• Coupled systems mechanics
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• v.7 no.5
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• pp.583-610
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• 2018

This paper discusses the issues associated with modeling frictional contact between solid bodies undergoing large deformations. The most common model for friction on contact interfaces in solid mechanics is the Coulomb friction model, in which two distinct responses are possible: stick and slip. Handling the transition between these two phases computationally has been a source of algorithmic instability, lack of convergence and non-unique solutions, particularly in the presence of large deformations. Most computational models for frictional contact have used penalty or updated Lagrangian approaches to enforce frictional contact conditions. These two approaches, however, present some computational challenges due to conditioning issues in penalty-type implementations and the iterative nature of the updated Lagrangian formulation, which, particularly in large simulations, may lead to relatively slow convergence. Alternatively, a plasticity-inspired implementation of frictional contact has been shown to handle the stick-slip conditions in a local, algorithmically efficient manner that substantially reduces computational cost and successfully avoids the issues of instability and lack of convergence often reported with other methods (Laursen and Simo 1993). The formulation of this approach, however, has been limited to the small deformations realm, a fact that severely limited its application to contact problems where large deformations are expected. In this paper, we present an algorithmically consistent formulation of this method that preserves its key advantages, while extending its application to the realm of large-deformation contact problems. We show that the method produces results similar to the augmented Lagrangian formulation at a reduced computational cost.

• Steel and Composite Structures
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• v.29 no.2
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• pp.243-256
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• 2018

This study conducts an optimization and sensitivity analysis on rectangular reinforced concrete (RC) beam using Lagrangian Multiplier Method (LMM) as programming optimization computer soft ware. The analysis is conducted to obtain the minimum design cost for both singly and doubly RC beams according to the specifications of three regulations of American concrete institute (ACI), British regulation (BS), and Iranian concrete regulation (ICS). Moreover, a sensitivity analysis on cost is performed with respect to the effective parameters such as length, width, and depth of beam, and area of reinforcement. Accordingly, various curves are developed to be feasibly utilized in design of RC beams. Numerical examples are also represented to better illustrate the design steps. The results indicate that instead of complex optimization relationships, the LMM can be used to minimize the cost of singly and doubly reinforced beams with different boundary conditions. The results of the sensitivity analysis on LMM indicate that each regulation can provide the most optimal values at specific situations. Therefore, using the graphs proposed for different design conditions can effectively help the designer (without necessity of primary optimization knowledge) choose the best regulation and values of design parameters.

• Computational Structural Engineering
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• v.10 no.2
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• pp.139-148
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• 1997

A finite element method is programmed to analyse the nonlinear behavior of axisymmetric structures. The lst order Mindlin shell theory which takes into account the transversal shear deformation is used to formulate a conical two node element with six degrees of freedom. To evade the shear locking phenomenon which arises in Mindlin type element when the effect of shear deformation tends to zero, the reduced integration of one point Gauss Quadrature at the center of element is employed. This method is the Updated Lagrangian formulation which refers the variables to the state of the most recent iteration. The solution is searched by Newton-Raphson iteration method. The tangent matrix of this method is obtained by a finite difference method by perturbating the degrees of freedom with small values. For the moment this program is limited to the analyses of non-linear elastic problems. For structures which could have elastic stability problem, the calculation is controled by displacement.

• Proceedings of the KSME Conference
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• 2004.11a
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• pp.474-479
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• 2004

According to our previous study, it is confirmed that the Petrov-Galerkin natural element method (PGNEM) completely resolves the numerical integration inaccuracy in the conventional Bubnov-Galerkin natural element method (BG-NEM). This paper is an extension of PG-NEM to two-dimensional nonlinear dynamic problem. For the analysis, a constant average acceleration method and a linearized total Lagrangian formulation is introduced with the PG-NEM. At every time step, the grid points are updated and the shape functions are reproduced from the relocated nodal distribution. This process enables the PG-NEM to provide more accurate and robust approximations. The representative numerical experiments performed by the test Fortran program, and the numerical results confirmed that the PG-NEM effectively and accurately approximates the nonlinear dynamic problem.

• Transactions of Materials Processing
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• v.4 no.4
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• pp.365-374
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• 1995

Numerical simulation of blade forming is carried out as stretch forming by an elasto-plastic finite element method. The method adopts a Lagrangian formulation, which incorporates large deformation and rotation, with a penalty method to treat the contact boundary condition. Numerical integration is done with a directional reduced integration scheme to avoid shear locking. The numerical results demonstrates various final shapes of blades which depend on the variation of the stretching force. The strain distributions in deformed blades are also obtained with the variation of the stretching force.

• Journal of computational fluids engineering
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• v.12 no.4
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• pp.14-19
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• 2007

We present a direct numerical simulation technique and some preliminary results of the capillary spreading of a droplet containing particles on the solid substrate. We used the level-set method with the continuous surface stress for description of droplet spreading with interfacial tension and employed the discontinuous Galerkin method for the stabilization of the interface advection equation. The distributed Lagrangian-multipliers method has been combined for the implicit treatment of rigid particles. We investigated the droplet spreading by the capillary force and discussed effects of the presence of particles on the spreading behavior. It has been observed that a particulate drop spreads less than the pure liquid drop. The amount of spread of a particulate drop has been found smaller than that of the liquid with effectively the same viscosity as the particulate drop.

• Steel and Composite Structures
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• v.32 no.3
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• pp.293-304
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• 2019

In the present paper, the element free Galerkin (EFG) method is developed for geometrically nonlinear analysis of deep beams considering small scale effect. To interpret the behavior of structure at the nano scale, the higher-order gradient elasticity nonlocal theory is taken into account. The radial point interpolation method with high order of continuity is used to construct the shape functions. The nonlinear equation of motion is derived using the principle of the minimization of total potential energy based on total Lagrangian approach. The Newmark method with the small time steps is used to solve the time dependent equations. At each time step, the iterative Newton-Raphson technique is applied to minimize the residential forces caused by the nonlinearity of the equations. The effects of nonlocal parameter and aspect ratio on stiffness and dynamic parameters are discussed by numerical examples. This paper furnishes a ground to develop the EFG method for large deformation analysis of structures considering small scale effects.

• Nonlinear Functional Analysis and Applications
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• v.28 no.3
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• pp.613-622
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• 2023

In this paper, finite element method is applied to solve boundary control problem governed by elliptic variational inequality with an infinite number of variables. First, we introduce some important features of the finite element method, boundary control problem governed by elliptic variational inequalities with an infinite number of variables in the case of the control and observation are on the boundary is introduced. We prove the existence of the solution by using the augmented Lagrangian multipliers method. A triangular type finite element method is used.

• Journal of Korean Society of Industrial and Systems Engineering
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• v.38 no.4
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• pp.109-116
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• 2015

This paper is concerned with the single vendor single buyer integrated production inventory problem. To make this problem more practical, space restriction and lead time proportional to lot size are considered. Since the space for the inventory is limited in most practical inventory system, the space restriction for the inventory of a vendor and a buyer is considered. As product's quantity to be manufactured by the vendor is increased, the lead time for the order is usually increased. Therefore, lead time for the product is proportional to the order quantity by the buyer. Demand is assumed to be stochastic and the continuous review inventory policy is used by the buyer. If the buyer places an order, then the vendor will start to manufacture products and the products will be transferred to the buyer with equal shipments many times. The mathematical formulation with space restriction for the inventory of a vendor and a buyer is suggested in this paper. This problem is constrained nonlinear integer programming problem. Order quantity, reorder points for the buyer, and the number of shipments are required to be determined. A Lagrangian relaxation approach, a popular solution method for constrained problem, is developed to find lower bound of this problem. Since a Lagrangian relaxation approach cannot guarantee the feasible solution, the solution method based on the Lagrangian relaxation approach is proposed to provide with a good feasible solution. Total costs by the proposed method are pretty close to those by the Lagrangian relaxation approach. Sensitivity analysis for space restriction for the vendor and the buyer is done to figure out the relationships between parameters.

• Transactions of the Korean Society of Mechanical Engineers
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• v.15 no.1
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• pp.241-251
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• 1991

The present simulation model relies on a new approach of the heavy particle motion in a turbulent flow considering the time and space correlation to the Lagrangian point of view. The turbulent field is, here, assumed that its characteristic scales are random and follow a Poisson's distribution. Using this model, we have computed the trajectory of each particle, that is, its velocity and position at each time in order to study the dispersion of particles in a grid turbulent flow. The computed results have been compared to the corresponding experimental data. Due to the complex mechanism of turbulence and the theoretically and experimentally lacking information, we had to make some assumptions for simplifying the situation, but we have found the good agreement between simulated and measured results. In particular, the application of the present method on the Lagrangian correlation of particle provides an interesting alternative to the usual computational methods.