each sub-domain. On the other hand, the proposed FETI-local analysis adopts localized Lagrange multipliers and the proposed FETI-mixed analysis uses both global and local Lagrange multipliers. The numerical analysis results by the proposed algorithms are compared with those obtained by dual-primal FETI method.

• Structural Engineering and Mechanics
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• v.26 no.6
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• pp.725-739
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• 2007

In recent years there are many plate bending elements that emerged for solving both thin and thick plates. The main features of these elements are that they are based on mix formulation interpolation with discrete collocation constraints. These elements passed the patch test for mix formulation and performed well for linear analysis of thin and thick plates. In this paper a member of this family of elements, namely, the Discrete Reissner-Mindlin (DRM) is further extended and developed to analyze both thin and thick plates with geometric nonlinearity. The Von K$\acute{a}$rm$\acute{a}$n's large displacement plate theory based on Lagrangian coordinate system is used. The Hu-Washizu variational principle is employed to formulate the stiffness matrix of the geometrically Nonlinear Discrete Reissner-Mindlin (NDRM). An iterative-incremental procedure is implemented to solve the nonlinear equations. The element is then tested for plates with simply supported and clamped edges under uniformly distributed transverse loads. The results obtained using the geometrically NDRM element is then compared with the results of available analytical solutions. It has been observed that the NDRM results agreed well with the analytical solutions results. Therefore, it is concluded that the NDRM element is both reliable and efficient in analyzing thin and thick plates with geometric non-linearity.

• Korean Journal of Mathematics
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• v.6 no.1
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• pp.27-46
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• 1998

We are concerned with an optimal control problem governed by a Poisson equation in which body force acts like a control parameter. The cost functional to be optimized is taken to represent the error from the desired observation and the cost due to the control. We recast the problem into the mixed formulation to take advantage of the minimax principle for the duality method. The existence of a saddle point for the Lagrangian shall be shown and the optimality system will be derived therein. Finally, to attain an optimal control, we combine the optimality system with an operational technique. By achieving the gradient of the cost functional, a convergent algorithm based on the projected gradient method is established.

• Transactions of the Korean Society of Mechanical Engineers
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• v.19 no.2
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• pp.354-362
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• 1995

The finite element modeling is used to study the buckling and postbuckling behavior of composite laminates with an embedded delamination. Degenerated shell element and rigid beam element are utilized for the finite element modeling. In the nonlinear finite element formulation, the updated Lagrangian description method based on the second Piola-Kirchhoff stress tensor and the Green strain tensor is used. The buckling and postbuckling behavior of composite laminates with a delamination are investigated for various delamination sizes, stacking sequences, and boundary conditions. It is shown that the buckling load and postbuckling behavior of composite laminates depend on the buckling model which is determined by the delamination size, stacking sequence and boundary condition. Also, results show that introduction of couplings can reduce greatly the buckling load.

• Journal of the Korean Society for Industrial and Applied Mathematics
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• v.18 no.1
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• pp.17-26
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• 2014

This paper presents development of an improved domain decomposition method for large scale structural problem that aims to provide high computational efficiency. In the previous researches, we developed the domain decomposition algorithm based on augmented Lagrangian formulation and proved numerical efficiency under both serial and parallel computing environment. In this paper, new computational analysis by the proposed domain decomposition method is performed. For this purpose, reduction in computational time achieved by the proposed algorithm is compared with that obtained by the dual-primal FETI method under serial computing condition. It is found that the proposed methods significantly accelerate the computational speed for a linear structural problem.

• Structural Engineering and Mechanics
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• v.24 no.2
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• pp.227-245
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• 2006

The performance of a three dimensional non-linear finite element model for hyperelastic material considering the effect of compressibility is studied by analyzing rubber blocks under different modes of deformation. It includes simple tension, pure shear, simple shear, pure bending and a mixed mode combining compression, shear and bending. The compressibility of the hyperelastic material is represented in the strain energy function. The nonlinear formulation is based on updated Lagrangian (UL) technique. The displacement model is implemented with a twenty node brick element having u, ${\nu}$ and w as the degrees of freedom at each node. The results obtained by the present numerical model are compared with the analytical solutions available for the basic modes of deformation where the agreement between the results is found to be satisfactory. In this context some new results are generated for future references since the number of available results on the present problem is not sufficient enough.

• Journal of the Computational Structural Engineering Institute of Korea
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• v.20 no.1
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• pp.29-37
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• 2007

This study deals with displacement analysis of anisotropic folded structures with triangular elements and quadrilateral elements. When folded plates are analyzed, triangular elements as well as quadrilateral elements are needed for conveniences of modelling. However, using triangular elements is not a simple problem. A simple formulation is presented which allows a quadrilateral element to degenerate into a triangular element. Therefore it can easily be used for computational simplicity and avoided complexities on mixed use of triangular element and quadrilateral element. In this paper, a high-order shear deformation theory using only Lagrangian interpolation functions and drilling degrees of freedom for folded plates are utilized for more accurate analysis. Especially, various results of anisotropic laminated and folded composite structures with triangular element and quadrilateral element show the structural behavior characteristics of them.