• 한국정밀공학회지
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• 제12권3호
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• pp.32-41
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• 1995

In the analysis of metal forming processes by the finite element method, there are many numerical instabilities such as element locking, hourglass mode and shear locking. These instabilities may have a bad effect upon accuracy and convergence. The present work is concerned with improvement of stability and efficiency in two-dimensional rigid-plastic finite element method using various type of elemenmts and numerical intergration schemes. As metal forming examples, upsetting and backward extrusion are taken for comparison among the methods: various element types and numerical integration schemes. Comparison is made in terms of stability and efficiency in element behavior and computational efficiency and a new scheme of adaptive directional reduced integration is introduced. As a result, the finite element computation has been stabilized from the viewpoint of computational time, convergency, and numerical instability.

• Earthquakes and Structures
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• 제11권2호
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• pp.297-313
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• 2016

A virtual parameter is introduced into the formulation of the previously published integration method to improve its stability properties. It seems that the numerical properties of this integration method are almost unaffected by this parameter except for the stability property. As a result, it can have second order accuracy, explicit formulation and controllable numerical dissipation in addition to the enhanced stability property. In fact, it can have unconditional stability for the system with the instantaneous degree of nonlinearity less than or equal to the specified value of the virtual parameter for the modes of interest for each time step.

• Structural Engineering and Mechanics
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• 제65권1호
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• pp.121-128
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• 2018

Three structure-dependent integration methods with no numerical dissipation have been successfully developed for time integration. Although these three integration methods generally have the same numerical properties, such as unconditional stability, second-order accuracy, explicit formulation, no overshoot and no numerical damping, there still exist some different numerical properties. It is found that TLM can only have unconditional stability for linear elastic and stiffness softening systems for zero viscous damping while for nonzero viscous damping it only has unconditional stability for linear elastic systems. Whereas, both CEM and CRM can have unconditional stability for linear elastic and stiffness softening systems for both zero and nonzero viscous damping. However, the most significantly different property among the three integration methods is a weak instability. In fact, both CRM and TLM have a weak instability, which will lead to an adverse overshoot or even a numerical instability in the high frequency responses to nonzero initial conditions. Whereas, CEM possesses no such an adverse weak instability. As a result, the performance of CEM is much better than for CRM and TLM. Notice that a weak instability property of CRM and TLM might severely limit its practical applications.

• Structural Engineering and Mechanics
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• 제75권3호
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• pp.357-367
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• 2020

The objective of this article is investigation of dynamic response of thick multilayer functionally graded (FG) beam under generalized dynamic forces. The plane stress problem is exploited to describe the constitutive equation of thick FG beam to get realistic and accurate response. Applied dynamic forces are assumed to be sinusoidal harmonic, sinusoidal pulse or triangle in time domain and point load. Equations of motion of deep FG beam are derived based on the Hamilton principle from kinematic relations and constitutive equations of plane stress problem. The numerical finite element procedure is adopted to discretize the space domain of structure and transform partial differential equations of motion to ordinary differential equations in time domain. Numerical time integration method is used to solve the system of equations in time domain and find the time responses. Numerical parametric studies are performed to illustrate effects of force type, graduation parameter, geometrical and stacking sequence of layers on the time response of deep multilayer FG beams.

• Structural Engineering and Mechanics
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• 제85권1호
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• pp.89-104
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• 2023

This work presents a proposal for employing reduced numerical integration in the formulation of the 4-node quadrilateral solid finite element. The use of these low-order integration rules leads to numerical instabilities such as those producing the hourglass effect. The proposed procedure allows evaluating a given constitutive model only in one integration point, achieving an attractive computational cost reduction and, also, successfully controls the hourglass effect. A validation of the proposal is included and discussed throughout the paper. To show the efficiency of the proposal, several application examples of masonry structures are studied and discussed. To represent the non-linear mechanical behaviour of masonry a plastic-damage model is implemented within the application of this sub-integration scheme. Also, in order to have a full and computationally efficient strategy to determine the behaviour of masonry structures, involving its evolution to collapse, a homogenization technique with a macro-modeling approach is used. The methodology discussed throughout this paper demonstrates a substantial computational cost reduction and an improved approximation of the non-linear problem evidenced by a reduction of up to 85% of the computational time for some cases.

• 대한기계학회:학술대회논문집
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• 대한기계학회 2004년도 추계학술대회
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• pp.453-458
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• 2004

This study proposes a new two-level condensation scheme for the construction of a reduced system. In the first step, the candidate area is selected for the construction of the reduced system by energy estimation in element-level. In the second step, primary degrees of freedom are selected by sequential elimination from the candidate degrees of freedom linked to the selected elements. Numerical examples demonstrate that the proposed method saves the computational cost effectively and provides a reduced system which predicts the eigenvalues accurately. Moreover, the well-constructed reduced system can present the reliable behavior of the structure under arbitrary dynamic loads comparing to that of global system. Time integration in a reduced system can save the computing time remarkably. Through a few numerical examples, the efficiency and reliability of the proposed scheme are verified.

• Structural Engineering and Mechanics
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• 제28권5호
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• pp.549-570
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• 2008

Numerical integration is an efficient approach for nonlinear dynamic analysis. In this paper, general category of the implicit integration errors will be discussed. In order to decrease the errors, Dynamic Relaxation method with modified time step (MFT) will be used. This procedure leads to an alternative algorithm which is very general and can be utilized with any implicit integration scheme. For numerical verification of the proposed technique, some single and multi degrees of freedom nonlinear dynamic systems will be analyzed. Moreover, results are compared with both exact and other available solutions. Suitable accuracy, high efficiency, simplicity, vector operations and automatic procedures are the main merits of the new algorithm in solving nonlinear dynamic problems.

• Structural Engineering and Mechanics
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• 제64권4호
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• pp.495-504
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• 2017

This paper proposes a new direct numerical integration algorithm for solving equation of motion in structural dynamics problems with nonlinear stiffness. The new implicit method's degree of accuracy is higher than that of existing methods due to the higher order of the acceleration. Two parameters are defined, leading to a new family of unconditionally stable methods, which helps to take greater time steps in integration and eliminate concerns about the duration of solving. The method developed can be utilized for a number of solid plane finite elements, examples of which are given to compare the proposed method with existing ones. The results indicate the superiority of the proposed method.

• Interaction and multiscale mechanics
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• 제6권2호
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• pp.107-125
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• 2013

A strain smoothing meshfree formulation with stabilized conforming nodal integration is presented for modeling the consolidation process in saturated porous media. In the present method, nodal strain smoothing is consistently introduced into the meshfree approximation of strain and pore pressure gradient variables associated with the saturated porous media. Meanwhile, in order to achieve a consistent numerical implementation, a smoothing approximation of the meshfree shape function within a nodal representative domain is also proposed in the stiffness construction. The resulting discrete system of equations is all expressed in smoothed nodal measures that are very efficient for numerical evaluation. Subsequently the space-time fully discrete equations are further established by the generalized trapezoidal rule for time integration. The effectiveness of the proposed meshfree consolidation analysis method is systematically illustrated by several benchmark problems.

• 대한기계학회논문집A
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• 제24권5호
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• pp.1075-1083
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• 2000

The prediction of the inelastic behavior of the structure is an essential part of reliability assessment procedure, because most of the failures are induced by the inelastic deformation, such as creep and plastic deformation. During decades, there has been much progress in understanding of the inelastic behavior of the materials and a lot of inelastic constitutive equations have been developed. These equations consist of the definition of inelastic strain and the evolution of the state variables introduced to quantify the irreversible processes occurred in the material. With respect to the definition of the inelastic strain, the inelastic constitutive models can be categorized into elastoplastic model, unified viscoplastic model and separated viscoplastic model and the different integration methods have been applied to each category. In the present investigation, the generalized integration method applicable for various types of constitutive equations is developed and implemented into ABAQUS by means of UMAT subroutine. The solution of the non-linear system of algebraic equations arising from time discretization with the generalized midpoint rule is determined using line-search technique in combination with Newton method. The strategy to control the time increment for the improvement of the accuracy of the numerical integration is proposed. Several numerical examples are considered to demonstrate the efficiency and applicability of the present method. The prediction of the inelastic behavior of the structure is an essential part of reliability assessment procedure, because most of the failures are induced by the inelastic deformation, such as creep and plastic deformation. During decades, there has been much progress in understanding of the inelastic behavior of the materials and a lot of inelastic constitutive equations have been developed. These equations consist of the definition of inelastic strain and the evolution of the state variables introduced to quantify the irreversible processes occurred in the material. With respect to the definition of the inelastic strain, the inelastic constitutive models can be categorized into elastoplastic model, unified viscoplastic model and separated viscoplastic model and the different integration methods have been applied to each category. In the present investigation, the generalized integration method applicable for various types of constitutive equations is developed and implemented into ABAQUS by means of UMAT subroutine. The solution of the non-linear system of algebraic equations arising from time discretization with the generalized midpoint rule is determined using line-search technique in combination with Newton method. The strategy to control the time increment for the improvement of the accuracy of the numerical integration is proposed. Several numerical examples are considered to demonstrate the efficiency and applicability of the present method.