Commenced in January 2007
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Edition: International
Paper Count: 13166

Search results for: numerical methods

13166 Improvement of the Numerical Integration's Quality in Meshless Methods

Authors: Ahlem Mougaida, Hedi Bel Hadj Salah


Several methods are suggested to improve the numerical integration in Galerkin weak form for Meshless methods. In fact, integrating without taking into account the characteristics of the shape functions reproduced by Meshless methods (rational functions, with compact support etc.), causes a large integration error that influences the PDE’s approximate solution. Comparisons between different methods of numerical integration for rational functions are discussed and compared. The algorithms are implemented in Matlab. Finally, numerical results were presented to prove the efficiency of our algorithms in improving results.

Keywords: adaptive methods, meshless, numerical integration, rational quadrature

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13165 Numerical Methods versus Bjerksund and Stensland Approximations for American Options Pricing

Authors: Marasovic Branka, Aljinovic Zdravka, Poklepovic Tea


Numerical methods like binomial and trinomial trees and finite difference methods can be used to price a wide range of options contracts for which there are no known analytical solutions. American options are the most famous of that kind of options. Besides numerical methods, American options can be valued with the approximation formulas, like Bjerksund-Stensland formulas from 1993 and 2002. When the value of American option is approximated by Bjerksund-Stensland formulas, the computer time spent to carry out that calculation is very short. The computer time spent using numerical methods can vary from less than one second to several minutes or even hours. However to be able to conduct a comparative analysis of numerical methods and Bjerksund-Stensland formulas, we will limit computer calculation time of numerical method to less than one second. Therefore, we ask the question: Which method will be most accurate at nearly the same computer calculation time?

Keywords: Bjerksund and Stensland approximations, computational analysis, finance, options pricing, numerical methods

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13164 Groundwater Seepage Estimation into Amirkabir Tunnel Using Analytical Methods and DEM and SGR Method

Authors: Hadi Farhadian, Homayoon Katibeh


In this paper, groundwater seepage into Amirkabir tunnel has been estimated using analytical and numerical methods for 14 different sections of the tunnel. Site Groundwater Rating (SGR) method also has been performed for qualitative and quantitative classification of the tunnel sections. The obtained results of above-mentioned methods were compared together. The study shows reasonable accordance with results of the all methods unless for two sections of tunnel. In these two sections there are some significant discrepancies between numerical and analytical results mainly originated from model geometry and high overburden. SGR and the analytical and numerical calculations, confirm the high concentration of seepage inflow in fault zones. Maximum seepage flow into tunnel has been estimated 0.425 lit/sec/m using analytical method and 0.628 lit/sec/m using numerical method occurred in crashed zone. Based on SGR method, six sections of 14 sections in Amirkabir tunnel axis are found to be in "No Risk" class that is supported by the analytical and numerical seepage value of less than 0.04 lit/sec/m.

Keywords: water Seepage, Amirkabir Tunnel, analytical method, DEM, SGR

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13163 Numerical and Analytical Approach for Film Condensation on Different Forms of Surfaces

Authors: A. Kazemi Jouybari, A. Mirabdolah Lavasani


This paper seeks to the solution of condensation around of a flat plate, circular and elliptical tube in way of numerical and analytical methods. Also, it calculates the entropy production rates. The first, problem was solved by using mesh dynamic and rational assumptions, next it was compared with the numerical solution that the result had acceptable errors. An additional supporting relation was applied based on a characteristic of condensation phenomenon for condensing elements. As it has been shown here, due to higher rates of heat transfer for elliptical tubes, they have more entropy production rates, in comparison to circular ones. Findings showed that two methods were efficient. Furthermore, analytical methods can be used to optimize the problem and reduce the entropy production rate.

Keywords: condensation, numerical solution, analytical solution, entropy rate

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13162 A Comparative Study between FEM and Meshless Methods

Authors: Jay N. Vyas, Sachin Daxini


Numerical simulation techniques are widely used now in product development and testing instead of expensive, time-consuming and sometimes dangerous laboratory experiments. Numerous numerical methods are available for performing simulation of physical problems of different engineering fields. Grid based methods, like Finite Element Method, are extensively used in performing various kinds of static, dynamic, structural and non-structural analysis during product development phase. Drawbacks of grid based methods in terms of discontinuous secondary field variable, dealing fracture mechanics and large deformation problems led to development of a relatively a new class of numerical simulation techniques in last few years, which are popular as Meshless methods or Meshfree Methods. Meshless Methods are expected to be more adaptive and flexible than Finite Element Method because domain descretization in Meshless Method requires only nodes. Present paper introduces Meshless Methods and differentiates it with Finite Element Method in terms of following aspects: Shape functions used, role of weight function, techniques to impose essential boundary conditions, integration techniques for discrete system equations, convergence rate, accuracy of solution and computational effort. Capabilities, benefits and limitations of Meshless Methods are discussed and concluded at the end of paper.

Keywords: numerical simulation, Grid-based methods, Finite Element Method, Meshless Methods

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13161 Assessment of Slope Stability by Continuum and Discontinuum Methods

Authors: Taleb Hosni Abderrahmane, Berga Abdelmadjid


The development of numerical analysis and its application to geomechanics problems have provided geotechnical engineers with extremely powerful tools. One of the most important problems in geotechnical engineering is the slope stability assessment. It is a very difficult task due to several aspects such the nature of the problem, experimental consideration, monitoring, controlling, and assessment. The main objective of this paper is to perform a comparative numerical study between the following methods: The Limit Equilibrium (LEM), Finite Element (FEM), Limit Analysis (LAM) and Distinct Element (DEM). The comparison is conducted in terms of the safety factors and the critical slip surfaces. Through the results, we see the feasibility to analyse slope stability by many methods.

Keywords: comparison, factor of safety, geomechanics, numerical methods, slope analysis, slip surfaces

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13160 Numerical Evolution Methods of Rational Form for Diffusion Equations

Authors: Said Algarni


The purpose of this study was to investigate selected numerical methods that demonstrate good performance in solving PDEs. We adapted alternative method that involve rational polynomials. Padé time stepping (PTS) method, which is highly stable for the purposes of the present application and is associated with lower computational costs, was applied. Furthermore, PTS was modified for our study which focused on diffusion equations. Numerical runs were conducted to obtain the optimal local error control threshold.

Keywords: Padé time stepping, finite difference, reaction diffusion equation, PDEs

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13159 Comparing Numerical Accuracy of Solutions of Ordinary Differential Equations (ODE) Using Taylor's Series Method, Euler's Method and Runge-Kutta (RK) Method

Authors: Palwinder Singh, Munish Sandhir, Tejinder Singh


The ordinary differential equations (ODE) represent a natural framework for mathematical modeling of many real-life situations in the field of engineering, control systems, physics, chemistry and astronomy etc. Such type of differential equations can be solved by analytical methods or by numerical methods. If the solution is calculated using analytical methods, it is done through calculus theories, and thus requires a longer time to solve. In this paper, we compare the numerical accuracy of the solutions given by the three main types of one-step initial value solvers: Taylor’s Series Method, Euler’s Method and Runge-Kutta Fourth Order Method (RK4). The comparison of accuracy is obtained through comparing the solutions of ordinary differential equation given by these three methods. Furthermore, to verify the accuracy; we compare these numerical solutions with the exact solutions.

Keywords: Ordinary differential equations (ODE), Taylor’s Series Method, Euler’s Method, Runge-Kutta Fourth Order Method

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13158 A Continuous Boundary Value Method of Order 8 for Solving the General Second Order Multipoint Boundary Value Problems

Authors: T. A. Biala


This paper deals with the numerical integration of the general second order multipoint boundary value problems. This has been achieved by the development of a continuous linear multistep method (LMM). The continuous LMM is used to construct a main discrete method to be used with some initial and final methods (also obtained from the continuous LMM) so that they form a discrete analogue of the continuous second order boundary value problems. These methods are used as boundary value methods and adapted to cope with the integration of the general second order multipoint boundary value problems. The convergence, the use and the region of absolute stability of the methods are discussed. Several numerical examples are implemented to elucidate our solution process.

Keywords: linear multistep methods, boundary value methods, second order multipoint boundary value problems, convergence

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13157 Induction Heating Process Design Using Comsol® Multiphysics Software Version 4.2a

Authors: K. Djellabi, M. E. H. Latreche


Induction heating computer simulation is a powerful tool for process design and optimization, induction coil design, equipment selection, as well as education and business presentations. The authors share their vast experience in the practical use of computer simulation for different induction heating and heat treating processes. In this paper deals with mathematical modeling and numerical simulation of induction heating furnaces with axisymmetric geometries. For the numerical solution, we propose finite element methods combined with boundary (FEM) for the electromagnetic model using COMSOL® Multiphysics Software. Some numerical results for an industrial furnace are shown with high frequency.

Keywords: numerical methods, induction furnaces, induction heating, finite element method, Comsol multiphysics software

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13156 Quartic Spline Method for Numerical Solution of Self-Adjoint Singularly Perturbed Boundary Value Problems

Authors: Reza Mohammadi


Using quartic spline, we develop a method for numerical solution of singularly perturbed two-point boundary-value problems. The purposed method is fourth-order accurate and applicable to problems both in singular and non-singular cases. The convergence analysis of the method is given. The resulting linear system of equations has been solved by using a tri-diagonal solver. We applied the presented method to test problems which have been solved by other existing methods in references, for comparison of presented method with the existing methods. Numerical results are given to illustrate the efficiency of our methods.

Keywords: second-order ordinary differential equation, singularly-perturbed, quartic spline, convergence analysis

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13155 Development of Extended Trapezoidal Method for Numerical Solution of Volterra Integro-Differential Equations

Authors: Fuziyah Ishak, Siti Norazura Ahmad


Volterra integro-differential equations appear in many models for real life phenomena. Since analytical solutions for this type of differential equations are hard and at times impossible to attain, engineers and scientists resort to numerical solutions that can be made as accurately as possible. Conventionally, numerical methods for ordinary differential equations are adapted to solve Volterra integro-differential equations. In this paper, numerical solution for solving Volterra integro-differential equation using extended trapezoidal method is described. Formulae for the integral and differential parts of the equation are presented. Numerical results show that the extended method is suitable for solving first order Volterra integro-differential equations.

Keywords: accuracy, extended trapezoidal method, numerical solution, Volterra integro-differential equations

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13154 Pricing European Options under Jump Diffusion Models with Fast L-stable Padé Scheme

Authors: Salah Alrabeei, Mohammad Yousuf


The goal of option pricing theory is to help the investors to manage their money, enhance returns and control their financial future by theoretically valuing their options. Modeling option pricing by Black-School models with jumps guarantees to consider the market movement. However, only numerical methods can solve this model. Furthermore, not all the numerical methods are efficient to solve these models because they have nonsmoothing payoffs or discontinuous derivatives at the exercise price. In this paper, the exponential time differencing (ETD) method is applied for solving partial integrodifferential equations arising in pricing European options under Merton’s and Kou’s jump-diffusion models. Fast Fourier Transform (FFT) algorithm is used as a matrix-vector multiplication solver, which reduces the complexity from O(M2) into O(M logM). A partial fraction form of Pad`e schemes is used to overcome the complexity of inverting polynomial of matrices. These two tools guarantee to get efficient and accurate numerical solutions. We construct a parallel and easy to implement a version of the numerical scheme. Numerical experiments are given to show how fast and accurate is our scheme.

Keywords: Integral differential equations, , L-stable methods, pricing European options, Jump–diffusion model

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13153 Three Dimensional Numerical Analysis for Longitudinal Seismic Response of Tunnels under Asynchronous Earthquake

Authors: Peng Li, Er-xiang Song


Numerical analysis of longitudinal tunnel seismic response due to spatial variation of earthquake ground motion is an important issue that cannot be ignored in the design and safety evaluation of tunnel structures. In this paper, numerical methods for analysis of tunnel longitudinal response under asynchronous seismic wave is extensively studied, including the improvement of the 1D time-domain finite element method, three dimensional numerical simulation technique for the site asynchronous earthquake response as well as the 3-D soil-tunnel structure interaction analysis. The study outcome will be beneficial to aid further research on the nonlinear meticulous numerical analysis and seismic response mechanism of tunnel structures under asynchronous earthquake motion.

Keywords: asynchronous input, longitudinal seismic response, tunnel structure, numerical simulation, traveling wave effect

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13152 Robust Numerical Method for Singularly Perturbed Semilinear Boundary Value Problem with Nonlocal Boundary Condition

Authors: Habtamu Garoma Debela, Gemechis File Duressa


In this work, our primary interest is to provide ε-uniformly convergent numerical techniques for solving singularly perturbed semilinear boundary value problems with non-local boundary condition. These singular perturbation problems are described by differential equations in which the highest-order derivative is multiplied by an arbitrarily small parameter ε (say) known as singular perturbation parameter. This leads to the existence of boundary layers, which are basically narrow regions in the neighborhood of the boundary of the domain, where the gradient of the solution becomes steep as the perturbation parameter tends to zero. Due to the appearance of the layer phenomena, it is a challenging task to provide ε-uniform numerical methods. The term 'ε-uniform' refers to identify those numerical methods in which the approximate solution converges to the corresponding exact solution (measured to the supremum norm) independently with respect to the perturbation parameter ε. Thus, the purpose of this work is to develop, analyze, and improve the ε-uniform numerical methods for solving singularly perturbed problems. These methods are based on nonstandard fitted finite difference method. The basic idea behind the fitted operator, finite difference method, is to replace the denominator functions of the classical derivatives with positive functions derived in such a way that they capture some notable properties of the governing differential equation. A uniformly convergent numerical method is constructed via nonstandard fitted operator numerical method and numerical integration methods to solve the problem. The non-local boundary condition is treated using numerical integration techniques. Additionally, Richardson extrapolation technique, which improves the first-order accuracy of the standard scheme to second-order convergence, is applied for singularly perturbed convection-diffusion problems using the proposed numerical method. Maximum absolute errors and rates of convergence for different values of perturbation parameter and mesh sizes are tabulated for the numerical example considered. The method is shown to be ε-uniformly convergent. Finally, extensive numerical experiments are conducted which support all of our theoretical findings. A concise conclusion is provided at the end of this work.

Keywords: nonlocal boundary condition, nonstandard fitted operator, semilinear problem, singular perturbation, uniformly convergent

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13151 Optimization of the Numerical Fracture Mechanics

Authors: H. Hentati, R. Abdelmoula, Li Jia, A. Maalej


In this work, we present numerical simulations of the quasi-static crack propagation based on the variation approach. We perform numerical simulations of a piece of brittle material without initial crack. An alternate minimization algorithm is used. Based on these numerical results, we determine the influence of numerical parameters on the location of crack. We show the importance of trying to optimize the time of numerical computation and we present the first attempt to develop a simple numerical method to optimize this time.

Keywords: fracture mechanics, optimization, variation approach, mechanic

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13150 A New Class of Conjugate Gradient Methods Based on a Modified Search Direction for Unconstrained Optimization

Authors: Belloufi Mohammed, Sellami Badreddine


Conjugate gradient methods have played a special role for solving large scale optimization problems due to the simplicity of their iteration, convergence properties and their low memory requirements. In this work, we propose a new class of conjugate gradient methods which ensures sufficient descent. Moreover, we propose a new search direction with the Wolfe line search technique for solving unconstrained optimization problems, a global convergence result for general functions is established provided that the line search satisfies the Wolfe conditions. Our numerical experiments indicate that our proposed methods are preferable and in general superior to the classical conjugate gradient methods in terms of efficiency and robustness.

Keywords: unconstrained optimization, conjugate gradient method, sufficient descent property, numerical comparisons

Procedia PDF Downloads 273
13149 A Family of Second Derivative Methods for Numerical Integration of Stiff Initial Value Problems in Ordinary Differential Equations

Authors: Luke Ukpebor, C. E. Abhulimen


Stiff initial value problems in ordinary differential equations are problems for which a typical solution is rapidly decaying exponentially, and their numerical investigations are very tedious. Conventional numerical integration solvers cannot cope effectively with stiff problems as they lack adequate stability characteristics. In this article, we developed a new family of four-step second derivative exponentially fitted method of order six for the numerical integration of stiff initial value problem of general first order differential equations. In deriving our method, we employed the idea of breaking down the general multi-derivative multistep method into predator and corrector schemes which possess free parameters that allow for automatic fitting into exponential functions. The stability analysis of the method was discussed and the method was implemented with numerical examples. The result shows that the method is A-stable and competes favorably with existing methods in terms of efficiency and accuracy.

Keywords: A-stable, exponentially fitted, four step, predator-corrector, second derivative, stiff initial value problems

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13148 FE Analysis of Blade-Disc Dovetail Joints Using Mortar Base Frictional Contact Formulation

Authors: Abbas Moradi, Mohsen Safajoy, Reza Yazdanparast


Analysis of blade-disc dovetail joints is one of the biggest challenges facing designers of aero-engines. To avoid comparatively expensive experimental full-scale tests, numerical methods can be used to simulate loaded disc-blades assembly. Mortar method provides a powerful and flexible tool for solving frictional contact problems. In this study, 2D frictional contact in dovetail has been analysed based on the mortar algorithm. In order to model the friction, the classical law of coulomb and moving friction cone algorithm is applied. The solution is then obtained by solving the resulting set of non-linear equations using an efficient numerical algorithm based on Newton–Raphson Method. The numerical results show that this approach has better convergence rate and accuracy than other proposed numerical methods.

Keywords: computational contact mechanics, dovetail joints, nonlinear FEM, mortar approach

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13147 The Impact of Cooperative Learning on Numerical Methods Course

Authors: Sara Bilal, Abdi Omar Shuriye, Raihan Othman


Numerical Methods is a course that can be conducted using workshops and group discussion. This study has been implemented on undergraduate students of level two at the Faculty of Engineering, International Islamic University Malaysia. The Numerical Method course has been delivered to two Sections 1 and 2 with 44 and 22 students in each section, respectively. Systematic steps have been followed to apply the student centered learning approach in teaching Numerical Method course. Initially, the instructor has chosen the topic which was Euler’s Method to solve Ordinary Differential Equations (ODE) to be learned. The students were then divided into groups with five members in each group. Initial instructions have been given to the group members to prepare their subtopics before meeting members from other groups to discuss the subtopics in an expert group inside the classroom. For the time assigned for the classroom discussion, the setting of the classroom was rearranged to accommodate the student centered learning approach. Teacher strength was by monitoring the process of learning inside and outside the class. The students have been assessed during the migrating to the expert groups, recording of a video explanation outside the classroom and during the final examination. Euler’s Method to solve the ODE was set as part of Question 3(b) in the final exam. It is observed that none of the students from both sections obtained a zero grade in Q3(b), compared to Q3(a) and Q3(c). Also, for Section 1(44 students), 29 students obtained the full mark of 7/7, while only 10 obtained 7/7 for Q3(a) and no students obtained 6/6 for Q3(c). Finally, we can recommend that the Numerical Method course be moved toward more student-centered Learning classrooms where the students will be engaged in group discussion rather than having a teacher one man show.

Keywords: teacher centered learning, student centered learning, mathematic, numerical methods

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13146 Generic Hybrid Models for Two-Dimensional Ultrasonic Guided Wave Problems

Authors: Manoj Reghu, Prabhu Rajagopal, C. V. Krishnamurthy, Krishnan Balasubramaniam


A thorough understanding of guided ultrasonic wave behavior in structures is essential for the application of existing Non Destructive Evaluation (NDE) technologies, as well as for the development of new methods. However, the analysis of guided wave phenomena is challenging because of their complex dispersive and multimodal nature. Although numerical solution procedures have proven to be very useful in this regard, the increasing complexity of features and defects to be considered, as well as the desire to improve the accuracy of inspection often imposes a large computational cost. Hybrid models that combine numerical solutions for wave scattering with faster alternative methods for wave propagation have long been considered as a solution to this problem. However usually such models require modification of the base code of the solution procedure. Here we aim to develop Generic Hybrid models that can be directly applied to any two different solution procedures. With this goal in mind, a Numerical Hybrid model and an Analytical-Numerical Hybrid model has been developed. The concept and implementation of these Hybrid models are discussed in this paper.

Keywords: guided ultrasonic waves, Finite Element Method (FEM), Hybrid model

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13145 Computational Approaches for Ballistic Impact Response of Stainless Steel 304

Authors: A. Mostafa


This paper presents a numerical study on determination of ballistic limit velocity (V50) of stainless steel 304 (SS 304) used in manufacturing security screens. The simulated ballistic impact tests were conducted on clamped sheets with different thicknesses using ABAQUS/Explicit nonlinear finite element (FE) package. The ballistic limit velocity was determined using three approaches, namely: numerical tests based on material properties, FE calculated residual velocities and FE calculated residual energies. Johnson-Cook plasticity and failure criterion were utilized to simulate the dynamic behaviour of the SS 304 under various strain rates, while the well-known Lambert-Jonas equation was used for the data regression for the residual velocity and energy model. Good agreement between the investigated numerical methods was achieved. Additionally, the dependence of the ballistic limit velocity on the sheet thickness was observed. The proposed approaches present viable and cost-effective assessment methods of the ballistic performance of SS 304, which will support the development of robust security screen systems.

Keywords: ballistic velocity, stainless steel, numerical approaches, security screen

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13144 Convergence of Sinc Methods Applied to Kuramoto-Sivashinsky Equation

Authors: Kamel Al-Khaled


A comparative study of the Sinc-Galerkin and Sinc-Collocation methods for solving the Kuramoto-Sivashinsky equation is given. Both approaches depend on using Sinc basis functions. Firstly, a numerical scheme using Sinc-Galerkin method is developed to approximate the solution of Kuramoto-Sivashinsky equation. Sinc approximations to both derivatives and indefinite integrals reduces the solution to an explicit system of algebraic equations. The error in the solution is shown to converge to the exact solution at an exponential. The convergence proof of the solution for the discrete system is given using fixed-point iteration. Secondly, a combination of a Crank-Nicolson formula in the time direction, with the Sinc-collocation in the space direction is presented, where the derivatives in the space variable are replaced by the necessary matrices to produce a system of algebraic equations. The methods are tested on two examples. The demonstrated results show that both of the presented methods more or less have the same accuracy.

Keywords: Sinc-Collocation, nonlinear PDEs, numerical methods, fixed-point

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13143 Spline Solution of Singularly Perturbed Boundary Value Problems

Authors: Reza Mohammadi


Using quartic spline, we develop a method for numerical solution of singularly perturbed two-point boundary-value problems. The purposed method is fourth-order accurate and applicable to problems both in singular and non-singular cases. The convergence analysis of the method is given. The resulting linear system of equations has been solved by using a tri-diagonal solver. We applied the presented method to test problems which have been solved by other existing methods in references, for comparison of presented method with the existing methods. Numerical results are given to illustrate the efficiency of our methods.

Keywords: second-order ordinary differential equation, singularly-perturbed, quartic spline, convergence analysis

Procedia PDF Downloads 205
13142 Numerical Modeling for Water Engineering and Obstacle Theory

Authors: Mounir Adal, Baalal Azeddine, Afifi Moulay Larbi


Numerical analysis is a branch of mathematics devoted to the development of iterative matrix calculation techniques. We are searching for operations optimization as objective to calculate and solve systems of equations of order n with time and energy saving for computers that are conducted to calculate and analyze big data by solving matrix equations. Furthermore, this scientific discipline is producing results with a margin of error of approximation called rates. Thus, the results obtained from the numerical analysis techniques that are held on computer software such as MATLAB or Simulink offers a preliminary diagnosis of the situation of the environment or space targets. By this we can offer technical procedures needed for engineering or scientific studies exploitable by engineers for water.

Keywords: numerical analysis methods, obstacles solving, engineering, simulation, numerical modeling, iteration, computer, MATLAB, water, underground, velocity

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13141 An Implementation of Meshless Method for Modeling an Elastoplasticity Coupled to Damage

Authors: Sendi Zohra, Belhadjsalah Hedi, Labergere Carl, Saanouni Khemais


The modeling of mechanical problems including both material and geometric nonlinearities with Finite Element Method (FEM) remains challenging. Meshless methods offer special properties to get rid of well-known drawbacks of the FEM. The main objective of Meshless Methods is to eliminate the difficulty of meshing and remeshing the entire structure by simply insertion or deletion of nodes, and alleviate other problems associated with the FEM, such as element distortion, locking and others. In this study, a robust numerical implementation of an Element Free Galerkin Method for an elastoplastic coupled to damage problem is presented. Several results issued from the numerical simulations by a DynamicExplicit resolution scheme are analyzed and critically compared with Element Finite Method results. Finally, different numerical examples are carried out to demonstrate the efficiency of this method.

Keywords: damage, dynamic explicit, elastoplasticity, isotropic hardening, meshless

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13140 Central Finite Volume Methods Applied in Relativistic Magnetohydrodynamics: Applications in Disks and Jets

Authors: Raphael de Oliveira Garcia, Samuel Rocha de Oliveira


We have developed a new computer program in Fortran 90, in order to obtain numerical solutions of a system of Relativistic Magnetohydrodynamics partial differential equations with predetermined gravitation (GRMHD), capable of simulating the formation of relativistic jets from the accretion disk of matter up to his ejection. Initially we carried out a study on numerical methods of unidimensional Finite Volume, namely Lax-Friedrichs, Lax-Wendroff, Nessyahu-Tadmor method and Godunov methods dependent on Riemann problems, applied to equations Euler in order to verify their main features and make comparisons among those methods. It was then implemented the method of Finite Volume Centered of Nessyahu-Tadmor, a numerical schemes that has a formulation free and without dimensional separation of Riemann problem solvers, even in two or more spatial dimensions, at this point, already applied in equations GRMHD. Finally, the Nessyahu-Tadmor method was possible to obtain stable numerical solutions - without spurious oscillations or excessive dissipation - from the magnetized accretion disk process in rotation with respect to a central black hole (BH) Schwarzschild and immersed in a magnetosphere, for the ejection of matter in the form of jet over a distance of fourteen times the radius of the BH, a record in terms of astrophysical simulation of this kind. Also in our simulations, we managed to get substructures jets. A great advantage obtained was that, with the our code, we got simulate GRMHD equations in a simple personal computer.

Keywords: finite volume methods, central schemes, fortran 90, relativistic astrophysics, jet

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13139 A Numerical Method for Diffusion and Cahn-Hilliard Equations on Evolving Spherical Surfaces

Authors: Jyh-Yang Wu, Sheng-Gwo Chen


In this paper, we present a simple effective numerical geometric method to estimate the divergence of a vector field over a curved surface. The conservation law is an important principle in physics and mathematics. However, many well-known numerical methods for solving diffusion equations do not obey conservation laws. Our presented method in this paper combines the divergence theorem with a generalized finite difference method and obeys the conservation law on discrete closed surfaces. We use the similar method to solve the Cahn-Hilliard equations on evolving spherical surfaces and observe stability results in our numerical simulations.

Keywords: conservation laws, diffusion equations, Cahn-Hilliard equations, evolving surfaces

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13138 Robust Numerical Scheme for Pricing American Options under Jump Diffusion Models

Authors: Salah Alrabeei, Mohammad Yousuf


The goal of option pricing theory is to help the investors to manage their money, enhance returns and control their financial future by theoretically valuing their options. However, most of the option pricing models have no analytical solution. Furthermore, not all the numerical methods are efficient to solve these models because they have nonsmoothing payoffs or discontinuous derivatives at the exercise price. In this paper, we solve the American option under jump diffusion models by using efficient time-dependent numerical methods. several techniques are integrated to reduced the overcome the computational complexity. Fast Fourier Transform (FFT) algorithm is used as a matrix-vector multiplication solver, which reduces the complexity from O(M2) into O(M logM). Partial fraction decomposition technique is applied to rational approximation schemes to overcome the complexity of inverting polynomial of matrices. The proposed method is easy to implement on serial or parallel versions. Numerical results are presented to prove the accuracy and efficiency of the proposed method.

Keywords: integral differential equations, jump–diffusion model, American options, rational approximation

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13137 A New Conjugate Gradient Method with Guaranteed Descent

Authors: B. Sellami, M. Belloufi


Conjugate gradient methods are an important class of methods for unconstrained optimization, especially for large-scale problems. Recently, they have been much studied. In this paper, we propose a new two-parameter family of conjugate gradient methods for unconstrained optimization. The two-parameter family of methods not only includes the already existing three practical nonlinear conjugate gradient methods, but also has other family of conjugate gradient methods as subfamily. The two-parameter family of methods with the Wolfe line search is shown to ensure the descent property of each search direction. Some general convergence results are also established for the two-parameter family of methods. The numerical results show that this method is efficient for the given test problems. In addition, the methods related to this family are uniformly discussed.

Keywords: unconstrained optimization, conjugate gradient method, line search, global convergence

Procedia PDF Downloads 318