• Structural Engineering and Mechanics
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• v.3 no.2
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• pp.145-162
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• 1995

A damped trapezoidal rule method for numerical time-integration is presented, and its application in analyses of dynamic response of damped structures is discussed. It is shown that the damped trapezoidal rule method has features that make it an attractive approach for applications in dynamic analyses of structures. Accuracy and stability analyses are developed for the damped single-degree-of-freedom systems. Error analyses are also performed for the Newmark beta method and compared with the damped trapezoidal rule method as a basis for discussion of the relative merits of the proposed method. The procedure is fully explicit and easy to implement. However, since the method is an explicit method, it is conditionally stable. The methodology is applied to several example problems to illustrate its strengths, limitations and inherent simplicity.

• Proceedings of the Korean Society for Noise and Vibration Engineering Conference
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• 1997.04a
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• pp.110-115
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• 1997

In this work, the constant average acceleration, which is a fundamental feature of the trapezoidal rule, is investigated and generalized. Using the generalization of average acceleration concept, a higher order accurate and unconditionally stable time-integration method is developed. The linear approximate of the present methods is exactly the same as the famous trapezoidal rule. To observe the accuracy and stability of the method, several numerical tests are performed and the results are compared with the results from the trapezoidal rule and the exact solution. From the numerical tests, it has been known that the present method has a higher order accuracy and unconditional stability.

• Transactions of the Korean Society of Mechanical Engineers A
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• v.21 no.10
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• pp.1712-1717
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• 1997

Stability and accuracy for the trapezoidal rule of the Newmark time integration method are analyzed when variable time step sizes are adopted. A new analytic approach to stability and accuracy analysis is also proposed for time integration methods with variable time step sizes. The trapezoidal rule with variable time step sizes has the "actual" unconditional stability which is the same as that of the method with constant time step sizes. However, the method with variable time step sizes is first-order accurate while the method with constant time step sizes is second-order accurate. accurate.

• Kyungpook Mathematical Journal
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• v.56 no.4
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• pp.1179-1189
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• 2016

We provide a combination of the forward Euler method and the trapezoidal quadrature rule leads to a two-step conservative numerical method which possesses TV-stable property together with consistency.

• Geotechnical Engineering
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• v.12 no.4
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• pp.145-156
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• 1996

An implicit stress integration algorithm was formulated for implementing an aiusotorpic hardening constitutive model which has been based op the generalized isotropic hardening rule in nonlinear finite element analysis technique. the rate form of stress tensor was implicitly integrated using the generalized trapezoidal rule and the tangent stress-strain modulus was evaluated consistently with the nonlinear solution technique. As a result, it has been found that the nonlinear analysis with the anisotropic hardening constitutive model might be performed accurately and efficiently.

• Transactions of the Korean Society of Mechanical Engineers A
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• v.27 no.11
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• pp.1907-1916
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• 2003

During decades, there has been much progress in understanding of the inelastic behavior of the materials and numerous inelastic constitutive equations have been developed. The complexity of these constitutive equations generally requires a stable and accurate numerical method. To obtain the increment of state variable, its evolution laws are linearized by several approximation methods, such as general midpoint rule(GMR) or general trapezoidal rule(GTR). In this investigation, semi-implicit integration schemes using GTR and GMR were developed and implemented into ABAQUS by means of UMAT subroutine. The comparison of integration schemes was conducted on the simple tension case, and simple shear case and nonproportional loading case. The fully implicit integration(FI) was the most stable but amplified the truncation error when the nonlinearity of state variable is strong. The semi-implicit integration using GTR gave the most accurate results at tension and shear problem. The numerical solutions with refined time increment were always placed between results of GTR and those of FI. GTR integration with adjusting midpoint parameter can be recommended as the best integration method for viscoplastic equation considering nonlinear kinematic hardening.

• Communications of the Korean Mathematical Society
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• v.33 no.3
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• pp.1025-1037
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• 2018

We consider an ill-posed problem for the heat equation $u_{xx}=u_t$ in the quarter plane {x > 0, t > 0}. We propose a new method to compute the heat flux $h(t)=u_x(1,t)$ from the boundary temperature g(t) = u(1, t). The operator $g{\mapsto}h=Hg$ is unbounded in $L^2({\mathbb{R}})$, so we approximate h(t) by $h_{\delta}(t)=u_x(1+{\delta},\;t)$, ${\delta}{\rightarrow}0$. When noise is present, the data is $g_{\epsilon}$ leading to a corresponding heat $h_{{\delta},{\epsilon}}$. We obtain an estimate of the error ${\parallel}h-h_{{\delta},{\epsilon}}{\parallel}$, as well as the error when $h_{{\delta},{\epsilon}}$ is approximated by the trapezoidal rule. With an a priori choice rule ${\delta}={\delta}({\epsilon})$ and ${\tau}={\tau}({\epsilon})$, the step size of the trapezoidal rule, the main theorem gives the error of the heat flux as a function of noise level ${\epsilon}$. Numerical examples show that the proposed method is effective and stable.

• Steel and Composite Structures
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• v.36 no.1
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• pp.47-62
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• 2020

In this study, free vibration behavior of trapezoidal sandwich plates with porous core and two graphene platelets (GPLs) reinforced nanocomposite outer layers are presented. The distribution of pores and GPLs are supposed to be functionally graded (FG) along the thickness of core and nanocomposite layers, respectively. The effective Young's modulus of the GPL-reinforced (GPLR) nanocomposite layers is determined using the modified Halpin-Tsai micromechanics model, while the Poisson's ratio and density are computed by the rule of mixtures. The FSDT plate theory is utilized to establish governing partial differential equations and boundary conditions (B.C.s) for trapezoidal plate. The governing equations together with related B.C.s are discretized using a mapping- generalized differential quadrature (GDQ) method in the spatial domain. Then natural frequencies of the trapezoidal sandwich plates are obtained by GDQ method. Validity of current study is evaluated by comparing its numerical results with those available in the literature. A special attention is drawn to the role of GPLs weight fraction, GPLs patterns of two faces through the thickness, porosity coefficient and distribution of porosity on natural frequencies characteristics. New results show the importance of this permeates on vibrational characteristics of porous/GPLR nanocomposite plates. Finally, the influences of B.C.s and dimension as well as the plate geometry such as face to core thickness ratio on the vibration behaviors of the trapezoidal plates are discussed.

• Journal of Korean Port Research
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• v.15 no.1
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• pp.87-97
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• 2001

The calculation of earthwork plays a major role in plan or design of many civil engineering projects, and thus it has become very important to advanced the accuracy of earthwork calculation. Current method used for estimating the volume of pit excavation assumes that the ground profile between the grid points is linear(trapezoidal rule), or nonlinear(simpson's formulas). In this paper the spot height method, least square method, and chamber formulas, Chen and Lin method are compared with the volumes of the pits in these examples. As a result of this study, algorithm of chen and Lin me쇙 by spline method should provide a better accuracy than the spot height method, least square method, chamber formulas. The Chen and Lin formulas can be used for estimating the excavation volume of a pit divide into a grid with unequal intervals. From the characteristics of the cubic spline polynomial, the modeling curve of the Chen and Lin method is smooth and matches the ground profile well. Generally speaking, the nonlinear profile formulas provide better accuracy than the linear profile formulas. The mathematical model mentioned make an offer maximum accuracy in estimating the volume of a pit excavation.

• Journal of the Korea Convergence Society
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• v.6 no.5
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• pp.227-232
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• 2015

The fuzzy linear programming(FLP) is the useful approach to many real world problems under uncertainty. This paper deals with a FLP whose objective value is fuzzy. And the right hand sides of convergent equality constraints are fuzzy numbers. We assume that the membership function of the objective value is piecewise linear and those of the right hand side are trapezoidal. Each of these trapezoidal functions can be algebraically replaced with three linear functions. Then the FLP problem is transformed into the Zimmermann's symmetric model. The fuzzy solution based on the max-min rule can be obtained by solving the crisp linear programming problem derived from the symmetric model. A numerical example has illustrated our approach. The application of our approach to the inconsistent linear system can enable generate us to get define the useful and flexible inexact solutions within acceptable tolerance. Further research is required to generalize the membership function.