• Title/Summary/Keyword: boundary condition

### FINITE DIFFERENCE METHOD FOR THE TWO-DIMENSIONAL BLACK-SCHOLES EQUATION WITH A HYBRID BOUNDARY CONDITION

• HEO, YOUNGJIN;HAN, HYUNSOO;JANG, HANBYEOL;CHOI, YONGHO;KIM, JUNSEOK
• Journal of the Korean Society for Industrial and Applied Mathematics
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• v.23 no.1
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• pp.19-30
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• 2019
• In this paper, we develop an accurate explicit finite difference method for the two-dimensional Black-Scholes equation with a hybrid boundary condition. In general, the correlation term in multi-asset options is problematic in numerical treatments partially due to cross derivatives and numerical boundary conditions at the far field domain corners. In the proposed hybrid boundary condition, we use a linear boundary condition at the boundaries where at least one asset is zero. After updating the numerical solution by one time step, we reduce the computational domain so that we do not need boundary conditions. To demonstrate the accuracy and efficiency of the proposed algorithm, we calculate option prices and their Greeks for the two-asset European call and cash-or-nothing options. Computational results show that the proposed method is accurate and is very useful for nonlinear boundary conditions.

### An Estimation on Failure Boundary Condition of Rocker Arm Shaft for 4-Cylinder SOHC Engine Using Orthogonal Array (직교배열표를 이용한 4기통 SOHC 엔진용 로커암 축의 파손경계조건 평가에 관한 연구)

• Lee, Soo-Jin;Lee, Dong-Woo;Hong, Soon-Hyeok;Cho, Seok-Swoo;Joo, Won-Sik
• Transactions of the Korean Society of Mechanical Engineers A
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• v.29 no.9 s.240
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• pp.1161-1168
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• 2005
• As a result of vehicle maintenance of rocker arm shaft for 4-cylinder SOHC engine, failure analysis of rocker arm shaft is needed. Because more than $30\%$ of vehicles investigated have been fractured. Failure analysis is classified into an naked eyes, microscope and X-ray fractography etc. It can predict applied load as well as load type. These methods are applicable to components with simple boundary condition but aren't applicable to components with complex boundary condition. The existing fractography don't catch hold of failure boundary condition quantitatively. Especially, in case that the components isn't fractured at same position. We must determine the most dangerous failure boundary condition to evaluate their operation mechanism. The effect of various factors on response should be estimated to solve this statical problem. This study presents the most dangerous failure boundary condition of rocker arm shaft using orthogonal array and ANOVA in order to assure its robustness.

### Consistent Boundary Condition for Horizontally-Polarized Shear (SH) Waves Propagated in Layered Waveguides (층상 waveguide에서의 SH파 전파 해석을 위한 경계조건)

• Lee, Jin Ho
• Journal of the Computational Structural Engineering Institute of Korea
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• v.34 no.2
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• pp.113-120
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• 2021
• The wave-propagation phenomenon in an infinite medium has been used to describe the physics in many fields of engineering and natural science. Analytical or numerical methods have been developed to obtain solutions to problems related to the wave-propagation phenomenon. Energy radiation into infinite regions must be accurately considered for accurate solutions to these problems; hence, various numerical and mechanical models as well as boundary conditions have been developed. This paper proposes a new boundary condition that can be applied to scalar-wave or horizontally-polarized shear-wave (or SH-wave) propagation problems in layered waveguides. A governing equation is obtained for the SH waves by applying finite-element discretization in the vertical direction of the waveguide and subsequently modified to derive the boundary condition for the infinite region of the waveguide. Using the orthogonality of the eigenmodes for the SH waves in a layered waveguide, the new boundary condition is shown to be equivalent to the existing root-finding absorbing boundary condition; further, the accuracy is shown to increase with the degree of the new boundary condition, and its stability can be proven. The accuracy and stability are then demonstrated by applying the proposed boundary condition to wave-propagation problems in layered waveguides.

### PERIODICITIES OF SOME ADDITIVE CELLULAR AUTOMATA

• Kim, Jae-Gyeom
• East Asian mathematical journal
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• v.31 no.1
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• pp.19-25
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• 2015
• We investigate periodicities of some additive cellular automata with null boundary condition.

### Second Order Bounce Back Boundary Condition for the Latice Boltzmann Fluid Simulation

• Kim, In-Chan
• Journal of Mechanical Science and Technology
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• v.14 no.1
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• pp.84-92
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• 2000
• A new bounce back boundary method of the second order in error is proposed for the lattice Boltzmann fluid simulation. This new method can be used for the arbitrarily irregular lattice geometry of a non-slip boundary. The traditional bounce back boundary condition for the lattice Boltzmann simulation is of the first order in error. Since the lattice Boltzmann method is the second order scheme by itself, a boundary technique of the second order has been desired to replace the first order bounce back method. This study shows that, contrary to the common belief that the bounce back boundary condition is unilaterally of the first order, the second order bounce back boundary condition can be realized. This study also shows that there exists a generalized bounce back technique that can be characterized by a single interpolation parameter. The second order bounce back method can be obtained by proper selection of this parameter in accordance with the detailed lattice geometry of the boundary. For an illustrative purpose, the transient Couette and the plane Poiseuille flows are solved by the lattice Boltzmann simulation with various boundary conditions. The results show that the generalized bounce back method yields the second order behavior in the error of the solution, provided that the interpolation parameter is properly selected. Coupled with its intuitive nature and the ease of implementation, the bounce back method can be as good as any second order boundary method.

### A Study on Shear Buckling Analysis and Boundary Condition of Sinusoidal Corrugated Plate by Using ABAQUS (ABAQUS를 이용한 정현파형 주름판의 전단좌굴해석과 경계조건에 관한 연구)

• Yoo, Mi-Na;Shon, Su-Deok;Lee, Seung-Jae;Kang, Joo-Won
• Journal of the Architectural Institute of Korea Structure & Construction
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• v.29 no.11
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• pp.3-11
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• 2013
• The characteristics of shear buckling about corrugated plates affect web height as well as the corrugated plate. Besides, there are complex phenomena, separated by local buckling, global buckling and ductility buckling. For this reason, the experimental and analytical study of corrugated webs are mainly conducted on shear buckling behavior and stability, whereas analytical study mainly uses commercial software. However, there are some differences between the application of boundary conditions and element modeling in the literature for each one. This study was performed to study about appropriate boundary condition in finite element analysis for sinusoidal corrugated plate. The analysis in this study was compared to shear buckling strength about each boundary condition with the plate shear theory, and shear buckling strength was obtained by converting the eigenvalues about primary mode. From the result of the analysis, the proposed boundary condition was demonstrated better condition.

### A solution method for the pressure-based boundary condition in the computation of two-dimensional incompressible viscous flow (2차원 비압축성 점성유동에 나타나는 압력 경계조건의 해결방안)

• 이재헌
• Transactions of the Korean Society of Mechanical Engineers
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• v.12 no.4
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• pp.926-933
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• 1988
• A Numerical method has been introduced to handle a pressure-based boundary condition of the incompressible viscous flow field. This method, based on SIMPLER algorithm, has been applied to analyze the flow characteristics within a two-dimensional duct of two-exit, as an example. From this, it is possible to determine the ratio of flow rate through two exits imposed on different static pressure. In order to check the validity of the present method, calculated velocity at the boundary imposed on pressure condition by the use of present method has been transferred to the velocity boundary condition of the conventional numerical method workable only with the velocity-based boundary condition. It is found that the calculated boundary pressure from conventional method are almost identical to those endowed originally. Present method, therefore will be widely applicable to the practical situations specified by the pressure-based boundary condition rather than the velocity one.

### A Non-Hydrostatic Pressure Model and its Implementation of the Dynamic Boundary Condition (동수압 모형의 동역학적 경계조건)

• Lee, Jong Wook;Lee, Jin Woo;Cho, Yong-Sik
• KSCE Journal of Civil and Environmental Engineering Research
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• v.28 no.6B
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• pp.691-696
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• 2008
• In this paper, a three-dimensional non-hydrostatic pressure model for free surface flows using a normalized vertical coordinate system is presented. To strongly couple the free surface and non-hydrostatic pressure in the momentum equations, a double predictor-corrector method is employed. This research is especially focused on implementing the dynamic boundary condition (a zero pressure condition) at the free surface. This boundary condition can be specified accurately with a small modification to existing models. Numerical results with and without this modification clearly show that a precise implementation of the dynamic boundary condition is paramountly important.

### Numerical simulation of coextrusion process of viscoelastic fluids using the open boundary condition method

• Park, Seung-Joon;Ahn, Kyung-Hyun;Lee, Seung-Jong
• Korea-Australia Rheology Journal
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• v.13 no.1
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• pp.37-45
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• 2001
• Numerical simulation of coextrusion process of viscoelastic fluids within a die has been carried out. In the coextrusion process velocity profile at the outflow boundary is not known a priori, which makes it difficult to impose the proper boundary condition at the outflow boundary. This difficulty has been avoided by using the open boundary condition (OBC) method. In this study, elastic viscous stress splitting (EVSS) formulation with streamline upwind (SU) method has been used in the finite element method. In order to test the validity of the OBC method, comparison between the results of fully developed condition at the outlet and those of OBC has been made for a Newtonian fluid. In the case of upper convected Maxwell (UCM) fluid, the effect of outflow boundary condition on the interface position has been investigated by using two meshes having different downstream lengths. In both cases, the results with the OBC method showed reasonable interface shape. In particular, for the UCM fluid the interface shape calculated with OBC was independent of the downstream length, while the results with the zero traction condition showed oscillation of interface position close to the outlet. Viscosity difference was found to be more important than elasticity difference in determining the final interface position. However, the overshoot of interface position near the con-fluent point increased with elasticity.

### 1D finite element artificial boundary method for layered half space site response from obliquely incident earthquake

• Zhao, Mi;Yin, Houquan;Du, Xiuli;Liu, Jingbo;Liang, Lingyu
• Earthquakes and Structures
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• v.9 no.1
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• pp.173-194
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• 2015
• Site response analysis is an important topic in earthquake engineering. A time-domain numerical method called as one-dimensional (1D) finite element artificial boundary method is proposed to simulate the homogeneous plane elastic wave propagation in a layered half space subjected to the obliquely incident plane body wave. In this method, an exact artificial boundary condition combining the absorbing boundary condition with the inputting boundary condition is developed to model the wave absorption and input effects of the truncated half space under layer system. The spatially two-dimensional (2D) problem consisting of the layer system with the artificial boundary condition is transformed equivalently into a 1D one along the vertical direction according to Snell's law. The resulting 1D problem is solved by the finite element method with a new explicit time integration algorithm. The 1D finite element artificial boundary method is verified by analyzing two engineering sites in time domain and by comparing with the frequency-domain transfer matrix method with fast Fourier transform.