• Journal of applied mathematics & informatics
• /
• v.31 no.3_4
• /
• pp.479-490
• /
• 2013

In this paper, we investigate a priori error estimates and superconvergence of varitional discretization for nonlinear parabolic optimal control problems with control constraints. The time discretization is based on the backward Euler method. The state and the adjoint state are approximated by piecewise linear functions and the control is not directly discretized. We derive a priori error estimates for the control and superconvergence between the numerical solution and elliptic projection for the state and the adjoint state and present a numerical example for illustrating our theoretical results.

• Journal of applied mathematics & informatics
• /
• v.32 no.5_6
• /
• pp.707-719
• /
• 2014

In this paper, we investigate a variational discretization approximation of elliptic optimal control problems with control constraints. The state and the co-state are approximated by piecewise linear functions, while the control is not directly discretized. By using some proper intermediate variables, we derive a second-order convergence in $L^2$-norm and superconvergence between the numerical solution and elliptic projection of the exact solution in $H^1$-norm or the gradient of the exact solution and recovery gradient in $L^2$-norm. Then we construct a posteriori error estimates by using the superconvergence results and do some numerical experiments to confirm our theoretical results.

• Nuclear Engineering and Technology
• /
• v.55 no.6
• /
• pp.2172-2194
• /
• 2023

A variational nodal method (VNM) with unstructured-mesh is presented for solving steady-state and dynamic neutron diffusion equations. Orthogonal polynomials are employed for spatial discretization, and the stiffness confinement method (SCM) is implemented for temporal discretization. Coordinate transformation relations are derived to map unstructured triangular nodes to a standard node. Methods for constructing triangular prism space trial functions and identifying unique nodes are elaborated. Additionally, the partitioned matrix (PM) and generalized partitioned matrix (GPM) methods are proposed to accelerate the within-group and power iterations. Neutron diffusion problems with different fuel assembly geometries validate the method. With less than 5 pcm eigenvalue (k_eff) error and 1% relative power error, the accuracy is comparable to reference methods. In addition, a test case based on the kilowatt heat pipe reactor, KRUSTY, is created, simulated, and evaluated to illustrate the method's precision and geometrical flexibility. The Dodds problem with a step transient perturbation proves that the SCM allows for sufficiently accurate power predictions even with a large time-step of approximately 0.1 s. In addition, combining the PM and GPM results in a speedup ratio of 2-3.

• The Journal of the Institute of Internet, Broadcasting and Communication
• /
• v.22 no.5
• /
• pp.75-86
• /
• 2022

This study aims to solve the problem of class imbalance in numerical data by using a deep learning-based Variational AutoEncoder and to improve the performance of the learning model by augmenting the learning data. We propose 'D-VAE' to artificially increase the number of records for a given table data. The main features of the proposed technique go through discretization and feature selection in the preprocessing process to optimize the data. In the discretization process, K-means are applied and grouped, and then converted into one-hot vectors by one-hot encoding technique. Subsequently, for memory efficiency, sample data are generated with Variational AutoEncoder using only features that help predict with RFECV among feature selection techniques. To verify the performance of the proposed model, we demonstrate its validity by conducting experiments by data augmentation ratio.

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

• Korean Journal of Computational Design and Engineering
• /
• v.12 no.4
• /
• pp.255-262
• /
• 2007

In the finite element analysis of forming process, objects are described with a finite number of elements and nodes and the approximated solutions can be obtained by the variational principle. One of the shortcomings of a finite element analysis is that the structure of mesh has become inefficient and unusable because discretization error increases as deformation proceeds due to severe distortion of elements. If the state of current mesh satisfies a certain remeshing criterion, analysis is stopped instantly and resumed with a reconstructed mesh. In the study, a new remeshing algorithm using tetrahedral elements has been developed, which is adapted to the desired mesh density. In order to reduce the discretization error, desired mesh sizes in each lesion of the workpiece are calculated using the Zinkiewicz and Zhu's a-posteriori error estimation scheme. The pre-constructed mesh is constructed based on the modified point insertion technique which is adapted to the density function. The object domain is divided into uniformly-sized sub-domains and the numbers of nodes in each sub-domain are redistributed, respectively. After finishing the redistribution process of nodes, a tetrahedral mesh is reconstructed with the redistributed nodes, which is adapted to the density map and resulting in good mesh quality. A goodness and adaptability of the constructed mesh is verified with a testing measure. The proposed remeshing technique is applied to the finite element analyses of forging processes.

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

• Journal of applied mathematics & informatics
• /
• v.31 no.1_2
• /
• pp.165-177
• /
• 2013

In this paper, we apply Homotopy perturbation transform method (HPTM) for solving singular fourth order parabolic partial differential equations with variable coefficients. This method is the combination of the Laplace transform method and Homotopy perturbation method. The nonlinear terms can be easily handled by the use of He's polynomials. The aim of using the Laplace transform is to overcome the deficiency that is mainly caused by unsatisfied conditions in other semi-analytical methods such as Homotopy perturbation method (HPM), Variational iteration method (VIM) and Adomain Decomposition method (ADM). The proposed scheme finds the solutions without any discretization or restrictive assumptions and avoids the round-off errors. The comparison shows a precise agreement between the results and introduces this method as an applicable one which it needs fewer computations and is much easier and more convenient than others, so it can be widely used in engineering too.

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

• Transactions of the Korean Society for Noise and Vibration Engineering
• /
• v.22 no.8
• /
• pp.791-799
• /
• 2012

In this paper free vibration analysis of symmetric and cross-ply elastic laminated shells based on FSDT was performed through discretization of equations of motion and boundary condition. Structural model of laminated composite cylindrical shells subjected to a combination of magnetic and thermal fields is developed via Hamilton's variational principle. These coupled equations of motion are based on the electromagnetic equations(Faraday, Ampere, Ohm, and Lorenz equations) and thermal equations which are involved in constitutive equations. Variations of dynamic characteristics of composite shells with applied magnetic field, temperature gradient, and stacking sequence are investigated and pertinent conclusions are derived.