Compare for both methods of simulations, we got the parallel computation with domain decomposition method is more successful for solve this problem with reducing about 7[r]

In the field of numerical analysis, ADI method is a **finite** **difference** **technique** which is used to solve parabolic and elliptic partial differential equations. It is particularly being used in solving heat conduction problem or diffusion equation in two or more dimensions [5]. Normally, Crank-Nicolson method is always being chosen as the method for solving heat conduction equation. However, this method is quite costly. The advantage of using ADI method is that, in every iteration, the equations that have to be solved in the ADI scheme provides a simple structure and therefore it is easier to be solved.

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In order to perform large scale numerical simulation of wave propagation in 3D heterogeneous multiscale viscoelastic media, **Finite** **Difference** **technique** and its parallel implementation based on domain decomposition is used. A couple of typical statements of borehole geophysics are dealt with—sonic log and cross well measurements. Both of them are essentially multiscales, which claims to take into account heterogeneities of very different sizes in order to provide re- liable results of simulations. Locally refined spatial grids help us to avoid the use of redundantly tiny grid cells in a tar- get area, but cause some troubles with uniform load of Processor Units involved in computations. We present results of scalability tests together with results of numerical simulations for both statements performed for some realistic models. Keywords: Seismic Wave propagation; Sonic Log; Numerical Simulation; Domain Decomposition

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proposed transient groundwater flow model is proposed. The computed hydraulic head is transformed to be the groundwa- ter flow velocity by using the second model. The results from the second model will be inputted into the third model as field data. The groundwater pollutant concentration is obtained by the third model. The hydraulic head of the first model is ap- proximated by an explicit **finite** **difference** method. An explicit **finite** **difference** **technique** is used to obtain the groundwater flow velocity of the second model. A forward time centered space **finite** **difference** **technique** is used to approximate the groundwater pollutant concentration. Long term groundwater quality around the landfill for 5, 10, 15, and 20 years are simu- lated. The groundwater quality is affected by the contaminated leachate pollutant release by the landfill. The proposed sim- ulations show that the different levels of hydraulic head have a small effect on the overall groundwater quality level. Our proposed simulations found that the main groundwater quality factor is the leachate pollutant concentration around the land- fill. Furthermore, a three-dimensional mathematical model for contaminated groundwater pollutant measurement should be introduced for a precise simulation.

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Over the past few years, **finite**-**difference** time-domain (FDTD) method [1] have become increasingly prevalent in the computational electromagnetic problems due to its simplicity, efficiency, robustness and versatility scheme for highly complex configuration in the computational domain. Generally, FDTD **technique** is the most well-known numerical method for the solution of problems in electromagnetic simulation ranging from RF to optical frequencies. It is considered to be one of the most powerful numerical techniques for solving partial differential equations of any kind. In addition, it can be utilized to solve the spatial as well as the temporal distributions of electric and magnetic fields in various media.

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Genetic algorithm (GA) is a global optimization **technique** based on the principles and concepts of natural selection and evolution. In GA, the population is the set of possible (trial) solutions, and each individual from the population is characterized by chromosome-like structures. The possibilities of survival for each individual are evaluated by the cost function, that is, function to optimize. The result of this evaluation, called fitness, plays an important role in selection and reproduction. Finally, evolution is achieved through the application of genetic operators which are: selection, crossover and mutation. It is worth noting that the GA operates on a coded version of the parameters rather than the parameters themselves. There are multiple versions of GAs according to the coding and/or genetic operators. In this work, the antenna design is carried out with an efficient mixed integer GA that is proposed in [Haupt (2007)]. This GA is versatile and more robust since it can work with real and/or binary values in the same chromosome. The fitness we’ve adopted is defined as:

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Non-standard **Finite** **DIFFERENCE** MODEL: Now we show an unreservedly convergent non-standard **finite** **difference**(NSFD) numerical model which be there describe on non-standard **finite** **difference** modeling concept introduced by Micken’s(Guidotti, 1999).Now show the covergenence scrutiny of the suggestedstructure.The NSFD model for the incessant dynamical system is given by:

The principal objective of this research work was to develop a **finite** element based **technique** to solve the electromagnetic time domain problem in a fast and efficient manner which can be considered as a realistic alternative to the standard FDTD method. All the objectives mentioned in the motivation section of the introductory chapter have been fulfilled in this research. To develop the **technique** the accuracy of result and CPU performance were given the top most priorities. To develop an explicit and data parallel formulation the Maxwell’s equations in their differential form was considered which is similar as in the FDTD method. The explicit formulation of the equations were discretised with using linear **finite** ele- ments (triangles for 2D and tetrahedrons for 3D). As the Maxwell’s equations are coupled a unique coupled dual mesh system was introduced in this thesis in Chapters 3 and 8. Hence the number of elements required were reduced to half for 2D and 1/5 times for 3D when compared with full mesh systems.

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The FDTD method is widely used in electromagnetic, microwave and photonic structures. The FDTD is more suitable for modeling dielectric and magnetic materials of **finite** regions. In structures with curved areas such as antipodal Vivaldi antennas, the accuracy of the FDTD results decrease. In order to overcome this problem, rather than increasing mesh size which requires too large a memory, we use conformal FDTD. The CFDTD is more accurate and numerically stable for modeling the PEC objects.

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A WAVEGUIDE irradiation system is a convenient way to study interaction of biological materials with microwave fields. Such a system can be used in a number of applications, such as irradiation of insects inside host bodies, investigation of tumor treatment inside healthy or infected tissues, and the like[1-3]. Fujiwara and Amemiya [1] used a waveguide section with reflection plate to study microwave power absorption in a biological specimen exposed to standing waves inside the waveguide, and analysis was made using a quasi-static approach. A similar analytical **technique** was used By Andreuccetti et. al.[2] to evaluate microwave power absorbed by woodworms and the surrounding wood. Quasi-static analysis consists of independent determination of the contribution of the electric and the magnetic fields separately on the total power absorption by the object under study. It gives reasonable results when the dimensions of the object are small compared to the wavelength of the incident field. Such analytical methods usually approximate the geometrical structure under investigation by known structures amendable to analytical descriptions, such as cubical or spheroid shape. In many practical situations, biological bodies and specimens have irregular shapes and any approximations could lead to inaccurate results.

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light propagation in longitudinally varying wavguides such as tapers, Y-junctions, and bends. The beam propagation method was first applied to optoelectronics in 1980 (Feit and Fleck 1980) and the solutions for the optical waveguides can be made to generate mode- related properties such as propagation constants, relative mode powers and group delays with high precision and considerable flexibility. The first reported BPM was based on the Fast Fourier Transform (FFT) and only solved the scalar wave equations under paraxial approximation. Therefore the FFT-BPM is only suitable for the case of weakly guiding structures, neglecting the vectorial properties of the field. Several numerical algorithms to treat the vectorial wave propagation (vector BPM) using the **finite** **difference** method, have been reported (Chung et al. 1991; Huang et al. 1992a; Huang and Xu 1992b). The VBPMs are capable of simulating polarized or even hybrid wave propagation in strongly guiding structures. Subsequently, the **finite** element method has been utilised to develop BPM approaches. A unified **finite** element beam propagation method has been reported (Tsuji and Koshiba 1996) for both TE and TM waves propagating in strongly guiding longitu- dinally varying optical waveguides. Obayya et al. (2000) has reported a full-vectorial BPM algorithm based on the **finite** element method to characterise 3-D optical guided wave devices.

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precisely the same as the set of subdirectly irreducibles in V . By Birkhoff’s Subdirect Representation Theorem, any variety is determined by its subdirectly irreducible members. Thus, this is evidently sufficient to get a **finite** basis result, provided V is locally **finite** and has a **finite** signature, in light of Birkhoff’s other work that showed how to find a **finite** basis for the N -variable laws of any such variety. Any variety that has this property is said to have a **finite** residual bound. Some who have used the **technique**—such as McKenzie (1987a), Willard (2000), and Kearnes, Szendrei, and Willard (2013+)—have actually added a **finite** residual bound as an hypothesis. However, as has been observed before, it is not a necessary hypothesis. It is well known that the 8-element quaternion group Q, generates a variety without a **finite** residual bound. On the other hand, it is indeed finitely based—or “doubly so,” to hyperbolize—as can be seen from Lyndon (1952), as Q forms a group of nilpotence class 2, as well as from Oates and Powell (1964), since they are **finite**. We have sought to study whether the results of Lyndon and Oates and Powell generalize further than is currently known.

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where E is unit operator. The **difference** Equation (3.6) contains unknown nonzero parameter ω and therefore it may be considered as a nonlinear equation with respect to the parameter ω and The series in (3.6) may be expressed through analytical functions depending on the sign of quantities ω and β and thereby the Equation (3.6) can be rewritten as

Given the mesh transformations for the problem under consideration, the solution scheme follows the basic pattern laid down in §3 .4 except that 1 the original graded meshes are picked b[r]

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In the flow model it was observed that water flows from region of high hydraulic head to region of low hy- draulic head and that an exact solution could be obtained if the grid spacing is small enough say 5 cm. Results show that in a steady-state flow field with no recharge, pathlines coincide with streamlines. It is therefore con- cluded that **Finite** **Difference** Method can be used to pre- dict the future direction of flow and particle location within a simulation domain.

In this paper, some new **finite** **difference** inequalities in two independent variables are established, which can be used as a handy tool in the study of boundedness, unique- ness, continuous dependence on initial data of solutions of certain **difference** equations. The established inequalities generalize some existing results in the literature.

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The numerical solution of nonlinear problems is a topic of basic importance in numerical mathematics, as stated in [1]. It has been a subject of extensive investigation in the past decades, thus having vast literature [2-5]. The most widespread way of finding numerical solutions is first discretizing the given problem, then solving the arising system of algebraic equations by a solver which is generally some iterative method. For nonlinear problems most often Newton’s method is used. However, when the work of compiling the Jacobians exceeds the advantage of quadratic convergence, one may prefer gradient type iterations including steepest descent or conjugate gradi- ents. An important example in this respect is the Sobolev gradient **technique**, which is relying on descent methods. The Sobolev gradient **technique** presents a general effi- cient preconditioning approach where the preconditioners are derived from the representation of the Sobolev inner product.

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Abstract—In this paper, a direct three dimensional **Finite**-**Difference** Time-Domain (3D-FDTD) approach is implemented to investigate the electromagnetic behavior of a Half Hollow Cylindrical Antenna. The conformal shape of this antenna is studied using the Conformal **Finite**- **Difference** Time-Domain (CFDTD). We shall prove that a variation of the antenna shape generates an important shift of the values of the resonant frequency (about 0.467 GHz). Compared with the planar shape, the geometrical shape reduces the space occupied by the antenna of about 36.28%.

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Conservation laws arise in many models in science and engineering. They are applied in fluid and gas dynamics, relativity theory, quantum mechanics, aerodynamics, meteorology and astrophysics (Eymard et al, 2003). Numerical methods for solving conservation laws include the **finite** **difference** method, **finite** element method and **finite** volume method. The **finite** volume method is now a popular choice for solving conservation laws because of its accuracy and ability to handle complex geometries as well as good approximations of boundary conditions (LeVeque, 2004, Hu & Joseph, 1990, Grigoryan, 2010, Moroney, 2006).

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